Research ArticleStability and Global Sensitivity Analysis for an Agree-DisagreeModel Partial Rank Correlation Coefficient and Latin HypercubeSampling Methods

Sara Bidah Omar Zakary and Mostafa Rachik1Laboratory of Analysis Modelling and Simulation Department of Mathematics and Informatics Faculty of Sciences Ben MrsquoSikHassan II University of Casablanca BP 7955 Sidi Othman Casablanca Morocco

Correspondence should be addressed to Omar Zakary zakarymagmailcom

Received 1 January 2020 Accepted 2 March 2020 Published 1 April 2020

Academic Editor Mayer Humi

Copyright copy 2020 Sara Bidah et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we present a new mathematical model that describes agree-disagree opinions during polls We first present themodel and its different compartments )en we use the next-generation matrix method to compute thresholds of equilibriumstability We perform the stability analysis of equilibria to determine under which conditions these equilibrium points are stable orunstable We show that the existence and stability of these equilibria are controlled by the calculated thresholds Finally we alsoperform several computational and statistical experiments to validate the theoretical results obtained in this work To study theinfluence of various parameters on these thresholds and to identify the most influential parameters a global sensitivity analysis iscarried out based on the partial rank correlation coefficient method and the Latin hypercube sampling

1 Introduction

Participation in political life requires citizensrsquo attentiontime knowledge money and motivation Citizens willparticipate if they receive benefits commensurate with thecost of participation Perhaps the debate on voting is thesubject of the most studied political behavior Individualeconomic social and psychological costs and voting benefitsare well known [1] Candidates and the media often use pollsin election campaigns to determine which candidates are inthe lead and who are likely to emerge victoriously Opinionpolls are surveys of intent for a sample of voters while futureresults allow participants to negotiate and discuss opinionsbased on a particular outcome Opinion polls revealed theexpected voting quotas Public opinion polls are widely usedto identify peoplersquos political positions vote and other be-haviors by asking questions about their opinions activitiesand personal characteristics Answers to these questions arethen counted statistically analyzed and interpreted Pollsrsquoresult also provides parties with the opportunity to refinetheir campaign strategies In the long run parties changetheir positions in public opinion [2]

Polls then play a key role in contemporary politicalcampaigns Party performance updates receive considerablemedia attention and often serve as a basis for politicalcommentary in the weeks leading up to the election day Weknow that knowing the positions of the electorate can affectvoter behavior [3 4] However it is unclear how votingshapes party leadersrsquo strategies during the election campaignPublic opinion is often criticized for influencing voterperceptions and focusing on popularity rather than politicsBy strategically communicating the poll results parties canshape votersrsquo vision of competition and encourage popularmobilization [5] )is explains why at the beginning of the20th century more than thirty democratic countries bannedthe publication of opinion polls near elections [6 7]

To fully understand the role of polls before the electionsit is necessary to take into account the offer of campaigns)is is particularly important in contexts of increased po-larization where holidays play an increasing role in deter-mining the type of information that reaches voters [8] Atany time party leaders may choose to speak of candidatesfrom their party or other candidates for election focusing onvarious political or gender issues Voter signals in opinion

HindawiInternational Journal of Differential EquationsVolume 2020 Article ID 5051248 14 pageshttpsdoiorg10115520205051248

polls can influence party leadersrsquo decisions to balance thesedifferent elements of campaign speech [9]

In the electoral process where there are only two po-litical parties the chances of creating strong political po-larization are increasing For example U S citizens face apolarizing scenario during a presidential election wherethey must vote for a two-party decision Republicans andDemocrats Previous studies have shown that U S electionsare attracting the Internet where both Twitter and blogs areshowing high political polarization [10 11]

Other political scenarios requiring a bilateral decisionare second-round electoral processes While citizens ofthe first round can vote from a wide range of differentpolitical parties in the last round they can vote only forthe two final candidates Voters may not be fully identifiedwith either side but they still have to take a side Previousarticles have shown that this second round increasespolitical polarization in the country [12] In [13] theauthors analyzed the 2017 presidential election in Chileand measured the resulting political polarization Of aminority of qualified users they could estimate theopinion of the majority

In this contribution we examine how votersrsquo preferencesand expectations compete with each other when dealing withvoter support information to candidates So we start bydeveloping a mathematical model to describe the evolutionof opinions to predict the probability of results and then wecalculate and analyze the equilibrium points of the model byderiving the important stability thresholds for each equi-librium state

)rough the development of more efficient modelsreliable statistical and mathematical methods are needed toimprove the accuracy of modeling One of the most recentlypopular ways is sensitivity analysis (SA) methods Sensitivityanalysis is used for a variety of reasons such as developingdecisions or recommendations communicating under-standing or quantifying the system and developing modelsIn the development of the model it can be used to verify thevalidity or accuracy of the model to simplify calibrate weakprocess or lost data and even to identify the importantparameter for other studies [14]

)e paper is organized as follows Section 2 introducesour new model giving some details about interactions be-tween different compartments and parameters of the modelIn Section 3 we derive basic reproduction numbers Section4 provides the results of the stability analysis of equilibria InSection 5 sensitivity analysis is performed to identify themost important parameter in the proposed model andSection 6 concludes the paper

2 Presentation of the Model

)ere are many scenarios involving a binary decision )epoll model that we present here describes the opinions ofagreement (or approval) and disagreement (or disapproval)regarding a candidate or idea in polls preceding the elec-tions Note that the model can also describe situations inwhich there are more than two candidate parties because wecan always reduce the situation to two decisions For

example if there are four parts A B C and D we wish tostudy the political position of party A and we can thenexamine the two subsets A and B C D of the investi-gation Votes for A are considered agree and votes for B Cor D are considered disagree More simply we consider theopinion poll of voters on the performance of the partystudied As an example we can cite the ldquoapprove or dis-approverdquo poll done by )omson Reuters on how DonaldTrump fulfills his role as the president [15]

Without loss of generality we devise a mathematicalmodel describing the evolution of agree and disagreeopinions during polls (surveys) and the types of surveys weconsider here are surveys that can be answered in agreementdisagreement or otherwise)us the population targeted bythe poll is regrouped into three groups agree disagree andignorant individuals

)e model herewith has been formulated using threecompartments Each of them has been described as follows

(1) Ignorant (I) people who do not know about the pollor those who abstain from voting for personalreasons

(2) Agree (A) people in agreement with the idea beingstudied

(3) Disagree (D) people in disagreement with the ideabeing studied

All contacts are modeled by the standard incidence rateFor the modeling processes a set of assumptions has beenused )ese are as follows

(1) )e targeted population is well mixed that is theignorant individuals are homogeneously spreadthroughout the population

(2) Recruitment and mortality are negligible under thetemporal scale consideration therefore no indi-vidual is recruited and no individual dies during thepoll

(3) Individuals have the right to communicate with eachother and can thus convince one another

(4) People who are unsure of their opinion are ignorant(5) People who abstain from voting are ignorant

Everyone has their reasons for agreement or dis-agreement An ignorant person can be persuaded bysomeone who agrees at a rate β1 or by someone who doesnot agree with the opinion at a rate β2 A person agreeingwith the opinion may be persuaded by someone who doesnot agree at a rate α2 or a person who disagrees with theopinion may be persuaded by someone who agrees at arate α1 People can abstain from voting or lose interestwithout any direct contact with individuals from theopposite opinion group then agree people become ig-norants at the rate c1 and disagree people become ig-norants at the rate c2 A flowchart describing differentinteractions between the compartments of the model ispresented in Figure 1

All these assumptions and considerations are written asthe following system of ordinary differential equations

2 International Journal of Differential Equations

Iprime minus β1AI

Nminus β2

DI

N+ c1A + c2D (1)

Aprime β1AI

N+ α1

AD

Nminus α2

AD

Nminus c1A (2)

Dprime β2DI

N+ α2

AD

Nminus α1

AD

Nminus c2D (3)

where I(0)ge 0 A(0)ge 0 D(0)ge 0 and N I + A + D Notethat Nprime Iprime + Aprime + Dprime 0 thus the population size N isconsidered as a constant in time We can easily prove thatfor nonnegative initial conditions the solutions of systems(1)ndash(3) are nonnegative To do this recall that by [16] thesystem of equation

xprime f x1 x2 xk( 1113857 (4)

with

x(0) x0 ge 0 (5)

is a positive system if and only if foralli 1 2 k

xiprime fi x1 ge 0 x2 ge 0 xi 0 xk ge 0( 1113857ge 0 (6)

)us for models (1)ndash(3) it is easy to verify that

I 0⟹Iprime ge 0

A 0⟹Aprime ge 0

D 0⟹Dprime ge 0

(7)

)erefore all the solutions of systems (1)ndash(3) arenonnegative

It is also clear that the solutions of models (1)ndash(3) arebounded based on the fact thatN I +A +D is constant andthen S le N A le N and D le N )erefore we will focus tostudy models (1)ndash(3) in the closed positively invariantfeasible set given by

Ω (I A D) isin R3

+

I + A + D N1113888 1113889 (8)

A summary of parameter description is given in Table 1

3 Thresholds Basic Reproductive Numbers

In epidemiology the basic reproductive number R0 (orepidemic threshold) is defined as the average number ofsecondary cases of an infection produced by a ldquotypicalrdquoinfected individual during hisher entire life as infectiouswhen introduced in a population of susceptibles [17ndash21])e threshold R0 is mathematically characterized in termsof infection transmission as a ldquodemographic processrdquo butoffspring production is not seen as giving birth in a de-mographic sense but it causes new infections throughtransmission [22ndash24] )us the infection process can beconsidered as a ldquoconsecutive generation of infected in-dividualsrdquo )e following growing generations indicate agrowing population (ie an epidemic) and the growthfactor for each generation indicates the potential forgrowth So the mathematical characterization of R0 is this

growth factor [22] Generally if R0 gt 1 an epidemicoccurs whereas if R0 lt 1 there will probably be noepidemic

Following this definition we will define our thresholdsas follows RD0 is the average number of new disagree-ments produced by an individual disagreeing introducedin a population of ignorant people during the period inwhich he or she was in this opinion And RA0 is theaverage number of new agreements produced by an in-dividual in agreement and that was introduced into apopulation of ignorant people during the period in whichhe or she was in that opinion

For the analysis of epidemic models the first step is tocalculate the disease-free equilibrium (DFE) )is equilib-rium point is then used to calculate the basic reproductivenumber using the next-generation matrix method )eobjective of this section is only the calculation of thethresholds and not the equilibrium states of the model Butin this method we have to determine the equilibrium stateswhen A 0 and when D 0

In this contribution let RX0 be the threshold of growingof the opinion X (either X A ldquoagreerdquo or X D ldquodisagreerdquo))en RD0 is the threshold associated with the disagree-freeequilibrium while RA0 is the one associated with the agree-free equilibrium

We calculate the equilibria for the aforementionedmodel and based on the next-generation matrix approachwe derive associated thresholds

)e points of equilibrium of systems (1)ndash(3) are thesolutions of

Iprime Aprime Dprime 0 (9)

for the disagree-free equilibrium when there is no negativeopinion ie if we put D 0 )is gives

Table 1 Parameter descriptions

Parameter Descriptionβ1 Ignorant to agree transmission rateβ2 Ignorant to disagree transmission rateα1 Disagree to agree transmission rateα2 Agree to disagree transmission ratec1 Interest loss factor of agree individualsc2 Interest loss factor of disagree individuals

I D

A

γ1A

β1(AIN)

β2(DIN)

γ2D

α1(ADN)

α2(ADN)

Figure 1 Flowchart for models (1)ndash(3)

International Journal of Differential Equations 3

Ilowast

c1

β1N

Alowast

N β1 minus c1( 1113857

β1

(10)

where Ilowast and Alowast represent the numbers of ignorant andagree individuals respectively in the absence of disagreepeople

)erefore for the system governed by (1)ndash(3) the dis-agree-free equilibrium is

e1 c1

β1N

N β1 minus c1( 1113857

β1 01113888 1113889 (11)

Following the second generation approach [22] wecompute the threshold RD0 associated to the disagree-freeequilibrium which is

RD0 α2β1 minus α2c1 + β2c1

α1β1 minus α1c1 + β1c2(12)

for the agree-free equilibrium when there is no positiveopinion ie if we put A 0 )is gives

Ilowast

c2

β2N

Dlowast

N β2 minus c2( 1113857

β2

(13)

where Ilowast and Dlowast represent the numbers of ignorant anddisagree individuals respectively in the absence of agreepeople

)erefore for the system governed by (1)ndash(3) the agree-free equilibrium is

e2 c2

β2N 0

N β2 minus c2( 1113857

β21113888 1113889 (14)

By also following the second generation approach [22]we compute the threshold RA0 associated to the agree-freeequilibrium which is

RA0 α1β2 minus α1c2 + β1c2

α2β2 minus α2c2 + β2c1 (15)

)roughout this paper we consider the following as-sumption the model parameters verify

β1 minus c1 gt minusβ1c2

α1

β1 minus c1 gt minusβ2c1

α2

β2 minus c2 gt minusβ2c1

α2

β2 minus c2 gt minusβ1c2

α1

(16)

)erefore from the above assumption we get

RD0 α2β1 minus α2c1 + β2c1

α1β1 minus α1c1 + β1c2gt 0

RA0 α1β2 minus α1c2 + β1c2

α2β2 minus α2c2 + β2c1gt 0

(17)

4 Stability Analysis

In order to analyze in terms of the proportions of ignorantagree and disagree individuals let i (IN) a (AN)and d (DN) denote the fraction of the classes I A and Din the population respectively After some calculations andreplacing I by i A by a and D by d equations (1)ndash(3) can bewritten as

iprime minus β1ai minus β2di + c1a + c2d (18)

aprime β1ai + α1ad minus α2ad minus c1a (19)

dprime β2di + α2ad minus α1ad minus c2d (20)

From the fact N I + A + D we have i + a + d 1 )enmodel systems (18)ndash(20) will be reduced to the following twodifferential equations

aprime β1a(1 minus a minus d) + α1ad minus α2ad minus c1a

dprime β2d(1 minus a minus d) + α2ad minus α1ad minus c2d(21)

which can be reduced to

aprime β1a(1 minus a minus d) + αa c1a

dprime β2d(1 minus a minus d) minus αad minus c2d(22)

where α α1 minus α2

41 Steady States )e steady states of system (22) are ob-tained by solving the system of equations

0 β1a(1 minus a minus d) + αad minus c1a

0 β2d(1 minus a minus d) minus αad minus c2d(23)

)is system has four equilibrium points and the trivialequilibrium E0 (0 0) is an equilibrium that exists alwayswithout any condition It means there is no survey and thereis no need to opinions

One disagree-free equilibrium E1 (1 minus (c1β1) 0) ex-ists if the condition β1 gt c1 holds and the agree-freeequilibrium E2 (0 1 minus (c2β2)) exists if β2 gt c2 holds

)e fourth and the positive equilibrium Elowast (alowast dlowast)where

alowast

αβ2 minus αc2 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

dlowast

minusαβ1 minus αc1 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

(24)

exists if one of the following conditions holdsldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo or ldquoRD0 lt 1

4 International Journal of Differential Equations

andRA0 lt 1 and α(α minus β1 + β2)lt 0rdquo In fact by a simplecalculation we get

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(25)

)us from (25) we deduce that alowast gt 0 and dlowast gt 0)roughout the article and without loss of generality we justconsider the first condition as a sufficient condition for theexistence of Elowast

42 Stability of Steady States )e Jacobian matrix of system(22) is

J J11 aα minus aβ1

minus αd minus β2d J221113888 1113889 (26)

where

J11 αd minus aβ1 minus c1 minus β1(a + d minus 1)

J22 minus c2 minus aα minus β2d minus β2(a + d minus 1)(27)

Proposition 1 1e equilibrium E0 (0 0) is unstable ifβ1 gt c1 or β2 gt c2 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E0( 1113857 β1 minus c1 0

0 β2 minus c21113888 1113889 (28)

It is clear that if β1 gt c1 or β2 gt c2 we get one positiveeigenvalue of J(E0) and then E0 is unstable Else we have alleigenvalues of J(E0) having the negative real part whichcompletes the proof

Remark 1 Note that the conditions in the previous prop-osition imply the existence of E1 or E2 )erefore E0 isunstable whenever there exists E1 or E2

Proposition 2 1e equilibrium E1 (1 minus (c1β1) 0) isunstable if RD0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E1( 1113857

c1 minus β1α minus β1( 1113857 β1 minus c1( 1113857

β1

0 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)

)e eigenvalues of J(E1) are

λ1 c1 minus β1

λ2 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

minusαβ1 minus αc1 + β1c2 minus β2c1

β1

(30)

It is clear from the existence condition of this equilib-rium that c1 minus β1 lt 0 )us the stability of the point ofequilibrium E1 is based on the eigenvalue λ2 of the matrixJ(E1) By a simple calculation we have

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1 (31)

)is implies that if RD0 gt 1 E1 is unstable else E1 isstable

Proposition 3 1e equilibrium E2 (0 1 minus (c2β2)) isunstable if RA0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E2( 1113857

β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1 0

minusα + β2( 1113857 β2 minus c2( 1113857

β2c2 minus β2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(32)

)e eigenvalues of J(E2) are

λ1 β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1

αβ2 minus αc2 + β1c2 minus β2c1

β2

λ2 c2 minus β2

(33)

It is clear from the existence condition of this equilib-rium that c2 minus β2 lt 0 )us the stability of the point ofequilibrium E2 is based on the eigenvalue λ1 of the matrixJ(E2) By a simple calculation we have

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1 (34)

)is implies that if RA0 gt 1 E2 is unstable else E2 isstable

Proposition 4 1e equilibrium Elowast (alowast dlowast) where

alowast

αβ2 minus αc2 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

dlowast

minusαβ1 minus αc1 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

(35)

is stable if ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gt c1 and c2 gtβ2 and α lt 0rdquo

Proof )e Jacobian matrix at this equilibrium is

J Elowast

( 1113857 J11 alowastα minus alowastβ1

minus αdlowast minus β2dlowast J221113888 1113889 (36)

where

J11 αdlowast

minus alowastβ1 minus c1 minus β1 a

lowast+ dlowast

minus 1( 1113857

J22 minus c2 minus alowastα minus β2d

lowastminus β2(a + d minus 1)

(37)

Using fact (23) we have

International Journal of Differential Equations 5

J11 minus β1alowast

J22 minus β2dlowast

(38)

)en

J Elowast

( 1113857 J1 J2

J3 J4

⎛⎝ ⎞⎠

J1 minusβ1 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J2 α minus β1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J3 α + β2( 1113857 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J4 β2 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

(39)

where the characteristic polynomial of is

λ2 + c1λ + c2 (40)

where

c1 minusβ1c2 minus β2c1

α

c2 minus 1

α α minus β1 + β2( 1113857

times α2β1β2 minus α2β1c2 minus α2β2c1 + α2c1c21113872

+ αβ21c2 minus αβ1β2c1 + αβ1β2c2 minus αβ1c1c2

minus αβ1c22 minus αβ22c1 + αβ2c

21 + αβ2c1c2

+ β21c22 minus 2β1β2c1c2 + β22c

211113873

minus 1

α α minus β1 + β2( 1113857

times αβ1 minus αc1 + β1c2 minus β2c1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

(41)

By the conditions ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gtc1 and c2 gt β2 and α lt 0rdquo we have minus (β1c2 minus β2c1α)gt 0 andby existence conditions and

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(42)

we have c1 gt 0 and c2 gt 0 Using the RouthndashHurwitz stabilitycriterion we conclude that the equilibrium point Elowast is lo-cally asymptotically stable

Remark 2

(1) Note that the conditions RD0 gt 1 and RA0 gt 1 lead to theinstability of E1 and E2 and help in the existence of Elowast

(2) Sufficient conditions in the above proposition couldbe reduced to α (β1c2 ndash β1c2) lt 0

A summary of sufficient existence and stability condi-tions of the steady states of model (11) is given in Table 2

43 Examples It can be seen from Table 3 that the equi-librium E0 exists and it is stable in the four cases example 1RD0 gt 1 and RA0 gt 1 example 2 RA0 lt 1 and RD0 lt 1 example3 RD0 lt 1 and RA0 gt 1 and example 4 RD0 gt 1 and RA0 lt 1

Example 5 shows the existence and the stability of thesteady state E1 in the case of RA0 lt 1 and RD0 lt 1 and in the caseofRA0gt 1 andRD0lt 1 in example 6 Examples 7 and 8 show theexistence and the stability of the equilibrium E2 in the cases ofldquoRA0 lt 1 and RD0 lt 1rdquo and ldquoRA0 lt 1 and RD0 gt 1rdquo respectively

Examples 9 and 10 show the possibility of the existence andstability of the two steady states E1 and E2 at the same time thatis ldquoβ1 gt c1 and β2 gt c2 and RA0 lt 1 and RD0 lt 1rdquo )eseexamples give an insight about the stability of each equilibriumfor given parametersrsquo values After some numerical calcula-tions we noted that in this situation the parameters c1 and c2may switch from E1 to E2 and the inverse while if c1 gt c2 thenE1 will be more attractive and if c2 gt c1 then E2 will be moreattractive and that with the same initial conditions We sim-ulate the model with different initial conditions to illustrate theimpact of initial conditions on the stability of E1 and E2 in thissituation Figure 2 depicts the stability of E1 and E2 at the sametime where we consider the same set of parameters fromexample 9 in Table 3 By changing the initial conditions we cansee that E1 is stable when I(0) 100 A(0) 100 andD(0) 80 while it can be seen thatE2 is also stablewith this setof parameters but after choosing I(0) 100 A(0) 80 andD(0) 100

Example 11 shows the existence and the stability of theequilibrium Elowast where ldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1+β2)gt 0rdquo and ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo while example 12shows the existence and the stability of Elowast whereldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo and ldquoβ1 gt c1and c2 gt β2 and α lt 0rdquo

Figure 3 depicts examples of the existence and stability ofthe equilibrium state E0 for different parametersrsquo values andthreshold RD0 and RA0 values simulated with initial con-ditions and parametersrsquo values from Table 3 Figure 4 depictsexamples of the existence and stability of the equilibriumstate E1 for different parametersrsquo values and threshold RD0and RA0 values simulated with initial conditions and pa-rametersrsquo values from Table 3 It can be seen fromFigure 4(b) that the functionD decreases towards zero but itwill take a long time In the Figure 4(c) we have considered1200 hours to show that the function D will go to zero butvery slowly Figure 5 depicts examples of the existence andstability of the equilibrium state E2 for different parametersrsquovalues and threshold RD0 and RA0 values simulated withinitial conditions and parametersrsquo values from Table 3 Wecan see from Figure 5(b) that the function A will also take avery long time to go to zero In Figure 5(c) we have con-sidered 1200 hours to show that function A will tend to zerobut very slowly Figure 6 depicts examples of the existenceand stability of the equilibrium state Elowast for different

6 International Journal of Differential Equations

parametersrsquo values and threshold RD0 and RA0 values sim-ulated with initial conditions and parametersrsquo values fromexamples 11 and 12 of Table 3

5 Analysis of Thresholds

51 No Abstention No Loss of Interest In most of the sit-uations people abstain from voting for personal reasonsHere we discuss the situation when there is no abstentionand there is no loss of interest of voting ie c1 c2 0)erefore RD0 and RA0 become

RD0 α2α1

RA0 α1α2

(43)

which means that thresholds RD0 and RA0 depend only on α1and α2 and the equilibria become

E1 (1 0)

E2 (0 1)(44)

and from the existence conditions in Table 2 we can deducethat there is no Elowast

E1 exists without any condition and it is stable if α2 lt α1and E2 exists without any condition and it is stable if α1 lt α2

)is result explains the effect of the polarization on theoutcome of polls When there is no abstention from votingand people keep their interest in voting then the mostinfluential parameters on the outcome of the polls are thepolarization factors α1 and α2 For example during a polldetermining the political position of one candidate Y byvoting with Approve or Disapprove or otherwise if can-didate Y can convince people by his political vision andorby other ways then he can change the course of events in hisfavor Mathematically he makes α2 lt α1 But if somehow hecontributes to making α1 lt α2 then things could spin out ofcontrol on the election day

Table 2 Summary of sufficient existence and stability conditions of equilibria

Equilibrium Existence conditions Stability conditionsE0 mdash β1 lt c1 and β2 lt c2E1 (1 minus (c1β1) 0) β1 gt c1 RD0 lt 1E2 (0 1 minus (c2β2)) β2 gt c2 RA0 lt 1Elowast (alowast dlowast) RD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0 β1 lt c1 and c2 lt β2 and α gt 0 or β1 gt c1 and c2 gt β2 and α lt 0

Table 3 Steady states and parametersrsquo values used in example simulations where I(0) A(0) D(0) 100

β1 β2 α1 α2 c1 c2 RD0 RA0

1 E0 (300 0 0) 00010 01010 01010 03010 05010 03010 19901 207302 E0 (300 0 0) 00010 02010 02010 03010 01010 05010 05000 085433 E0 (300 0 0) 00010 00010 02010 01010 01010 01010 05000 200014 E0 (300 0 0) 00010 00010 01010 02010 01010 01010 20001 050005 E1 (29703 2970297 0) 01010 00010 02010 02010 00010 00010 06634 049256 E1 (29703 2970297 0) 01010 00010 00010 00010 00010 00010 05025 1017 E2 (29703 0 2970297) 00010 01010 02010 02010 01010 00010 04925 066348 E2 (29703 0 2970297) 00010 01010 00010 00010 00010 00010 101 050259 Elowast1 (1203593 1796407 0) 05010 01010 01010 00010 02010 00010 06688 0519610 Elowast2 (1203593 0 1796407) 01010 05010 00010 01010 00010 02010 05196 0668811 Elowast (1333333 105556 1561111) 01010 07010 03010 00010 02010 03010 4677774 1067212 Elowast (2142857 804643 52500) 03010 04010 00010 08010 02010 05010 10649 3008004

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E1 = (112 168 0)

IAD

(a)

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E2 = (2 0 278)

IAD

(b)

Figure 2 Illustration of the bistability of E1 and E2 at the same time using the set of parameters of example 9 from Table 3 Stability of E1 (a)when I(0) 100 A(0) 100 and D(0) 80 Stability of E2 (b) when I(0) 100 A(0) 80 and D(0) 100

International Journal of Differential Equations 7

0 20 40 60 80 100 120Time (hours)

050

100150200250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(b)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(c)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(d)

Figure 3 Examples of the steady state E0 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 1 (b) Example 2 (c) Example 3 (d) Example 4

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(b)

Figure 4 Continued

8 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

Iprime minus β1AI

Nminus β2

DI

N+ c1A + c2D (1)

Aprime β1AI

N+ α1

AD

Nminus α2

AD

Nminus c1A (2)

Dprime β2DI

N+ α2

AD

Nminus α1

AD

Nminus c2D (3)

where I(0)ge 0 A(0)ge 0 D(0)ge 0 and N I + A + D Notethat Nprime Iprime + Aprime + Dprime 0 thus the population size N isconsidered as a constant in time We can easily prove thatfor nonnegative initial conditions the solutions of systems(1)ndash(3) are nonnegative To do this recall that by [16] thesystem of equation

xprime f x1 x2 xk( 1113857 (4)

with

x(0) x0 ge 0 (5)

is a positive system if and only if foralli 1 2 k

xiprime fi x1 ge 0 x2 ge 0 xi 0 xk ge 0( 1113857ge 0 (6)

)us for models (1)ndash(3) it is easy to verify that

I 0⟹Iprime ge 0

A 0⟹Aprime ge 0

D 0⟹Dprime ge 0

(7)

)erefore all the solutions of systems (1)ndash(3) arenonnegative

It is also clear that the solutions of models (1)ndash(3) arebounded based on the fact thatN I +A +D is constant andthen S le N A le N and D le N )erefore we will focus tostudy models (1)ndash(3) in the closed positively invariantfeasible set given by

Ω (I A D) isin R3

+

I + A + D N1113888 1113889 (8)

A summary of parameter description is given in Table 1

3 Thresholds Basic Reproductive Numbers

In epidemiology the basic reproductive number R0 (orepidemic threshold) is defined as the average number ofsecondary cases of an infection produced by a ldquotypicalrdquoinfected individual during hisher entire life as infectiouswhen introduced in a population of susceptibles [17ndash21])e threshold R0 is mathematically characterized in termsof infection transmission as a ldquodemographic processrdquo butoffspring production is not seen as giving birth in a de-mographic sense but it causes new infections throughtransmission [22ndash24] )us the infection process can beconsidered as a ldquoconsecutive generation of infected in-dividualsrdquo )e following growing generations indicate agrowing population (ie an epidemic) and the growthfactor for each generation indicates the potential forgrowth So the mathematical characterization of R0 is this

growth factor [22] Generally if R0 gt 1 an epidemicoccurs whereas if R0 lt 1 there will probably be noepidemic

Following this definition we will define our thresholdsas follows RD0 is the average number of new disagree-ments produced by an individual disagreeing introducedin a population of ignorant people during the period inwhich he or she was in this opinion And RA0 is theaverage number of new agreements produced by an in-dividual in agreement and that was introduced into apopulation of ignorant people during the period in whichhe or she was in that opinion

For the analysis of epidemic models the first step is tocalculate the disease-free equilibrium (DFE) )is equilib-rium point is then used to calculate the basic reproductivenumber using the next-generation matrix method )eobjective of this section is only the calculation of thethresholds and not the equilibrium states of the model Butin this method we have to determine the equilibrium stateswhen A 0 and when D 0

In this contribution let RX0 be the threshold of growingof the opinion X (either X A ldquoagreerdquo or X D ldquodisagreerdquo))en RD0 is the threshold associated with the disagree-freeequilibrium while RA0 is the one associated with the agree-free equilibrium

We calculate the equilibria for the aforementionedmodel and based on the next-generation matrix approachwe derive associated thresholds

)e points of equilibrium of systems (1)ndash(3) are thesolutions of

Iprime Aprime Dprime 0 (9)

for the disagree-free equilibrium when there is no negativeopinion ie if we put D 0 )is gives

Table 1 Parameter descriptions

Parameter Descriptionβ1 Ignorant to agree transmission rateβ2 Ignorant to disagree transmission rateα1 Disagree to agree transmission rateα2 Agree to disagree transmission ratec1 Interest loss factor of agree individualsc2 Interest loss factor of disagree individuals

I D

A

γ1A

β1(AIN)

β2(DIN)

γ2D

α1(ADN)

α2(ADN)

Figure 1 Flowchart for models (1)ndash(3)

International Journal of Differential Equations 3

Ilowast

c1

β1N

Alowast

N β1 minus c1( 1113857

β1

(10)

where Ilowast and Alowast represent the numbers of ignorant andagree individuals respectively in the absence of disagreepeople

)erefore for the system governed by (1)ndash(3) the dis-agree-free equilibrium is

e1 c1

β1N

N β1 minus c1( 1113857

β1 01113888 1113889 (11)

Following the second generation approach [22] wecompute the threshold RD0 associated to the disagree-freeequilibrium which is

RD0 α2β1 minus α2c1 + β2c1

α1β1 minus α1c1 + β1c2(12)

for the agree-free equilibrium when there is no positiveopinion ie if we put A 0 )is gives

Ilowast

c2

β2N

Dlowast

N β2 minus c2( 1113857

β2

(13)

where Ilowast and Dlowast represent the numbers of ignorant anddisagree individuals respectively in the absence of agreepeople

)erefore for the system governed by (1)ndash(3) the agree-free equilibrium is

e2 c2

β2N 0

N β2 minus c2( 1113857

β21113888 1113889 (14)

By also following the second generation approach [22]we compute the threshold RA0 associated to the agree-freeequilibrium which is

RA0 α1β2 minus α1c2 + β1c2

α2β2 minus α2c2 + β2c1 (15)

)roughout this paper we consider the following as-sumption the model parameters verify

β1 minus c1 gt minusβ1c2

α1

β1 minus c1 gt minusβ2c1

α2

β2 minus c2 gt minusβ2c1

α2

β2 minus c2 gt minusβ1c2

α1

(16)

)erefore from the above assumption we get

RD0 α2β1 minus α2c1 + β2c1

α1β1 minus α1c1 + β1c2gt 0

RA0 α1β2 minus α1c2 + β1c2

α2β2 minus α2c2 + β2c1gt 0

(17)

4 Stability Analysis

In order to analyze in terms of the proportions of ignorantagree and disagree individuals let i (IN) a (AN)and d (DN) denote the fraction of the classes I A and Din the population respectively After some calculations andreplacing I by i A by a and D by d equations (1)ndash(3) can bewritten as

iprime minus β1ai minus β2di + c1a + c2d (18)

aprime β1ai + α1ad minus α2ad minus c1a (19)

dprime β2di + α2ad minus α1ad minus c2d (20)

From the fact N I + A + D we have i + a + d 1 )enmodel systems (18)ndash(20) will be reduced to the following twodifferential equations

aprime β1a(1 minus a minus d) + α1ad minus α2ad minus c1a

dprime β2d(1 minus a minus d) + α2ad minus α1ad minus c2d(21)

which can be reduced to

aprime β1a(1 minus a minus d) + αa c1a

dprime β2d(1 minus a minus d) minus αad minus c2d(22)

where α α1 minus α2

41 Steady States )e steady states of system (22) are ob-tained by solving the system of equations

0 β1a(1 minus a minus d) + αad minus c1a

0 β2d(1 minus a minus d) minus αad minus c2d(23)

)is system has four equilibrium points and the trivialequilibrium E0 (0 0) is an equilibrium that exists alwayswithout any condition It means there is no survey and thereis no need to opinions

One disagree-free equilibrium E1 (1 minus (c1β1) 0) ex-ists if the condition β1 gt c1 holds and the agree-freeequilibrium E2 (0 1 minus (c2β2)) exists if β2 gt c2 holds

)e fourth and the positive equilibrium Elowast (alowast dlowast)where

alowast

αβ2 minus αc2 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

dlowast

minusαβ1 minus αc1 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

(24)

exists if one of the following conditions holdsldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo or ldquoRD0 lt 1

4 International Journal of Differential Equations

andRA0 lt 1 and α(α minus β1 + β2)lt 0rdquo In fact by a simplecalculation we get

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(25)

)us from (25) we deduce that alowast gt 0 and dlowast gt 0)roughout the article and without loss of generality we justconsider the first condition as a sufficient condition for theexistence of Elowast

42 Stability of Steady States )e Jacobian matrix of system(22) is

J J11 aα minus aβ1

minus αd minus β2d J221113888 1113889 (26)

where

J11 αd minus aβ1 minus c1 minus β1(a + d minus 1)

J22 minus c2 minus aα minus β2d minus β2(a + d minus 1)(27)

Proposition 1 1e equilibrium E0 (0 0) is unstable ifβ1 gt c1 or β2 gt c2 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E0( 1113857 β1 minus c1 0

0 β2 minus c21113888 1113889 (28)

It is clear that if β1 gt c1 or β2 gt c2 we get one positiveeigenvalue of J(E0) and then E0 is unstable Else we have alleigenvalues of J(E0) having the negative real part whichcompletes the proof

Remark 1 Note that the conditions in the previous prop-osition imply the existence of E1 or E2 )erefore E0 isunstable whenever there exists E1 or E2

Proposition 2 1e equilibrium E1 (1 minus (c1β1) 0) isunstable if RD0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E1( 1113857

c1 minus β1α minus β1( 1113857 β1 minus c1( 1113857

β1

0 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)

)e eigenvalues of J(E1) are

λ1 c1 minus β1

λ2 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

minusαβ1 minus αc1 + β1c2 minus β2c1

β1

(30)

It is clear from the existence condition of this equilib-rium that c1 minus β1 lt 0 )us the stability of the point ofequilibrium E1 is based on the eigenvalue λ2 of the matrixJ(E1) By a simple calculation we have

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1 (31)

)is implies that if RD0 gt 1 E1 is unstable else E1 isstable

Proposition 3 1e equilibrium E2 (0 1 minus (c2β2)) isunstable if RA0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E2( 1113857

β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1 0

minusα + β2( 1113857 β2 minus c2( 1113857

β2c2 minus β2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(32)

)e eigenvalues of J(E2) are

λ1 β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1

αβ2 minus αc2 + β1c2 minus β2c1

β2

λ2 c2 minus β2

(33)

It is clear from the existence condition of this equilib-rium that c2 minus β2 lt 0 )us the stability of the point ofequilibrium E2 is based on the eigenvalue λ1 of the matrixJ(E2) By a simple calculation we have

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1 (34)

)is implies that if RA0 gt 1 E2 is unstable else E2 isstable

Proposition 4 1e equilibrium Elowast (alowast dlowast) where

alowast

αβ2 minus αc2 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

dlowast

minusαβ1 minus αc1 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

(35)

is stable if ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gt c1 and c2 gtβ2 and α lt 0rdquo

Proof )e Jacobian matrix at this equilibrium is

J Elowast

( 1113857 J11 alowastα minus alowastβ1

minus αdlowast minus β2dlowast J221113888 1113889 (36)

where

J11 αdlowast

minus alowastβ1 minus c1 minus β1 a

lowast+ dlowast

minus 1( 1113857

J22 minus c2 minus alowastα minus β2d

lowastminus β2(a + d minus 1)

(37)

Using fact (23) we have

International Journal of Differential Equations 5

J11 minus β1alowast

J22 minus β2dlowast

(38)

)en

J Elowast

( 1113857 J1 J2

J3 J4

⎛⎝ ⎞⎠

J1 minusβ1 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J2 α minus β1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J3 α + β2( 1113857 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J4 β2 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

(39)

where the characteristic polynomial of is

λ2 + c1λ + c2 (40)

where

c1 minusβ1c2 minus β2c1

α

c2 minus 1

α α minus β1 + β2( 1113857

times α2β1β2 minus α2β1c2 minus α2β2c1 + α2c1c21113872

+ αβ21c2 minus αβ1β2c1 + αβ1β2c2 minus αβ1c1c2

minus αβ1c22 minus αβ22c1 + αβ2c

21 + αβ2c1c2

+ β21c22 minus 2β1β2c1c2 + β22c

211113873

minus 1

α α minus β1 + β2( 1113857

times αβ1 minus αc1 + β1c2 minus β2c1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

(41)

By the conditions ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gtc1 and c2 gt β2 and α lt 0rdquo we have minus (β1c2 minus β2c1α)gt 0 andby existence conditions and

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(42)

we have c1 gt 0 and c2 gt 0 Using the RouthndashHurwitz stabilitycriterion we conclude that the equilibrium point Elowast is lo-cally asymptotically stable

Remark 2

(1) Note that the conditions RD0 gt 1 and RA0 gt 1 lead to theinstability of E1 and E2 and help in the existence of Elowast

(2) Sufficient conditions in the above proposition couldbe reduced to α (β1c2 ndash β1c2) lt 0

A summary of sufficient existence and stability condi-tions of the steady states of model (11) is given in Table 2

43 Examples It can be seen from Table 3 that the equi-librium E0 exists and it is stable in the four cases example 1RD0 gt 1 and RA0 gt 1 example 2 RA0 lt 1 and RD0 lt 1 example3 RD0 lt 1 and RA0 gt 1 and example 4 RD0 gt 1 and RA0 lt 1

Example 5 shows the existence and the stability of thesteady state E1 in the case of RA0 lt 1 and RD0 lt 1 and in the caseofRA0gt 1 andRD0lt 1 in example 6 Examples 7 and 8 show theexistence and the stability of the equilibrium E2 in the cases ofldquoRA0 lt 1 and RD0 lt 1rdquo and ldquoRA0 lt 1 and RD0 gt 1rdquo respectively

Examples 9 and 10 show the possibility of the existence andstability of the two steady states E1 and E2 at the same time thatis ldquoβ1 gt c1 and β2 gt c2 and RA0 lt 1 and RD0 lt 1rdquo )eseexamples give an insight about the stability of each equilibriumfor given parametersrsquo values After some numerical calcula-tions we noted that in this situation the parameters c1 and c2may switch from E1 to E2 and the inverse while if c1 gt c2 thenE1 will be more attractive and if c2 gt c1 then E2 will be moreattractive and that with the same initial conditions We sim-ulate the model with different initial conditions to illustrate theimpact of initial conditions on the stability of E1 and E2 in thissituation Figure 2 depicts the stability of E1 and E2 at the sametime where we consider the same set of parameters fromexample 9 in Table 3 By changing the initial conditions we cansee that E1 is stable when I(0) 100 A(0) 100 andD(0) 80 while it can be seen thatE2 is also stablewith this setof parameters but after choosing I(0) 100 A(0) 80 andD(0) 100

Example 11 shows the existence and the stability of theequilibrium Elowast where ldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1+β2)gt 0rdquo and ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo while example 12shows the existence and the stability of Elowast whereldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo and ldquoβ1 gt c1and c2 gt β2 and α lt 0rdquo

Figure 3 depicts examples of the existence and stability ofthe equilibrium state E0 for different parametersrsquo values andthreshold RD0 and RA0 values simulated with initial con-ditions and parametersrsquo values from Table 3 Figure 4 depictsexamples of the existence and stability of the equilibriumstate E1 for different parametersrsquo values and threshold RD0and RA0 values simulated with initial conditions and pa-rametersrsquo values from Table 3 It can be seen fromFigure 4(b) that the functionD decreases towards zero but itwill take a long time In the Figure 4(c) we have considered1200 hours to show that the function D will go to zero butvery slowly Figure 5 depicts examples of the existence andstability of the equilibrium state E2 for different parametersrsquovalues and threshold RD0 and RA0 values simulated withinitial conditions and parametersrsquo values from Table 3 Wecan see from Figure 5(b) that the function A will also take avery long time to go to zero In Figure 5(c) we have con-sidered 1200 hours to show that function A will tend to zerobut very slowly Figure 6 depicts examples of the existenceand stability of the equilibrium state Elowast for different

6 International Journal of Differential Equations

parametersrsquo values and threshold RD0 and RA0 values sim-ulated with initial conditions and parametersrsquo values fromexamples 11 and 12 of Table 3

5 Analysis of Thresholds

51 No Abstention No Loss of Interest In most of the sit-uations people abstain from voting for personal reasonsHere we discuss the situation when there is no abstentionand there is no loss of interest of voting ie c1 c2 0)erefore RD0 and RA0 become

RD0 α2α1

RA0 α1α2

(43)

which means that thresholds RD0 and RA0 depend only on α1and α2 and the equilibria become

E1 (1 0)

E2 (0 1)(44)

and from the existence conditions in Table 2 we can deducethat there is no Elowast

E1 exists without any condition and it is stable if α2 lt α1and E2 exists without any condition and it is stable if α1 lt α2

)is result explains the effect of the polarization on theoutcome of polls When there is no abstention from votingand people keep their interest in voting then the mostinfluential parameters on the outcome of the polls are thepolarization factors α1 and α2 For example during a polldetermining the political position of one candidate Y byvoting with Approve or Disapprove or otherwise if can-didate Y can convince people by his political vision andorby other ways then he can change the course of events in hisfavor Mathematically he makes α2 lt α1 But if somehow hecontributes to making α1 lt α2 then things could spin out ofcontrol on the election day

Table 2 Summary of sufficient existence and stability conditions of equilibria

Equilibrium Existence conditions Stability conditionsE0 mdash β1 lt c1 and β2 lt c2E1 (1 minus (c1β1) 0) β1 gt c1 RD0 lt 1E2 (0 1 minus (c2β2)) β2 gt c2 RA0 lt 1Elowast (alowast dlowast) RD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0 β1 lt c1 and c2 lt β2 and α gt 0 or β1 gt c1 and c2 gt β2 and α lt 0

Table 3 Steady states and parametersrsquo values used in example simulations where I(0) A(0) D(0) 100

β1 β2 α1 α2 c1 c2 RD0 RA0

1 E0 (300 0 0) 00010 01010 01010 03010 05010 03010 19901 207302 E0 (300 0 0) 00010 02010 02010 03010 01010 05010 05000 085433 E0 (300 0 0) 00010 00010 02010 01010 01010 01010 05000 200014 E0 (300 0 0) 00010 00010 01010 02010 01010 01010 20001 050005 E1 (29703 2970297 0) 01010 00010 02010 02010 00010 00010 06634 049256 E1 (29703 2970297 0) 01010 00010 00010 00010 00010 00010 05025 1017 E2 (29703 0 2970297) 00010 01010 02010 02010 01010 00010 04925 066348 E2 (29703 0 2970297) 00010 01010 00010 00010 00010 00010 101 050259 Elowast1 (1203593 1796407 0) 05010 01010 01010 00010 02010 00010 06688 0519610 Elowast2 (1203593 0 1796407) 01010 05010 00010 01010 00010 02010 05196 0668811 Elowast (1333333 105556 1561111) 01010 07010 03010 00010 02010 03010 4677774 1067212 Elowast (2142857 804643 52500) 03010 04010 00010 08010 02010 05010 10649 3008004

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E1 = (112 168 0)

IAD

(a)

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E2 = (2 0 278)

IAD

(b)

Figure 2 Illustration of the bistability of E1 and E2 at the same time using the set of parameters of example 9 from Table 3 Stability of E1 (a)when I(0) 100 A(0) 100 and D(0) 80 Stability of E2 (b) when I(0) 100 A(0) 80 and D(0) 100

International Journal of Differential Equations 7

0 20 40 60 80 100 120Time (hours)

050

100150200250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(b)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(c)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(d)

Figure 3 Examples of the steady state E0 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 1 (b) Example 2 (c) Example 3 (d) Example 4

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(b)

Figure 4 Continued

8 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

Ilowast

c1

β1N

Alowast

N β1 minus c1( 1113857

β1

(10)

where Ilowast and Alowast represent the numbers of ignorant andagree individuals respectively in the absence of disagreepeople

)erefore for the system governed by (1)ndash(3) the dis-agree-free equilibrium is

e1 c1

β1N

N β1 minus c1( 1113857

β1 01113888 1113889 (11)

Following the second generation approach [22] wecompute the threshold RD0 associated to the disagree-freeequilibrium which is

RD0 α2β1 minus α2c1 + β2c1

α1β1 minus α1c1 + β1c2(12)

for the agree-free equilibrium when there is no positiveopinion ie if we put A 0 )is gives

Ilowast

c2

β2N

Dlowast

N β2 minus c2( 1113857

β2

(13)

where Ilowast and Dlowast represent the numbers of ignorant anddisagree individuals respectively in the absence of agreepeople

)erefore for the system governed by (1)ndash(3) the agree-free equilibrium is

e2 c2

β2N 0

N β2 minus c2( 1113857

β21113888 1113889 (14)

By also following the second generation approach [22]we compute the threshold RA0 associated to the agree-freeequilibrium which is

RA0 α1β2 minus α1c2 + β1c2

α2β2 minus α2c2 + β2c1 (15)

)roughout this paper we consider the following as-sumption the model parameters verify

β1 minus c1 gt minusβ1c2

α1

β1 minus c1 gt minusβ2c1

α2

β2 minus c2 gt minusβ2c1

α2

β2 minus c2 gt minusβ1c2

α1

(16)

)erefore from the above assumption we get

RD0 α2β1 minus α2c1 + β2c1

α1β1 minus α1c1 + β1c2gt 0

RA0 α1β2 minus α1c2 + β1c2

α2β2 minus α2c2 + β2c1gt 0

(17)

4 Stability Analysis

In order to analyze in terms of the proportions of ignorantagree and disagree individuals let i (IN) a (AN)and d (DN) denote the fraction of the classes I A and Din the population respectively After some calculations andreplacing I by i A by a and D by d equations (1)ndash(3) can bewritten as

iprime minus β1ai minus β2di + c1a + c2d (18)

aprime β1ai + α1ad minus α2ad minus c1a (19)

dprime β2di + α2ad minus α1ad minus c2d (20)

From the fact N I + A + D we have i + a + d 1 )enmodel systems (18)ndash(20) will be reduced to the following twodifferential equations

aprime β1a(1 minus a minus d) + α1ad minus α2ad minus c1a

dprime β2d(1 minus a minus d) + α2ad minus α1ad minus c2d(21)

which can be reduced to

aprime β1a(1 minus a minus d) + αa c1a

dprime β2d(1 minus a minus d) minus αad minus c2d(22)

where α α1 minus α2

41 Steady States )e steady states of system (22) are ob-tained by solving the system of equations

0 β1a(1 minus a minus d) + αad minus c1a

0 β2d(1 minus a minus d) minus αad minus c2d(23)

)is system has four equilibrium points and the trivialequilibrium E0 (0 0) is an equilibrium that exists alwayswithout any condition It means there is no survey and thereis no need to opinions

One disagree-free equilibrium E1 (1 minus (c1β1) 0) ex-ists if the condition β1 gt c1 holds and the agree-freeequilibrium E2 (0 1 minus (c2β2)) exists if β2 gt c2 holds

)e fourth and the positive equilibrium Elowast (alowast dlowast)where

alowast

αβ2 minus αc2 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

dlowast

minusαβ1 minus αc1 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

(24)

exists if one of the following conditions holdsldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo or ldquoRD0 lt 1

4 International Journal of Differential Equations

andRA0 lt 1 and α(α minus β1 + β2)lt 0rdquo In fact by a simplecalculation we get

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(25)

)us from (25) we deduce that alowast gt 0 and dlowast gt 0)roughout the article and without loss of generality we justconsider the first condition as a sufficient condition for theexistence of Elowast

42 Stability of Steady States )e Jacobian matrix of system(22) is

J J11 aα minus aβ1

minus αd minus β2d J221113888 1113889 (26)

where

J11 αd minus aβ1 minus c1 minus β1(a + d minus 1)

J22 minus c2 minus aα minus β2d minus β2(a + d minus 1)(27)

Proposition 1 1e equilibrium E0 (0 0) is unstable ifβ1 gt c1 or β2 gt c2 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E0( 1113857 β1 minus c1 0

0 β2 minus c21113888 1113889 (28)

It is clear that if β1 gt c1 or β2 gt c2 we get one positiveeigenvalue of J(E0) and then E0 is unstable Else we have alleigenvalues of J(E0) having the negative real part whichcompletes the proof

Remark 1 Note that the conditions in the previous prop-osition imply the existence of E1 or E2 )erefore E0 isunstable whenever there exists E1 or E2

Proposition 2 1e equilibrium E1 (1 minus (c1β1) 0) isunstable if RD0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E1( 1113857

c1 minus β1α minus β1( 1113857 β1 minus c1( 1113857

β1

0 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)

)e eigenvalues of J(E1) are

λ1 c1 minus β1

λ2 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

minusαβ1 minus αc1 + β1c2 minus β2c1

β1

(30)

It is clear from the existence condition of this equilib-rium that c1 minus β1 lt 0 )us the stability of the point ofequilibrium E1 is based on the eigenvalue λ2 of the matrixJ(E1) By a simple calculation we have

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1 (31)

)is implies that if RD0 gt 1 E1 is unstable else E1 isstable

Proposition 3 1e equilibrium E2 (0 1 minus (c2β2)) isunstable if RA0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E2( 1113857

β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1 0

minusα + β2( 1113857 β2 minus c2( 1113857

β2c2 minus β2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(32)

)e eigenvalues of J(E2) are

λ1 β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1

αβ2 minus αc2 + β1c2 minus β2c1

β2

λ2 c2 minus β2

(33)

It is clear from the existence condition of this equilib-rium that c2 minus β2 lt 0 )us the stability of the point ofequilibrium E2 is based on the eigenvalue λ1 of the matrixJ(E2) By a simple calculation we have

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1 (34)

)is implies that if RA0 gt 1 E2 is unstable else E2 isstable

Proposition 4 1e equilibrium Elowast (alowast dlowast) where

alowast

αβ2 minus αc2 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

dlowast

minusαβ1 minus αc1 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

(35)

is stable if ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gt c1 and c2 gtβ2 and α lt 0rdquo

Proof )e Jacobian matrix at this equilibrium is

J Elowast

( 1113857 J11 alowastα minus alowastβ1

minus αdlowast minus β2dlowast J221113888 1113889 (36)

where

J11 αdlowast

minus alowastβ1 minus c1 minus β1 a

lowast+ dlowast

minus 1( 1113857

J22 minus c2 minus alowastα minus β2d

lowastminus β2(a + d minus 1)

(37)

Using fact (23) we have

International Journal of Differential Equations 5

J11 minus β1alowast

J22 minus β2dlowast

(38)

)en

J Elowast

( 1113857 J1 J2

J3 J4

⎛⎝ ⎞⎠

J1 minusβ1 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J2 α minus β1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J3 α + β2( 1113857 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J4 β2 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

(39)

where the characteristic polynomial of is

λ2 + c1λ + c2 (40)

where

c1 minusβ1c2 minus β2c1

α

c2 minus 1

α α minus β1 + β2( 1113857

times α2β1β2 minus α2β1c2 minus α2β2c1 + α2c1c21113872

+ αβ21c2 minus αβ1β2c1 + αβ1β2c2 minus αβ1c1c2

minus αβ1c22 minus αβ22c1 + αβ2c

21 + αβ2c1c2

+ β21c22 minus 2β1β2c1c2 + β22c

211113873

minus 1

α α minus β1 + β2( 1113857

times αβ1 minus αc1 + β1c2 minus β2c1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

(41)

By the conditions ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gtc1 and c2 gt β2 and α lt 0rdquo we have minus (β1c2 minus β2c1α)gt 0 andby existence conditions and

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(42)

we have c1 gt 0 and c2 gt 0 Using the RouthndashHurwitz stabilitycriterion we conclude that the equilibrium point Elowast is lo-cally asymptotically stable

Remark 2

(1) Note that the conditions RD0 gt 1 and RA0 gt 1 lead to theinstability of E1 and E2 and help in the existence of Elowast

(2) Sufficient conditions in the above proposition couldbe reduced to α (β1c2 ndash β1c2) lt 0

A summary of sufficient existence and stability condi-tions of the steady states of model (11) is given in Table 2

43 Examples It can be seen from Table 3 that the equi-librium E0 exists and it is stable in the four cases example 1RD0 gt 1 and RA0 gt 1 example 2 RA0 lt 1 and RD0 lt 1 example3 RD0 lt 1 and RA0 gt 1 and example 4 RD0 gt 1 and RA0 lt 1

Example 5 shows the existence and the stability of thesteady state E1 in the case of RA0 lt 1 and RD0 lt 1 and in the caseofRA0gt 1 andRD0lt 1 in example 6 Examples 7 and 8 show theexistence and the stability of the equilibrium E2 in the cases ofldquoRA0 lt 1 and RD0 lt 1rdquo and ldquoRA0 lt 1 and RD0 gt 1rdquo respectively

Examples 9 and 10 show the possibility of the existence andstability of the two steady states E1 and E2 at the same time thatis ldquoβ1 gt c1 and β2 gt c2 and RA0 lt 1 and RD0 lt 1rdquo )eseexamples give an insight about the stability of each equilibriumfor given parametersrsquo values After some numerical calcula-tions we noted that in this situation the parameters c1 and c2may switch from E1 to E2 and the inverse while if c1 gt c2 thenE1 will be more attractive and if c2 gt c1 then E2 will be moreattractive and that with the same initial conditions We sim-ulate the model with different initial conditions to illustrate theimpact of initial conditions on the stability of E1 and E2 in thissituation Figure 2 depicts the stability of E1 and E2 at the sametime where we consider the same set of parameters fromexample 9 in Table 3 By changing the initial conditions we cansee that E1 is stable when I(0) 100 A(0) 100 andD(0) 80 while it can be seen thatE2 is also stablewith this setof parameters but after choosing I(0) 100 A(0) 80 andD(0) 100

Example 11 shows the existence and the stability of theequilibrium Elowast where ldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1+β2)gt 0rdquo and ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo while example 12shows the existence and the stability of Elowast whereldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo and ldquoβ1 gt c1and c2 gt β2 and α lt 0rdquo

Figure 3 depicts examples of the existence and stability ofthe equilibrium state E0 for different parametersrsquo values andthreshold RD0 and RA0 values simulated with initial con-ditions and parametersrsquo values from Table 3 Figure 4 depictsexamples of the existence and stability of the equilibriumstate E1 for different parametersrsquo values and threshold RD0and RA0 values simulated with initial conditions and pa-rametersrsquo values from Table 3 It can be seen fromFigure 4(b) that the functionD decreases towards zero but itwill take a long time In the Figure 4(c) we have considered1200 hours to show that the function D will go to zero butvery slowly Figure 5 depicts examples of the existence andstability of the equilibrium state E2 for different parametersrsquovalues and threshold RD0 and RA0 values simulated withinitial conditions and parametersrsquo values from Table 3 Wecan see from Figure 5(b) that the function A will also take avery long time to go to zero In Figure 5(c) we have con-sidered 1200 hours to show that function A will tend to zerobut very slowly Figure 6 depicts examples of the existenceand stability of the equilibrium state Elowast for different

6 International Journal of Differential Equations

parametersrsquo values and threshold RD0 and RA0 values sim-ulated with initial conditions and parametersrsquo values fromexamples 11 and 12 of Table 3

5 Analysis of Thresholds

51 No Abstention No Loss of Interest In most of the sit-uations people abstain from voting for personal reasonsHere we discuss the situation when there is no abstentionand there is no loss of interest of voting ie c1 c2 0)erefore RD0 and RA0 become

RD0 α2α1

RA0 α1α2

(43)

which means that thresholds RD0 and RA0 depend only on α1and α2 and the equilibria become

E1 (1 0)

E2 (0 1)(44)

and from the existence conditions in Table 2 we can deducethat there is no Elowast

E1 exists without any condition and it is stable if α2 lt α1and E2 exists without any condition and it is stable if α1 lt α2

)is result explains the effect of the polarization on theoutcome of polls When there is no abstention from votingand people keep their interest in voting then the mostinfluential parameters on the outcome of the polls are thepolarization factors α1 and α2 For example during a polldetermining the political position of one candidate Y byvoting with Approve or Disapprove or otherwise if can-didate Y can convince people by his political vision andorby other ways then he can change the course of events in hisfavor Mathematically he makes α2 lt α1 But if somehow hecontributes to making α1 lt α2 then things could spin out ofcontrol on the election day

Table 2 Summary of sufficient existence and stability conditions of equilibria

Equilibrium Existence conditions Stability conditionsE0 mdash β1 lt c1 and β2 lt c2E1 (1 minus (c1β1) 0) β1 gt c1 RD0 lt 1E2 (0 1 minus (c2β2)) β2 gt c2 RA0 lt 1Elowast (alowast dlowast) RD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0 β1 lt c1 and c2 lt β2 and α gt 0 or β1 gt c1 and c2 gt β2 and α lt 0

Table 3 Steady states and parametersrsquo values used in example simulations where I(0) A(0) D(0) 100

β1 β2 α1 α2 c1 c2 RD0 RA0

1 E0 (300 0 0) 00010 01010 01010 03010 05010 03010 19901 207302 E0 (300 0 0) 00010 02010 02010 03010 01010 05010 05000 085433 E0 (300 0 0) 00010 00010 02010 01010 01010 01010 05000 200014 E0 (300 0 0) 00010 00010 01010 02010 01010 01010 20001 050005 E1 (29703 2970297 0) 01010 00010 02010 02010 00010 00010 06634 049256 E1 (29703 2970297 0) 01010 00010 00010 00010 00010 00010 05025 1017 E2 (29703 0 2970297) 00010 01010 02010 02010 01010 00010 04925 066348 E2 (29703 0 2970297) 00010 01010 00010 00010 00010 00010 101 050259 Elowast1 (1203593 1796407 0) 05010 01010 01010 00010 02010 00010 06688 0519610 Elowast2 (1203593 0 1796407) 01010 05010 00010 01010 00010 02010 05196 0668811 Elowast (1333333 105556 1561111) 01010 07010 03010 00010 02010 03010 4677774 1067212 Elowast (2142857 804643 52500) 03010 04010 00010 08010 02010 05010 10649 3008004

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E1 = (112 168 0)

IAD

(a)

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E2 = (2 0 278)

IAD

(b)

Figure 2 Illustration of the bistability of E1 and E2 at the same time using the set of parameters of example 9 from Table 3 Stability of E1 (a)when I(0) 100 A(0) 100 and D(0) 80 Stability of E2 (b) when I(0) 100 A(0) 80 and D(0) 100

International Journal of Differential Equations 7

0 20 40 60 80 100 120Time (hours)

050

100150200250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(b)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(c)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(d)

Figure 3 Examples of the steady state E0 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 1 (b) Example 2 (c) Example 3 (d) Example 4

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(b)

Figure 4 Continued

8 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

andRA0 lt 1 and α(α minus β1 + β2)lt 0rdquo In fact by a simplecalculation we get

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(25)

)us from (25) we deduce that alowast gt 0 and dlowast gt 0)roughout the article and without loss of generality we justconsider the first condition as a sufficient condition for theexistence of Elowast

42 Stability of Steady States )e Jacobian matrix of system(22) is

J J11 aα minus aβ1

minus αd minus β2d J221113888 1113889 (26)

where

J11 αd minus aβ1 minus c1 minus β1(a + d minus 1)

J22 minus c2 minus aα minus β2d minus β2(a + d minus 1)(27)

Proposition 1 1e equilibrium E0 (0 0) is unstable ifβ1 gt c1 or β2 gt c2 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E0( 1113857 β1 minus c1 0

0 β2 minus c21113888 1113889 (28)

It is clear that if β1 gt c1 or β2 gt c2 we get one positiveeigenvalue of J(E0) and then E0 is unstable Else we have alleigenvalues of J(E0) having the negative real part whichcompletes the proof

Remark 1 Note that the conditions in the previous prop-osition imply the existence of E1 or E2 )erefore E0 isunstable whenever there exists E1 or E2

Proposition 2 1e equilibrium E1 (1 minus (c1β1) 0) isunstable if RD0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E1( 1113857

c1 minus β1α minus β1( 1113857 β1 minus c1( 1113857

β1

0 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)

)e eigenvalues of J(E1) are

λ1 c1 minus β1

λ2 αc1

β1minus 11113888 1113889 minus c2 +

β2c1

β1

minusαβ1 minus αc1 + β1c2 minus β2c1

β1

(30)

It is clear from the existence condition of this equilib-rium that c1 minus β1 lt 0 )us the stability of the point ofequilibrium E1 is based on the eigenvalue λ2 of the matrixJ(E1) By a simple calculation we have

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1 (31)

)is implies that if RD0 gt 1 E1 is unstable else E1 isstable

Proposition 3 1e equilibrium E2 (0 1 minus (c2β2)) isunstable if RA0 gt 1 Otherwise it is stable

Proof )e Jacobian matrix at this equilibrium is

J E2( 1113857

β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1 0

minusα + β2( 1113857 β2 minus c2( 1113857

β2c2 minus β2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(32)

)e eigenvalues of J(E2) are

λ1 β1c2

β2minus α

c2

β2minus 11113888 1113889 minus c1

αβ2 minus αc2 + β1c2 minus β2c1

β2

λ2 c2 minus β2

(33)

It is clear from the existence condition of this equilib-rium that c2 minus β2 lt 0 )us the stability of the point ofequilibrium E2 is based on the eigenvalue λ1 of the matrixJ(E2) By a simple calculation we have

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1 (34)

)is implies that if RA0 gt 1 E2 is unstable else E2 isstable

Proposition 4 1e equilibrium Elowast (alowast dlowast) where

alowast

αβ2 minus αc2 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

dlowast

minusαβ1 minus αc1 + β1c2 minus β2c1

α α minus β1 + β2( 1113857

(35)

is stable if ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gt c1 and c2 gtβ2 and α lt 0rdquo

Proof )e Jacobian matrix at this equilibrium is

J Elowast

( 1113857 J11 alowastα minus alowastβ1

minus αdlowast minus β2dlowast J221113888 1113889 (36)

where

J11 αdlowast

minus alowastβ1 minus c1 minus β1 a

lowast+ dlowast

minus 1( 1113857

J22 minus c2 minus alowastα minus β2d

lowastminus β2(a + d minus 1)

(37)

Using fact (23) we have

International Journal of Differential Equations 5

J11 minus β1alowast

J22 minus β2dlowast

(38)

)en

J Elowast

( 1113857 J1 J2

J3 J4

⎛⎝ ⎞⎠

J1 minusβ1 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J2 α minus β1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J3 α + β2( 1113857 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J4 β2 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

(39)

where the characteristic polynomial of is

λ2 + c1λ + c2 (40)

where

c1 minusβ1c2 minus β2c1

α

c2 minus 1

α α minus β1 + β2( 1113857

times α2β1β2 minus α2β1c2 minus α2β2c1 + α2c1c21113872

+ αβ21c2 minus αβ1β2c1 + αβ1β2c2 minus αβ1c1c2

minus αβ1c22 minus αβ22c1 + αβ2c

21 + αβ2c1c2

+ β21c22 minus 2β1β2c1c2 + β22c

211113873

minus 1

α α minus β1 + β2( 1113857

times αβ1 minus αc1 + β1c2 minus β2c1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

(41)

By the conditions ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gtc1 and c2 gt β2 and α lt 0rdquo we have minus (β1c2 minus β2c1α)gt 0 andby existence conditions and

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(42)

we have c1 gt 0 and c2 gt 0 Using the RouthndashHurwitz stabilitycriterion we conclude that the equilibrium point Elowast is lo-cally asymptotically stable

Remark 2

(1) Note that the conditions RD0 gt 1 and RA0 gt 1 lead to theinstability of E1 and E2 and help in the existence of Elowast

(2) Sufficient conditions in the above proposition couldbe reduced to α (β1c2 ndash β1c2) lt 0

A summary of sufficient existence and stability condi-tions of the steady states of model (11) is given in Table 2

43 Examples It can be seen from Table 3 that the equi-librium E0 exists and it is stable in the four cases example 1RD0 gt 1 and RA0 gt 1 example 2 RA0 lt 1 and RD0 lt 1 example3 RD0 lt 1 and RA0 gt 1 and example 4 RD0 gt 1 and RA0 lt 1

Example 5 shows the existence and the stability of thesteady state E1 in the case of RA0 lt 1 and RD0 lt 1 and in the caseofRA0gt 1 andRD0lt 1 in example 6 Examples 7 and 8 show theexistence and the stability of the equilibrium E2 in the cases ofldquoRA0 lt 1 and RD0 lt 1rdquo and ldquoRA0 lt 1 and RD0 gt 1rdquo respectively

Examples 9 and 10 show the possibility of the existence andstability of the two steady states E1 and E2 at the same time thatis ldquoβ1 gt c1 and β2 gt c2 and RA0 lt 1 and RD0 lt 1rdquo )eseexamples give an insight about the stability of each equilibriumfor given parametersrsquo values After some numerical calcula-tions we noted that in this situation the parameters c1 and c2may switch from E1 to E2 and the inverse while if c1 gt c2 thenE1 will be more attractive and if c2 gt c1 then E2 will be moreattractive and that with the same initial conditions We sim-ulate the model with different initial conditions to illustrate theimpact of initial conditions on the stability of E1 and E2 in thissituation Figure 2 depicts the stability of E1 and E2 at the sametime where we consider the same set of parameters fromexample 9 in Table 3 By changing the initial conditions we cansee that E1 is stable when I(0) 100 A(0) 100 andD(0) 80 while it can be seen thatE2 is also stablewith this setof parameters but after choosing I(0) 100 A(0) 80 andD(0) 100

Example 11 shows the existence and the stability of theequilibrium Elowast where ldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1+β2)gt 0rdquo and ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo while example 12shows the existence and the stability of Elowast whereldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo and ldquoβ1 gt c1and c2 gt β2 and α lt 0rdquo

Figure 3 depicts examples of the existence and stability ofthe equilibrium state E0 for different parametersrsquo values andthreshold RD0 and RA0 values simulated with initial con-ditions and parametersrsquo values from Table 3 Figure 4 depictsexamples of the existence and stability of the equilibriumstate E1 for different parametersrsquo values and threshold RD0and RA0 values simulated with initial conditions and pa-rametersrsquo values from Table 3 It can be seen fromFigure 4(b) that the functionD decreases towards zero but itwill take a long time In the Figure 4(c) we have considered1200 hours to show that the function D will go to zero butvery slowly Figure 5 depicts examples of the existence andstability of the equilibrium state E2 for different parametersrsquovalues and threshold RD0 and RA0 values simulated withinitial conditions and parametersrsquo values from Table 3 Wecan see from Figure 5(b) that the function A will also take avery long time to go to zero In Figure 5(c) we have con-sidered 1200 hours to show that function A will tend to zerobut very slowly Figure 6 depicts examples of the existenceand stability of the equilibrium state Elowast for different

6 International Journal of Differential Equations

parametersrsquo values and threshold RD0 and RA0 values sim-ulated with initial conditions and parametersrsquo values fromexamples 11 and 12 of Table 3

5 Analysis of Thresholds

51 No Abstention No Loss of Interest In most of the sit-uations people abstain from voting for personal reasonsHere we discuss the situation when there is no abstentionand there is no loss of interest of voting ie c1 c2 0)erefore RD0 and RA0 become

RD0 α2α1

RA0 α1α2

(43)

which means that thresholds RD0 and RA0 depend only on α1and α2 and the equilibria become

E1 (1 0)

E2 (0 1)(44)

and from the existence conditions in Table 2 we can deducethat there is no Elowast

E1 exists without any condition and it is stable if α2 lt α1and E2 exists without any condition and it is stable if α1 lt α2

)is result explains the effect of the polarization on theoutcome of polls When there is no abstention from votingand people keep their interest in voting then the mostinfluential parameters on the outcome of the polls are thepolarization factors α1 and α2 For example during a polldetermining the political position of one candidate Y byvoting with Approve or Disapprove or otherwise if can-didate Y can convince people by his political vision andorby other ways then he can change the course of events in hisfavor Mathematically he makes α2 lt α1 But if somehow hecontributes to making α1 lt α2 then things could spin out ofcontrol on the election day

Table 2 Summary of sufficient existence and stability conditions of equilibria

Equilibrium Existence conditions Stability conditionsE0 mdash β1 lt c1 and β2 lt c2E1 (1 minus (c1β1) 0) β1 gt c1 RD0 lt 1E2 (0 1 minus (c2β2)) β2 gt c2 RA0 lt 1Elowast (alowast dlowast) RD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0 β1 lt c1 and c2 lt β2 and α gt 0 or β1 gt c1 and c2 gt β2 and α lt 0

Table 3 Steady states and parametersrsquo values used in example simulations where I(0) A(0) D(0) 100

β1 β2 α1 α2 c1 c2 RD0 RA0

1 E0 (300 0 0) 00010 01010 01010 03010 05010 03010 19901 207302 E0 (300 0 0) 00010 02010 02010 03010 01010 05010 05000 085433 E0 (300 0 0) 00010 00010 02010 01010 01010 01010 05000 200014 E0 (300 0 0) 00010 00010 01010 02010 01010 01010 20001 050005 E1 (29703 2970297 0) 01010 00010 02010 02010 00010 00010 06634 049256 E1 (29703 2970297 0) 01010 00010 00010 00010 00010 00010 05025 1017 E2 (29703 0 2970297) 00010 01010 02010 02010 01010 00010 04925 066348 E2 (29703 0 2970297) 00010 01010 00010 00010 00010 00010 101 050259 Elowast1 (1203593 1796407 0) 05010 01010 01010 00010 02010 00010 06688 0519610 Elowast2 (1203593 0 1796407) 01010 05010 00010 01010 00010 02010 05196 0668811 Elowast (1333333 105556 1561111) 01010 07010 03010 00010 02010 03010 4677774 1067212 Elowast (2142857 804643 52500) 03010 04010 00010 08010 02010 05010 10649 3008004

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E1 = (112 168 0)

IAD

(a)

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E2 = (2 0 278)

IAD

(b)

Figure 2 Illustration of the bistability of E1 and E2 at the same time using the set of parameters of example 9 from Table 3 Stability of E1 (a)when I(0) 100 A(0) 100 and D(0) 80 Stability of E2 (b) when I(0) 100 A(0) 80 and D(0) 100

International Journal of Differential Equations 7

0 20 40 60 80 100 120Time (hours)

050

100150200250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(b)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(c)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(d)

Figure 3 Examples of the steady state E0 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 1 (b) Example 2 (c) Example 3 (d) Example 4

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(b)

Figure 4 Continued

8 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

J11 minus β1alowast

J22 minus β2dlowast

(38)

)en

J Elowast

( 1113857 J1 J2

J3 J4

⎛⎝ ⎞⎠

J1 minusβ1 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J2 α minus β1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J3 α + β2( 1113857 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

J4 β2 αβ1 minus αc1 + β1c2 minus β2c1( 1113857

α α minus β1 + β2( 1113857

(39)

where the characteristic polynomial of is

λ2 + c1λ + c2 (40)

where

c1 minusβ1c2 minus β2c1

α

c2 minus 1

α α minus β1 + β2( 1113857

times α2β1β2 minus α2β1c2 minus α2β2c1 + α2c1c21113872

+ αβ21c2 minus αβ1β2c1 + αβ1β2c2 minus αβ1c1c2

minus αβ1c22 minus αβ22c1 + αβ2c

21 + αβ2c1c2

+ β21c22 minus 2β1β2c1c2 + β22c

211113873

minus 1

α α minus β1 + β2( 1113857

times αβ1 minus αc1 + β1c2 minus β2c1( 1113857 αβ2 minus αc2 + β1c2 minus β2c1( 1113857

(41)

By the conditions ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo or ldquoβ1 gtc1 and c2 gt β2 and α lt 0rdquo we have minus (β1c2 minus β2c1α)gt 0 andby existence conditions and

αβ2 minus αc2 + β1c2 minus β2c1 gt 0⟺RA0 gt 1

αβ1 minus αc1 + β1c2 minus β2c1 lt 0⟺RD0 gt 1(42)

we have c1 gt 0 and c2 gt 0 Using the RouthndashHurwitz stabilitycriterion we conclude that the equilibrium point Elowast is lo-cally asymptotically stable

Remark 2

(1) Note that the conditions RD0 gt 1 and RA0 gt 1 lead to theinstability of E1 and E2 and help in the existence of Elowast

(2) Sufficient conditions in the above proposition couldbe reduced to α (β1c2 ndash β1c2) lt 0

A summary of sufficient existence and stability condi-tions of the steady states of model (11) is given in Table 2

43 Examples It can be seen from Table 3 that the equi-librium E0 exists and it is stable in the four cases example 1RD0 gt 1 and RA0 gt 1 example 2 RA0 lt 1 and RD0 lt 1 example3 RD0 lt 1 and RA0 gt 1 and example 4 RD0 gt 1 and RA0 lt 1

Example 5 shows the existence and the stability of thesteady state E1 in the case of RA0 lt 1 and RD0 lt 1 and in the caseofRA0gt 1 andRD0lt 1 in example 6 Examples 7 and 8 show theexistence and the stability of the equilibrium E2 in the cases ofldquoRA0 lt 1 and RD0 lt 1rdquo and ldquoRA0 lt 1 and RD0 gt 1rdquo respectively

Examples 9 and 10 show the possibility of the existence andstability of the two steady states E1 and E2 at the same time thatis ldquoβ1 gt c1 and β2 gt c2 and RA0 lt 1 and RD0 lt 1rdquo )eseexamples give an insight about the stability of each equilibriumfor given parametersrsquo values After some numerical calcula-tions we noted that in this situation the parameters c1 and c2may switch from E1 to E2 and the inverse while if c1 gt c2 thenE1 will be more attractive and if c2 gt c1 then E2 will be moreattractive and that with the same initial conditions We sim-ulate the model with different initial conditions to illustrate theimpact of initial conditions on the stability of E1 and E2 in thissituation Figure 2 depicts the stability of E1 and E2 at the sametime where we consider the same set of parameters fromexample 9 in Table 3 By changing the initial conditions we cansee that E1 is stable when I(0) 100 A(0) 100 andD(0) 80 while it can be seen thatE2 is also stablewith this setof parameters but after choosing I(0) 100 A(0) 80 andD(0) 100

Example 11 shows the existence and the stability of theequilibrium Elowast where ldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1+β2)gt 0rdquo and ldquoβ1 lt c1 and c2 lt β2 and α gt 0rdquo while example 12shows the existence and the stability of Elowast whereldquoRD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0rdquo and ldquoβ1 gt c1and c2 gt β2 and α lt 0rdquo

Figure 3 depicts examples of the existence and stability ofthe equilibrium state E0 for different parametersrsquo values andthreshold RD0 and RA0 values simulated with initial con-ditions and parametersrsquo values from Table 3 Figure 4 depictsexamples of the existence and stability of the equilibriumstate E1 for different parametersrsquo values and threshold RD0and RA0 values simulated with initial conditions and pa-rametersrsquo values from Table 3 It can be seen fromFigure 4(b) that the functionD decreases towards zero but itwill take a long time In the Figure 4(c) we have considered1200 hours to show that the function D will go to zero butvery slowly Figure 5 depicts examples of the existence andstability of the equilibrium state E2 for different parametersrsquovalues and threshold RD0 and RA0 values simulated withinitial conditions and parametersrsquo values from Table 3 Wecan see from Figure 5(b) that the function A will also take avery long time to go to zero In Figure 5(c) we have con-sidered 1200 hours to show that function A will tend to zerobut very slowly Figure 6 depicts examples of the existenceand stability of the equilibrium state Elowast for different

6 International Journal of Differential Equations

parametersrsquo values and threshold RD0 and RA0 values sim-ulated with initial conditions and parametersrsquo values fromexamples 11 and 12 of Table 3

5 Analysis of Thresholds

51 No Abstention No Loss of Interest In most of the sit-uations people abstain from voting for personal reasonsHere we discuss the situation when there is no abstentionand there is no loss of interest of voting ie c1 c2 0)erefore RD0 and RA0 become

RD0 α2α1

RA0 α1α2

(43)

which means that thresholds RD0 and RA0 depend only on α1and α2 and the equilibria become

E1 (1 0)

E2 (0 1)(44)

and from the existence conditions in Table 2 we can deducethat there is no Elowast

E1 exists without any condition and it is stable if α2 lt α1and E2 exists without any condition and it is stable if α1 lt α2

)is result explains the effect of the polarization on theoutcome of polls When there is no abstention from votingand people keep their interest in voting then the mostinfluential parameters on the outcome of the polls are thepolarization factors α1 and α2 For example during a polldetermining the political position of one candidate Y byvoting with Approve or Disapprove or otherwise if can-didate Y can convince people by his political vision andorby other ways then he can change the course of events in hisfavor Mathematically he makes α2 lt α1 But if somehow hecontributes to making α1 lt α2 then things could spin out ofcontrol on the election day

Table 2 Summary of sufficient existence and stability conditions of equilibria

Equilibrium Existence conditions Stability conditionsE0 mdash β1 lt c1 and β2 lt c2E1 (1 minus (c1β1) 0) β1 gt c1 RD0 lt 1E2 (0 1 minus (c2β2)) β2 gt c2 RA0 lt 1Elowast (alowast dlowast) RD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0 β1 lt c1 and c2 lt β2 and α gt 0 or β1 gt c1 and c2 gt β2 and α lt 0

Table 3 Steady states and parametersrsquo values used in example simulations where I(0) A(0) D(0) 100

β1 β2 α1 α2 c1 c2 RD0 RA0

1 E0 (300 0 0) 00010 01010 01010 03010 05010 03010 19901 207302 E0 (300 0 0) 00010 02010 02010 03010 01010 05010 05000 085433 E0 (300 0 0) 00010 00010 02010 01010 01010 01010 05000 200014 E0 (300 0 0) 00010 00010 01010 02010 01010 01010 20001 050005 E1 (29703 2970297 0) 01010 00010 02010 02010 00010 00010 06634 049256 E1 (29703 2970297 0) 01010 00010 00010 00010 00010 00010 05025 1017 E2 (29703 0 2970297) 00010 01010 02010 02010 01010 00010 04925 066348 E2 (29703 0 2970297) 00010 01010 00010 00010 00010 00010 101 050259 Elowast1 (1203593 1796407 0) 05010 01010 01010 00010 02010 00010 06688 0519610 Elowast2 (1203593 0 1796407) 01010 05010 00010 01010 00010 02010 05196 0668811 Elowast (1333333 105556 1561111) 01010 07010 03010 00010 02010 03010 4677774 1067212 Elowast (2142857 804643 52500) 03010 04010 00010 08010 02010 05010 10649 3008004

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E1 = (112 168 0)

IAD

(a)

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E2 = (2 0 278)

IAD

(b)

Figure 2 Illustration of the bistability of E1 and E2 at the same time using the set of parameters of example 9 from Table 3 Stability of E1 (a)when I(0) 100 A(0) 100 and D(0) 80 Stability of E2 (b) when I(0) 100 A(0) 80 and D(0) 100

International Journal of Differential Equations 7

0 20 40 60 80 100 120Time (hours)

050

100150200250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(b)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(c)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(d)

Figure 3 Examples of the steady state E0 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 1 (b) Example 2 (c) Example 3 (d) Example 4

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(b)

Figure 4 Continued

8 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

parametersrsquo values and threshold RD0 and RA0 values sim-ulated with initial conditions and parametersrsquo values fromexamples 11 and 12 of Table 3

5 Analysis of Thresholds

51 No Abstention No Loss of Interest In most of the sit-uations people abstain from voting for personal reasonsHere we discuss the situation when there is no abstentionand there is no loss of interest of voting ie c1 c2 0)erefore RD0 and RA0 become

RD0 α2α1

RA0 α1α2

(43)

which means that thresholds RD0 and RA0 depend only on α1and α2 and the equilibria become

E1 (1 0)

E2 (0 1)(44)

and from the existence conditions in Table 2 we can deducethat there is no Elowast

E1 exists without any condition and it is stable if α2 lt α1and E2 exists without any condition and it is stable if α1 lt α2

)is result explains the effect of the polarization on theoutcome of polls When there is no abstention from votingand people keep their interest in voting then the mostinfluential parameters on the outcome of the polls are thepolarization factors α1 and α2 For example during a polldetermining the political position of one candidate Y byvoting with Approve or Disapprove or otherwise if can-didate Y can convince people by his political vision andorby other ways then he can change the course of events in hisfavor Mathematically he makes α2 lt α1 But if somehow hecontributes to making α1 lt α2 then things could spin out ofcontrol on the election day

Table 2 Summary of sufficient existence and stability conditions of equilibria

Equilibrium Existence conditions Stability conditionsE0 mdash β1 lt c1 and β2 lt c2E1 (1 minus (c1β1) 0) β1 gt c1 RD0 lt 1E2 (0 1 minus (c2β2)) β2 gt c2 RA0 lt 1Elowast (alowast dlowast) RD0 gt 1 andRA0 gt 1 and α(α minus β1 + β2)gt 0 β1 lt c1 and c2 lt β2 and α gt 0 or β1 gt c1 and c2 gt β2 and α lt 0

Table 3 Steady states and parametersrsquo values used in example simulations where I(0) A(0) D(0) 100

β1 β2 α1 α2 c1 c2 RD0 RA0

1 E0 (300 0 0) 00010 01010 01010 03010 05010 03010 19901 207302 E0 (300 0 0) 00010 02010 02010 03010 01010 05010 05000 085433 E0 (300 0 0) 00010 00010 02010 01010 01010 01010 05000 200014 E0 (300 0 0) 00010 00010 01010 02010 01010 01010 20001 050005 E1 (29703 2970297 0) 01010 00010 02010 02010 00010 00010 06634 049256 E1 (29703 2970297 0) 01010 00010 00010 00010 00010 00010 05025 1017 E2 (29703 0 2970297) 00010 01010 02010 02010 01010 00010 04925 066348 E2 (29703 0 2970297) 00010 01010 00010 00010 00010 00010 101 050259 Elowast1 (1203593 1796407 0) 05010 01010 01010 00010 02010 00010 06688 0519610 Elowast2 (1203593 0 1796407) 01010 05010 00010 01010 00010 02010 05196 0668811 Elowast (1333333 105556 1561111) 01010 07010 03010 00010 02010 03010 4677774 1067212 Elowast (2142857 804643 52500) 03010 04010 00010 08010 02010 05010 10649 3008004

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E1 = (112 168 0)

IAD

(a)

0 100 200 300 400 500 600Time (hours)

050

100150200250300

Num

ber o

f peo

ple

E2 = (2 0 278)

IAD

(b)

Figure 2 Illustration of the bistability of E1 and E2 at the same time using the set of parameters of example 9 from Table 3 Stability of E1 (a)when I(0) 100 A(0) 100 and D(0) 80 Stability of E2 (b) when I(0) 100 A(0) 80 and D(0) 100

International Journal of Differential Equations 7

0 20 40 60 80 100 120Time (hours)

050

100150200250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(b)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(c)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(d)

Figure 3 Examples of the steady state E0 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 1 (b) Example 2 (c) Example 3 (d) Example 4

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(b)

Figure 4 Continued

8 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

0 20 40 60 80 100 120Time (hours)

050

100150200250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(b)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(c)

0 20 40 60 80 100 120Time (hours)

IAD

050

100150200250300

Num

ber o

f peo

ple

(d)

Figure 3 Examples of the steady state E0 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 1 (b) Example 2 (c) Example 3 (d) Example 4

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(a)

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250300

Num

ber o

f peo

ple

IAD

(b)

Figure 4 Continued

8 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

52 One Chance In some situations voters are not allowedto modify their choice then they could take one side untilthe end of the survey In this section we discuss the sit-uation when there is no persuasion and then there is nopolarization ie α1 α2 0 )erefore RD0 and RA0become

RD0 β2c1

β1c2

RA0 β1c2

β2c1

(45)

0 200 400 600 800 1000Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(c)

Figure 4 Examples of the steady state E1 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 5 (b) Example 6 (c) Example 9

0 20 40 60 80 100 120Time (hours)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 200 400 600 800 1000Time (hours)

IAD

(c)

Figure 5 Examples of the steady state E2 simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 7 (b) Example 8 (c) Example 10

International Journal of Differential Equations 9

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

whichmeans that thresholds RD0 and RA0 depend only on β1β2 c1 and c2 and there is no change in the equilibria E1 andE2 From the conditions of the existence in Table 2 we candeduce that there is no Elowast A sufficient condition to make E1stable is ldquoβ2 lt β1 and c1 lt c2rdquo and the sufficient condition tomake E2 stable is ldquoβ2 gt β1 and c1 gt c2rdquo )is result explainsthe efficiency of election campaigns For instance if there aretwo candidate parts X and Y and the poll is carried out tostudy the political position of Y then votes for Y are con-sidered as agree and votes for X are considered as disagree Ifcandidate Y presents a successful election campaign it willattract more people alongside him and increase the numberof people agreeing (ie β2 lt β1) which may lead disagreepeople to lose interest or abstain from voting (ie c1 lt c2)

53 Statistical Analysis Here we use probability distribu-tion functions of the six parameters given in Table 4 sampledby using the Latin hypercube sampling see Figure 7 Wecompute the probabilities of equilibria existence and stabilityconditions It can be seen from Table 5 that the probability ofE1 to exist is about 05960 while its probability of stability isabout 053 )e probability of E1 to exist and to be stable is04040)e probability of the existence of E2 is about 06250while the probability of its stability is about 05370 )eprobability of E1 to exist and to be stable is 043 )eequilibrium Elowast has a probability of existence about 00240and a probability of stability about 02190 while Elowast existsand it is stable with a probability of 0001

6 Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identifythe effectiveness of model parameters or inputs and thusprovides essential information about the model perfor-mance Out of the many methods of carrying out sensitivityanalysis is the partial rank correlation coefficient (PRCC)method that has been used in this paper )e PRCC is amethod based on sampling One of the most efficientmethods used for sampling is the Latin hypercube sampling(LHS) which is a type of Monte Carlo sampling [25] as it

densely stratifies the input parameters As the name suggeststhe PRCC measures the strength between the inputs andoutputs of the model using correlation through the samplingdone by the LHS method [26ndash28]

Parameters β1 and β2 follow normal distribution withmean and standard deviation 05 and 001 respectively whileparameters α1 α2 c1 and c2 follow triangular distributionwith minimum maximum and mode as 002 08 and 051respectively A summary of probability distribution functionsis given in Table 4 In Latin hypercube sampling approachprobability density function (given in Table 4) for each pa-rameter is stratified into 100 equiprobable (1100) serial in-tervals )en a single value is chosen randomly from eachinterval)is produces 100 sets of values of RD0 and RA0 from100 sets of different parameter values mixed randomly cal-culated by using equations (4) and (5) respectively

Sensitivity analysis has been done concerning the basicreproduction numbers RD0 and RA0 and the main objectiveof this section is to identify which parameters are importantin contributing variability to the outcome of basic repro-duction numbers based on their estimation uncertainty

To sort the model parameters according to the size oftheir effect on RD0 and RA0 a partial rank correlationcoefficient is calculated between the values of each of the sixparameters and the values of RD0 and RA0 in order toidentify and measure the statistical influence of any one ofthe six input parameters on thresholds RD0 and RA0 )elarger the partial rank correlation coefficient the larger theinfluence is on the input parameter affecting the magnitudeof RD0 and RA0

0

50

100

150

200

250

300N

umbe

r of p

eopl

e

0 20 40 60 80 100 120Time (hours)

IAD

(a)

0

50

100

150

200

250

300

Num

ber o

f peo

ple

0 20 40 60 80 100 120Time (hours)

IAD

(b)

Figure 6 Examples of the steady state Elowast simulated with initial conditions I(0) 100 A(0) 100 and D(0) 100 and parametersrsquo valuesfrom Table 3 (a) Example 11 (b) Example 12

Table 4 Six input parametersrsquo probability distribution functions

Parameters Pdfβ1 Normalβ2 Normalα1 Triangularα2 Triangularc1 Triangularc2 Triangular

10 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

As shown in Table 6 the transmission rate β1 andparameters c1 and c2 are highly correlated with the thresholdRD0 with corresponding values minus 01374 and 01613 andminus 02101 respectively Moderate correlation exists betweentransmission rates α1 and α2 and RD0 with correspondingvalues as minus 00633 and 00669 respectively Weak correla-tions have been observed between the transmission rate β2and RD0 with the corresponding value 00025

As shown in the sensitivity index column of Table 6 theparameter c2 accounts for themaximum variability 08499 inthe outcome of basic reproduction number RD0 )e pa-rameter c1 is the next to account for the variability 07492 inthe outcome of RD0 )en the transmission rate β1 accountsfor the variability 05596 in the outcome of RD0 followed bythe transmission rate α2 that accounts for the variability05269 in the outcome of RD0 Transmission parameters α1and β2 account for the least variability 01065 and 00192 inthe outcome of basic reproduction number RD0 respectivelyHence parameters c1 and c2 and the transmission rate β1 arethe most influential parameters in determining RD0

It can be seen from Table 7 that the parameter c1 is highlycorrelated with the threshold RA0 with the corresponding value

0 200 400 600 800 100002

03

04

05

06

07

08

(a)

0 200 400 600 800 10000

02

04

06

08

1

(b)

0 200 400 600 800 10000

2

4

6

8

(c)

0 200 400 600 800 10000

2

4

6

8

(d)

0 200 400 600 800 10000

2

4

6

8

(e)

0 200 400 600 800 10000

2

4

6

8

(f )

Figure 7 Probability distribution functions of the six sampled input parametersrsquo values )ese results have been obtained from Latinhypercube sampling using a sample size of 1000 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

Table 5 Summary of probabilities of existence and stability conditions of equilibria

Equilibrium Probability of existence Probability of stability Probability of existence and stabilityE1 05960 05300 04040E2 06250 05370 04300Elowast 00240 02190 0001

Table 6 PRCCs for RD0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS minus 01374 01728 05596β2 LHS 00025 09800 00192α1 LHS minus 00633 05318 01065α2 LHS 00669 05085 05269c1 LHS 01613 01090 07492c2 LHS minus 02101 00359 08499

Table 7 PRCCs for RA0 and six input parameters and the cor-responding sensitivity index

Parameters Sampling PRCCs p value Sensitivity indexβ1 LHS 02137 00328 05753β2 LHS minus 01928 00546 02421α1 LHS 00857 03967 00159α2 LHS minus 02076 00382 03872c1 LHS minus 03343 67425eminus 04 08004c2 LHS 02523 00113 07122

International Journal of Differential Equations 11

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

02 04 06 08ndash10

0

10

20

30

40R D

0

(a)

020 04 06 08ndash10

0

10

20

30

40

R D0

(b)

020 04 06 08ndash10

0

10

20

30

40

R D0

(c)

020 04 06 08ndash10

0

10

20

30

40

R D0

(d)

020 04 06 08ndash10

0

10

20

30

40

R D0

(e)

020 04 06 08ndash10

0

10

20

30

40

R D0

(f )

Figure 8 Scatter plots for the basic reproduction number RD0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

02 04 06 08ndash5

0

5

10

15

R A0

(a)

ndash5

0

5

10

15

R A0

020 04 06 08

(b)

ndash5

0

5

10

15

R A0

020 04 06 08

(c)

Figure 9 Continued

12 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

minus 03343Moderate correlation exists between transmission ratesβ1 β2 and α2 and the parameter c2 and RA0 with correspondingvalues as 02137 minus 01928 minus 02076 and 02523 respectivelyWeak correlations have been observed between the transmis-sion rate α1 and RA0 with the corresponding value 00857

One can see in the sensitivity index column of Table 7that the parameter c1 accounts for the maximum variability08004 in the outcome of basic reproduction number RA0)e parameter c2 is the next to account for the variability07122 in the outcome of RA0 )e transmission rate β1accounts for the variability 05753 in the outcome of thisthreshold followed by the transmission rate α2 that accountsfor the variability 03872 in the outcome of RA0 Trans-mission parameters β2 and α1 account for the least variability02421 and 00159 in the outcome of basic reproductionnumber RA0 respectively Hence parameters c1 and c2 andthe transmission rate β1 are also the most influential pa-rameters in determining RA0

Scatter plots comparing the basic reproduction numbersRD0 and RA0 against each of the six parameters β1 β2 α1 α2c1 and c2 are shown in Figures 8 and 9 respectively basedon Latin hypercube sampling with a sample size of 100)ese scatter plots clearly show the linear relationshipsbetween outcome of RD0 and RA0 and input parameters

7 Conclusion

In this paper an IAD model-type compartmental model hasbeen considered to explore agree-disagree opinions duringpolls )e equations governing the system have been solvedto compute equilibrium states and the next-generationmatrix method is used to derive basic reproduction numbersRA0 and RD0

)e model exhibits four feasible points of equilibriumnamely the trivial equilibrium agree-free equilibriumdisagree-free equilibrium and the positive equilibriumSufficient equilibrium conditions of existence are given andstability analysis is performed to show under which con-ditions equilibrium states are stable or unstable )e stabilityof these points of equilibrium is controlled by the threshold

number RA0 and RD0 If the threshold RD0 is less than onethe disagree opinion dies out and the disagree-free equi-librium is stable If RD0 is greater than one the disagreeopinion persists and the disagree-free equilibrium is un-stable If the threshold RA0 is less than one the agreeopinion dies out and the agree-free equilibrium is stable IfRA0 is greater than one the agree opinion persists and thedisagree-free equilibrium is unstable We simulated someexamples with different parameter values to show the ex-istence and the stability of such equilibria

)e probabilities of the existence and probabilities of thestability of equilibria are computed based on the parametersrsquodistribution function sampled with the Latin hypercubesamplingmethod To identify themost influential parameter inthe proposed model global sensitivity analysis is carried outbased on the partial rank correlation coefficient method andLatin hypercube sampling )is statistical study shows that themost influential parameters in the determination of thresholdsof equilibria stability are β1 the polarization parameter ofignorant people by people agreeing the parameter c1 of theloss of interest of the people agreeing and finally c2 the pa-rameter of loss of interest of disagreeing people

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors would like to thank all the members of theEditorial Board who were responsible for this paper

References

[1] S J Rosenstone and J M HansenMobilization Participationand Democracy in America Macmillan New York NY USA2003

ndash5

0

5

10

15R A

0

020 04 06 08

(d)

ndash5

0

5

10

15

R A0

020 04 06 08

(e)

ndash5

0

5

10

15

R A0

020 04 06 08

(f )

Figure 9 Scatter plots for the basic reproduction number RA0 and six sampled input parametersrsquo values )ese results have been obtainedfrom Latin hypercube sampling using a sample size of 100 (a) β1 (b) β2 (c) α1 (d) α2 (e) c1 (f ) c2

International Journal of Differential Equations 13

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations

[2] L Ezrow C De Vries M Steenbergen and E EdwardsldquoMean voter representation and partisan constituency rep-resentation do parties respond to the mean voter position orto their supportersrdquo Party Politics vol 17 no 3 pp 275ndash3012011

[3] J Duffy and M Tavits ldquoBeliefs and voting decisions a test ofthe pivotal voter modelrdquoAmerican Journal of Political Sciencevol 52 no 3 pp 603ndash618 2008

[4] M F Meffert and T Gschwend ldquoPolls coalition signals andstrategic voting an experimental investigation of perceptionsand effectsrdquo European Journal of Political Research vol 50no 5 pp 636ndash667 2011

[5] R B Morton D Muller L Page and B Torgler ldquoExit pollsturnout and bandwagon voting evidence from a naturalexperimentrdquo European Economic Review vol 77 pp 65ndash812015

[6] T E Patterson ldquoOf polls mountains US Journalists andtheir use of election surveysrdquo Public Opinion Quarterlyvol 69 no 5 pp 716ndash724 2005

[7] R Chung1e Freedom to Publish Opinion Polls AWorldwideUpdate of 2012 ESOMAR Amsterdam Netherlands 2012

[8] R Lachat ldquo)e impact of party polarization on ideologicalvotingrdquo Electoral Studies vol 27 no 4 pp 687ndash698 2008

[9] M M Pereira ldquoDo parties respond strategically to opinionpolls Evidence from campaign statementsrdquo Electoral Studiesvol 59 pp 78ndash86 2019

[10] L A Adamic and N Glance ldquoLeft Te political blogosphereand the 2004 US Election divided they blogrdquo in Proceedingsof the 3rd International Workshop on Link Discovery(LinkKDD rsquo05) pp 36ndash43 ACM New York NY USAAugust 2005

[11] M D Conover J Ratkiewicz M Francisco B GoncA Flammini and F Menczer ldquoLeft Political polarization ontwitterrdquo in Proceedings of the Fifh International AAAI Con-ference on Weblogs and Social Media pp 89ndash96 BarcelonaCatalonia Spain July 2011

[12] G Serra ldquoLeft Should we expect primary elections to createpolarization a robust median voter theorem with rationalpartiesrdquo in Routledge Handbook of Primary ElectionsR G Boatright Ed pp 72ndash94 Routledge New York NYUSA 2018

[13] G Olivares ldquoOpinion polarization during a dichotomouselectoral processrdquo Complexity vol 2019 2019

[14] D J Pannell ldquoSensitivity analysis of normative economicmodels theoretical framework and practical strategiesrdquo Ag-ricultural Economics vol 16 no 2 pp 139ndash152 1997

[15] httpspollingreuterscomresponseCP3_2typeweekdates20170201-20190409collapsedtrue

[16] M Ludovic Applications en Biologie Ecologie Environne-ment Universite Nice Sophia Antipolis Nice France 2004

[17] O Diekmann and A J Johan ldquoOn the definition and thecomputation of the basic reproduction ratio R 0 in models forinfectious diseases in heterogeneous populationsrdquo Journal ofMathematical Biology vol 28 no 4 pp 365ndash382 1990

[18] P Lou ldquoModelling seasonal brucellosis epidemics in bayin-golin mongol autonomous prefecture of Xinjiang China2010ndash2014rdquo BioMed Research International vol 2016 2016

[19] S Liu ldquoGlobal analysis of a model of viral infection with latentstage and two types of target cellsrdquo Journal of AppliedMathematics vol 2013 2013

[20] P Padmanabhan and S Padmanabhan ldquoComputational andmathematical methods to estimate the basic reproductionnumber and final size for single-stage and multistage pro-gression disease models for zika with preventative measuresrdquo

Computational and Mathematical Methods in Medicinevol 2017 2017

[21] M A Khan S W Shah S Ullah and J F Gomez-Aguilar ldquoAdynamical model of asymptomatic carrier zika virus withoptimal control strategiesrdquo Nonlinear Analysis Real WorldApplications vol 50 pp 144ndash170 2019

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquo)econstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of 1e Royal Society Interfacevol 7 no 47 pp 873ndash885 2010

[23] R Jan M A Khan and J F Gomez-Aguilar ldquoAsymptomaticcarriers in transmission dynamics of dengue with controlinterventionsrdquo Optimal Control Applications and Methodsvol 2019 2019

[24] S Ullah M A Khan and J F Gomez-Aguilar ldquoMathematicalformulation of hepatitis B virus with optimal control analy-sisrdquo Optimal Control Applications and Methods vol 40 no 3pp 529ndash544 2019

[25] J C Helton F J Davis and J D Johnson ldquoA comparison ofuncertainty and sensitivity analysis results obtained withrandom and Latin Hypercube samplingrdquo Reliability Engi-neering amp System Safety vol 89 no 3 pp 305ndash330 2005

[26] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensi-tivity analysis in systems biologyrdquo Journal of 1eoretical Bi-ology vol 254 no 1 pp 178ndash196 2008

[27] D M Hamby A Comparison of Sensitivity Analysis Tech-niques Westinghouse Savannah River Company Aiken SCUSA 1994

[28] J C )iele W Kurth and V Grimm ldquoFacilitating parameterestimation and sensitivity analysis of agent-based models acookbook using NetLogo and Rrdquo Journal of Artificial Societiesand Social Simulation vol 17 pp 30ndash47 2014

14 International Journal of Differential Equations