Research ArticleA Discrete Mathematical Modeling and Optimal Control of theRumor Propagation in Online Social Network

Amine El Bhih 1 Rachid Ghazzali 1 Soukaina Ben Rhila 1 Mostafa Rachik 1

and Adil El Alami Laaroussi 12

1Laboratory of Analysis Modeling and Simulation Department of Mathematics and Computer ScienceFaculty of Sciences Ben MrsquoSik Hassan II University Casablanca BP 7955 Sidi Othman Casablanca Morocco2Laboratory of Applied Sciences and Didactics Higher Normal School Tetouan Abdelmalek Essaadi UniversityTetouan Morocco

Correspondence should be addressed to Amine El Bhih elbhihaminegmailcom

Received 9 January 2020 Revised 28 March 2020 Accepted 30 March 2020 Published 1 May 2020

Academic Editor Rodica Luca

Copyright copy 2020 Amine El Bhih et alampis 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 a new rumor spreadingmodel in social networks has been investigatedWe propose a new version primarily based on thecholeramodel in order to take into account the expert pages specialized in the dissemination of rumors from an existing IRCSSmodel Inthe second part we recommend an optimal control strategy to fight against the spread of the rumor and the study aims at characterizingthe three optimal controls which minimize the number of spreader users fake pages and corresponding costs theoretically we haveproved the existence of optimal controls and we have given a characterization of controls in terms of states and adjoint functions basedon a discrete version of Pontryaginrsquos maximum principle To illustrate the theoretical results obtained we propose numerical simulationsfor several scenarios applying the forward-backward sweep method (FBSM) to solve our optimality system in an iterative process

1 Introduction

ampe spread of rumors is a complicated phenomenon thathas occupied a large place in human life since ancient timescivilizations have studied and analyzed this phenomenonwhere many elements overlap and intervene includingwhat is natural sociological economic and psychologicalHistory has recognized over the years the emergence ofmany rumors that have spread extensively among com-munities it was once the center of interaction and eval-uation by company commanders throughout history andpeople invent rumors and spread them so that they can beused for political or financial purposes Rumors can makean industry prosper while it can also wrest victory from thejaws of defeat ampe rumor phenomenon has recognizedmany changes in its composition with the existence andincreasing use of technological capabilities and modernonline technologies that companies recognize ampis phe-nomenon has given another dimension since it has beenused using the media and genius in the competition be-tween countries and what is recognized as buzz or

propaganda and controversy using the publication of falsenews in whole or partly to influence power with opponentsof power Confined to the elections between Trump andHillary where Hillary was the most popular and was thefavorite of political researchers [1] until the closing monthbefore the presidential election where some of the com-petent communication agencies published many newsabout Hillary this news played an important and funda-mental role in influencing public opinion and turning thescales in favor of Trump who eventually won Jennifer et alin their article [2] gave a contribution to understand thedynamics of this unusual campaign in which social mediaplayed a major role In [3] we can find a collection ofTrump tweets divided by subjects (people places andthings Trump has insulted on twitter) who ultimately re-ceived mathematical modeling is one of the most necessaryfunctions of arithmetic that make a contribution to therepresentation and simulation of social economic organicand ecological phenomena and convert them into math-ematical equations that are formulated studied analyzedand interpreted their results as an example Cazzoli et al

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 4386476 12 pageshttpsdoiorg10115520204386476

analyzed the impact of tweets on the financial market [4]for example Vilas et al analyzed the interference betweencryptocurrencies and company tickers in the London stock [5]

In this context many researchers have developed specificmathematical fashions representing the dynamics of the rumorand the factors interfering with its spread [6ndash9] In this workand primarily based on the model they proposed we introducea new approach by taking into account the spread of rumorsthrough social media We will describe and explain the modelwe will use here As we mentioned many organizations whichspecialized in propaganda dissemination have grown to be theusage of social media to facilitate unfold of the rumor and largevolume of users For this purpose extraordinary pages arecreated to unfold a rumor about a unique situation or targetpersonampis page is promoted by means of fictitious customerscreated for this purpose ampey create a private network offriends friends instead are the first victims each and every timethey like or comment onwhat is posted on the page or fictitiouspeople this pastime is displayed to all their pals or possibly palsof their buddies inadvertently which is promoted by using thisrumor passively by them while research indicates that thenumber of users of the networks is rising at a magnificentcharge and it has come to be one of the basics in the disciplineof communication and publicity and in accordance to sta-tistics what is dealt with millions of rumors spread every dayand in view that in 2019 the entire of Russia is considering thedecision to attend some web sites as Telegram because of thethreat that brings countrywide security And if some rumorsarouse ridicule such as announcing that Nicolas Cage wasdracula others were ofmagnificent danger to see greater in thisregard we refer the reader to the interesting book [10]

In 1927 Kermack and McKendrick [11] were the firstresearchers on mathematical epidemiology to suggest thesusceptible-infected-removed (SIR) model that describes therapid explosion of an infectious disease for a short time In1964 Goffman and Newill developed in their article titledldquoGeneralization of epidemic principle an application to thetransmission of thoughtsrdquo [12] a new notion for modeling thetransmission of ideas within a society based on the mathe-matical model SIR due to the terrific similarity between the twophenomena ampis model was earlier used to model thetransmission of diseases and epidemics inside communities inthe introduction of their work the authors cited that themethod which has been already described does not take intoaccount the almost endless variety of complexities which inreality arise [12] Based on the preceding work Daley andKendall in their letter titled ldquoEpidemics and Rumoursrdquocounseled making use of the preceding thought to modelingthe unfold of rumors inside communities [13] With the de-velopment of societies and the emergence of contemporarytechnological means (transport-communication) new ele-ments have emerged that similarly complicate the phenome-non of rumor and contribute to the massive spread of rumorsthis has led many researchers to think about growing thepreviousmodel As an example the work that has been done byBettencourt et al [14] authors proposed a new model takinginto account new factors by extending the SIR model to aSEIZR model with two additional compartments In the samecontext Laarabi et al [15] developed and analyzed the delayed

model for the transmission dynamics of rumor with constantrecruitment and incubation periodampis is in addition to manycurrent works that have currently been produced that take intoaccount quite a few factors concerned in the development of athought that truly simulates the dynamics of hearsay propa-gation to take a broader view the reader is referred to thearticle [16] ampere are other works which have modeled thephenomenon of rumor propagation using other approachesfor example complex-network opinion dynamics modelsamong which Wang et al [17] established a new compre-hensive model this model captures the role of memory theeffects of conformity the differences in the subjective pro-pensity to produce distortions and the variations in the degreeof trust that people have in each other through the concept ofentropy of information In addition they have developed thismodel in [18] by taking into account information distortionand polarization effects in order to modeling imprecision inhuman memory and communication and the consequentprogressive drift of information Based on the single-issueopinion dynamics model Tan and Cheong [19] incorporated ageneral model of multi-issue opinion dynamics where theagreement on one issue can promote greater inclusivity indiscussing other issues With the emergence of social networksand their impact on verbal exchange within communities theplace they are taking extra and greater area within the com-munity it grew to be clear that they should be taken intoaccount as a principal intervening in the unfold of rumors inthis context many research studies which adopt this hypothesishave been carried out to take thinking on some of these workssee the article [20] for instance in the work [21] authors hadimplemented a mathematical model in order to modeling thedynamics of a rumor in the social community with the aid ofadding three new compartments reviewers sharers and col-lectors people who review the message accumulate themessage share the message or do not respond to the messagerespectively however in the work [21] authors have beenrestricted to highlight the function of customers of the com-munity and omit the effect of the network itself in particularthe function of pages that spread the rumor inside the networkthe loading of false information in these pages is a source ofhearsay between browsers and is considered as a big elementwhich helps in the rapid unfold of rumors such as rivers andvalleys which store microorganism and microbes and are ahotbed for the multiplication and growth of bacteria thattransmit diseases to human beings through the use of water ofthose rivers a proper instance of this similarity is the choleraepidemic In this work we propose a new model which de-scribes the dynamics of rumor spread through social mediabased on the cholera model [22] and combine it with theprevious workmodel by adding new cubicles Pwhich representthe pagersquos rumor Moreover we apply an optimal controlstrategy in order to fight against the spread of the rumorthrough social media regarding to this we use theoreticalresults provided by Balatif et al [23] where authors imple-mented a discrete time model that describes the dynamics ofvoters and they proposed an optimal control strategy the sameidea and strategy were applied by Labzai et al [24] and in orderto modeling and control smoking Kouidere et al [25] sug-gested a model of the evolution from prediabetes to diabetes

2 Discrete Dynamics in Nature and Society

with optimal control approach Other models from optimalcontrol problems and population dynamics can be found in[26ndash30]

In this paper in Section 2 we propose a discrete ISpStP

mathematical model that describes the dynamic of a pop-ulation that reacts in the spread of the rumor in a socialnetwork In Section 3 we present an optimal controlproblem for the proposed model where we give some resultsconcerning the existence of the optimal control and wecharacterize the optimal controls using the Pontryaginmaximum principle in discrete time Numerical simulationsthrough MATLAB are given in Section 4 Finally we con-clude the paper in Section 5

2 Mathematical Model andSimulation without Controls

In this section we consider a discrete mathematical modelISpStP that will describe the dynamics of a population ofrumors our model consists of four compartments repre-senting the subdivision of the population that reacts in thespread of the rumor in a social network

I ignorant users who do not know the rumor andsusceptible to be informedSp spreader users who spread the rumorSt stifler individuals who refuse to spread the rumorP rumorrsquos page the page specialized in spreading therumor

ampe compartment I represents the number of users who donot understand the rumor and who are susceptible to be in-formed and this population increases with the charge μN

which represents the new customers created an ignorant in-quires about the rumor through two ways either by using

consulting a specialized page in the diffusion of the rumor or atonce by means of the contact with a spreader some of theseusers deactivate their account at a rate μI ampus in thiscompartment we have an incoming flux which equals to μN

and an outgoing flux which equals toαhSpI + αeI(Pκ + P) + μI

ampe compartment Sp this compartment represents thenumber of humans who spread the rumor either immedi-ately or by way of sharing one-page publications or viadeveloping new publications ampus we have an incomingflux which equals to θ(αhSpI + αeI(Pκ + P)) which rep-resents the proportion of the new users who will spread therumor After the contact between two spreaders one of themdecides not to diffuse the information at a rate cS2p and afterthe contact of a spreader and a stifler the stifler succeeds toconvince him that the information is false at a rate λStSpafter a certain period a portion of the spreaders decide not tospread the rumor at a rate βSp

ampe compartment St this compartment represents thenumber of stiflers who refuse to spread the rumor ampisnumber increases at a rate (1 minus θ)(αhSpI + αeI(Pκ + P))

which represents the portion of users who knew that theinformation is wrong in addition to the flux that left the Sp

compartment Sp(cSp + λSt) + βSp and decreases with therate μ1St of stiflers who have deactivated their accounts

ampe compartment P this compartment represents thepage specialized in the diffusion of the rumor ampis pageconsists of malicious publications about the rumor in thispage Sp has the proper right to publish and share thesepublications at a rate δ1Sp and ε2Sp respectively and theignorant who consults the page also shares these publica-tions at a rate ε1I

ampe diagram will demonstrate the flux directions ofindividuals among the compartments Figure 1

I(n + 1) I(n) + μN minus αhSp(n)I(n) minus αeI(n)P(n)

κ + P(n)minus μ1I(n)

Sp(n + 1) Sp(n) + θ αhSp(n)I(n) + αeI(n)P(n)

κ + P(n)1113888 1113889 minus μ1 + β( 1113857Sp(n) minus cSp(n) Sp(n) + St(n)1113872 1113873

St(n + 1) St(n) +(1 minus θ) αhSp(n)I(n) + αeI(n)P(n)

κ + P(n)1113888 1113889 + cSp(n) Sp(n) + St(n)1113872 1113873 + βSp(n) minus μ1St(n)

P(n + 1) P(n) + ε1N + ε2Sp(n) minus μ2P(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

with initial values I(0) Sp(0) St(0) and P(0) which arenonnegatives

In order to demonstrate the effectiveness of themodel wehave proposed we will present a numerical simulation withFigure 2 so that we can see how well the model adapts toreality Initial values are approximate data that we suggestedafter studying and researching some statistics about the usersof social networks and the values are attached in the table

From Figure 2 we note that there is no significanteffect until the 30th day 30 days after the launch of therumor the number of ignorants decreases sharply and incontrast there is a significant rise of spreader people andthe number of stiflers and pages are rising on averagethese changes indicate that after 30 days trading rumorhas become more and more due to the continuous pub-lication of it

Discrete Dynamics in Nature and Society 3

3 The Model with Controls

Now we introduce our controls into system (1) as thecontrol measures to fight the spread of the rumor we extendour system by including three kinds of controls u v and wampe first control u is to tell users that the information orpublication is false and contains a malicious rumor thesecond control v is through the admin where he deactivatesan account after learning that it is fake or aimed at spreading

the rumor and the last one w is also applied by the adminthis time by deactivating the page intended to spread therumor after the arrival of a certain number of complaints

In the aim of better understanding the effects of anycontrol measure of these strategies we introduce three newvariables πi where i 1 2 3 πi 0 in the absence of thecontrol and πi 1 in the presence of the control

I(n + 1) I(n) + μN minus αhSp(n)I(n) minus αeI(n)P(n)

κ + P(n)minus μ1I(n) minus π1unI(n)

Sp(n + 1) Sp(n) + θ αhSp(n)I(n) + αeI(n)P(n)

κ + P(n)1113888 1113889 minus μ1 + β( 1113857Sp(n) minus cSp(n) Sp(n) + St(n)1113872 1113873 minus π2vnSp(n)

St(n + 1) St(n) +(1 minus θ) αhSp(n)I(n) + αeIn

Pn

κ + Pn

1113888 1113889 + cSp(n) Sp(n) + St(n)1113872 1113873 + βSp(n) minus μ1St(n) + π1unIn

P(n + 1) Pn + ε1N + ε2Sp(n) minus μ2Pn minus π3wnPn

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

P

I Sp St

αe (Pκ + P)

θ (αh + αe)

γSPSP

γSPSt

β SPμ1

μ1

μ1

μN

(1 ndash θ) (αh + αe (Pκ + P))

Figure 1 Description diagram of the rumor dynamics

Ignorant without controlSpreader without control

Stifler without controlPage without control

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

100

Figure 2 Dynamics without control strategy

4 Discrete Dynamics in Nature and Society

4 An Optimal Control Approach

ampe problem is to minimize the objective functional

J(u v w) MNSp(N) + 1113944Nminus 1

k0MkSp(k) +

12Akπ1u

2k1113874

+12Bkπ2v

2k +

12Ckπ3w

2k1113875

(3)

where Ak gt 0 Bk gt 0 Ck gt 0 and Mk gt 0 for i isin 0 N

are the cost coefficients ampey are selected to weigh therelative importance of Sp(k) uk vk and wk at time kNis thefinal time

In other words we seek the optimal controls uk vk andwk such that

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (4)

where Uad is the set of admissible controls defined by

Uad uk vk wk( 1113857

ale uk le b cle vk le d elewk lef1113896 1113897 k 0 N minus 1

(5)

and (a b c d e f) isin (]0 1[)6ampe sufficient condition for the existence of an optimal

control (ulowastk vlowastk wlowastk ) for problem (3) comes from the fol-lowing theorem

Theorem 1 7ere exists an optimal control (ulowast vlowast wlowast) suchthat

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (6)

subject to control system (2) with initial conditions

Proof ampere are a finite number of time steps S

(S0 S1 Sn) I (I0 I1 IN) and P (P0 P1

PN) which are uniformly bounded for all (u v w) isin Uadampus J(u v w) is uniformly bounded for all (u v w) in thecontrol set Uad Since J(u v w) isboundedinf(uvw)isinUad

J(u v w) is finite and there exists asequence u

j

k vj

k wj

k isin Uad such that limj⟶+infinJ(uj

k vj

k wj

k)

inf(uvw)isinUadJ(u v w) and the corresponding sequences of

states Ii Sit Si

p and Bi Since there is a finite number ofuniformly bounded sequences there exist(ulowastk vlowastk wlowastk ) isin U3

ad and S I P R isin RN+1 such that on asubsequence u

j

k⟶ ulowastk vj

k⟶ vlowastk wj

k⟶ w lowastk Sj⟶ SlowastIj⟶ Ilowast and Pj⟶ Plowast Finally due to the finite di-mensional structure of system (2) and the objective functionJ(u v w) (ulowast vlowast wlowast) is an optimal control with corre-sponding states Ilowast Slowastt Slowastp Plowast ampereforeinf(uvw)isinUad

J(u v w) is achievedIn order to derive the necessary condition for the optimal

control Pontryaginrsquos maximum principle in discrete timegiven in [31 32] was used ampis principle converts into aproblem of minimizing a Hamiltonian at the time stepdefined by

Hk MkSp(k) +12Akπ1u

2k +

12Bkπ2v

2k

+12Ckπ3w

2k + 1113944

4

j1λjk+1fjk+1

(7)

where fjk+1 is the right side of the system of differenceequation (2) of the jth state variable at time step k + 1

Using Pontryaginrsquos maximum principle in discrete time[31ndash33] we can say the following theorem

Theorem 2 Let Ilowast Slowastp Slowastt and Plowast be optimal state solutionswith an associated optimal control (ulowast vlowast wlowast) for optimalcontrol problem (3) 7en there exist adjoint variablesλ1k λ2k λ3k λ4k satisfying

Δλ1k λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k 1 + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk(k)1113858 1113859 + λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k λ2k+1 minus cSp(k)Sp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

Discrete Dynamics in Nature and Society 5

with transversality conditions at time

λ1N AN

λ2N λ3N λ4N 0(9)

Furthermore for k 0 1 N minus 1 and forπ1 π2 π3 1 the optimal controls ulowastk vlowastk and wlowastk aregiven by

ulowastk min b max a

λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

1113888 11138891113888 1113889

vlowastk min d max c

minus λ2k+1c Sp(k)1113872 11138732St(k)

Bk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

wlowastk min f max e

λ4k+1P(k)

Ck

1113888 11138891113888 1113889

(10)

Proof ampe Hamiltonian of the optimal problem is given by

Hk MkSp(k) +12Aku

2k +

12Bkv

2k +

12Ckw

2k + λ1k+1 I(k) + μN minus αhSp(k)I(k) minus αeI(k)

P(k)

κ + P(k)minus μ1I(k) minus π1ukI1113888 1113889

+ λ2k+1 Sp(k) + θ αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 minus (μ + β)Sp(k) minus cSp(k) Sp(k) + St(k)1113872 1113873 minus π2vkSp(k)1113888 1113889

+ λ3k+1 St(k) +(1 minus θ) αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 + cSp(k) Sp(k) + St(k)1113872 1113873 + βSp(k) minus μSt(k) + π1ukI1113888 1113889

+ λ4k+1 P(k) + ε1N(k) + ε2Sp(k) minus μ2P(k) minus π3wkP(k)1113872 1113873

(11)

For k 0 1 N minus 1 the adjoint equations andtransversality conditions can be obtained by using

Pontryaginrsquos maximum principle in discrete time given in[31ndash33] such that

Δλ1k zHk

zIk

λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k zHk

zSp(k) Mk + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk1113858 1113859

+λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k zHk

zSt(k) λ2k+1 minus cSp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k zHk

zPk

λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

6 Discrete Dynamics in Nature and Society

Put Φ(N) ANIN

λ1N zΦ(N)

zI AN

λ2N zΦ(N)

zSp

0

λ3N zΦ(N)

zSt

0

λ4N zΦ(N)

zP 0

(13)

For k 0 1 N minus 1 the optimal controls ulowastk vlowastk andwlowastk can be solved from the following optimality condition

zH

zuk

Akuk minus λ1k+1π1I(k) + λ3k+1cπ1Sp(k)I(k) 0⟺

uk λ1k+1π1I(k) minus λ3k+1cπ1Sp(k)I(k)

Ak

zH

zvk

Bkvk + λ2k+1c Sp(k)1113872 11138732π2(k) 0⟺

vk minus λ2k+1c Sp(k)1113872 1113873

2π2St(k)

Bk

zH

zwk

Ckwk minus λ4k+1π3P(k) 0⟺

wk λ4k+1π3P(k)

Ck

(14)

So for π1 π2 π3 1 we have

uk λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

vk minus λ2k+1c Sp(k)1113872 1113873

2St(k)

Bk

wk λ4k+1P(k)

Ck

(15)

However if πi 0 for i 1 2 3 the control attached tothis case will be eliminated and removed By the bounds inUad of the controls it is easy to obtain uk

lowast vklowast and wk

lowast in theform of (10)

41 Algorithm In this section we present the results ob-tained by solving numerically the optimality system ampissystem consists of the state system adjoint system initialand final time conditions and the control characterizationSo the optimality system is given by the following

Step 1 I0 i0 Sp0 sp0 St0 st0 P0 p0λ2N λ3N λ4N 0 λ1N AN and given ulowastk0 vlowastk0and wlowastk0

Step 2 for k 0 1 N minus 1 do

Ik+1 μN minus αhSpkIk minus αeIk

Pk

κ + Pk

minus μ1Ik

Sp(k+1) θ αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 minus (μ + β)Spk minus cSpk Spk + Stk1113872 1113873

Sp(k+1) (1 minus θ) αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 + cSpk Spk + Stk1113872 1113873 + βSpk minus μStk

Pk+1 ε1N + ε2Spk minus μ2Pk

⋮ ⋮

⋮ ⋮

λ1Tminus k λ1Tminus k+1 1 minus αhSpk minus αe

Pk

κ + Pk

minus μ11113888 1113889 + λ2Tminus k+1θ αhSpk + αe

Pk

κ + Pk

1113888 1113889 + λ3Tminus k+1(1 minus θ) αhSpk + αe

Pk

κ + Pk

1113888 1113889

λ2Tminus k 1 + λ1Tminus k+1 minus αhIk( 1113857 + λ2Tminus k+1 1 + θαhIk minus (μ + β) minus 2c( 1113857 + λ3Tminus k+1 (1 minus θ) αhSpk1113872 1113873 + 2c + β1113960 1113961 + λ4Tminus k+1ε2

Discrete Dynamics in Nature and Society 7

λ3Tminus k λ2Tminus k+1 minus cSpkSpk1113960 1113961 + λ3Tminus k+1 1 + cSpk minus μ1113960 1113961

λ4Tminus k λ1Tminus k+1 minus αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ2Tminus k+1θ αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ3Tminus k+1(1 minus θ) αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ4Tminus k+1 1 minus μ2( 1113857

uk+1 min b max a1

Ak

λ1Tminus k+1Ik minus λ3Tminus k+1cSpkIk1113872 11138731113888 11138891113890 1113891

vk+1 min d max c1

Bk

minus λ2Tminus k+1c Spk1113872 11138732Stk1113874 11138751113888 11138891113890 1113891

wk+1 min f max e1

Ck

minus λ4Tminus k+1Pk1113872 11138731113888 11138891113890 1113891 (16)

end forStep 3 for k 0 1 N write

Ilowastk Ik

Slowastpk Spk

Slowasttk Stk

Plowastk Pk

ulowastk uk

vlowastk vk

wlowastk wk

(17)

end forIn this formulation there were initial conditions for the

state variables and terminal conditions for the adjoints ampatis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k N We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

5 Numerical Simulation

In this paragraph we give numerical simulation to highlightthe effectiveness of the strategy that we have developed in theframework of eliminating the rumor and limit its spreadand the initial values are the same as in Table 1 with regardto other initial values they are proposed values after astatistical study

In this section we introduce our control strategy whichconsists in using three kinds of controls the first one un

where an ignorant is informed by a stifler that the infor-mation is false or a rumor the second noted vn which isapplied by a professional admin and which consists ofdeactivating or blocking a fake account that specializes in

spreading the rumor and the third noted wn which is alsoapplied by an admin by blocking the pages created in orderto spread the rumor and this after reaching a significantnumber of signals and after the diagnosis of the content ofthe concerned page We apply our strategy for a period offifteen weeks that we assume an average duration of thespread of a rumor Figure 3 illustrates the results foundFrom Figure 3 we see that the effect of the strategy begins asearly as the second week and the number of the ignorantdecreases from 2000 in a fatal way until reaching almost 100this number is recovered in the number of stiflers whichincreases significantly from zero until reaching 2000 ampesame remark for the number of pages we notice that be-tween the first and the second week the number increasesstep by step however after the second week of the appli-cation of controls the number decreases until less than 50As for the number of spreaders the number is decreasingfrom 50 to zero

In the next section and in order to give more detailsabout our strategy as well as the effect of each control wechoose to apply three scenarios in each of these scenarioswe apply separately one control Figures 4ndash6 illustrate theresult found in each scenario for the three cases

(a) Apply only the control u

(b) Apply only the control v

(c) Apply only the control w

51 Apply Only the Control u In this scenario we simulatethe case where we apply a single control u with which weinform a portion of the ignorants by the false information sowe win this proportion in favor of stiflers and the control isapplied over a period of 100 days From the figure we seethat after 10 days the number of spreaders falls and does notexceed 3000 people throughout the period and the numberof stiflers meanwhile increases sharply to almost 20000which shows the effectiveness of our control

52 Apply Only the Control v In the second scenario weapply a single control v but this time a control that focuses

8 Discrete Dynamics in Nature and Society

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

with optimal control approach Other models from optimalcontrol problems and population dynamics can be found in[26ndash30]

In this paper in Section 2 we propose a discrete ISpStP

mathematical model that describes the dynamic of a pop-ulation that reacts in the spread of the rumor in a socialnetwork In Section 3 we present an optimal controlproblem for the proposed model where we give some resultsconcerning the existence of the optimal control and wecharacterize the optimal controls using the Pontryaginmaximum principle in discrete time Numerical simulationsthrough MATLAB are given in Section 4 Finally we con-clude the paper in Section 5

2 Mathematical Model andSimulation without Controls

In this section we consider a discrete mathematical modelISpStP that will describe the dynamics of a population ofrumors our model consists of four compartments repre-senting the subdivision of the population that reacts in thespread of the rumor in a social network

I ignorant users who do not know the rumor andsusceptible to be informedSp spreader users who spread the rumorSt stifler individuals who refuse to spread the rumorP rumorrsquos page the page specialized in spreading therumor

ampe compartment I represents the number of users who donot understand the rumor and who are susceptible to be in-formed and this population increases with the charge μN

which represents the new customers created an ignorant in-quires about the rumor through two ways either by using

consulting a specialized page in the diffusion of the rumor or atonce by means of the contact with a spreader some of theseusers deactivate their account at a rate μI ampus in thiscompartment we have an incoming flux which equals to μN

and an outgoing flux which equals toαhSpI + αeI(Pκ + P) + μI

ampe compartment Sp this compartment represents thenumber of humans who spread the rumor either immedi-ately or by way of sharing one-page publications or viadeveloping new publications ampus we have an incomingflux which equals to θ(αhSpI + αeI(Pκ + P)) which rep-resents the proportion of the new users who will spread therumor After the contact between two spreaders one of themdecides not to diffuse the information at a rate cS2p and afterthe contact of a spreader and a stifler the stifler succeeds toconvince him that the information is false at a rate λStSpafter a certain period a portion of the spreaders decide not tospread the rumor at a rate βSp

ampe compartment St this compartment represents thenumber of stiflers who refuse to spread the rumor ampisnumber increases at a rate (1 minus θ)(αhSpI + αeI(Pκ + P))

which represents the portion of users who knew that theinformation is wrong in addition to the flux that left the Sp

compartment Sp(cSp + λSt) + βSp and decreases with therate μ1St of stiflers who have deactivated their accounts

ampe compartment P this compartment represents thepage specialized in the diffusion of the rumor ampis pageconsists of malicious publications about the rumor in thispage Sp has the proper right to publish and share thesepublications at a rate δ1Sp and ε2Sp respectively and theignorant who consults the page also shares these publica-tions at a rate ε1I

ampe diagram will demonstrate the flux directions ofindividuals among the compartments Figure 1

I(n + 1) I(n) + μN minus αhSp(n)I(n) minus αeI(n)P(n)

κ + P(n)minus μ1I(n)

Sp(n + 1) Sp(n) + θ αhSp(n)I(n) + αeI(n)P(n)

κ + P(n)1113888 1113889 minus μ1 + β( 1113857Sp(n) minus cSp(n) Sp(n) + St(n)1113872 1113873

St(n + 1) St(n) +(1 minus θ) αhSp(n)I(n) + αeI(n)P(n)

κ + P(n)1113888 1113889 + cSp(n) Sp(n) + St(n)1113872 1113873 + βSp(n) minus μ1St(n)

P(n + 1) P(n) + ε1N + ε2Sp(n) minus μ2P(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

with initial values I(0) Sp(0) St(0) and P(0) which arenonnegatives

In order to demonstrate the effectiveness of themodel wehave proposed we will present a numerical simulation withFigure 2 so that we can see how well the model adapts toreality Initial values are approximate data that we suggestedafter studying and researching some statistics about the usersof social networks and the values are attached in the table

From Figure 2 we note that there is no significanteffect until the 30th day 30 days after the launch of therumor the number of ignorants decreases sharply and incontrast there is a significant rise of spreader people andthe number of stiflers and pages are rising on averagethese changes indicate that after 30 days trading rumorhas become more and more due to the continuous pub-lication of it

Discrete Dynamics in Nature and Society 3

3 The Model with Controls

Now we introduce our controls into system (1) as thecontrol measures to fight the spread of the rumor we extendour system by including three kinds of controls u v and wampe first control u is to tell users that the information orpublication is false and contains a malicious rumor thesecond control v is through the admin where he deactivatesan account after learning that it is fake or aimed at spreading

the rumor and the last one w is also applied by the adminthis time by deactivating the page intended to spread therumor after the arrival of a certain number of complaints

In the aim of better understanding the effects of anycontrol measure of these strategies we introduce three newvariables πi where i 1 2 3 πi 0 in the absence of thecontrol and πi 1 in the presence of the control

I(n + 1) I(n) + μN minus αhSp(n)I(n) minus αeI(n)P(n)

κ + P(n)minus μ1I(n) minus π1unI(n)

Sp(n + 1) Sp(n) + θ αhSp(n)I(n) + αeI(n)P(n)

κ + P(n)1113888 1113889 minus μ1 + β( 1113857Sp(n) minus cSp(n) Sp(n) + St(n)1113872 1113873 minus π2vnSp(n)

St(n + 1) St(n) +(1 minus θ) αhSp(n)I(n) + αeIn

Pn

κ + Pn

1113888 1113889 + cSp(n) Sp(n) + St(n)1113872 1113873 + βSp(n) minus μ1St(n) + π1unIn

P(n + 1) Pn + ε1N + ε2Sp(n) minus μ2Pn minus π3wnPn

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

P

I Sp St

αe (Pκ + P)

θ (αh + αe)

γSPSP

γSPSt

β SPμ1

μ1

μ1

μN

(1 ndash θ) (αh + αe (Pκ + P))

Figure 1 Description diagram of the rumor dynamics

Ignorant without controlSpreader without control

Stifler without controlPage without control

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

100

Figure 2 Dynamics without control strategy

4 Discrete Dynamics in Nature and Society

4 An Optimal Control Approach

ampe problem is to minimize the objective functional

J(u v w) MNSp(N) + 1113944Nminus 1

k0MkSp(k) +

12Akπ1u

2k1113874

+12Bkπ2v

2k +

12Ckπ3w

2k1113875

(3)

where Ak gt 0 Bk gt 0 Ck gt 0 and Mk gt 0 for i isin 0 N

are the cost coefficients ampey are selected to weigh therelative importance of Sp(k) uk vk and wk at time kNis thefinal time

In other words we seek the optimal controls uk vk andwk such that

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (4)

where Uad is the set of admissible controls defined by

Uad uk vk wk( 1113857

ale uk le b cle vk le d elewk lef1113896 1113897 k 0 N minus 1

(5)

and (a b c d e f) isin (]0 1[)6ampe sufficient condition for the existence of an optimal

control (ulowastk vlowastk wlowastk ) for problem (3) comes from the fol-lowing theorem

Theorem 1 7ere exists an optimal control (ulowast vlowast wlowast) suchthat

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (6)

subject to control system (2) with initial conditions

Proof ampere are a finite number of time steps S

(S0 S1 Sn) I (I0 I1 IN) and P (P0 P1

PN) which are uniformly bounded for all (u v w) isin Uadampus J(u v w) is uniformly bounded for all (u v w) in thecontrol set Uad Since J(u v w) isboundedinf(uvw)isinUad

J(u v w) is finite and there exists asequence u

j

k vj

k wj

k isin Uad such that limj⟶+infinJ(uj

k vj

k wj

k)

inf(uvw)isinUadJ(u v w) and the corresponding sequences of

states Ii Sit Si

p and Bi Since there is a finite number ofuniformly bounded sequences there exist(ulowastk vlowastk wlowastk ) isin U3

ad and S I P R isin RN+1 such that on asubsequence u

j

k⟶ ulowastk vj

k⟶ vlowastk wj

k⟶ w lowastk Sj⟶ SlowastIj⟶ Ilowast and Pj⟶ Plowast Finally due to the finite di-mensional structure of system (2) and the objective functionJ(u v w) (ulowast vlowast wlowast) is an optimal control with corre-sponding states Ilowast Slowastt Slowastp Plowast ampereforeinf(uvw)isinUad

J(u v w) is achievedIn order to derive the necessary condition for the optimal

control Pontryaginrsquos maximum principle in discrete timegiven in [31 32] was used ampis principle converts into aproblem of minimizing a Hamiltonian at the time stepdefined by

Hk MkSp(k) +12Akπ1u

2k +

12Bkπ2v

2k

+12Ckπ3w

2k + 1113944

4

j1λjk+1fjk+1

(7)

where fjk+1 is the right side of the system of differenceequation (2) of the jth state variable at time step k + 1

Using Pontryaginrsquos maximum principle in discrete time[31ndash33] we can say the following theorem

Theorem 2 Let Ilowast Slowastp Slowastt and Plowast be optimal state solutionswith an associated optimal control (ulowast vlowast wlowast) for optimalcontrol problem (3) 7en there exist adjoint variablesλ1k λ2k λ3k λ4k satisfying

Δλ1k λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k 1 + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk(k)1113858 1113859 + λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k λ2k+1 minus cSp(k)Sp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

Discrete Dynamics in Nature and Society 5

with transversality conditions at time

λ1N AN

λ2N λ3N λ4N 0(9)

Furthermore for k 0 1 N minus 1 and forπ1 π2 π3 1 the optimal controls ulowastk vlowastk and wlowastk aregiven by

ulowastk min b max a

λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

1113888 11138891113888 1113889

vlowastk min d max c

minus λ2k+1c Sp(k)1113872 11138732St(k)

Bk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

wlowastk min f max e

λ4k+1P(k)

Ck

1113888 11138891113888 1113889

(10)

Proof ampe Hamiltonian of the optimal problem is given by

Hk MkSp(k) +12Aku

2k +

12Bkv

2k +

12Ckw

2k + λ1k+1 I(k) + μN minus αhSp(k)I(k) minus αeI(k)

P(k)

κ + P(k)minus μ1I(k) minus π1ukI1113888 1113889

+ λ2k+1 Sp(k) + θ αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 minus (μ + β)Sp(k) minus cSp(k) Sp(k) + St(k)1113872 1113873 minus π2vkSp(k)1113888 1113889

+ λ3k+1 St(k) +(1 minus θ) αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 + cSp(k) Sp(k) + St(k)1113872 1113873 + βSp(k) minus μSt(k) + π1ukI1113888 1113889

+ λ4k+1 P(k) + ε1N(k) + ε2Sp(k) minus μ2P(k) minus π3wkP(k)1113872 1113873

(11)

For k 0 1 N minus 1 the adjoint equations andtransversality conditions can be obtained by using

Pontryaginrsquos maximum principle in discrete time given in[31ndash33] such that

Δλ1k zHk

zIk

λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k zHk

zSp(k) Mk + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk1113858 1113859

+λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k zHk

zSt(k) λ2k+1 minus cSp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k zHk

zPk

λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

6 Discrete Dynamics in Nature and Society

Put Φ(N) ANIN

λ1N zΦ(N)

zI AN

λ2N zΦ(N)

zSp

0

λ3N zΦ(N)

zSt

0

λ4N zΦ(N)

zP 0

(13)

For k 0 1 N minus 1 the optimal controls ulowastk vlowastk andwlowastk can be solved from the following optimality condition

zH

zuk

Akuk minus λ1k+1π1I(k) + λ3k+1cπ1Sp(k)I(k) 0⟺

uk λ1k+1π1I(k) minus λ3k+1cπ1Sp(k)I(k)

Ak

zH

zvk

Bkvk + λ2k+1c Sp(k)1113872 11138732π2(k) 0⟺

vk minus λ2k+1c Sp(k)1113872 1113873

2π2St(k)

Bk

zH

zwk

Ckwk minus λ4k+1π3P(k) 0⟺

wk λ4k+1π3P(k)

Ck

(14)

So for π1 π2 π3 1 we have

uk λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

vk minus λ2k+1c Sp(k)1113872 1113873

2St(k)

Bk

wk λ4k+1P(k)

Ck

(15)

However if πi 0 for i 1 2 3 the control attached tothis case will be eliminated and removed By the bounds inUad of the controls it is easy to obtain uk

lowast vklowast and wk

lowast in theform of (10)

41 Algorithm In this section we present the results ob-tained by solving numerically the optimality system ampissystem consists of the state system adjoint system initialand final time conditions and the control characterizationSo the optimality system is given by the following

Step 1 I0 i0 Sp0 sp0 St0 st0 P0 p0λ2N λ3N λ4N 0 λ1N AN and given ulowastk0 vlowastk0and wlowastk0

Step 2 for k 0 1 N minus 1 do

Ik+1 μN minus αhSpkIk minus αeIk

Pk

κ + Pk

minus μ1Ik

Sp(k+1) θ αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 minus (μ + β)Spk minus cSpk Spk + Stk1113872 1113873

Sp(k+1) (1 minus θ) αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 + cSpk Spk + Stk1113872 1113873 + βSpk minus μStk

Pk+1 ε1N + ε2Spk minus μ2Pk

⋮ ⋮

⋮ ⋮

λ1Tminus k λ1Tminus k+1 1 minus αhSpk minus αe

Pk

κ + Pk

minus μ11113888 1113889 + λ2Tminus k+1θ αhSpk + αe

Pk

κ + Pk

1113888 1113889 + λ3Tminus k+1(1 minus θ) αhSpk + αe

Pk

κ + Pk

1113888 1113889

λ2Tminus k 1 + λ1Tminus k+1 minus αhIk( 1113857 + λ2Tminus k+1 1 + θαhIk minus (μ + β) minus 2c( 1113857 + λ3Tminus k+1 (1 minus θ) αhSpk1113872 1113873 + 2c + β1113960 1113961 + λ4Tminus k+1ε2

Discrete Dynamics in Nature and Society 7

λ3Tminus k λ2Tminus k+1 minus cSpkSpk1113960 1113961 + λ3Tminus k+1 1 + cSpk minus μ1113960 1113961

λ4Tminus k λ1Tminus k+1 minus αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ2Tminus k+1θ αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ3Tminus k+1(1 minus θ) αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ4Tminus k+1 1 minus μ2( 1113857

uk+1 min b max a1

Ak

λ1Tminus k+1Ik minus λ3Tminus k+1cSpkIk1113872 11138731113888 11138891113890 1113891

vk+1 min d max c1

Bk

minus λ2Tminus k+1c Spk1113872 11138732Stk1113874 11138751113888 11138891113890 1113891

wk+1 min f max e1

Ck

minus λ4Tminus k+1Pk1113872 11138731113888 11138891113890 1113891 (16)

end forStep 3 for k 0 1 N write

Ilowastk Ik

Slowastpk Spk

Slowasttk Stk

Plowastk Pk

ulowastk uk

vlowastk vk

wlowastk wk

(17)

end forIn this formulation there were initial conditions for the

state variables and terminal conditions for the adjoints ampatis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k N We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

5 Numerical Simulation

In this paragraph we give numerical simulation to highlightthe effectiveness of the strategy that we have developed in theframework of eliminating the rumor and limit its spreadand the initial values are the same as in Table 1 with regardto other initial values they are proposed values after astatistical study

In this section we introduce our control strategy whichconsists in using three kinds of controls the first one un

where an ignorant is informed by a stifler that the infor-mation is false or a rumor the second noted vn which isapplied by a professional admin and which consists ofdeactivating or blocking a fake account that specializes in

spreading the rumor and the third noted wn which is alsoapplied by an admin by blocking the pages created in orderto spread the rumor and this after reaching a significantnumber of signals and after the diagnosis of the content ofthe concerned page We apply our strategy for a period offifteen weeks that we assume an average duration of thespread of a rumor Figure 3 illustrates the results foundFrom Figure 3 we see that the effect of the strategy begins asearly as the second week and the number of the ignorantdecreases from 2000 in a fatal way until reaching almost 100this number is recovered in the number of stiflers whichincreases significantly from zero until reaching 2000 ampesame remark for the number of pages we notice that be-tween the first and the second week the number increasesstep by step however after the second week of the appli-cation of controls the number decreases until less than 50As for the number of spreaders the number is decreasingfrom 50 to zero

In the next section and in order to give more detailsabout our strategy as well as the effect of each control wechoose to apply three scenarios in each of these scenarioswe apply separately one control Figures 4ndash6 illustrate theresult found in each scenario for the three cases

(a) Apply only the control u

(b) Apply only the control v

(c) Apply only the control w

51 Apply Only the Control u In this scenario we simulatethe case where we apply a single control u with which weinform a portion of the ignorants by the false information sowe win this proportion in favor of stiflers and the control isapplied over a period of 100 days From the figure we seethat after 10 days the number of spreaders falls and does notexceed 3000 people throughout the period and the numberof stiflers meanwhile increases sharply to almost 20000which shows the effectiveness of our control

52 Apply Only the Control v In the second scenario weapply a single control v but this time a control that focuses

8 Discrete Dynamics in Nature and Society

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

3 The Model with Controls

Now we introduce our controls into system (1) as thecontrol measures to fight the spread of the rumor we extendour system by including three kinds of controls u v and wampe first control u is to tell users that the information orpublication is false and contains a malicious rumor thesecond control v is through the admin where he deactivatesan account after learning that it is fake or aimed at spreading

the rumor and the last one w is also applied by the adminthis time by deactivating the page intended to spread therumor after the arrival of a certain number of complaints

In the aim of better understanding the effects of anycontrol measure of these strategies we introduce three newvariables πi where i 1 2 3 πi 0 in the absence of thecontrol and πi 1 in the presence of the control

I(n + 1) I(n) + μN minus αhSp(n)I(n) minus αeI(n)P(n)

κ + P(n)minus μ1I(n) minus π1unI(n)

Sp(n + 1) Sp(n) + θ αhSp(n)I(n) + αeI(n)P(n)

κ + P(n)1113888 1113889 minus μ1 + β( 1113857Sp(n) minus cSp(n) Sp(n) + St(n)1113872 1113873 minus π2vnSp(n)

St(n + 1) St(n) +(1 minus θ) αhSp(n)I(n) + αeIn

Pn

κ + Pn

1113888 1113889 + cSp(n) Sp(n) + St(n)1113872 1113873 + βSp(n) minus μ1St(n) + π1unIn

P(n + 1) Pn + ε1N + ε2Sp(n) minus μ2Pn minus π3wnPn

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

P

I Sp St

αe (Pκ + P)

θ (αh + αe)

γSPSP

γSPSt

β SPμ1

μ1

μ1

μN

(1 ndash θ) (αh + αe (Pκ + P))

Figure 1 Description diagram of the rumor dynamics

Ignorant without controlSpreader without control

Stifler without controlPage without control

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

100

Figure 2 Dynamics without control strategy

4 Discrete Dynamics in Nature and Society

4 An Optimal Control Approach

ampe problem is to minimize the objective functional

J(u v w) MNSp(N) + 1113944Nminus 1

k0MkSp(k) +

12Akπ1u

2k1113874

+12Bkπ2v

2k +

12Ckπ3w

2k1113875

(3)

where Ak gt 0 Bk gt 0 Ck gt 0 and Mk gt 0 for i isin 0 N

are the cost coefficients ampey are selected to weigh therelative importance of Sp(k) uk vk and wk at time kNis thefinal time

In other words we seek the optimal controls uk vk andwk such that

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (4)

where Uad is the set of admissible controls defined by

Uad uk vk wk( 1113857

ale uk le b cle vk le d elewk lef1113896 1113897 k 0 N minus 1

(5)

and (a b c d e f) isin (]0 1[)6ampe sufficient condition for the existence of an optimal

control (ulowastk vlowastk wlowastk ) for problem (3) comes from the fol-lowing theorem

Theorem 1 7ere exists an optimal control (ulowast vlowast wlowast) suchthat

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (6)

subject to control system (2) with initial conditions

Proof ampere are a finite number of time steps S

(S0 S1 Sn) I (I0 I1 IN) and P (P0 P1

PN) which are uniformly bounded for all (u v w) isin Uadampus J(u v w) is uniformly bounded for all (u v w) in thecontrol set Uad Since J(u v w) isboundedinf(uvw)isinUad

J(u v w) is finite and there exists asequence u

j

k vj

k wj

k isin Uad such that limj⟶+infinJ(uj

k vj

k wj

k)

inf(uvw)isinUadJ(u v w) and the corresponding sequences of

states Ii Sit Si

p and Bi Since there is a finite number ofuniformly bounded sequences there exist(ulowastk vlowastk wlowastk ) isin U3

ad and S I P R isin RN+1 such that on asubsequence u

j

k⟶ ulowastk vj

k⟶ vlowastk wj

k⟶ w lowastk Sj⟶ SlowastIj⟶ Ilowast and Pj⟶ Plowast Finally due to the finite di-mensional structure of system (2) and the objective functionJ(u v w) (ulowast vlowast wlowast) is an optimal control with corre-sponding states Ilowast Slowastt Slowastp Plowast ampereforeinf(uvw)isinUad

J(u v w) is achievedIn order to derive the necessary condition for the optimal

control Pontryaginrsquos maximum principle in discrete timegiven in [31 32] was used ampis principle converts into aproblem of minimizing a Hamiltonian at the time stepdefined by

Hk MkSp(k) +12Akπ1u

2k +

12Bkπ2v

2k

+12Ckπ3w

2k + 1113944

4

j1λjk+1fjk+1

(7)

where fjk+1 is the right side of the system of differenceequation (2) of the jth state variable at time step k + 1

Using Pontryaginrsquos maximum principle in discrete time[31ndash33] we can say the following theorem

Theorem 2 Let Ilowast Slowastp Slowastt and Plowast be optimal state solutionswith an associated optimal control (ulowast vlowast wlowast) for optimalcontrol problem (3) 7en there exist adjoint variablesλ1k λ2k λ3k λ4k satisfying

Δλ1k λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k 1 + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk(k)1113858 1113859 + λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k λ2k+1 minus cSp(k)Sp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

Discrete Dynamics in Nature and Society 5

with transversality conditions at time

λ1N AN

λ2N λ3N λ4N 0(9)

Furthermore for k 0 1 N minus 1 and forπ1 π2 π3 1 the optimal controls ulowastk vlowastk and wlowastk aregiven by

ulowastk min b max a

λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

1113888 11138891113888 1113889

vlowastk min d max c

minus λ2k+1c Sp(k)1113872 11138732St(k)

Bk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

wlowastk min f max e

λ4k+1P(k)

Ck

1113888 11138891113888 1113889

(10)

Proof ampe Hamiltonian of the optimal problem is given by

Hk MkSp(k) +12Aku

2k +

12Bkv

2k +

12Ckw

2k + λ1k+1 I(k) + μN minus αhSp(k)I(k) minus αeI(k)

P(k)

κ + P(k)minus μ1I(k) minus π1ukI1113888 1113889

+ λ2k+1 Sp(k) + θ αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 minus (μ + β)Sp(k) minus cSp(k) Sp(k) + St(k)1113872 1113873 minus π2vkSp(k)1113888 1113889

+ λ3k+1 St(k) +(1 minus θ) αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 + cSp(k) Sp(k) + St(k)1113872 1113873 + βSp(k) minus μSt(k) + π1ukI1113888 1113889

+ λ4k+1 P(k) + ε1N(k) + ε2Sp(k) minus μ2P(k) minus π3wkP(k)1113872 1113873

(11)

For k 0 1 N minus 1 the adjoint equations andtransversality conditions can be obtained by using

Pontryaginrsquos maximum principle in discrete time given in[31ndash33] such that

Δλ1k zHk

zIk

λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k zHk

zSp(k) Mk + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk1113858 1113859

+λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k zHk

zSt(k) λ2k+1 minus cSp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k zHk

zPk

λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

6 Discrete Dynamics in Nature and Society

Put Φ(N) ANIN

λ1N zΦ(N)

zI AN

λ2N zΦ(N)

zSp

0

λ3N zΦ(N)

zSt

0

λ4N zΦ(N)

zP 0

(13)

For k 0 1 N minus 1 the optimal controls ulowastk vlowastk andwlowastk can be solved from the following optimality condition

zH

zuk

Akuk minus λ1k+1π1I(k) + λ3k+1cπ1Sp(k)I(k) 0⟺

uk λ1k+1π1I(k) minus λ3k+1cπ1Sp(k)I(k)

Ak

zH

zvk

Bkvk + λ2k+1c Sp(k)1113872 11138732π2(k) 0⟺

vk minus λ2k+1c Sp(k)1113872 1113873

2π2St(k)

Bk

zH

zwk

Ckwk minus λ4k+1π3P(k) 0⟺

wk λ4k+1π3P(k)

Ck

(14)

So for π1 π2 π3 1 we have

uk λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

vk minus λ2k+1c Sp(k)1113872 1113873

2St(k)

Bk

wk λ4k+1P(k)

Ck

(15)

However if πi 0 for i 1 2 3 the control attached tothis case will be eliminated and removed By the bounds inUad of the controls it is easy to obtain uk

lowast vklowast and wk

lowast in theform of (10)

41 Algorithm In this section we present the results ob-tained by solving numerically the optimality system ampissystem consists of the state system adjoint system initialand final time conditions and the control characterizationSo the optimality system is given by the following

Step 1 I0 i0 Sp0 sp0 St0 st0 P0 p0λ2N λ3N λ4N 0 λ1N AN and given ulowastk0 vlowastk0and wlowastk0

Step 2 for k 0 1 N minus 1 do

Ik+1 μN minus αhSpkIk minus αeIk

Pk

κ + Pk

minus μ1Ik

Sp(k+1) θ αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 minus (μ + β)Spk minus cSpk Spk + Stk1113872 1113873

Sp(k+1) (1 minus θ) αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 + cSpk Spk + Stk1113872 1113873 + βSpk minus μStk

Pk+1 ε1N + ε2Spk minus μ2Pk

⋮ ⋮

⋮ ⋮

λ1Tminus k λ1Tminus k+1 1 minus αhSpk minus αe

Pk

κ + Pk

minus μ11113888 1113889 + λ2Tminus k+1θ αhSpk + αe

Pk

κ + Pk

1113888 1113889 + λ3Tminus k+1(1 minus θ) αhSpk + αe

Pk

κ + Pk

1113888 1113889

λ2Tminus k 1 + λ1Tminus k+1 minus αhIk( 1113857 + λ2Tminus k+1 1 + θαhIk minus (μ + β) minus 2c( 1113857 + λ3Tminus k+1 (1 minus θ) αhSpk1113872 1113873 + 2c + β1113960 1113961 + λ4Tminus k+1ε2

Discrete Dynamics in Nature and Society 7

λ3Tminus k λ2Tminus k+1 minus cSpkSpk1113960 1113961 + λ3Tminus k+1 1 + cSpk minus μ1113960 1113961

λ4Tminus k λ1Tminus k+1 minus αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ2Tminus k+1θ αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ3Tminus k+1(1 minus θ) αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ4Tminus k+1 1 minus μ2( 1113857

uk+1 min b max a1

Ak

λ1Tminus k+1Ik minus λ3Tminus k+1cSpkIk1113872 11138731113888 11138891113890 1113891

vk+1 min d max c1

Bk

minus λ2Tminus k+1c Spk1113872 11138732Stk1113874 11138751113888 11138891113890 1113891

wk+1 min f max e1

Ck

minus λ4Tminus k+1Pk1113872 11138731113888 11138891113890 1113891 (16)

end forStep 3 for k 0 1 N write

Ilowastk Ik

Slowastpk Spk

Slowasttk Stk

Plowastk Pk

ulowastk uk

vlowastk vk

wlowastk wk

(17)

end forIn this formulation there were initial conditions for the

state variables and terminal conditions for the adjoints ampatis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k N We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

5 Numerical Simulation

In this paragraph we give numerical simulation to highlightthe effectiveness of the strategy that we have developed in theframework of eliminating the rumor and limit its spreadand the initial values are the same as in Table 1 with regardto other initial values they are proposed values after astatistical study

In this section we introduce our control strategy whichconsists in using three kinds of controls the first one un

where an ignorant is informed by a stifler that the infor-mation is false or a rumor the second noted vn which isapplied by a professional admin and which consists ofdeactivating or blocking a fake account that specializes in

spreading the rumor and the third noted wn which is alsoapplied by an admin by blocking the pages created in orderto spread the rumor and this after reaching a significantnumber of signals and after the diagnosis of the content ofthe concerned page We apply our strategy for a period offifteen weeks that we assume an average duration of thespread of a rumor Figure 3 illustrates the results foundFrom Figure 3 we see that the effect of the strategy begins asearly as the second week and the number of the ignorantdecreases from 2000 in a fatal way until reaching almost 100this number is recovered in the number of stiflers whichincreases significantly from zero until reaching 2000 ampesame remark for the number of pages we notice that be-tween the first and the second week the number increasesstep by step however after the second week of the appli-cation of controls the number decreases until less than 50As for the number of spreaders the number is decreasingfrom 50 to zero

In the next section and in order to give more detailsabout our strategy as well as the effect of each control wechoose to apply three scenarios in each of these scenarioswe apply separately one control Figures 4ndash6 illustrate theresult found in each scenario for the three cases

(a) Apply only the control u

(b) Apply only the control v

(c) Apply only the control w

51 Apply Only the Control u In this scenario we simulatethe case where we apply a single control u with which weinform a portion of the ignorants by the false information sowe win this proportion in favor of stiflers and the control isapplied over a period of 100 days From the figure we seethat after 10 days the number of spreaders falls and does notexceed 3000 people throughout the period and the numberof stiflers meanwhile increases sharply to almost 20000which shows the effectiveness of our control

52 Apply Only the Control v In the second scenario weapply a single control v but this time a control that focuses

8 Discrete Dynamics in Nature and Society

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

4 An Optimal Control Approach

ampe problem is to minimize the objective functional

J(u v w) MNSp(N) + 1113944Nminus 1

k0MkSp(k) +

12Akπ1u

2k1113874

+12Bkπ2v

2k +

12Ckπ3w

2k1113875

(3)

where Ak gt 0 Bk gt 0 Ck gt 0 and Mk gt 0 for i isin 0 N

are the cost coefficients ampey are selected to weigh therelative importance of Sp(k) uk vk and wk at time kNis thefinal time

In other words we seek the optimal controls uk vk andwk such that

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (4)

where Uad is the set of admissible controls defined by

Uad uk vk wk( 1113857

ale uk le b cle vk le d elewk lef1113896 1113897 k 0 N minus 1

(5)

and (a b c d e f) isin (]0 1[)6ampe sufficient condition for the existence of an optimal

control (ulowastk vlowastk wlowastk ) for problem (3) comes from the fol-lowing theorem

Theorem 1 7ere exists an optimal control (ulowast vlowast wlowast) suchthat

J ulowast vlowast wlowast

( 1113857 min(uvw)

J(u v w)

(u v w)isin Uad1113896 1113897 (6)

subject to control system (2) with initial conditions

Proof ampere are a finite number of time steps S

(S0 S1 Sn) I (I0 I1 IN) and P (P0 P1

PN) which are uniformly bounded for all (u v w) isin Uadampus J(u v w) is uniformly bounded for all (u v w) in thecontrol set Uad Since J(u v w) isboundedinf(uvw)isinUad

J(u v w) is finite and there exists asequence u

j

k vj

k wj

k isin Uad such that limj⟶+infinJ(uj

k vj

k wj

k)

inf(uvw)isinUadJ(u v w) and the corresponding sequences of

states Ii Sit Si

p and Bi Since there is a finite number ofuniformly bounded sequences there exist(ulowastk vlowastk wlowastk ) isin U3

ad and S I P R isin RN+1 such that on asubsequence u

j

k⟶ ulowastk vj

k⟶ vlowastk wj

k⟶ w lowastk Sj⟶ SlowastIj⟶ Ilowast and Pj⟶ Plowast Finally due to the finite di-mensional structure of system (2) and the objective functionJ(u v w) (ulowast vlowast wlowast) is an optimal control with corre-sponding states Ilowast Slowastt Slowastp Plowast ampereforeinf(uvw)isinUad

J(u v w) is achievedIn order to derive the necessary condition for the optimal

control Pontryaginrsquos maximum principle in discrete timegiven in [31 32] was used ampis principle converts into aproblem of minimizing a Hamiltonian at the time stepdefined by

Hk MkSp(k) +12Akπ1u

2k +

12Bkπ2v

2k

+12Ckπ3w

2k + 1113944

4

j1λjk+1fjk+1

(7)

where fjk+1 is the right side of the system of differenceequation (2) of the jth state variable at time step k + 1

Using Pontryaginrsquos maximum principle in discrete time[31ndash33] we can say the following theorem

Theorem 2 Let Ilowast Slowastp Slowastt and Plowast be optimal state solutionswith an associated optimal control (ulowast vlowast wlowast) for optimalcontrol problem (3) 7en there exist adjoint variablesλ1k λ2k λ3k λ4k satisfying

Δλ1k λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k 1 + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk(k)1113858 1113859 + λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k λ2k+1 minus cSp(k)Sp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

Discrete Dynamics in Nature and Society 5

with transversality conditions at time

λ1N AN

λ2N λ3N λ4N 0(9)

Furthermore for k 0 1 N minus 1 and forπ1 π2 π3 1 the optimal controls ulowastk vlowastk and wlowastk aregiven by

ulowastk min b max a

λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

1113888 11138891113888 1113889

vlowastk min d max c

minus λ2k+1c Sp(k)1113872 11138732St(k)

Bk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

wlowastk min f max e

λ4k+1P(k)

Ck

1113888 11138891113888 1113889

(10)

Proof ampe Hamiltonian of the optimal problem is given by

Hk MkSp(k) +12Aku

2k +

12Bkv

2k +

12Ckw

2k + λ1k+1 I(k) + μN minus αhSp(k)I(k) minus αeI(k)

P(k)

κ + P(k)minus μ1I(k) minus π1ukI1113888 1113889

+ λ2k+1 Sp(k) + θ αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 minus (μ + β)Sp(k) minus cSp(k) Sp(k) + St(k)1113872 1113873 minus π2vkSp(k)1113888 1113889

+ λ3k+1 St(k) +(1 minus θ) αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 + cSp(k) Sp(k) + St(k)1113872 1113873 + βSp(k) minus μSt(k) + π1ukI1113888 1113889

+ λ4k+1 P(k) + ε1N(k) + ε2Sp(k) minus μ2P(k) minus π3wkP(k)1113872 1113873

(11)

For k 0 1 N minus 1 the adjoint equations andtransversality conditions can be obtained by using

Pontryaginrsquos maximum principle in discrete time given in[31ndash33] such that

Δλ1k zHk

zIk

λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k zHk

zSp(k) Mk + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk1113858 1113859

+λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k zHk

zSt(k) λ2k+1 minus cSp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k zHk

zPk

λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

6 Discrete Dynamics in Nature and Society

Put Φ(N) ANIN

λ1N zΦ(N)

zI AN

λ2N zΦ(N)

zSp

0

λ3N zΦ(N)

zSt

0

λ4N zΦ(N)

zP 0

(13)

For k 0 1 N minus 1 the optimal controls ulowastk vlowastk andwlowastk can be solved from the following optimality condition

zH

zuk

Akuk minus λ1k+1π1I(k) + λ3k+1cπ1Sp(k)I(k) 0⟺

uk λ1k+1π1I(k) minus λ3k+1cπ1Sp(k)I(k)

Ak

zH

zvk

Bkvk + λ2k+1c Sp(k)1113872 11138732π2(k) 0⟺

vk minus λ2k+1c Sp(k)1113872 1113873

2π2St(k)

Bk

zH

zwk

Ckwk minus λ4k+1π3P(k) 0⟺

wk λ4k+1π3P(k)

Ck

(14)

So for π1 π2 π3 1 we have

uk λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

vk minus λ2k+1c Sp(k)1113872 1113873

2St(k)

Bk

wk λ4k+1P(k)

Ck

(15)

However if πi 0 for i 1 2 3 the control attached tothis case will be eliminated and removed By the bounds inUad of the controls it is easy to obtain uk

lowast vklowast and wk

lowast in theform of (10)

41 Algorithm In this section we present the results ob-tained by solving numerically the optimality system ampissystem consists of the state system adjoint system initialand final time conditions and the control characterizationSo the optimality system is given by the following

Step 1 I0 i0 Sp0 sp0 St0 st0 P0 p0λ2N λ3N λ4N 0 λ1N AN and given ulowastk0 vlowastk0and wlowastk0

Step 2 for k 0 1 N minus 1 do

Ik+1 μN minus αhSpkIk minus αeIk

Pk

κ + Pk

minus μ1Ik

Sp(k+1) θ αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 minus (μ + β)Spk minus cSpk Spk + Stk1113872 1113873

Sp(k+1) (1 minus θ) αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 + cSpk Spk + Stk1113872 1113873 + βSpk minus μStk

Pk+1 ε1N + ε2Spk minus μ2Pk

⋮ ⋮

⋮ ⋮

λ1Tminus k λ1Tminus k+1 1 minus αhSpk minus αe

Pk

κ + Pk

minus μ11113888 1113889 + λ2Tminus k+1θ αhSpk + αe

Pk

κ + Pk

1113888 1113889 + λ3Tminus k+1(1 minus θ) αhSpk + αe

Pk

κ + Pk

1113888 1113889

λ2Tminus k 1 + λ1Tminus k+1 minus αhIk( 1113857 + λ2Tminus k+1 1 + θαhIk minus (μ + β) minus 2c( 1113857 + λ3Tminus k+1 (1 minus θ) αhSpk1113872 1113873 + 2c + β1113960 1113961 + λ4Tminus k+1ε2

Discrete Dynamics in Nature and Society 7

λ3Tminus k λ2Tminus k+1 minus cSpkSpk1113960 1113961 + λ3Tminus k+1 1 + cSpk minus μ1113960 1113961

λ4Tminus k λ1Tminus k+1 minus αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ2Tminus k+1θ αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ3Tminus k+1(1 minus θ) αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ4Tminus k+1 1 minus μ2( 1113857

uk+1 min b max a1

Ak

λ1Tminus k+1Ik minus λ3Tminus k+1cSpkIk1113872 11138731113888 11138891113890 1113891

vk+1 min d max c1

Bk

minus λ2Tminus k+1c Spk1113872 11138732Stk1113874 11138751113888 11138891113890 1113891

wk+1 min f max e1

Ck

minus λ4Tminus k+1Pk1113872 11138731113888 11138891113890 1113891 (16)

end forStep 3 for k 0 1 N write

Ilowastk Ik

Slowastpk Spk

Slowasttk Stk

Plowastk Pk

ulowastk uk

vlowastk vk

wlowastk wk

(17)

end forIn this formulation there were initial conditions for the

state variables and terminal conditions for the adjoints ampatis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k N We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

5 Numerical Simulation

In this paragraph we give numerical simulation to highlightthe effectiveness of the strategy that we have developed in theframework of eliminating the rumor and limit its spreadand the initial values are the same as in Table 1 with regardto other initial values they are proposed values after astatistical study

In this section we introduce our control strategy whichconsists in using three kinds of controls the first one un

where an ignorant is informed by a stifler that the infor-mation is false or a rumor the second noted vn which isapplied by a professional admin and which consists ofdeactivating or blocking a fake account that specializes in

spreading the rumor and the third noted wn which is alsoapplied by an admin by blocking the pages created in orderto spread the rumor and this after reaching a significantnumber of signals and after the diagnosis of the content ofthe concerned page We apply our strategy for a period offifteen weeks that we assume an average duration of thespread of a rumor Figure 3 illustrates the results foundFrom Figure 3 we see that the effect of the strategy begins asearly as the second week and the number of the ignorantdecreases from 2000 in a fatal way until reaching almost 100this number is recovered in the number of stiflers whichincreases significantly from zero until reaching 2000 ampesame remark for the number of pages we notice that be-tween the first and the second week the number increasesstep by step however after the second week of the appli-cation of controls the number decreases until less than 50As for the number of spreaders the number is decreasingfrom 50 to zero

In the next section and in order to give more detailsabout our strategy as well as the effect of each control wechoose to apply three scenarios in each of these scenarioswe apply separately one control Figures 4ndash6 illustrate theresult found in each scenario for the three cases

(a) Apply only the control u

(b) Apply only the control v

(c) Apply only the control w

51 Apply Only the Control u In this scenario we simulatethe case where we apply a single control u with which weinform a portion of the ignorants by the false information sowe win this proportion in favor of stiflers and the control isapplied over a period of 100 days From the figure we seethat after 10 days the number of spreaders falls and does notexceed 3000 people throughout the period and the numberof stiflers meanwhile increases sharply to almost 20000which shows the effectiveness of our control

52 Apply Only the Control v In the second scenario weapply a single control v but this time a control that focuses

8 Discrete Dynamics in Nature and Society

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

with transversality conditions at time

λ1N AN

λ2N λ3N λ4N 0(9)

Furthermore for k 0 1 N minus 1 and forπ1 π2 π3 1 the optimal controls ulowastk vlowastk and wlowastk aregiven by

ulowastk min b max a

λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

1113888 11138891113888 1113889

vlowastk min d max c

minus λ2k+1c Sp(k)1113872 11138732St(k)

Bk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

wlowastk min f max e

λ4k+1P(k)

Ck

1113888 11138891113888 1113889

(10)

Proof ampe Hamiltonian of the optimal problem is given by

Hk MkSp(k) +12Aku

2k +

12Bkv

2k +

12Ckw

2k + λ1k+1 I(k) + μN minus αhSp(k)I(k) minus αeI(k)

P(k)

κ + P(k)minus μ1I(k) minus π1ukI1113888 1113889

+ λ2k+1 Sp(k) + θ αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 minus (μ + β)Sp(k) minus cSp(k) Sp(k) + St(k)1113872 1113873 minus π2vkSp(k)1113888 1113889

+ λ3k+1 St(k) +(1 minus θ) αhSp(k)I(k) + αeI(k)P(k)

κ + P(k)1113888 1113889 + cSp(k) Sp(k) + St(k)1113872 1113873 + βSp(k) minus μSt(k) + π1ukI1113888 1113889

+ λ4k+1 P(k) + ε1N(k) + ε2Sp(k) minus μ2P(k) minus π3wkP(k)1113872 1113873

(11)

For k 0 1 N minus 1 the adjoint equations andtransversality conditions can be obtained by using

Pontryaginrsquos maximum principle in discrete time given in[31ndash33] such that

Δλ1k zHk

zIk

λ1k+1 1 minus αhSp(k) minus αe

P(k)

κ + P(k)minus μ1 minus π1uk1113890 1113891 + λ2k+1θ αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

+λ3k+1(1 minus θ) αhSp(k) + αe

P(k)

κ + P(k)1113890 1113891

Δλ2k zHk

zSp(k) Mk + λ1k+1 minus αhI(k)1113858 1113859 + λ2k+1 1 + θαhI(k) minus (μ + β) minus 2c minus π2vk1113858 1113859

+λ3k+1 (1 minus θ) αhSp(k)1113872 1113873 + 2c + β1113960 1113961 + λ4k+1ε2

Δλ3k zHk

zSt(k) λ2k+1 minus cSp(k)1113960 1113961 + λ3k+1 1 + cSp(k) minus μ1113960 1113961

Δλ4k zHk

zPk

λ1k+1 minus αeI(k)κ

(κ + P(k))21113890 1113891 + λ2k+1θ αeI(k)

κ(κ + P(k))2

1113890 1113891 + λ3k+1(1 minus θ) αeI(k)κ

(κ + P(k))21113890 1113891

+λ4k+1 1 minus μ2 minus π3wk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

6 Discrete Dynamics in Nature and Society

Put Φ(N) ANIN

λ1N zΦ(N)

zI AN

λ2N zΦ(N)

zSp

0

λ3N zΦ(N)

zSt

0

λ4N zΦ(N)

zP 0

(13)

For k 0 1 N minus 1 the optimal controls ulowastk vlowastk andwlowastk can be solved from the following optimality condition

zH

zuk

Akuk minus λ1k+1π1I(k) + λ3k+1cπ1Sp(k)I(k) 0⟺

uk λ1k+1π1I(k) minus λ3k+1cπ1Sp(k)I(k)

Ak

zH

zvk

Bkvk + λ2k+1c Sp(k)1113872 11138732π2(k) 0⟺

vk minus λ2k+1c Sp(k)1113872 1113873

2π2St(k)

Bk

zH

zwk

Ckwk minus λ4k+1π3P(k) 0⟺

wk λ4k+1π3P(k)

Ck

(14)

So for π1 π2 π3 1 we have

uk λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

vk minus λ2k+1c Sp(k)1113872 1113873

2St(k)

Bk

wk λ4k+1P(k)

Ck

(15)

However if πi 0 for i 1 2 3 the control attached tothis case will be eliminated and removed By the bounds inUad of the controls it is easy to obtain uk

lowast vklowast and wk

lowast in theform of (10)

41 Algorithm In this section we present the results ob-tained by solving numerically the optimality system ampissystem consists of the state system adjoint system initialand final time conditions and the control characterizationSo the optimality system is given by the following

Step 1 I0 i0 Sp0 sp0 St0 st0 P0 p0λ2N λ3N λ4N 0 λ1N AN and given ulowastk0 vlowastk0and wlowastk0

Step 2 for k 0 1 N minus 1 do

Ik+1 μN minus αhSpkIk minus αeIk

Pk

κ + Pk

minus μ1Ik

Sp(k+1) θ αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 minus (μ + β)Spk minus cSpk Spk + Stk1113872 1113873

Sp(k+1) (1 minus θ) αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 + cSpk Spk + Stk1113872 1113873 + βSpk minus μStk

Pk+1 ε1N + ε2Spk minus μ2Pk

⋮ ⋮

⋮ ⋮

λ1Tminus k λ1Tminus k+1 1 minus αhSpk minus αe

Pk

κ + Pk

minus μ11113888 1113889 + λ2Tminus k+1θ αhSpk + αe

Pk

κ + Pk

1113888 1113889 + λ3Tminus k+1(1 minus θ) αhSpk + αe

Pk

κ + Pk

1113888 1113889

λ2Tminus k 1 + λ1Tminus k+1 minus αhIk( 1113857 + λ2Tminus k+1 1 + θαhIk minus (μ + β) minus 2c( 1113857 + λ3Tminus k+1 (1 minus θ) αhSpk1113872 1113873 + 2c + β1113960 1113961 + λ4Tminus k+1ε2

Discrete Dynamics in Nature and Society 7

λ3Tminus k λ2Tminus k+1 minus cSpkSpk1113960 1113961 + λ3Tminus k+1 1 + cSpk minus μ1113960 1113961

λ4Tminus k λ1Tminus k+1 minus αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ2Tminus k+1θ αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ3Tminus k+1(1 minus θ) αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ4Tminus k+1 1 minus μ2( 1113857

uk+1 min b max a1

Ak

λ1Tminus k+1Ik minus λ3Tminus k+1cSpkIk1113872 11138731113888 11138891113890 1113891

vk+1 min d max c1

Bk

minus λ2Tminus k+1c Spk1113872 11138732Stk1113874 11138751113888 11138891113890 1113891

wk+1 min f max e1

Ck

minus λ4Tminus k+1Pk1113872 11138731113888 11138891113890 1113891 (16)

end forStep 3 for k 0 1 N write

Ilowastk Ik

Slowastpk Spk

Slowasttk Stk

Plowastk Pk

ulowastk uk

vlowastk vk

wlowastk wk

(17)

end forIn this formulation there were initial conditions for the

state variables and terminal conditions for the adjoints ampatis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k N We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

5 Numerical Simulation

In this paragraph we give numerical simulation to highlightthe effectiveness of the strategy that we have developed in theframework of eliminating the rumor and limit its spreadand the initial values are the same as in Table 1 with regardto other initial values they are proposed values after astatistical study

In this section we introduce our control strategy whichconsists in using three kinds of controls the first one un

where an ignorant is informed by a stifler that the infor-mation is false or a rumor the second noted vn which isapplied by a professional admin and which consists ofdeactivating or blocking a fake account that specializes in

spreading the rumor and the third noted wn which is alsoapplied by an admin by blocking the pages created in orderto spread the rumor and this after reaching a significantnumber of signals and after the diagnosis of the content ofthe concerned page We apply our strategy for a period offifteen weeks that we assume an average duration of thespread of a rumor Figure 3 illustrates the results foundFrom Figure 3 we see that the effect of the strategy begins asearly as the second week and the number of the ignorantdecreases from 2000 in a fatal way until reaching almost 100this number is recovered in the number of stiflers whichincreases significantly from zero until reaching 2000 ampesame remark for the number of pages we notice that be-tween the first and the second week the number increasesstep by step however after the second week of the appli-cation of controls the number decreases until less than 50As for the number of spreaders the number is decreasingfrom 50 to zero

In the next section and in order to give more detailsabout our strategy as well as the effect of each control wechoose to apply three scenarios in each of these scenarioswe apply separately one control Figures 4ndash6 illustrate theresult found in each scenario for the three cases

(a) Apply only the control u

(b) Apply only the control v

(c) Apply only the control w

51 Apply Only the Control u In this scenario we simulatethe case where we apply a single control u with which weinform a portion of the ignorants by the false information sowe win this proportion in favor of stiflers and the control isapplied over a period of 100 days From the figure we seethat after 10 days the number of spreaders falls and does notexceed 3000 people throughout the period and the numberof stiflers meanwhile increases sharply to almost 20000which shows the effectiveness of our control

52 Apply Only the Control v In the second scenario weapply a single control v but this time a control that focuses

8 Discrete Dynamics in Nature and Society

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

Put Φ(N) ANIN

λ1N zΦ(N)

zI AN

λ2N zΦ(N)

zSp

0

λ3N zΦ(N)

zSt

0

λ4N zΦ(N)

zP 0

(13)

For k 0 1 N minus 1 the optimal controls ulowastk vlowastk andwlowastk can be solved from the following optimality condition

zH

zuk

Akuk minus λ1k+1π1I(k) + λ3k+1cπ1Sp(k)I(k) 0⟺

uk λ1k+1π1I(k) minus λ3k+1cπ1Sp(k)I(k)

Ak

zH

zvk

Bkvk + λ2k+1c Sp(k)1113872 11138732π2(k) 0⟺

vk minus λ2k+1c Sp(k)1113872 1113873

2π2St(k)

Bk

zH

zwk

Ckwk minus λ4k+1π3P(k) 0⟺

wk λ4k+1π3P(k)

Ck

(14)

So for π1 π2 π3 1 we have

uk λ1k+1I(k) minus λ3k+1cSp(k)I(k)

Ak

vk minus λ2k+1c Sp(k)1113872 1113873

2St(k)

Bk

wk λ4k+1P(k)

Ck

(15)

However if πi 0 for i 1 2 3 the control attached tothis case will be eliminated and removed By the bounds inUad of the controls it is easy to obtain uk

lowast vklowast and wk

lowast in theform of (10)

41 Algorithm In this section we present the results ob-tained by solving numerically the optimality system ampissystem consists of the state system adjoint system initialand final time conditions and the control characterizationSo the optimality system is given by the following

Step 1 I0 i0 Sp0 sp0 St0 st0 P0 p0λ2N λ3N λ4N 0 λ1N AN and given ulowastk0 vlowastk0and wlowastk0

Step 2 for k 0 1 N minus 1 do

Ik+1 μN minus αhSpkIk minus αeIk

Pk

κ + Pk

minus μ1Ik

Sp(k+1) θ αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 minus (μ + β)Spk minus cSpk Spk + Stk1113872 1113873

Sp(k+1) (1 minus θ) αhSpkIk + αeIk

Pk

κ + Pk

1113888 1113889 + cSpk Spk + Stk1113872 1113873 + βSpk minus μStk

Pk+1 ε1N + ε2Spk minus μ2Pk

⋮ ⋮

⋮ ⋮

λ1Tminus k λ1Tminus k+1 1 minus αhSpk minus αe

Pk

κ + Pk

minus μ11113888 1113889 + λ2Tminus k+1θ αhSpk + αe

Pk

κ + Pk

1113888 1113889 + λ3Tminus k+1(1 minus θ) αhSpk + αe

Pk

κ + Pk

1113888 1113889

λ2Tminus k 1 + λ1Tminus k+1 minus αhIk( 1113857 + λ2Tminus k+1 1 + θαhIk minus (μ + β) minus 2c( 1113857 + λ3Tminus k+1 (1 minus θ) αhSpk1113872 1113873 + 2c + β1113960 1113961 + λ4Tminus k+1ε2

Discrete Dynamics in Nature and Society 7

λ3Tminus k λ2Tminus k+1 minus cSpkSpk1113960 1113961 + λ3Tminus k+1 1 + cSpk minus μ1113960 1113961

λ4Tminus k λ1Tminus k+1 minus αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ2Tminus k+1θ αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ3Tminus k+1(1 minus θ) αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ4Tminus k+1 1 minus μ2( 1113857

uk+1 min b max a1

Ak

λ1Tminus k+1Ik minus λ3Tminus k+1cSpkIk1113872 11138731113888 11138891113890 1113891

vk+1 min d max c1

Bk

minus λ2Tminus k+1c Spk1113872 11138732Stk1113874 11138751113888 11138891113890 1113891

wk+1 min f max e1

Ck

minus λ4Tminus k+1Pk1113872 11138731113888 11138891113890 1113891 (16)

end forStep 3 for k 0 1 N write

Ilowastk Ik

Slowastpk Spk

Slowasttk Stk

Plowastk Pk

ulowastk uk

vlowastk vk

wlowastk wk

(17)

end forIn this formulation there were initial conditions for the

state variables and terminal conditions for the adjoints ampatis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k N We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

5 Numerical Simulation

In this paragraph we give numerical simulation to highlightthe effectiveness of the strategy that we have developed in theframework of eliminating the rumor and limit its spreadand the initial values are the same as in Table 1 with regardto other initial values they are proposed values after astatistical study

In this section we introduce our control strategy whichconsists in using three kinds of controls the first one un

where an ignorant is informed by a stifler that the infor-mation is false or a rumor the second noted vn which isapplied by a professional admin and which consists ofdeactivating or blocking a fake account that specializes in

spreading the rumor and the third noted wn which is alsoapplied by an admin by blocking the pages created in orderto spread the rumor and this after reaching a significantnumber of signals and after the diagnosis of the content ofthe concerned page We apply our strategy for a period offifteen weeks that we assume an average duration of thespread of a rumor Figure 3 illustrates the results foundFrom Figure 3 we see that the effect of the strategy begins asearly as the second week and the number of the ignorantdecreases from 2000 in a fatal way until reaching almost 100this number is recovered in the number of stiflers whichincreases significantly from zero until reaching 2000 ampesame remark for the number of pages we notice that be-tween the first and the second week the number increasesstep by step however after the second week of the appli-cation of controls the number decreases until less than 50As for the number of spreaders the number is decreasingfrom 50 to zero

In the next section and in order to give more detailsabout our strategy as well as the effect of each control wechoose to apply three scenarios in each of these scenarioswe apply separately one control Figures 4ndash6 illustrate theresult found in each scenario for the three cases

(a) Apply only the control u

(b) Apply only the control v

(c) Apply only the control w

51 Apply Only the Control u In this scenario we simulatethe case where we apply a single control u with which weinform a portion of the ignorants by the false information sowe win this proportion in favor of stiflers and the control isapplied over a period of 100 days From the figure we seethat after 10 days the number of spreaders falls and does notexceed 3000 people throughout the period and the numberof stiflers meanwhile increases sharply to almost 20000which shows the effectiveness of our control

52 Apply Only the Control v In the second scenario weapply a single control v but this time a control that focuses

8 Discrete Dynamics in Nature and Society

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

λ3Tminus k λ2Tminus k+1 minus cSpkSpk1113960 1113961 + λ3Tminus k+1 1 + cSpk minus μ1113960 1113961

λ4Tminus k λ1Tminus k+1 minus αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ2Tminus k+1θ αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ3Tminus k+1(1 minus θ) αeIk

κκ + Pk( 1113857

2⎡⎣ ⎤⎦ + λ4Tminus k+1 1 minus μ2( 1113857

uk+1 min b max a1

Ak

λ1Tminus k+1Ik minus λ3Tminus k+1cSpkIk1113872 11138731113888 11138891113890 1113891

vk+1 min d max c1

Bk

minus λ2Tminus k+1c Spk1113872 11138732Stk1113874 11138751113888 11138891113890 1113891

wk+1 min f max e1

Ck

minus λ4Tminus k+1Pk1113872 11138731113888 11138891113890 1113891 (16)

end forStep 3 for k 0 1 N write

Ilowastk Ik

Slowastpk Spk

Slowasttk Stk

Plowastk Pk

ulowastk uk

vlowastk vk

wlowastk wk

(17)

end forIn this formulation there were initial conditions for the

state variables and terminal conditions for the adjoints ampatis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k N We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

5 Numerical Simulation

In this paragraph we give numerical simulation to highlightthe effectiveness of the strategy that we have developed in theframework of eliminating the rumor and limit its spreadand the initial values are the same as in Table 1 with regardto other initial values they are proposed values after astatistical study

In this section we introduce our control strategy whichconsists in using three kinds of controls the first one un

where an ignorant is informed by a stifler that the infor-mation is false or a rumor the second noted vn which isapplied by a professional admin and which consists ofdeactivating or blocking a fake account that specializes in

spreading the rumor and the third noted wn which is alsoapplied by an admin by blocking the pages created in orderto spread the rumor and this after reaching a significantnumber of signals and after the diagnosis of the content ofthe concerned page We apply our strategy for a period offifteen weeks that we assume an average duration of thespread of a rumor Figure 3 illustrates the results foundFrom Figure 3 we see that the effect of the strategy begins asearly as the second week and the number of the ignorantdecreases from 2000 in a fatal way until reaching almost 100this number is recovered in the number of stiflers whichincreases significantly from zero until reaching 2000 ampesame remark for the number of pages we notice that be-tween the first and the second week the number increasesstep by step however after the second week of the appli-cation of controls the number decreases until less than 50As for the number of spreaders the number is decreasingfrom 50 to zero

In the next section and in order to give more detailsabout our strategy as well as the effect of each control wechoose to apply three scenarios in each of these scenarioswe apply separately one control Figures 4ndash6 illustrate theresult found in each scenario for the three cases

(a) Apply only the control u

(b) Apply only the control v

(c) Apply only the control w

51 Apply Only the Control u In this scenario we simulatethe case where we apply a single control u with which weinform a portion of the ignorants by the false information sowe win this proportion in favor of stiflers and the control isapplied over a period of 100 days From the figure we seethat after 10 days the number of spreaders falls and does notexceed 3000 people throughout the period and the numberof stiflers meanwhile increases sharply to almost 20000which shows the effectiveness of our control

52 Apply Only the Control v In the second scenario weapply a single control v but this time a control that focuses

8 Discrete Dynamics in Nature and Society

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

on the spreaders by disabling their accounts and the periodis always 100 days the figure shows us that the number ofspreaders decreases since the 3rd day but this time we see

that the number Sp tends to zero however there is a smallimprovement in the number of stiflers and we notice thatthe number grows slowly and reaches 10000 people

Table 1 Rumor model parameters and values

Parameter Description Valueαh ampe proportion of ignorants who become the spreader after discussing with a spreader (00005day)αe ampe proportion of ignorants who become the spreader after consulting a page of rumor (00007day)θ ampe proportion of the ignorant who become the spreader (065day)(1 minus θ) ampe proportion of the ignorant who become the stifler (035day)μ New users and deactivated users 00008dayβ ampe proportion of spreaders who become stiflers (0002day)c ampe proportion of spreaders who become stiflers after contacting another spreader (000005day)ε1 ampe rate of shared publications of a rumor page by ignorants (0000005day)ε2 ampe rate of shared publications of a rumor page by spreaders 000001dayδ1 New rumor pages created by spreaders 004day

50 10 15

Spreader with controlStifler with control

Ignorant with controlPage with control

0

2

4

6

8

10

12

14

16

18

20

Figure 3 Dynamics with controls u v and w

0

2

4

6

8

10

12

14

16

18

20

Spreader without controlStifler without controlIgnorant without control

Spreader with control (u)Stifler with control (u)Ignorant with control (u)

0 10 20 30 40 50 60 70 80 90 100

Figure 4 Dynamics with control u

Discrete Dynamics in Nature and Society 9

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

53 Apply Only the Control w In the last scenario we applythe only remaining control w by disabling the pages reservedfor the diffusion of the rumor From the figure we see thatthe numbers of stiflers and spreaders have undergone thesame changes as in the previous scenarios and the numberof pages has fallen dramatically which shows the effec-tiveness of this strategy because it gives similar results to theprevious ones and more which is logical since this strategyaims and attacks the source of the rumor and the necessarytool for spreaders in order to spread the rumor

6 Conclusion

In this work we propose a new model which describes thedynamics of rumor spread through social media based onthe cholera model this allows us to introduce a new

compartment representing pages or tweets or any other kindof publication that disseminates the rumor Moreover weapply a control strategy in order to fight against the spread ofthe rumor since there is an overlap between both phe-nomena the strategy is also based on the cholera model andwhich proved its effectiveness against this disease ampreecontrol strategies were introduced and by the introductionof three new variables πi i 1 2 3 we could study andcombine several scenarios in order to see the impact and theeffect of each one of these controls on the reduction of therumor spread ampe numerical resolution of the system withdifference equations as well as the numerical simulationsenabled us to compare and see the difference between eachscenario in a concrete way Numerical results prove theeffectiveness of our strategy ampe purpose of this work isachieved and we have proved the effectiveness of our

2

4

6

8

Spreader with control (v)Stifler with control (v)Ignorant with control (v)

Spreader without controlStifler without controlIgnorant without control

0 10 20 30 40 50 60 70 80 90 1000

10

12

14

16

18

20

Figure 5 Dynamics with control v

0

5

10

15

20

25

30

Spreader with control (w)Stifler without controlIgnorant with control (w)Page with control (w)

Spreader without controlStifler with control (w)Ignorant without controlPage without control

0 10 20 30 40 50 60 70 80 90 100

Figure 6 Dynamics with control w

10 Discrete Dynamics in Nature and Society

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

strategy and its importance in fighting the spread of anyrumor throughout any social network

Data Availability

ampe disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpwwwnetworkrepositorycom)

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

Research reported in this paper was supported by theMoroccan Systems ampeory Network

References

[1] L J Sabato K Kondik and G Skelley ldquoRepublicans 2016what to do with the Donaldrdquo 2015 httpcenterforpoliticsorgcrystalballarticlesrepublicans-2016-what-to-do-with-the-donald

[2] J Fromm S Melzer B Ross and S Stieglitz Trump versusClintonmdashTwitter Communication during the US PrimariesPalgrave Macmillan London UK 2016

[3] J C Lee and K Quealy ldquoampe 337 people places and thingsdonald trump has insulted on twitter a complete listrdquo 2016httpswwwnytimescominteractive20160128upshotdonald-trump-twitter-insultshtml

[4] L Cazzoli R Sharma M Treccani and F Lillo ldquoA large scalestudy to understand the relation between twitter and financialmarketrdquo in Proceedings of the 2016 7ird European NetworkIntelligence Conference IEEE Wroclaw Poland September2016

[5] A N Vilas R P D Redondo and A L Garcia ldquoampe irruptionof cryptocurrencies into Twitter cashtags a classifying solu-tionrdquo IEEE Access vol 8 pp 32698ndash32713 2020

[6] L Zhu H Zhao and H Wang ldquoComplex dynamic behaviorof a rumor propagation model with spatial-temporal diffusiontermsrdquo Information Sciences vol 349-350 pp 119ndash136 2016

[7] J Ma D Li and Z Tian ldquoRumor spreading in online socialnetworks by considering the bipolar social reinforcementrdquoPhysica A Statistical Mechanics and Its Applications vol 447pp 108ndash115 2016

[8] Q Liu T Li and M Sun ldquoampe analysis of an SEIR rumorpropagation model on heterogeneous networkrdquo Physica AStatistical Mechanics and Its Applications vol 469 pp 372ndash380 2017

[9] W Huang ldquoOn rumour spreading with skepticism and de-nialrdquo Technical Report 2011

[10] RE Bartholomew and P HassallAColorful History of PopularDelusions Prometheus Books New York NY USA 2015

[11] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemicsrdquo Proceedings of the RoyalSociety of London Series A vol 115 no 772 pp 700ndash7211927

[12] W Goffman and V A Newill ldquoGeneralisation of epidemictheory an application to the transmission of ideasrdquo Naturevol 204 no 4955 pp 225ndash228 1964

[13] D J Daley and D G Kendall ldquoEpidemics and rumoursrdquoNature vol 204 no 4963 p 1118 1964

[14] L M A Bettencourt A Cintron-Arias D I Kaiser andC Castillo-Chavez ldquoampe power of a good idea quantitativemodeling of the spread of ideas from epidemiologicalmodelsrdquo Physica A Statistical Mechanics and Its Applicationsvol 364 pp 513ndash536 2006

[15] H Laarabi A Abta M Rachik and J BouyaghroumnildquoStability analysis of a delayed rumor propagation modelrdquoDifferential Equations and Dynamical Systems vol 24 no 4pp 407ndash415 2016

[16] J Jia and W Wu ldquoA rumor transmission model with in-cubation in social networksrdquo Physica A Statistical Mechanicsand Its Applications vol 491 pp 453ndash462 2018

[17] C Wang Z X Tan Y Ye L Wang K H Cheong andN-G Xie ldquoA rumor spreading model based on informationentropyrdquo Scientific Reports vol 7 no 1 2017

[18] C Wang J M Koh K H Cheong and N-G Xie ldquoPro-gressive information polarization in a complex-network en-tropic social dynamics modelrdquo IEEE Access vol 7pp 35394ndash35404 2019

[19] Z X Tan and K H Cheong ldquoCross-issue solidarity and truthconvergence in opinion dynamicsrdquo Journal of Physics AMathematical and 7eoretical vol 51 no 35 Article ID355101 2018

[20] Y Hu Q Pan W Hou and M He ldquoRumor spreading modelwith the different attitudes towards rumorsrdquo Physica AStatistical Mechanics and Its Applications vol 502 pp 331ndash344 2018

[21] S Dong F-H Fan and Y-C Huang ldquoStudies on the pop-ulation dynamics of a rumor-spreading model in online socialnetworksrdquo Physica A Statistical Mechanics and Its Applica-tions vol 492 pp 10ndash20 2018

[22] A Guille H Hacid C Favre and D A Zighed ldquoInformationdiffusion in online social networksrdquo ACM Sigmod Recordvol 42 no 2 pp 17ndash28 2013

[23] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[24] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[25] A Kouidere O Balatif H Ferjouchia A Boutayeb andM Rachik ldquoOptimal control strategy for a discrete time to thedynamics of a population of diabetics with highlighting theimpact of living environmentrdquo Discrete Dynamics in Natureand Society vol 2019 Article ID 6342169 8 pages 2019

[26] P Georgescu and Y H Hsieh ldquoGlobal dynamics for apredator-prey model with stage structure for predatorrdquo SIAMJournal on Applied Mathematics vol 67 no 5 pp 379ndash13952007

[27] C H Jia and D X Feng ldquoAn optimal control problem of acoupled nonlinear parabolic population systemrdquo ActaMathematicae Applicatae Sinica English Series vol 23 no 3pp 377ndash388 2007

[28] A El Alami Laaroussi andM Rachik ldquoOn the regional controlof a reaction-diffusion system SIRrdquo Bulletin of MathematicalBiology vol 82 no 1 2019

[29] R Ghazzali A E A Laaroussi A El Bhih and M RachikldquoOn the control of a reaction-diffusion system a class of SIRdistributed parameter systemsrdquo International Journal ofDynamics and Control vol 7 no 3 pp 1021ndash1034 2019

[30] B R Soukaina G Rachid A El Bhih and R MostafaldquoOptimal control problem of a quarantine model in multi

Discrete Dynamics in Nature and Society 11

region with spatial dynamicsrdquo Communications in Mathe-matical Biology and Neuroscience vol 10 2020

[31] L S Pontryagin V G Boltyanskii R V Gamkrelidze andEMishchenko7eMathematical7eory of Optimal Processes(International Series of Monographs in Pure and AppliedMathematics) Interscience New York NY USA 1962

[32] V Guibout and A M Bloch ldquoA discrete maximum principlefor solving optimal control problemsrdquo in Proceedings of the2004 43rd IEEE Conference on Decision and Control (CDC)vol 2 IEEE Nassau Bahamas pp 1806ndash1811 December2004

[33] W Ding R Hendon B Cathey E Lancaster and R GermickldquoDiscrete time optimal control applied to pest controlproblemsrdquo Involve A Journal of Mathematics vol 7 no 4pp 479ndash489 2014

12 Discrete Dynamics in Nature and Society