Research ArticleSelf-Awareness-Based Resource Allocation Strategy forContainment of Epidemic Spreading

Xiaolong Chen1 Quanhui Liu2 Ruijie Wang 3 Qing Li1 and Wei Wang 4

1Financial Intelligence and Financial Engineering Key Laboratory of Sichuan Province School of Economic InformationEngineering Southwestern University of Finance and Economics Chengdu 611130 China2College of Computer Science Sichuan University Chengdu 610065 China3Aba Teachers University Aba 623002 China4Cybersecurity Research Institute Sichuan University Chengdu 610065 China

Correspondence should be addressed to Ruijie Wang ruijiewang001163com

Received 7 February 2020 Revised 26 April 2020 Accepted 9 May 2020 Published 23 May 2020

Academic Editor Mahdi Jalili

Copyright copy 2020 XiaolongChen et al+is is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Resource support between individuals is of particular importance in controlling or mitigating epidemic spreading especiallyduring pandemics However there remains the question of how we can protect ourselves from being infected while helping othersby donating resources in fighting against the epidemic To answer the question we propose a novel resource allocation model byconsidering the awareness of self-protection of individuals In the model a tuning parameter is introduced to quantify the reactionstrength of individuals when they are aware of the disease And then a coupled model of resource allocation and disease spreadingis proposed to study the impact of self-awareness on resource allocation and its impact on the dynamics of epidemic spreading+rough theoretical analysis and extensive Monte Carlo simulations we find that in the stationary state the system converges totwo states the whole healthy or the completely infected which indicates an abrupt increase in the prevalence when there is ashortage of resources More importantly we find that too cautious and too selfless for the people during the outbreak of anepidemic are both not suitable for disease control +rough extensive simulations we locate the optimal point at which there is amaximum value of the epidemic threshold and an outbreak can be delayed to the greatest extent At last we study further theeffects of the network structure on the coupled dynamics We find that the degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phenomenon in behavior response Based on the results of this study aconstructive suggestion is that in the face of a global pandemic individuals or countries should strengthen mutual support andcooperation while doing their own prevention to suppress the epidemic optimally

1 Introduction

Controlling the outbreak of epidemic is one of the mostimportant topics in human history During the past decadesthe onset of several major global health threats such as the2003 spread of SARS the H1N1 influenza pandemic in 2009and the western Africa Ebola outbreaks in 2014 have de-prived tens of thousands of lives all around the world [1ndash3]In 2019 a novel coronavirus causing severe acute respiratorydisease (COVID-19) emerged By the end of April 2020there have been more than two million confirmed COVID-19 infections reported all over the world [4] +e surge ininfections has led to a severe shortage of medical resources

+ousands of confirmed and suspected cases await treat-ment [5] Facing the rapid outbreak of the disease thecontribution of resources from healthy individuals is nec-essary but the self-protection of the susceptible individualscannot be ignored +us the immediate problem is how canwe protect ourselves from being infected while helpingothers in fighting against the epidemic

A large number of researchers from various disciplineshave made efforts to study the topic of optimal resourceallocation in disease suppressing in the past years [6ndash10] Forexample Preciado et al [11] studied the problem of theoptimal distribution of vaccination resources to controlepidemic spreading based on complex networks+ey found

HindawiComplexityVolume 2020 Article ID 3256415 12 pageshttpsdoiorg10115520203256415

the cost-optimal distribution of the vaccination resourcewhen different levels of vaccination are allowed through aconvex framework Further they studied the optimal allo-cation of the preventive and corrective resources to achievethe highest level of containment when the budget is given inadvance or finding the minimum budget required when thebudget is not specified [12] Chen et al [13] solved theproblem of optimal allocation of a limited medical resourcebased on mean-field theory +e above works considered theproblem from a mathematical perspective and were solvedfrom the premise that both the number of resources and thespreading state of the epidemic are fixed

However the real scenario is more complicated than astatic mathematical problem Multiple dynamical processesalways interact and coevolve [14 15] forming a more realisticstarting point For example during the outbreak of an epi-demic the quantity of available resources is largely affected bythe propagation of the disease and in turn the dynamicalchange of the necessary resources influences the dynamicalproperty of the epidemic spreading +e coevolution ofmultiple dynamical processes has attracted extensive researchin recent years [16] Bottcher [17] studied the coevolution ofresource and epidemics and they found a critical recoverycost that if the cost is above the critical value epidemics spiralout of control into ldquoexplosiverdquo spread Chen et al [18] studiedthe effects of social support from local connections on thespreading dynamics of the epidemic +ey proposed a co-evolution spreading model on multiplex networks and founda hybrid phase transition on networks with heterogeneousdegree distribution In this multiplex network frameworkChen et al [19] further studied the impact of preferentialresource allocation on the social subnetwork on the spreadingdynamics of the epidemic

In addition to the physical resources that can directlymitigate or control the epidemic spreading the awareness ofthe epidemic in populations is another type of resource +epublic can perceive the threat of the epidemic social networkthrough a platform andmassmedia and then takemeasures toprotect themselves +erefore the interplay between aware-ness diffusion and epidemic spreading is another topic thathave attracted extensive research [20ndash22] A mass of worksaddressed the problem from different perspectives consid-ering for example the risk perception and behavioral changes[23ndash25] Granell et al [14] studied the interplay between theprocesses of epidemic spreading and awareness diffusion ontop of multiplex networks and found a metacritical Wanget al [26] studied the coevolution mechanisms using both realonline and offline data Based on the empirical analysis theyproposed a coupled model on a multiplex network to studythe coupled dynamics of these two processes

Although a mass of works on the coevolution ofawareness diffusion and disease spreading have been carriedout a question that needs to be addressed is how doesawareness affect the behavior of resource donation inpopulations To answer this question a novel resource al-location model that incorporates the influence of theawareness is proposed in this paper We further considerthat the healthy individuals in an outbreak of disease areproviders of essential resources and they can not only

produce resources but also donate them to those who need+e donation behavior in turn leads to less resource for self-protection and a more significant probability to be infected+us when the healthy individuals are aware of the threat ofthe disease and refuse to donate resources they would havemore resources for self-protection and a lower probability ofbeing infected however this is not conducive to the overallprevention and control of the disease +us we are facedwith a game-like problem between resource contributionand self-protection [27] To solve the problem optimally weshould study the complex dynamic mechanisms among self-awareness resource donation and epidemic spreading+us a coupled dynamical model on top of the complexnetworks is proposed and a dynamic message-passingmethod is adopted for the theoretical analysis in this paper

Based on the model we first investigate the influence ofawareness on the coupled dynamics of resource allocation anddisease spreading on scale-free networks To quantify theawareness for self-protection a parameter α is assigned to eachindividual +rough theoretical analysis and numeric simu-lations we find that the system has only two stationary statesnamely the absorb state and the globe outbreak stateWith theincrease of α the epidemic threshold first increases and thendecreases which indicates that there is an optimal value of αFurther we locate the optimal value αopt through extensivesimulations under which the disease can be suppressed to thegreatest extent +en we explain qualitatively the optimalphenomenon And then we investigate the impact of degreeheterogeneity on the coupled dynamics and find that thedegree heterogeneity does not alter the optimal phenomenonand the abrupt increase in prevalence with a shortage of re-source At last we find that the epidemic threshold increaseswith the decrease in degree heterogeneity which suggests thatnetwork heterogeneity promotes the outbreak of disease

2 Model Description

21 Epidemic Model A resource-based epidemiologicalsusceptible-infected-susceptible model (r-SIS) is proposed todescribe the coupled dynamics of epidemic spreading andresource allocation on a complex network Individuals arerepresented by nodes in the network and an adjacencymatrix A is introduced to represent the connection betweennodes If there is an edge between nodes i and j the elementaij 1 otherwise aij 0 According to this scheme anyindividual can be in two different states susceptible (S) andinfected (I) +e infection propagates between each pair ofI-state and S-state neighbors with an infection rate 1113957λ in onecontact which is assumed to depend on whether the S-statenodes donate resources see the details in Section 22 At anytime t each I-state node i recovers with a recovery rate ri(t)Resources including medical funds and food can promoterecovery of patients from disease [28 29] +us we definethe recovery rate of each I-state node as a function of theresource quantity received from healthy neighbors in thispaper As each I-state node may get a different amount ofresource the recovery rate varies from node to nodeConsequently the recovery rate of any node i at time t can bedefined as

2 Complexity

ri(t) 1 minus (1 minus μ)εωi(t)

(1)

where ωi(t) is the resource quantity of node i received fromhealthy neighbors at time t and μ is the basic recovery rate Aparameter ε isin [0 1] is introduced in our model to representthe resource utilization rate [30] Since in the real scenariothere is the common phenomenon of the waste on resource[31 32] in medical and other service systems implying theresource received from healthy neighbors may not be fullyutilized on curing and recovery Without loss of generalitywe set ϵ 06 throughout this work ie only 60 of theresources received are used

For the r-SIS model the value ρi(t) is defined as theprobability that any node i is in the infected state +efraction of infected nodes in a network of size N at time t canbe calculated by averaging the overall N nodes

ρ(t) 1N

1113944

N

i1ρi(t) (2)

Further we define the prevalence of the disease in thestationary state as ρ equiv ρ(infin)

22 Resource Allocation Model Based on Behavior ResponseIn the real scenario healthy individuals can produce re-sources For simplicity we consider that each individual(node) in the network can generate one unit resource at atime step During an outbreak of a disease the susceptibleindividuals can perceive the threat of the disease intuitivelyby acquiring the information from neighbors Generally themore infected neighbors of an individual the deeper hesheis aware of the disease [33 34] People aware of the diseasemay have different reactions [33] Also to quantify thereaction strength of an individual to the local information ofdisease a tuning parameter α is introduced Based on thedescription above we can define the probability that ahealthy individual with m infected neighbors donates re-source as

q(m) q0(1 minus α)m

(3)

where q0 is a basic donation probability When α 0 allhealthy nodes have the same donation probability q0 Be-sides we consider that a healthy node can donate one unitresource equally to its I-state neighbors at a time Based onthe resource allocation scheme the amount of resourceωj⟶i that node j with m infected neighbors donate to one ofits I-state neighbor i can be expressed as

ωj⟶i q(m)1m

(4)

When disease breaks out in the human populationpeople aware of the disease in their proximity can takemeasures to reduce their susceptibility leading to a re-duction in the effective rate of infection [14 35] We con-sider that if an individual is aware and refuses to donateresource for self-protection it reduces its infectivity by afactor c +e basic infection rate is denoted as λ and the

annealing infection rate is denoted as 1113957λ which can thus beexpressed as

1113957λ λ if distribute the resources

cλ else1113896 (5)

If a healthy individual donates resource to infectedneighbors the individual has a larger probability to be in-fected on the contrary if the individual does not donateresource there is a relatively smaller probability to be in-fected +e annealing infection rate of any node i can also beexpressed as a function of q

1113957λi q(m)λ +[1 minus q(m)]cλ (6)

3 Dynamic Message-Passing Method

In order to theoretically analyze the dynamic processes wedevelop a generated dynamic message-passing method(GDMP) [36 37] In this method the message θj⟶i isdefined on the directed edges of a network to carry causalinformation of the flow of contagion which can onlytransfer one way along directed links θj⟶i represents theprobability that node j is infectious because it was infectedby one of its neighbors other than node i In computingθj⟶i we only take into account the contributions to ρj thatcome from the neighbors other than i +e higher orderprocess of j being infected by i and then passes the infectionback to i is neglected Combining θj⟶i and equation (3) forresource allocations the resources ωi(t) that an infectednode i receive from its healthy neighbors can be expressed as

ωi(t) 1113944j

aij 1 minus θj⟶i(t)1113960 1113961q mj(t)1113960 1113961

mj(t) (7)

where mj(t) is the expected number of I-state neighbors ofnode j at time t which is expressed as

mj(t) 1113944hnei

ajhθh⟶j(t) + 1 (8)

where the plus one takes into account that node i is infectedat this moment +e factor (1 minus θj⟶i(t)) in equation (7)stands for the probability that node j is susceptible at time tWith the definition above the discrete-time version of theevolution of ρi(t) in a time interval Δt reads [38]

ρi(t + Δt) 1 minus ρi(t)( 1113857 1 minus Ωi(t)( 1113857 + 1 minus ri(t)Δt1113858 1113859ρi(t)

(9)

whereΩi(t) is the probability that the node is not infected byany neighbor with the product being over the set Ni of theneighbors of node i +e expression of Ωi(t) is as follows

Ωi(t) 1113945jisinNi

1 minus Δt1113957λi(t)θj⟶i(t)1113960 1113961(10)

Note that the first term on the right-hand side ofequation (9) stands for the probability that node i is inS-state and infected by at least one of its neighbors +esecond term is the probability that node i is in I-state and

Complexity 3

does not recover Similarly we can get the time evolution ofθj⟶i(t) as

θj⟶i(t + Δt) 1 minus θj⟶i(t)1113872 1113873 1 minus ϕj⟶i(t)1113872 1113873 + 1 minus rj(t)Δt1113872 1113873θj⟶i(t)

(11)

where ϕj⟶i(t) is the probability that node j is not infectedby any of its neighbors excluding node i which can beexpressed as

ϕj⟶i(t) 1113945ℓisinNji

1 minus Δt1113957λj(t)θℓ⟶j(t)1113960 1113961(12)

+e product in equation (12) is over the set Nji of theneighbors of j excluding i Further by setting Δt 1 andconsidering situation in a stationary state equations (9) and(11) become

ρi 1 minus ρi( 1113857 1 minus Ωi( 1113857 + 1 minus ri( 1113857ρi (13)

θj⟶i 1 minus θj⟶i1113872 1113873 1 minus ϕj⟶i1113872 1113873 + 1 minus rj1113872 1113873θj⟶i (14)

+rough numerical iteration we can compute the in-fection probability of any node at any time ρi(t) andprevalence ρ in the stationary state for different values of αand λ However the equations can only be solved numer-ically except for the trivial solutions of ρi 0 and θj⟶i 0for all i 1 N which leads to an overall ρ 0 phase ofan all-healthy population

Due to nonlinearities in equations (7)ndash(12) they do nothave a closed analytic form and this disallows obtaining theepidemic threshold λc for fixed values of α such that ρgt 0 ifλgt λc and ρ 0 when λlt λc +e calculation of λc can beperformed by considering that when λ⟶ λc ρi⟶ 0 andθj⟶i⟶ 0 and the number of infected neighbors of anyhealthy node is approximately zero in the thermodynamiclimit +en prior to reaching λc the expression(1 minus θj⟶i)⟶ 1 is valid We can get a physical picture thatthe isolated infected nodes are well separated and sur-rounded by healthy nodes and any infected node i canreceive all the resource from each of its neighbors By addingthese assumptions to equation (7) resource ωi becomesωi kiq0(1 minus α) By linearizing equation (1) and neglectingthe second-order terms for small μ we obtain

ri(t) asymp ϵμωi(t)

ϵμkiq0(1 minus α)(15)

Equation (15) suggests that the recovery rate is pro-portional to the node degree and inversely proportional to αwhen λ⟶ λc For the sake of clarity the basic recovery rateis set at μ 001 in this paper Further equations (10) and(12) can also be linearized using θj⟶i asymp 0 as

qi asymp 1 minus 1113957λi 1113944

N

j1ajiθj⟶i (16)

ϕj⟶i asymp 1 minus 1113957λj 1113944l⟶ hisinVE

Mj⟶il⟶hθl⟶h(17)

where VE is the set of directed edges andM is the |VE| times |VE|

nonbacktracking matrix [39] of the network with the ele-ments labelled by the edges

Mj⟶il⟶h δjh 1 minus δil( 1113857 (18)

with δil being the Dirac delta function Substituting equation(17) into equation (14) and ignoring the higher order termsof θj⟶i give

1113944 minus δljδihrj + 1113957λjMj⟶il⟶h1113872 1113873θl⟶h 0 (19)

Finally considering that mj 1 when λ⟶ λc equation(6) becomes

1113954λ equiv 1113957λj (1 minus c)q0(1 minus α) + c1113858 1113859λ (20)

To estimate the epidemic threshold we calculate theaverage recovery rate as

langrrang ϵμlangkrangq0(1 minus α) (21)

By inserting equations (20) and (21) into equation (19)we get

1113944 minus δljδihlangrrang + 1113954λMj⟶il⟶h1113872 1113873θl⟶h 0 (22)

+e system of equations in equation (22) has a nontrivialsolution if and only if langrrang1113954λ is an eigenvalue of the matrix M

[38] +e lowest value 1113954λc is then given by

1113954λc langrrang

Λmax (23)

where Λmax is the largest eigenvalue of M [15 37 40]

4 Numerical Verification andSimulation Results

In this section we study systematically the effects of self-awareness and the network structure on the coupled dy-namics of resource allocation and disease spreading re-spectively through numerical verification and Monte Carlosimulations In the simulation the synchronous updatingmethod [41 42] is applied to the disease infection and re-source allocation processes Within each time increment Δtwhere Δt 1 in this paper infection propagates from anyI-state node j to S-state node i with probability 1113957λiΔt and anyI-state node j recovers to S-state with a probability rjΔtWith the spreading of disease the resource allocationprocess co-occurs +e dynamics terminate once it enters asteady state in which the number of infected nodes onlyfluctuates within a small range Note that we fix the factor c

at a constant value c 005 throughout the paper such thatif any healthy individual j chooses to reserve their resourcethe probability that they are infected in one contact with aninfected neighbor reduces to 1113957λj 005λ

41 Effects of Self-Awareness on the Spreading DynamicsIn this section we investigate the effects of awareness forself-protection on the spreading dynamics We consider that

4 Complexity

the coupled processes of resource allocation and diseasespreading takes place on a scale-free network as many real-world networks have skewed degree distributions [43ndash46]To build the network we adopt the uncorrelated configu-ration model (UCM) [47 48] according to a given degreedistribution P(k) sim kminus c with maximum degree kmax

N

radic

[49] and minimum degree kmin 3 which assures no degreecorrelation of the network when N is sufficiently large Toavoid the influence of the network structure on the resultthe degree exponent is set at c 24 the network size is set atN 10000 and the average degree is set at langkrang 8 in thesimulations In addition we leverage the susceptibilitymeasure χ to determine the epidemic threshold throughsimulations [50] which is expressed as

χ Nlangρ2rang minus langρrang2

langρrang (24)

where lang rang represents the ensemble average over all real-izations +e epidemic threshold can then be determinedwhen the value of χ exhibits diverging peaks at the certaininfection rate [50 51]

We first investigate the effects of self-awareness on thespreading dynamics using Monte Carlo simulations Ini-tially a fraction of ρ(0) 01 nodes are selected randomly asseeds and the remaining nodes are in the susceptible stateTo present different reaction strength of individuals whenthey are aware of a certain disease from local information weselect eight typical values of α from α 01 to α 09 in thesimulation In Figures 1(a) and 1(c) we plot the prevalence ρin the stationary state as a function of basic infection rate λfor different α Symbols in Figures 1(a) and 1(c) representthe results obtained by Monte Carlo simulations and linesare the theoretical results obtained from numeric iterationsrespectively From the curves in Figures 1(a) and 1(c) weobserve that the system converges to two possible stationarystates either the whole population is healthy or it becomescompletely infected for any α which tells us that when thereis a shortage in resource the disease breaks out abruptly

Besides we can observe from Figures 1(a) and 1(b) thatwith the increase of α from α 01 to α 05 the epidemicthreshold increases gradually see the peaks of χ for thecorresponding α It reveals that the stronger the individualrsquossense of self-protection themore delayed the outbreak of thedisease within this parameter interval (see the right arrow)On the contrary we observe from Figures 1(c) and 1(d) thatwhen α increases from α 06 to α 09 the thresholddecreases gradually which reveals that the disease breaks outmore easily with a stronger sense of self-protection withinthis parameter interval (see the left arrow)+e phenomenonsuggests that too cautious or too selfless for the peopleduring the outbreak of an epidemic are both not suitable fordisease control and there is an optimal value of the reactionstrength at which an epidemic outbreak is postponed to thegreatest extent

We further study systematically the effects of behaviorresponse and the basic infection rate on the spreading dy-namics In Figure 2 we exhibit the full phase diagram (α minus

λ) of the coupled dynamics of resource allocation and

disease spreading Colors in Figure 2(a) encode the fractionof infected nodes ρ in the stationary state +e epidemicthreshold λc marked by red circles rises monotonically untilit reaches the maximum at αopt (indicated by the blue dottedline) and then falls gradually with the increase of α Besideswe observe that there are only two possible stationary statesthe whole healthy (marked by blue color) and the wholeinfected of the population (marked by yellow color)

Figure 3(a) plots the time evolution of ρ(t) for six typicalvalues of α when the basic infection rate is fixed at λ 004We find that when the value of α is small the systemconverges to a stationary state rapidly such as ρ(infin) 10for α 01 With the increase of α it takes a longer time forthe system to reach a stationary state Further to exhibit theeffects of α on the dynamics more intuitively we plot thefraction of infected nodes at a fixed time t 200 as afunction of α in Figure 3(b) which is denoted as ρ(α) for thesake of clarity We observe that the value of ρ(α) decreasescontinuously with α until reaching the minimum value atαopt asymp 048 (marked by red circle in Figure 3(b)) and thenincreases gradually with α

Next we qualitatively explain the optimal phenomena bystudying the time evolution of the critical quantities

We begin by studying the case when α is small forexample α 01 We observe in Figure 4 that in the initialstage the donation probability for α 01 is the highest (seethe blue line in the top panel of Figure 4(a)) since a smallervalue of α means a higher willingness of healthy individualsto allocate resources Although the resource of healthy in-dividuals can improve the recovery probability of infectedneighbors to a certain extent it also makes themselves morelikely to be infected We can observe in Figures 4(b) and 4(c)that the average recovery rate langrrang and the infection rate lang1113957λrang

is the highest for α 01 meanwhile there is a lowest valueof the effective infection rate lang1113957λranglangrrang as shown inFigure 4(d) However with the high probability of beinginfected for the healthy nodes the number of infected in-dividuals increases at a high rate (see the blue line in thebottom pane of Figure 4(a)) When people are aware of theincrement of the infected neighbors they reduce their do-nation willingness which leads to a reduction in infectionrate lang1113957λrang as shown in Figures 4(a) and 4(b) Consequentlywith less resource received from healthy neighbors therecovery rate of infected nodes reduces accordingly seeFigure 4(c) which leads to an increase of the effective in-fection rate lang1113957λranglangrrang [15]+e increase in lang1113957λranglangrrang has led to afurther increase in the number of infected nodes +enpeople become more aware of the threat of disease and thusreduce the probability of resource donation further whichleads to a further decrease in the infection rate lang1113957λrang and therecovery rate langrrang and finally the increase of the effectiveinfection rate lang1113957λranglangrrang

Specifically we observe from Figure 4(d) that when itsurpasses a critical time tlowast indicated by the dotted line in thefigure the value of lang1113957λranglangrrang proliferates which suggests thatin this stage the infection of healthy individuals is muchfaster than the recovery of infected individuals With morenewly infected nodes the donation probability langqrang and theinfection rate lang1113957λrang decrease further which results in less

Complexity 5

003 004 005002λ

0

02

04

06

08

1

ρ

α = 01α = 02

α = 03α = 05

(a)

0

4000

8000

χ

003 004 005002λ

(b)

001 002 003 0040λ

0

02

04

06

08

1

ρ

α = 06α = 07

α = 08α = 09

(c)

0

4000

8000

χ

001 002 003 0040λ

(d)

Figure 1 Effects of self-awareness on the dynamics of disease spreading on a scale-free network (a) and (c) +e prevalence ρ in thestationary state as a function of basic infection rate λ for varieties of reaction strength α Symbols represent the results obtained fromMonteCarlo simulations and lines represent the results of the GDMPmethod (b) and (d)+e corresponding susceptibility measure χ as a functionof λ Data are obtained by averaging over 500 independent simulations

0

02

04

06

08

1

0

001

002

003

004

005

λ

02 04 06 08 100α

Figure 2 +e phase diagram in the parameter plane (α minus λ) on a scale-free network Colors encode the value of ρ obtained from MonteCarlo simulations Red circles connected by dotted lines represent theoretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained by averaging 50 Monte Carlo simulations for each point in the grid 200 times 200

6 Complexity

ρ (t)

λ = 004

α = 01α = 03α = 05

α = 07α = 08α = 09

200 400 600 8000t

0

02

04

06

08

1

(a)

ρ (α

)

αopt = 048

λ = 004t = 200

0

02

04

06

08

1

02 04 05 06 08 10α

(b)

Figure 3 Effects of behavior response on evolution of the fraction of infected nodes ρ(t) (a) +e time evolution of ρ(t) for varieties of αusing Monte Carlo simulations for a fixed value of λ 004 (b) Plot of the fraction of infected nodes versus the change in α at a fixed timet 200 and infection rate λ 004 +e results of the simulations are obtained by averaging over 300 realizations

0

200 400 6000t

0

5

⟨m⟩

05

1

⟨q⟩

α = 01α = 03α = 05

α = 07α = 09

(a)

300 600 9000t

α = 01α = 03α = 05

α = 07α = 09

0

001

002

003

004

⟨λ~ ⟩

(b)

Figure 4 Continued

Complexity 7

resources donated to support the recovery of infected nodes+us the recovery rate of infected nodes langrrang drops abruptlywhich in turn promotes the increases of the effective in-fection rate lang1113957λranglangrrang further and then more and number ofinfected nodes appear Consequently the cascading failureof the entire system occurs

Based on the above analysis for a small value of α ieα 01 we can reasonably explain why people are morewilling to contribute a resource while the disease is morelikely to break out

Secondly we study the case when α is significant forexample α 09 As a larger value of α means more sensitiveof the individuals to the disease and a lower willingness toallocate resources +us we observe from Figure 4(a) thatinitially there is a smallest value of langqrang (see the green stars intop pane of Figure 4(a)) and the infection rate lang1113957λrang Conse-quently the infected nodes receive the lowest value of theresource to recover which leads to the smallest value of therecovery rate langrrang as shown by the green stars in Figure 4(c)+en the recovery of infected nodes is delayed leading to ahigh effective infection rate We can observe in Figure 4(d)that when α 09 there is a highest value of lang1113957λranglangrrang +ehigh effective infection rate leads to a rapid increase in thenumber of infected nodes We can observe in the bottom paneof Figure 4(a) that in the early stage there is a second largestvalue of langmrang for α 09 as denoted by the green stars +elarge value of langmrang can further reduce the willingness of re-source donation for the healthy individuals thus we canobserve a continuous decline in langqrang and lang1113957λrang +e worse thingis that the recovery rate of infected nodes keeps declining withless and less resource (see the curve in Figure 4(c)) which leadsto a rapid growth of lang1113957λranglangrrang (see the curve in Figure 4(d))

+us we can explain the reason why a higher sense ofself-protection of the population cannot suppress the diseaseeffectively

At last we observe in Figure 4 that when the value ofα is around the optimal value αopt there is a relativelylower value of langqrang comparing to the case of α 01 in theinitial stage which results in a lower value of lang1113957λrang (see theyellow squares in Figures 4(a) and 4(b)) +e lowerwillingness of resource donation induces to a relativelysmaller value of the recovery rate langrrang as shown inFigure 4(c) However we can observe from Figure 4(d)that the effective infection rate lang1113957λranglangrrang keeps the lowestvalue in the early stage which suggests that the diseasepropagates slowly in the population and the number ofinfected nodes increases slowly which is verified by thecurve in the bottom pane of Figure 4(a) Further thesmall value of langmrang promotes the increase of langqrang (see thecurve in the top pane of Figure 4(a)) which results in theincrease of the recovery rate langrrang And finally the effectiveinfection rate lang1113957λranglangrrang decreases further as shown inFigure 4(d) +us the disease can be suppressed to thegreatest extend

+rough the three steps we explain the optimal phe-nomena in the coupled dynamics of resource allocation anddisease spreading

Finally we further verify our explanation by studying thecritical quantities as the function of parameter α at a fixedtime t and basic infection rate λ Figures 5(a) to 5(d) plot thevalues of langqrang langmrang lang1113957λrang langrrang and lang1113957λranglangrrang as a function of αwhen t 200 and λ 004 For the sake of clarity we denotethe local minimum and maximum value as XLmin and Xmaxand the global minimum and maximum value as Xmin andXmax respectively where X isin [langqrang langmrang lang1113957λrang lang1113957λranglangrrang] Weobserve that although when α is around αopt there is a localmaximum of langqrangLmax and lang1113957λrangLmax+e recovery rate reachesmaximum langqrangmax and the effective infection rate reaches thelowest (lang1113957λranglangrrang)min which indicates that the disease can beoptimally suppressed at this point

200 400 6000t

0

005

01

015

02

⟨r⟩

α = 01α = 03α = 05

α = 07α = 09

(c)

100

105

⟨λ~ ⟩⟨r⟩

102 103tlowast

t

α = 01α = 03α = 05

α = 07α = 09

(d)

Figure 4 Plots of the critical parameters versus t for typical values of α (a) Top pane time evolution of the average donation rate langqrangBottom pane the evolution of average number of infected neighbors of all nodes langmrang (b) Time evolution of the average infection rate lang1113957λrang(c) +e complete evolution of the average recovery rate langrrang (d) Log-log plots of the average effective infection rate lang1113957λranglangrrang Basic infectionrate is fixed at λ 004 +e results of the simulations are obtained by averaging over 300 realizations

8 Complexity

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

ri(t) 1 minus (1 minus μ)εωi(t)

(1)

where ωi(t) is the resource quantity of node i received fromhealthy neighbors at time t and μ is the basic recovery rate Aparameter ε isin [0 1] is introduced in our model to representthe resource utilization rate [30] Since in the real scenariothere is the common phenomenon of the waste on resource[31 32] in medical and other service systems implying theresource received from healthy neighbors may not be fullyutilized on curing and recovery Without loss of generalitywe set ϵ 06 throughout this work ie only 60 of theresources received are used

For the r-SIS model the value ρi(t) is defined as theprobability that any node i is in the infected state +efraction of infected nodes in a network of size N at time t canbe calculated by averaging the overall N nodes

ρ(t) 1N

1113944

N

i1ρi(t) (2)

Further we define the prevalence of the disease in thestationary state as ρ equiv ρ(infin)

22 Resource Allocation Model Based on Behavior ResponseIn the real scenario healthy individuals can produce re-sources For simplicity we consider that each individual(node) in the network can generate one unit resource at atime step During an outbreak of a disease the susceptibleindividuals can perceive the threat of the disease intuitivelyby acquiring the information from neighbors Generally themore infected neighbors of an individual the deeper hesheis aware of the disease [33 34] People aware of the diseasemay have different reactions [33] Also to quantify thereaction strength of an individual to the local information ofdisease a tuning parameter α is introduced Based on thedescription above we can define the probability that ahealthy individual with m infected neighbors donates re-source as

q(m) q0(1 minus α)m

(3)

where q0 is a basic donation probability When α 0 allhealthy nodes have the same donation probability q0 Be-sides we consider that a healthy node can donate one unitresource equally to its I-state neighbors at a time Based onthe resource allocation scheme the amount of resourceωj⟶i that node j with m infected neighbors donate to one ofits I-state neighbor i can be expressed as

ωj⟶i q(m)1m

(4)

When disease breaks out in the human populationpeople aware of the disease in their proximity can takemeasures to reduce their susceptibility leading to a re-duction in the effective rate of infection [14 35] We con-sider that if an individual is aware and refuses to donateresource for self-protection it reduces its infectivity by afactor c +e basic infection rate is denoted as λ and the

annealing infection rate is denoted as 1113957λ which can thus beexpressed as

1113957λ λ if distribute the resources

cλ else1113896 (5)

If a healthy individual donates resource to infectedneighbors the individual has a larger probability to be in-fected on the contrary if the individual does not donateresource there is a relatively smaller probability to be in-fected +e annealing infection rate of any node i can also beexpressed as a function of q

1113957λi q(m)λ +[1 minus q(m)]cλ (6)

3 Dynamic Message-Passing Method

In order to theoretically analyze the dynamic processes wedevelop a generated dynamic message-passing method(GDMP) [36 37] In this method the message θj⟶i isdefined on the directed edges of a network to carry causalinformation of the flow of contagion which can onlytransfer one way along directed links θj⟶i represents theprobability that node j is infectious because it was infectedby one of its neighbors other than node i In computingθj⟶i we only take into account the contributions to ρj thatcome from the neighbors other than i +e higher orderprocess of j being infected by i and then passes the infectionback to i is neglected Combining θj⟶i and equation (3) forresource allocations the resources ωi(t) that an infectednode i receive from its healthy neighbors can be expressed as

ωi(t) 1113944j

aij 1 minus θj⟶i(t)1113960 1113961q mj(t)1113960 1113961

mj(t) (7)

where mj(t) is the expected number of I-state neighbors ofnode j at time t which is expressed as

mj(t) 1113944hnei

ajhθh⟶j(t) + 1 (8)

where the plus one takes into account that node i is infectedat this moment +e factor (1 minus θj⟶i(t)) in equation (7)stands for the probability that node j is susceptible at time tWith the definition above the discrete-time version of theevolution of ρi(t) in a time interval Δt reads [38]

ρi(t + Δt) 1 minus ρi(t)( 1113857 1 minus Ωi(t)( 1113857 + 1 minus ri(t)Δt1113858 1113859ρi(t)

(9)

whereΩi(t) is the probability that the node is not infected byany neighbor with the product being over the set Ni of theneighbors of node i +e expression of Ωi(t) is as follows

Ωi(t) 1113945jisinNi

1 minus Δt1113957λi(t)θj⟶i(t)1113960 1113961(10)

Note that the first term on the right-hand side ofequation (9) stands for the probability that node i is inS-state and infected by at least one of its neighbors +esecond term is the probability that node i is in I-state and

Complexity 3

does not recover Similarly we can get the time evolution ofθj⟶i(t) as

θj⟶i(t + Δt) 1 minus θj⟶i(t)1113872 1113873 1 minus ϕj⟶i(t)1113872 1113873 + 1 minus rj(t)Δt1113872 1113873θj⟶i(t)

(11)

where ϕj⟶i(t) is the probability that node j is not infectedby any of its neighbors excluding node i which can beexpressed as

ϕj⟶i(t) 1113945ℓisinNji

1 minus Δt1113957λj(t)θℓ⟶j(t)1113960 1113961(12)

+e product in equation (12) is over the set Nji of theneighbors of j excluding i Further by setting Δt 1 andconsidering situation in a stationary state equations (9) and(11) become

ρi 1 minus ρi( 1113857 1 minus Ωi( 1113857 + 1 minus ri( 1113857ρi (13)

θj⟶i 1 minus θj⟶i1113872 1113873 1 minus ϕj⟶i1113872 1113873 + 1 minus rj1113872 1113873θj⟶i (14)

+rough numerical iteration we can compute the in-fection probability of any node at any time ρi(t) andprevalence ρ in the stationary state for different values of αand λ However the equations can only be solved numer-ically except for the trivial solutions of ρi 0 and θj⟶i 0for all i 1 N which leads to an overall ρ 0 phase ofan all-healthy population

Due to nonlinearities in equations (7)ndash(12) they do nothave a closed analytic form and this disallows obtaining theepidemic threshold λc for fixed values of α such that ρgt 0 ifλgt λc and ρ 0 when λlt λc +e calculation of λc can beperformed by considering that when λ⟶ λc ρi⟶ 0 andθj⟶i⟶ 0 and the number of infected neighbors of anyhealthy node is approximately zero in the thermodynamiclimit +en prior to reaching λc the expression(1 minus θj⟶i)⟶ 1 is valid We can get a physical picture thatthe isolated infected nodes are well separated and sur-rounded by healthy nodes and any infected node i canreceive all the resource from each of its neighbors By addingthese assumptions to equation (7) resource ωi becomesωi kiq0(1 minus α) By linearizing equation (1) and neglectingthe second-order terms for small μ we obtain

ri(t) asymp ϵμωi(t)

ϵμkiq0(1 minus α)(15)

Equation (15) suggests that the recovery rate is pro-portional to the node degree and inversely proportional to αwhen λ⟶ λc For the sake of clarity the basic recovery rateis set at μ 001 in this paper Further equations (10) and(12) can also be linearized using θj⟶i asymp 0 as

qi asymp 1 minus 1113957λi 1113944

N

j1ajiθj⟶i (16)

ϕj⟶i asymp 1 minus 1113957λj 1113944l⟶ hisinVE

Mj⟶il⟶hθl⟶h(17)

where VE is the set of directed edges andM is the |VE| times |VE|

nonbacktracking matrix [39] of the network with the ele-ments labelled by the edges

Mj⟶il⟶h δjh 1 minus δil( 1113857 (18)

with δil being the Dirac delta function Substituting equation(17) into equation (14) and ignoring the higher order termsof θj⟶i give

1113944 minus δljδihrj + 1113957λjMj⟶il⟶h1113872 1113873θl⟶h 0 (19)

Finally considering that mj 1 when λ⟶ λc equation(6) becomes

1113954λ equiv 1113957λj (1 minus c)q0(1 minus α) + c1113858 1113859λ (20)

To estimate the epidemic threshold we calculate theaverage recovery rate as

langrrang ϵμlangkrangq0(1 minus α) (21)

By inserting equations (20) and (21) into equation (19)we get

1113944 minus δljδihlangrrang + 1113954λMj⟶il⟶h1113872 1113873θl⟶h 0 (22)

+e system of equations in equation (22) has a nontrivialsolution if and only if langrrang1113954λ is an eigenvalue of the matrix M

[38] +e lowest value 1113954λc is then given by

1113954λc langrrang

Λmax (23)

where Λmax is the largest eigenvalue of M [15 37 40]

4 Numerical Verification andSimulation Results

In this section we study systematically the effects of self-awareness and the network structure on the coupled dy-namics of resource allocation and disease spreading re-spectively through numerical verification and Monte Carlosimulations In the simulation the synchronous updatingmethod [41 42] is applied to the disease infection and re-source allocation processes Within each time increment Δtwhere Δt 1 in this paper infection propagates from anyI-state node j to S-state node i with probability 1113957λiΔt and anyI-state node j recovers to S-state with a probability rjΔtWith the spreading of disease the resource allocationprocess co-occurs +e dynamics terminate once it enters asteady state in which the number of infected nodes onlyfluctuates within a small range Note that we fix the factor c

at a constant value c 005 throughout the paper such thatif any healthy individual j chooses to reserve their resourcethe probability that they are infected in one contact with aninfected neighbor reduces to 1113957λj 005λ

41 Effects of Self-Awareness on the Spreading DynamicsIn this section we investigate the effects of awareness forself-protection on the spreading dynamics We consider that

4 Complexity

the coupled processes of resource allocation and diseasespreading takes place on a scale-free network as many real-world networks have skewed degree distributions [43ndash46]To build the network we adopt the uncorrelated configu-ration model (UCM) [47 48] according to a given degreedistribution P(k) sim kminus c with maximum degree kmax

N

radic

[49] and minimum degree kmin 3 which assures no degreecorrelation of the network when N is sufficiently large Toavoid the influence of the network structure on the resultthe degree exponent is set at c 24 the network size is set atN 10000 and the average degree is set at langkrang 8 in thesimulations In addition we leverage the susceptibilitymeasure χ to determine the epidemic threshold throughsimulations [50] which is expressed as

χ Nlangρ2rang minus langρrang2

langρrang (24)

where lang rang represents the ensemble average over all real-izations +e epidemic threshold can then be determinedwhen the value of χ exhibits diverging peaks at the certaininfection rate [50 51]

We first investigate the effects of self-awareness on thespreading dynamics using Monte Carlo simulations Ini-tially a fraction of ρ(0) 01 nodes are selected randomly asseeds and the remaining nodes are in the susceptible stateTo present different reaction strength of individuals whenthey are aware of a certain disease from local information weselect eight typical values of α from α 01 to α 09 in thesimulation In Figures 1(a) and 1(c) we plot the prevalence ρin the stationary state as a function of basic infection rate λfor different α Symbols in Figures 1(a) and 1(c) representthe results obtained by Monte Carlo simulations and linesare the theoretical results obtained from numeric iterationsrespectively From the curves in Figures 1(a) and 1(c) weobserve that the system converges to two possible stationarystates either the whole population is healthy or it becomescompletely infected for any α which tells us that when thereis a shortage in resource the disease breaks out abruptly

Besides we can observe from Figures 1(a) and 1(b) thatwith the increase of α from α 01 to α 05 the epidemicthreshold increases gradually see the peaks of χ for thecorresponding α It reveals that the stronger the individualrsquossense of self-protection themore delayed the outbreak of thedisease within this parameter interval (see the right arrow)On the contrary we observe from Figures 1(c) and 1(d) thatwhen α increases from α 06 to α 09 the thresholddecreases gradually which reveals that the disease breaks outmore easily with a stronger sense of self-protection withinthis parameter interval (see the left arrow)+e phenomenonsuggests that too cautious or too selfless for the peopleduring the outbreak of an epidemic are both not suitable fordisease control and there is an optimal value of the reactionstrength at which an epidemic outbreak is postponed to thegreatest extent

We further study systematically the effects of behaviorresponse and the basic infection rate on the spreading dy-namics In Figure 2 we exhibit the full phase diagram (α minus

λ) of the coupled dynamics of resource allocation and

disease spreading Colors in Figure 2(a) encode the fractionof infected nodes ρ in the stationary state +e epidemicthreshold λc marked by red circles rises monotonically untilit reaches the maximum at αopt (indicated by the blue dottedline) and then falls gradually with the increase of α Besideswe observe that there are only two possible stationary statesthe whole healthy (marked by blue color) and the wholeinfected of the population (marked by yellow color)

Figure 3(a) plots the time evolution of ρ(t) for six typicalvalues of α when the basic infection rate is fixed at λ 004We find that when the value of α is small the systemconverges to a stationary state rapidly such as ρ(infin) 10for α 01 With the increase of α it takes a longer time forthe system to reach a stationary state Further to exhibit theeffects of α on the dynamics more intuitively we plot thefraction of infected nodes at a fixed time t 200 as afunction of α in Figure 3(b) which is denoted as ρ(α) for thesake of clarity We observe that the value of ρ(α) decreasescontinuously with α until reaching the minimum value atαopt asymp 048 (marked by red circle in Figure 3(b)) and thenincreases gradually with α

Next we qualitatively explain the optimal phenomena bystudying the time evolution of the critical quantities

We begin by studying the case when α is small forexample α 01 We observe in Figure 4 that in the initialstage the donation probability for α 01 is the highest (seethe blue line in the top panel of Figure 4(a)) since a smallervalue of α means a higher willingness of healthy individualsto allocate resources Although the resource of healthy in-dividuals can improve the recovery probability of infectedneighbors to a certain extent it also makes themselves morelikely to be infected We can observe in Figures 4(b) and 4(c)that the average recovery rate langrrang and the infection rate lang1113957λrang

is the highest for α 01 meanwhile there is a lowest valueof the effective infection rate lang1113957λranglangrrang as shown inFigure 4(d) However with the high probability of beinginfected for the healthy nodes the number of infected in-dividuals increases at a high rate (see the blue line in thebottom pane of Figure 4(a)) When people are aware of theincrement of the infected neighbors they reduce their do-nation willingness which leads to a reduction in infectionrate lang1113957λrang as shown in Figures 4(a) and 4(b) Consequentlywith less resource received from healthy neighbors therecovery rate of infected nodes reduces accordingly seeFigure 4(c) which leads to an increase of the effective in-fection rate lang1113957λranglangrrang [15]+e increase in lang1113957λranglangrrang has led to afurther increase in the number of infected nodes +enpeople become more aware of the threat of disease and thusreduce the probability of resource donation further whichleads to a further decrease in the infection rate lang1113957λrang and therecovery rate langrrang and finally the increase of the effectiveinfection rate lang1113957λranglangrrang

Specifically we observe from Figure 4(d) that when itsurpasses a critical time tlowast indicated by the dotted line in thefigure the value of lang1113957λranglangrrang proliferates which suggests thatin this stage the infection of healthy individuals is muchfaster than the recovery of infected individuals With morenewly infected nodes the donation probability langqrang and theinfection rate lang1113957λrang decrease further which results in less

Complexity 5

003 004 005002λ

0

02

04

06

08

1

ρ

α = 01α = 02

α = 03α = 05

(a)

0

4000

8000

χ

003 004 005002λ

(b)

001 002 003 0040λ

0

02

04

06

08

1

ρ

α = 06α = 07

α = 08α = 09

(c)

0

4000

8000

χ

001 002 003 0040λ

(d)

Figure 1 Effects of self-awareness on the dynamics of disease spreading on a scale-free network (a) and (c) +e prevalence ρ in thestationary state as a function of basic infection rate λ for varieties of reaction strength α Symbols represent the results obtained fromMonteCarlo simulations and lines represent the results of the GDMPmethod (b) and (d)+e corresponding susceptibility measure χ as a functionof λ Data are obtained by averaging over 500 independent simulations

0

02

04

06

08

1

0

001

002

003

004

005

λ

02 04 06 08 100α

Figure 2 +e phase diagram in the parameter plane (α minus λ) on a scale-free network Colors encode the value of ρ obtained from MonteCarlo simulations Red circles connected by dotted lines represent theoretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained by averaging 50 Monte Carlo simulations for each point in the grid 200 times 200

6 Complexity

ρ (t)

λ = 004

α = 01α = 03α = 05

α = 07α = 08α = 09

200 400 600 8000t

0

02

04

06

08

1

(a)

ρ (α

)

αopt = 048

λ = 004t = 200

0

02

04

06

08

1

02 04 05 06 08 10α

(b)

Figure 3 Effects of behavior response on evolution of the fraction of infected nodes ρ(t) (a) +e time evolution of ρ(t) for varieties of αusing Monte Carlo simulations for a fixed value of λ 004 (b) Plot of the fraction of infected nodes versus the change in α at a fixed timet 200 and infection rate λ 004 +e results of the simulations are obtained by averaging over 300 realizations

0

200 400 6000t

0

5

⟨m⟩

05

1

⟨q⟩

α = 01α = 03α = 05

α = 07α = 09

(a)

300 600 9000t

α = 01α = 03α = 05

α = 07α = 09

0

001

002

003

004

⟨λ~ ⟩

(b)

Figure 4 Continued

Complexity 7

resources donated to support the recovery of infected nodes+us the recovery rate of infected nodes langrrang drops abruptlywhich in turn promotes the increases of the effective in-fection rate lang1113957λranglangrrang further and then more and number ofinfected nodes appear Consequently the cascading failureof the entire system occurs

Based on the above analysis for a small value of α ieα 01 we can reasonably explain why people are morewilling to contribute a resource while the disease is morelikely to break out

Secondly we study the case when α is significant forexample α 09 As a larger value of α means more sensitiveof the individuals to the disease and a lower willingness toallocate resources +us we observe from Figure 4(a) thatinitially there is a smallest value of langqrang (see the green stars intop pane of Figure 4(a)) and the infection rate lang1113957λrang Conse-quently the infected nodes receive the lowest value of theresource to recover which leads to the smallest value of therecovery rate langrrang as shown by the green stars in Figure 4(c)+en the recovery of infected nodes is delayed leading to ahigh effective infection rate We can observe in Figure 4(d)that when α 09 there is a highest value of lang1113957λranglangrrang +ehigh effective infection rate leads to a rapid increase in thenumber of infected nodes We can observe in the bottom paneof Figure 4(a) that in the early stage there is a second largestvalue of langmrang for α 09 as denoted by the green stars +elarge value of langmrang can further reduce the willingness of re-source donation for the healthy individuals thus we canobserve a continuous decline in langqrang and lang1113957λrang +e worse thingis that the recovery rate of infected nodes keeps declining withless and less resource (see the curve in Figure 4(c)) which leadsto a rapid growth of lang1113957λranglangrrang (see the curve in Figure 4(d))

+us we can explain the reason why a higher sense ofself-protection of the population cannot suppress the diseaseeffectively

At last we observe in Figure 4 that when the value ofα is around the optimal value αopt there is a relativelylower value of langqrang comparing to the case of α 01 in theinitial stage which results in a lower value of lang1113957λrang (see theyellow squares in Figures 4(a) and 4(b)) +e lowerwillingness of resource donation induces to a relativelysmaller value of the recovery rate langrrang as shown inFigure 4(c) However we can observe from Figure 4(d)that the effective infection rate lang1113957λranglangrrang keeps the lowestvalue in the early stage which suggests that the diseasepropagates slowly in the population and the number ofinfected nodes increases slowly which is verified by thecurve in the bottom pane of Figure 4(a) Further thesmall value of langmrang promotes the increase of langqrang (see thecurve in the top pane of Figure 4(a)) which results in theincrease of the recovery rate langrrang And finally the effectiveinfection rate lang1113957λranglangrrang decreases further as shown inFigure 4(d) +us the disease can be suppressed to thegreatest extend

+rough the three steps we explain the optimal phe-nomena in the coupled dynamics of resource allocation anddisease spreading

Finally we further verify our explanation by studying thecritical quantities as the function of parameter α at a fixedtime t and basic infection rate λ Figures 5(a) to 5(d) plot thevalues of langqrang langmrang lang1113957λrang langrrang and lang1113957λranglangrrang as a function of αwhen t 200 and λ 004 For the sake of clarity we denotethe local minimum and maximum value as XLmin and Xmaxand the global minimum and maximum value as Xmin andXmax respectively where X isin [langqrang langmrang lang1113957λrang lang1113957λranglangrrang] Weobserve that although when α is around αopt there is a localmaximum of langqrangLmax and lang1113957λrangLmax+e recovery rate reachesmaximum langqrangmax and the effective infection rate reaches thelowest (lang1113957λranglangrrang)min which indicates that the disease can beoptimally suppressed at this point

200 400 6000t

0

005

01

015

02

⟨r⟩

α = 01α = 03α = 05

α = 07α = 09

(c)

100

105

⟨λ~ ⟩⟨r⟩

102 103tlowast

t

α = 01α = 03α = 05

α = 07α = 09

(d)

Figure 4 Plots of the critical parameters versus t for typical values of α (a) Top pane time evolution of the average donation rate langqrangBottom pane the evolution of average number of infected neighbors of all nodes langmrang (b) Time evolution of the average infection rate lang1113957λrang(c) +e complete evolution of the average recovery rate langrrang (d) Log-log plots of the average effective infection rate lang1113957λranglangrrang Basic infectionrate is fixed at λ 004 +e results of the simulations are obtained by averaging over 300 realizations

8 Complexity

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

does not recover Similarly we can get the time evolution ofθj⟶i(t) as

θj⟶i(t + Δt) 1 minus θj⟶i(t)1113872 1113873 1 minus ϕj⟶i(t)1113872 1113873 + 1 minus rj(t)Δt1113872 1113873θj⟶i(t)

(11)

where ϕj⟶i(t) is the probability that node j is not infectedby any of its neighbors excluding node i which can beexpressed as

ϕj⟶i(t) 1113945ℓisinNji

1 minus Δt1113957λj(t)θℓ⟶j(t)1113960 1113961(12)

+e product in equation (12) is over the set Nji of theneighbors of j excluding i Further by setting Δt 1 andconsidering situation in a stationary state equations (9) and(11) become

ρi 1 minus ρi( 1113857 1 minus Ωi( 1113857 + 1 minus ri( 1113857ρi (13)

θj⟶i 1 minus θj⟶i1113872 1113873 1 minus ϕj⟶i1113872 1113873 + 1 minus rj1113872 1113873θj⟶i (14)

+rough numerical iteration we can compute the in-fection probability of any node at any time ρi(t) andprevalence ρ in the stationary state for different values of αand λ However the equations can only be solved numer-ically except for the trivial solutions of ρi 0 and θj⟶i 0for all i 1 N which leads to an overall ρ 0 phase ofan all-healthy population

Due to nonlinearities in equations (7)ndash(12) they do nothave a closed analytic form and this disallows obtaining theepidemic threshold λc for fixed values of α such that ρgt 0 ifλgt λc and ρ 0 when λlt λc +e calculation of λc can beperformed by considering that when λ⟶ λc ρi⟶ 0 andθj⟶i⟶ 0 and the number of infected neighbors of anyhealthy node is approximately zero in the thermodynamiclimit +en prior to reaching λc the expression(1 minus θj⟶i)⟶ 1 is valid We can get a physical picture thatthe isolated infected nodes are well separated and sur-rounded by healthy nodes and any infected node i canreceive all the resource from each of its neighbors By addingthese assumptions to equation (7) resource ωi becomesωi kiq0(1 minus α) By linearizing equation (1) and neglectingthe second-order terms for small μ we obtain

ri(t) asymp ϵμωi(t)

ϵμkiq0(1 minus α)(15)

Equation (15) suggests that the recovery rate is pro-portional to the node degree and inversely proportional to αwhen λ⟶ λc For the sake of clarity the basic recovery rateis set at μ 001 in this paper Further equations (10) and(12) can also be linearized using θj⟶i asymp 0 as

qi asymp 1 minus 1113957λi 1113944

N

j1ajiθj⟶i (16)

ϕj⟶i asymp 1 minus 1113957λj 1113944l⟶ hisinVE

Mj⟶il⟶hθl⟶h(17)

where VE is the set of directed edges andM is the |VE| times |VE|

nonbacktracking matrix [39] of the network with the ele-ments labelled by the edges

Mj⟶il⟶h δjh 1 minus δil( 1113857 (18)

with δil being the Dirac delta function Substituting equation(17) into equation (14) and ignoring the higher order termsof θj⟶i give

1113944 minus δljδihrj + 1113957λjMj⟶il⟶h1113872 1113873θl⟶h 0 (19)

Finally considering that mj 1 when λ⟶ λc equation(6) becomes

1113954λ equiv 1113957λj (1 minus c)q0(1 minus α) + c1113858 1113859λ (20)

To estimate the epidemic threshold we calculate theaverage recovery rate as

langrrang ϵμlangkrangq0(1 minus α) (21)

By inserting equations (20) and (21) into equation (19)we get

1113944 minus δljδihlangrrang + 1113954λMj⟶il⟶h1113872 1113873θl⟶h 0 (22)

+e system of equations in equation (22) has a nontrivialsolution if and only if langrrang1113954λ is an eigenvalue of the matrix M

[38] +e lowest value 1113954λc is then given by

1113954λc langrrang

Λmax (23)

where Λmax is the largest eigenvalue of M [15 37 40]

4 Numerical Verification andSimulation Results

In this section we study systematically the effects of self-awareness and the network structure on the coupled dy-namics of resource allocation and disease spreading re-spectively through numerical verification and Monte Carlosimulations In the simulation the synchronous updatingmethod [41 42] is applied to the disease infection and re-source allocation processes Within each time increment Δtwhere Δt 1 in this paper infection propagates from anyI-state node j to S-state node i with probability 1113957λiΔt and anyI-state node j recovers to S-state with a probability rjΔtWith the spreading of disease the resource allocationprocess co-occurs +e dynamics terminate once it enters asteady state in which the number of infected nodes onlyfluctuates within a small range Note that we fix the factor c

at a constant value c 005 throughout the paper such thatif any healthy individual j chooses to reserve their resourcethe probability that they are infected in one contact with aninfected neighbor reduces to 1113957λj 005λ

41 Effects of Self-Awareness on the Spreading DynamicsIn this section we investigate the effects of awareness forself-protection on the spreading dynamics We consider that

4 Complexity

the coupled processes of resource allocation and diseasespreading takes place on a scale-free network as many real-world networks have skewed degree distributions [43ndash46]To build the network we adopt the uncorrelated configu-ration model (UCM) [47 48] according to a given degreedistribution P(k) sim kminus c with maximum degree kmax

N

radic

[49] and minimum degree kmin 3 which assures no degreecorrelation of the network when N is sufficiently large Toavoid the influence of the network structure on the resultthe degree exponent is set at c 24 the network size is set atN 10000 and the average degree is set at langkrang 8 in thesimulations In addition we leverage the susceptibilitymeasure χ to determine the epidemic threshold throughsimulations [50] which is expressed as

χ Nlangρ2rang minus langρrang2

langρrang (24)

where lang rang represents the ensemble average over all real-izations +e epidemic threshold can then be determinedwhen the value of χ exhibits diverging peaks at the certaininfection rate [50 51]

We first investigate the effects of self-awareness on thespreading dynamics using Monte Carlo simulations Ini-tially a fraction of ρ(0) 01 nodes are selected randomly asseeds and the remaining nodes are in the susceptible stateTo present different reaction strength of individuals whenthey are aware of a certain disease from local information weselect eight typical values of α from α 01 to α 09 in thesimulation In Figures 1(a) and 1(c) we plot the prevalence ρin the stationary state as a function of basic infection rate λfor different α Symbols in Figures 1(a) and 1(c) representthe results obtained by Monte Carlo simulations and linesare the theoretical results obtained from numeric iterationsrespectively From the curves in Figures 1(a) and 1(c) weobserve that the system converges to two possible stationarystates either the whole population is healthy or it becomescompletely infected for any α which tells us that when thereis a shortage in resource the disease breaks out abruptly

Besides we can observe from Figures 1(a) and 1(b) thatwith the increase of α from α 01 to α 05 the epidemicthreshold increases gradually see the peaks of χ for thecorresponding α It reveals that the stronger the individualrsquossense of self-protection themore delayed the outbreak of thedisease within this parameter interval (see the right arrow)On the contrary we observe from Figures 1(c) and 1(d) thatwhen α increases from α 06 to α 09 the thresholddecreases gradually which reveals that the disease breaks outmore easily with a stronger sense of self-protection withinthis parameter interval (see the left arrow)+e phenomenonsuggests that too cautious or too selfless for the peopleduring the outbreak of an epidemic are both not suitable fordisease control and there is an optimal value of the reactionstrength at which an epidemic outbreak is postponed to thegreatest extent

We further study systematically the effects of behaviorresponse and the basic infection rate on the spreading dy-namics In Figure 2 we exhibit the full phase diagram (α minus

λ) of the coupled dynamics of resource allocation and

disease spreading Colors in Figure 2(a) encode the fractionof infected nodes ρ in the stationary state +e epidemicthreshold λc marked by red circles rises monotonically untilit reaches the maximum at αopt (indicated by the blue dottedline) and then falls gradually with the increase of α Besideswe observe that there are only two possible stationary statesthe whole healthy (marked by blue color) and the wholeinfected of the population (marked by yellow color)

Figure 3(a) plots the time evolution of ρ(t) for six typicalvalues of α when the basic infection rate is fixed at λ 004We find that when the value of α is small the systemconverges to a stationary state rapidly such as ρ(infin) 10for α 01 With the increase of α it takes a longer time forthe system to reach a stationary state Further to exhibit theeffects of α on the dynamics more intuitively we plot thefraction of infected nodes at a fixed time t 200 as afunction of α in Figure 3(b) which is denoted as ρ(α) for thesake of clarity We observe that the value of ρ(α) decreasescontinuously with α until reaching the minimum value atαopt asymp 048 (marked by red circle in Figure 3(b)) and thenincreases gradually with α

Next we qualitatively explain the optimal phenomena bystudying the time evolution of the critical quantities

We begin by studying the case when α is small forexample α 01 We observe in Figure 4 that in the initialstage the donation probability for α 01 is the highest (seethe blue line in the top panel of Figure 4(a)) since a smallervalue of α means a higher willingness of healthy individualsto allocate resources Although the resource of healthy in-dividuals can improve the recovery probability of infectedneighbors to a certain extent it also makes themselves morelikely to be infected We can observe in Figures 4(b) and 4(c)that the average recovery rate langrrang and the infection rate lang1113957λrang

is the highest for α 01 meanwhile there is a lowest valueof the effective infection rate lang1113957λranglangrrang as shown inFigure 4(d) However with the high probability of beinginfected for the healthy nodes the number of infected in-dividuals increases at a high rate (see the blue line in thebottom pane of Figure 4(a)) When people are aware of theincrement of the infected neighbors they reduce their do-nation willingness which leads to a reduction in infectionrate lang1113957λrang as shown in Figures 4(a) and 4(b) Consequentlywith less resource received from healthy neighbors therecovery rate of infected nodes reduces accordingly seeFigure 4(c) which leads to an increase of the effective in-fection rate lang1113957λranglangrrang [15]+e increase in lang1113957λranglangrrang has led to afurther increase in the number of infected nodes +enpeople become more aware of the threat of disease and thusreduce the probability of resource donation further whichleads to a further decrease in the infection rate lang1113957λrang and therecovery rate langrrang and finally the increase of the effectiveinfection rate lang1113957λranglangrrang

Specifically we observe from Figure 4(d) that when itsurpasses a critical time tlowast indicated by the dotted line in thefigure the value of lang1113957λranglangrrang proliferates which suggests thatin this stage the infection of healthy individuals is muchfaster than the recovery of infected individuals With morenewly infected nodes the donation probability langqrang and theinfection rate lang1113957λrang decrease further which results in less

Complexity 5

003 004 005002λ

0

02

04

06

08

1

ρ

α = 01α = 02

α = 03α = 05

(a)

0

4000

8000

χ

003 004 005002λ

(b)

001 002 003 0040λ

0

02

04

06

08

1

ρ

α = 06α = 07

α = 08α = 09

(c)

0

4000

8000

χ

001 002 003 0040λ

(d)

Figure 1 Effects of self-awareness on the dynamics of disease spreading on a scale-free network (a) and (c) +e prevalence ρ in thestationary state as a function of basic infection rate λ for varieties of reaction strength α Symbols represent the results obtained fromMonteCarlo simulations and lines represent the results of the GDMPmethod (b) and (d)+e corresponding susceptibility measure χ as a functionof λ Data are obtained by averaging over 500 independent simulations

0

02

04

06

08

1

0

001

002

003

004

005

λ

02 04 06 08 100α

Figure 2 +e phase diagram in the parameter plane (α minus λ) on a scale-free network Colors encode the value of ρ obtained from MonteCarlo simulations Red circles connected by dotted lines represent theoretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained by averaging 50 Monte Carlo simulations for each point in the grid 200 times 200

6 Complexity

ρ (t)

λ = 004

α = 01α = 03α = 05

α = 07α = 08α = 09

200 400 600 8000t

0

02

04

06

08

1

(a)

ρ (α

)

αopt = 048

λ = 004t = 200

0

02

04

06

08

1

02 04 05 06 08 10α

(b)

Figure 3 Effects of behavior response on evolution of the fraction of infected nodes ρ(t) (a) +e time evolution of ρ(t) for varieties of αusing Monte Carlo simulations for a fixed value of λ 004 (b) Plot of the fraction of infected nodes versus the change in α at a fixed timet 200 and infection rate λ 004 +e results of the simulations are obtained by averaging over 300 realizations

0

200 400 6000t

0

5

⟨m⟩

05

1

⟨q⟩

α = 01α = 03α = 05

α = 07α = 09

(a)

300 600 9000t

α = 01α = 03α = 05

α = 07α = 09

0

001

002

003

004

⟨λ~ ⟩

(b)

Figure 4 Continued

Complexity 7

resources donated to support the recovery of infected nodes+us the recovery rate of infected nodes langrrang drops abruptlywhich in turn promotes the increases of the effective in-fection rate lang1113957λranglangrrang further and then more and number ofinfected nodes appear Consequently the cascading failureof the entire system occurs

Based on the above analysis for a small value of α ieα 01 we can reasonably explain why people are morewilling to contribute a resource while the disease is morelikely to break out

Secondly we study the case when α is significant forexample α 09 As a larger value of α means more sensitiveof the individuals to the disease and a lower willingness toallocate resources +us we observe from Figure 4(a) thatinitially there is a smallest value of langqrang (see the green stars intop pane of Figure 4(a)) and the infection rate lang1113957λrang Conse-quently the infected nodes receive the lowest value of theresource to recover which leads to the smallest value of therecovery rate langrrang as shown by the green stars in Figure 4(c)+en the recovery of infected nodes is delayed leading to ahigh effective infection rate We can observe in Figure 4(d)that when α 09 there is a highest value of lang1113957λranglangrrang +ehigh effective infection rate leads to a rapid increase in thenumber of infected nodes We can observe in the bottom paneof Figure 4(a) that in the early stage there is a second largestvalue of langmrang for α 09 as denoted by the green stars +elarge value of langmrang can further reduce the willingness of re-source donation for the healthy individuals thus we canobserve a continuous decline in langqrang and lang1113957λrang +e worse thingis that the recovery rate of infected nodes keeps declining withless and less resource (see the curve in Figure 4(c)) which leadsto a rapid growth of lang1113957λranglangrrang (see the curve in Figure 4(d))

+us we can explain the reason why a higher sense ofself-protection of the population cannot suppress the diseaseeffectively

At last we observe in Figure 4 that when the value ofα is around the optimal value αopt there is a relativelylower value of langqrang comparing to the case of α 01 in theinitial stage which results in a lower value of lang1113957λrang (see theyellow squares in Figures 4(a) and 4(b)) +e lowerwillingness of resource donation induces to a relativelysmaller value of the recovery rate langrrang as shown inFigure 4(c) However we can observe from Figure 4(d)that the effective infection rate lang1113957λranglangrrang keeps the lowestvalue in the early stage which suggests that the diseasepropagates slowly in the population and the number ofinfected nodes increases slowly which is verified by thecurve in the bottom pane of Figure 4(a) Further thesmall value of langmrang promotes the increase of langqrang (see thecurve in the top pane of Figure 4(a)) which results in theincrease of the recovery rate langrrang And finally the effectiveinfection rate lang1113957λranglangrrang decreases further as shown inFigure 4(d) +us the disease can be suppressed to thegreatest extend

+rough the three steps we explain the optimal phe-nomena in the coupled dynamics of resource allocation anddisease spreading

Finally we further verify our explanation by studying thecritical quantities as the function of parameter α at a fixedtime t and basic infection rate λ Figures 5(a) to 5(d) plot thevalues of langqrang langmrang lang1113957λrang langrrang and lang1113957λranglangrrang as a function of αwhen t 200 and λ 004 For the sake of clarity we denotethe local minimum and maximum value as XLmin and Xmaxand the global minimum and maximum value as Xmin andXmax respectively where X isin [langqrang langmrang lang1113957λrang lang1113957λranglangrrang] Weobserve that although when α is around αopt there is a localmaximum of langqrangLmax and lang1113957λrangLmax+e recovery rate reachesmaximum langqrangmax and the effective infection rate reaches thelowest (lang1113957λranglangrrang)min which indicates that the disease can beoptimally suppressed at this point

200 400 6000t

0

005

01

015

02

⟨r⟩

α = 01α = 03α = 05

α = 07α = 09

(c)

100

105

⟨λ~ ⟩⟨r⟩

102 103tlowast

t

α = 01α = 03α = 05

α = 07α = 09

(d)

Figure 4 Plots of the critical parameters versus t for typical values of α (a) Top pane time evolution of the average donation rate langqrangBottom pane the evolution of average number of infected neighbors of all nodes langmrang (b) Time evolution of the average infection rate lang1113957λrang(c) +e complete evolution of the average recovery rate langrrang (d) Log-log plots of the average effective infection rate lang1113957λranglangrrang Basic infectionrate is fixed at λ 004 +e results of the simulations are obtained by averaging over 300 realizations

8 Complexity

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

the coupled processes of resource allocation and diseasespreading takes place on a scale-free network as many real-world networks have skewed degree distributions [43ndash46]To build the network we adopt the uncorrelated configu-ration model (UCM) [47 48] according to a given degreedistribution P(k) sim kminus c with maximum degree kmax

N

radic

[49] and minimum degree kmin 3 which assures no degreecorrelation of the network when N is sufficiently large Toavoid the influence of the network structure on the resultthe degree exponent is set at c 24 the network size is set atN 10000 and the average degree is set at langkrang 8 in thesimulations In addition we leverage the susceptibilitymeasure χ to determine the epidemic threshold throughsimulations [50] which is expressed as

χ Nlangρ2rang minus langρrang2

langρrang (24)

where lang rang represents the ensemble average over all real-izations +e epidemic threshold can then be determinedwhen the value of χ exhibits diverging peaks at the certaininfection rate [50 51]

We first investigate the effects of self-awareness on thespreading dynamics using Monte Carlo simulations Ini-tially a fraction of ρ(0) 01 nodes are selected randomly asseeds and the remaining nodes are in the susceptible stateTo present different reaction strength of individuals whenthey are aware of a certain disease from local information weselect eight typical values of α from α 01 to α 09 in thesimulation In Figures 1(a) and 1(c) we plot the prevalence ρin the stationary state as a function of basic infection rate λfor different α Symbols in Figures 1(a) and 1(c) representthe results obtained by Monte Carlo simulations and linesare the theoretical results obtained from numeric iterationsrespectively From the curves in Figures 1(a) and 1(c) weobserve that the system converges to two possible stationarystates either the whole population is healthy or it becomescompletely infected for any α which tells us that when thereis a shortage in resource the disease breaks out abruptly

Besides we can observe from Figures 1(a) and 1(b) thatwith the increase of α from α 01 to α 05 the epidemicthreshold increases gradually see the peaks of χ for thecorresponding α It reveals that the stronger the individualrsquossense of self-protection themore delayed the outbreak of thedisease within this parameter interval (see the right arrow)On the contrary we observe from Figures 1(c) and 1(d) thatwhen α increases from α 06 to α 09 the thresholddecreases gradually which reveals that the disease breaks outmore easily with a stronger sense of self-protection withinthis parameter interval (see the left arrow)+e phenomenonsuggests that too cautious or too selfless for the peopleduring the outbreak of an epidemic are both not suitable fordisease control and there is an optimal value of the reactionstrength at which an epidemic outbreak is postponed to thegreatest extent

We further study systematically the effects of behaviorresponse and the basic infection rate on the spreading dy-namics In Figure 2 we exhibit the full phase diagram (α minus

λ) of the coupled dynamics of resource allocation and

disease spreading Colors in Figure 2(a) encode the fractionof infected nodes ρ in the stationary state +e epidemicthreshold λc marked by red circles rises monotonically untilit reaches the maximum at αopt (indicated by the blue dottedline) and then falls gradually with the increase of α Besideswe observe that there are only two possible stationary statesthe whole healthy (marked by blue color) and the wholeinfected of the population (marked by yellow color)

Figure 3(a) plots the time evolution of ρ(t) for six typicalvalues of α when the basic infection rate is fixed at λ 004We find that when the value of α is small the systemconverges to a stationary state rapidly such as ρ(infin) 10for α 01 With the increase of α it takes a longer time forthe system to reach a stationary state Further to exhibit theeffects of α on the dynamics more intuitively we plot thefraction of infected nodes at a fixed time t 200 as afunction of α in Figure 3(b) which is denoted as ρ(α) for thesake of clarity We observe that the value of ρ(α) decreasescontinuously with α until reaching the minimum value atαopt asymp 048 (marked by red circle in Figure 3(b)) and thenincreases gradually with α

Next we qualitatively explain the optimal phenomena bystudying the time evolution of the critical quantities

We begin by studying the case when α is small forexample α 01 We observe in Figure 4 that in the initialstage the donation probability for α 01 is the highest (seethe blue line in the top panel of Figure 4(a)) since a smallervalue of α means a higher willingness of healthy individualsto allocate resources Although the resource of healthy in-dividuals can improve the recovery probability of infectedneighbors to a certain extent it also makes themselves morelikely to be infected We can observe in Figures 4(b) and 4(c)that the average recovery rate langrrang and the infection rate lang1113957λrang

is the highest for α 01 meanwhile there is a lowest valueof the effective infection rate lang1113957λranglangrrang as shown inFigure 4(d) However with the high probability of beinginfected for the healthy nodes the number of infected in-dividuals increases at a high rate (see the blue line in thebottom pane of Figure 4(a)) When people are aware of theincrement of the infected neighbors they reduce their do-nation willingness which leads to a reduction in infectionrate lang1113957λrang as shown in Figures 4(a) and 4(b) Consequentlywith less resource received from healthy neighbors therecovery rate of infected nodes reduces accordingly seeFigure 4(c) which leads to an increase of the effective in-fection rate lang1113957λranglangrrang [15]+e increase in lang1113957λranglangrrang has led to afurther increase in the number of infected nodes +enpeople become more aware of the threat of disease and thusreduce the probability of resource donation further whichleads to a further decrease in the infection rate lang1113957λrang and therecovery rate langrrang and finally the increase of the effectiveinfection rate lang1113957λranglangrrang

Specifically we observe from Figure 4(d) that when itsurpasses a critical time tlowast indicated by the dotted line in thefigure the value of lang1113957λranglangrrang proliferates which suggests thatin this stage the infection of healthy individuals is muchfaster than the recovery of infected individuals With morenewly infected nodes the donation probability langqrang and theinfection rate lang1113957λrang decrease further which results in less

Complexity 5

003 004 005002λ

0

02

04

06

08

1

ρ

α = 01α = 02

α = 03α = 05

(a)

0

4000

8000

χ

003 004 005002λ

(b)

001 002 003 0040λ

0

02

04

06

08

1

ρ

α = 06α = 07

α = 08α = 09

(c)

0

4000

8000

χ

001 002 003 0040λ

(d)

Figure 1 Effects of self-awareness on the dynamics of disease spreading on a scale-free network (a) and (c) +e prevalence ρ in thestationary state as a function of basic infection rate λ for varieties of reaction strength α Symbols represent the results obtained fromMonteCarlo simulations and lines represent the results of the GDMPmethod (b) and (d)+e corresponding susceptibility measure χ as a functionof λ Data are obtained by averaging over 500 independent simulations

0

02

04

06

08

1

0

001

002

003

004

005

λ

02 04 06 08 100α

Figure 2 +e phase diagram in the parameter plane (α minus λ) on a scale-free network Colors encode the value of ρ obtained from MonteCarlo simulations Red circles connected by dotted lines represent theoretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained by averaging 50 Monte Carlo simulations for each point in the grid 200 times 200

6 Complexity

ρ (t)

λ = 004

α = 01α = 03α = 05

α = 07α = 08α = 09

200 400 600 8000t

0

02

04

06

08

1

(a)

ρ (α

)

αopt = 048

λ = 004t = 200

0

02

04

06

08

1

02 04 05 06 08 10α

(b)

Figure 3 Effects of behavior response on evolution of the fraction of infected nodes ρ(t) (a) +e time evolution of ρ(t) for varieties of αusing Monte Carlo simulations for a fixed value of λ 004 (b) Plot of the fraction of infected nodes versus the change in α at a fixed timet 200 and infection rate λ 004 +e results of the simulations are obtained by averaging over 300 realizations

0

200 400 6000t

0

5

⟨m⟩

05

1

⟨q⟩

α = 01α = 03α = 05

α = 07α = 09

(a)

300 600 9000t

α = 01α = 03α = 05

α = 07α = 09

0

001

002

003

004

⟨λ~ ⟩

(b)

Figure 4 Continued

Complexity 7

resources donated to support the recovery of infected nodes+us the recovery rate of infected nodes langrrang drops abruptlywhich in turn promotes the increases of the effective in-fection rate lang1113957λranglangrrang further and then more and number ofinfected nodes appear Consequently the cascading failureof the entire system occurs

Based on the above analysis for a small value of α ieα 01 we can reasonably explain why people are morewilling to contribute a resource while the disease is morelikely to break out

Secondly we study the case when α is significant forexample α 09 As a larger value of α means more sensitiveof the individuals to the disease and a lower willingness toallocate resources +us we observe from Figure 4(a) thatinitially there is a smallest value of langqrang (see the green stars intop pane of Figure 4(a)) and the infection rate lang1113957λrang Conse-quently the infected nodes receive the lowest value of theresource to recover which leads to the smallest value of therecovery rate langrrang as shown by the green stars in Figure 4(c)+en the recovery of infected nodes is delayed leading to ahigh effective infection rate We can observe in Figure 4(d)that when α 09 there is a highest value of lang1113957λranglangrrang +ehigh effective infection rate leads to a rapid increase in thenumber of infected nodes We can observe in the bottom paneof Figure 4(a) that in the early stage there is a second largestvalue of langmrang for α 09 as denoted by the green stars +elarge value of langmrang can further reduce the willingness of re-source donation for the healthy individuals thus we canobserve a continuous decline in langqrang and lang1113957λrang +e worse thingis that the recovery rate of infected nodes keeps declining withless and less resource (see the curve in Figure 4(c)) which leadsto a rapid growth of lang1113957λranglangrrang (see the curve in Figure 4(d))

+us we can explain the reason why a higher sense ofself-protection of the population cannot suppress the diseaseeffectively

At last we observe in Figure 4 that when the value ofα is around the optimal value αopt there is a relativelylower value of langqrang comparing to the case of α 01 in theinitial stage which results in a lower value of lang1113957λrang (see theyellow squares in Figures 4(a) and 4(b)) +e lowerwillingness of resource donation induces to a relativelysmaller value of the recovery rate langrrang as shown inFigure 4(c) However we can observe from Figure 4(d)that the effective infection rate lang1113957λranglangrrang keeps the lowestvalue in the early stage which suggests that the diseasepropagates slowly in the population and the number ofinfected nodes increases slowly which is verified by thecurve in the bottom pane of Figure 4(a) Further thesmall value of langmrang promotes the increase of langqrang (see thecurve in the top pane of Figure 4(a)) which results in theincrease of the recovery rate langrrang And finally the effectiveinfection rate lang1113957λranglangrrang decreases further as shown inFigure 4(d) +us the disease can be suppressed to thegreatest extend

+rough the three steps we explain the optimal phe-nomena in the coupled dynamics of resource allocation anddisease spreading

Finally we further verify our explanation by studying thecritical quantities as the function of parameter α at a fixedtime t and basic infection rate λ Figures 5(a) to 5(d) plot thevalues of langqrang langmrang lang1113957λrang langrrang and lang1113957λranglangrrang as a function of αwhen t 200 and λ 004 For the sake of clarity we denotethe local minimum and maximum value as XLmin and Xmaxand the global minimum and maximum value as Xmin andXmax respectively where X isin [langqrang langmrang lang1113957λrang lang1113957λranglangrrang] Weobserve that although when α is around αopt there is a localmaximum of langqrangLmax and lang1113957λrangLmax+e recovery rate reachesmaximum langqrangmax and the effective infection rate reaches thelowest (lang1113957λranglangrrang)min which indicates that the disease can beoptimally suppressed at this point

200 400 6000t

0

005

01

015

02

⟨r⟩

α = 01α = 03α = 05

α = 07α = 09

(c)

100

105

⟨λ~ ⟩⟨r⟩

102 103tlowast

t

α = 01α = 03α = 05

α = 07α = 09

(d)

Figure 4 Plots of the critical parameters versus t for typical values of α (a) Top pane time evolution of the average donation rate langqrangBottom pane the evolution of average number of infected neighbors of all nodes langmrang (b) Time evolution of the average infection rate lang1113957λrang(c) +e complete evolution of the average recovery rate langrrang (d) Log-log plots of the average effective infection rate lang1113957λranglangrrang Basic infectionrate is fixed at λ 004 +e results of the simulations are obtained by averaging over 300 realizations

8 Complexity

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

003 004 005002λ

0

02

04

06

08

1

ρ

α = 01α = 02

α = 03α = 05

(a)

0

4000

8000

χ

003 004 005002λ

(b)

001 002 003 0040λ

0

02

04

06

08

1

ρ

α = 06α = 07

α = 08α = 09

(c)

0

4000

8000

χ

001 002 003 0040λ

(d)

Figure 1 Effects of self-awareness on the dynamics of disease spreading on a scale-free network (a) and (c) +e prevalence ρ in thestationary state as a function of basic infection rate λ for varieties of reaction strength α Symbols represent the results obtained fromMonteCarlo simulations and lines represent the results of the GDMPmethod (b) and (d)+e corresponding susceptibility measure χ as a functionof λ Data are obtained by averaging over 500 independent simulations

0

02

04

06

08

1

0

001

002

003

004

005

λ

02 04 06 08 100α

Figure 2 +e phase diagram in the parameter plane (α minus λ) on a scale-free network Colors encode the value of ρ obtained from MonteCarlo simulations Red circles connected by dotted lines represent theoretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained by averaging 50 Monte Carlo simulations for each point in the grid 200 times 200

6 Complexity

ρ (t)

λ = 004

α = 01α = 03α = 05

α = 07α = 08α = 09

200 400 600 8000t

0

02

04

06

08

1

(a)

ρ (α

)

αopt = 048

λ = 004t = 200

0

02

04

06

08

1

02 04 05 06 08 10α

(b)

Figure 3 Effects of behavior response on evolution of the fraction of infected nodes ρ(t) (a) +e time evolution of ρ(t) for varieties of αusing Monte Carlo simulations for a fixed value of λ 004 (b) Plot of the fraction of infected nodes versus the change in α at a fixed timet 200 and infection rate λ 004 +e results of the simulations are obtained by averaging over 300 realizations

0

200 400 6000t

0

5

⟨m⟩

05

1

⟨q⟩

α = 01α = 03α = 05

α = 07α = 09

(a)

300 600 9000t

α = 01α = 03α = 05

α = 07α = 09

0

001

002

003

004

⟨λ~ ⟩

(b)

Figure 4 Continued

Complexity 7

resources donated to support the recovery of infected nodes+us the recovery rate of infected nodes langrrang drops abruptlywhich in turn promotes the increases of the effective in-fection rate lang1113957λranglangrrang further and then more and number ofinfected nodes appear Consequently the cascading failureof the entire system occurs

Based on the above analysis for a small value of α ieα 01 we can reasonably explain why people are morewilling to contribute a resource while the disease is morelikely to break out

Secondly we study the case when α is significant forexample α 09 As a larger value of α means more sensitiveof the individuals to the disease and a lower willingness toallocate resources +us we observe from Figure 4(a) thatinitially there is a smallest value of langqrang (see the green stars intop pane of Figure 4(a)) and the infection rate lang1113957λrang Conse-quently the infected nodes receive the lowest value of theresource to recover which leads to the smallest value of therecovery rate langrrang as shown by the green stars in Figure 4(c)+en the recovery of infected nodes is delayed leading to ahigh effective infection rate We can observe in Figure 4(d)that when α 09 there is a highest value of lang1113957λranglangrrang +ehigh effective infection rate leads to a rapid increase in thenumber of infected nodes We can observe in the bottom paneof Figure 4(a) that in the early stage there is a second largestvalue of langmrang for α 09 as denoted by the green stars +elarge value of langmrang can further reduce the willingness of re-source donation for the healthy individuals thus we canobserve a continuous decline in langqrang and lang1113957λrang +e worse thingis that the recovery rate of infected nodes keeps declining withless and less resource (see the curve in Figure 4(c)) which leadsto a rapid growth of lang1113957λranglangrrang (see the curve in Figure 4(d))

+us we can explain the reason why a higher sense ofself-protection of the population cannot suppress the diseaseeffectively

At last we observe in Figure 4 that when the value ofα is around the optimal value αopt there is a relativelylower value of langqrang comparing to the case of α 01 in theinitial stage which results in a lower value of lang1113957λrang (see theyellow squares in Figures 4(a) and 4(b)) +e lowerwillingness of resource donation induces to a relativelysmaller value of the recovery rate langrrang as shown inFigure 4(c) However we can observe from Figure 4(d)that the effective infection rate lang1113957λranglangrrang keeps the lowestvalue in the early stage which suggests that the diseasepropagates slowly in the population and the number ofinfected nodes increases slowly which is verified by thecurve in the bottom pane of Figure 4(a) Further thesmall value of langmrang promotes the increase of langqrang (see thecurve in the top pane of Figure 4(a)) which results in theincrease of the recovery rate langrrang And finally the effectiveinfection rate lang1113957λranglangrrang decreases further as shown inFigure 4(d) +us the disease can be suppressed to thegreatest extend

+rough the three steps we explain the optimal phe-nomena in the coupled dynamics of resource allocation anddisease spreading

Finally we further verify our explanation by studying thecritical quantities as the function of parameter α at a fixedtime t and basic infection rate λ Figures 5(a) to 5(d) plot thevalues of langqrang langmrang lang1113957λrang langrrang and lang1113957λranglangrrang as a function of αwhen t 200 and λ 004 For the sake of clarity we denotethe local minimum and maximum value as XLmin and Xmaxand the global minimum and maximum value as Xmin andXmax respectively where X isin [langqrang langmrang lang1113957λrang lang1113957λranglangrrang] Weobserve that although when α is around αopt there is a localmaximum of langqrangLmax and lang1113957λrangLmax+e recovery rate reachesmaximum langqrangmax and the effective infection rate reaches thelowest (lang1113957λranglangrrang)min which indicates that the disease can beoptimally suppressed at this point

200 400 6000t

0

005

01

015

02

⟨r⟩

α = 01α = 03α = 05

α = 07α = 09

(c)

100

105

⟨λ~ ⟩⟨r⟩

102 103tlowast

t

α = 01α = 03α = 05

α = 07α = 09

(d)

Figure 4 Plots of the critical parameters versus t for typical values of α (a) Top pane time evolution of the average donation rate langqrangBottom pane the evolution of average number of infected neighbors of all nodes langmrang (b) Time evolution of the average infection rate lang1113957λrang(c) +e complete evolution of the average recovery rate langrrang (d) Log-log plots of the average effective infection rate lang1113957λranglangrrang Basic infectionrate is fixed at λ 004 +e results of the simulations are obtained by averaging over 300 realizations

8 Complexity

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

ρ (t)

λ = 004

α = 01α = 03α = 05

α = 07α = 08α = 09

200 400 600 8000t

0

02

04

06

08

1

(a)

ρ (α

)

αopt = 048

λ = 004t = 200

0

02

04

06

08

1

02 04 05 06 08 10α

(b)

Figure 3 Effects of behavior response on evolution of the fraction of infected nodes ρ(t) (a) +e time evolution of ρ(t) for varieties of αusing Monte Carlo simulations for a fixed value of λ 004 (b) Plot of the fraction of infected nodes versus the change in α at a fixed timet 200 and infection rate λ 004 +e results of the simulations are obtained by averaging over 300 realizations

0

200 400 6000t

0

5

⟨m⟩

05

1

⟨q⟩

α = 01α = 03α = 05

α = 07α = 09

(a)

300 600 9000t

α = 01α = 03α = 05

α = 07α = 09

0

001

002

003

004

⟨λ~ ⟩

(b)

Figure 4 Continued

Complexity 7

resources donated to support the recovery of infected nodes+us the recovery rate of infected nodes langrrang drops abruptlywhich in turn promotes the increases of the effective in-fection rate lang1113957λranglangrrang further and then more and number ofinfected nodes appear Consequently the cascading failureof the entire system occurs

Based on the above analysis for a small value of α ieα 01 we can reasonably explain why people are morewilling to contribute a resource while the disease is morelikely to break out

Secondly we study the case when α is significant forexample α 09 As a larger value of α means more sensitiveof the individuals to the disease and a lower willingness toallocate resources +us we observe from Figure 4(a) thatinitially there is a smallest value of langqrang (see the green stars intop pane of Figure 4(a)) and the infection rate lang1113957λrang Conse-quently the infected nodes receive the lowest value of theresource to recover which leads to the smallest value of therecovery rate langrrang as shown by the green stars in Figure 4(c)+en the recovery of infected nodes is delayed leading to ahigh effective infection rate We can observe in Figure 4(d)that when α 09 there is a highest value of lang1113957λranglangrrang +ehigh effective infection rate leads to a rapid increase in thenumber of infected nodes We can observe in the bottom paneof Figure 4(a) that in the early stage there is a second largestvalue of langmrang for α 09 as denoted by the green stars +elarge value of langmrang can further reduce the willingness of re-source donation for the healthy individuals thus we canobserve a continuous decline in langqrang and lang1113957λrang +e worse thingis that the recovery rate of infected nodes keeps declining withless and less resource (see the curve in Figure 4(c)) which leadsto a rapid growth of lang1113957λranglangrrang (see the curve in Figure 4(d))

+us we can explain the reason why a higher sense ofself-protection of the population cannot suppress the diseaseeffectively

At last we observe in Figure 4 that when the value ofα is around the optimal value αopt there is a relativelylower value of langqrang comparing to the case of α 01 in theinitial stage which results in a lower value of lang1113957λrang (see theyellow squares in Figures 4(a) and 4(b)) +e lowerwillingness of resource donation induces to a relativelysmaller value of the recovery rate langrrang as shown inFigure 4(c) However we can observe from Figure 4(d)that the effective infection rate lang1113957λranglangrrang keeps the lowestvalue in the early stage which suggests that the diseasepropagates slowly in the population and the number ofinfected nodes increases slowly which is verified by thecurve in the bottom pane of Figure 4(a) Further thesmall value of langmrang promotes the increase of langqrang (see thecurve in the top pane of Figure 4(a)) which results in theincrease of the recovery rate langrrang And finally the effectiveinfection rate lang1113957λranglangrrang decreases further as shown inFigure 4(d) +us the disease can be suppressed to thegreatest extend

+rough the three steps we explain the optimal phe-nomena in the coupled dynamics of resource allocation anddisease spreading

Finally we further verify our explanation by studying thecritical quantities as the function of parameter α at a fixedtime t and basic infection rate λ Figures 5(a) to 5(d) plot thevalues of langqrang langmrang lang1113957λrang langrrang and lang1113957λranglangrrang as a function of αwhen t 200 and λ 004 For the sake of clarity we denotethe local minimum and maximum value as XLmin and Xmaxand the global minimum and maximum value as Xmin andXmax respectively where X isin [langqrang langmrang lang1113957λrang lang1113957λranglangrrang] Weobserve that although when α is around αopt there is a localmaximum of langqrangLmax and lang1113957λrangLmax+e recovery rate reachesmaximum langqrangmax and the effective infection rate reaches thelowest (lang1113957λranglangrrang)min which indicates that the disease can beoptimally suppressed at this point

200 400 6000t

0

005

01

015

02

⟨r⟩

α = 01α = 03α = 05

α = 07α = 09

(c)

100

105

⟨λ~ ⟩⟨r⟩

102 103tlowast

t

α = 01α = 03α = 05

α = 07α = 09

(d)

Figure 4 Plots of the critical parameters versus t for typical values of α (a) Top pane time evolution of the average donation rate langqrangBottom pane the evolution of average number of infected neighbors of all nodes langmrang (b) Time evolution of the average infection rate lang1113957λrang(c) +e complete evolution of the average recovery rate langrrang (d) Log-log plots of the average effective infection rate lang1113957λranglangrrang Basic infectionrate is fixed at λ 004 +e results of the simulations are obtained by averaging over 300 realizations

8 Complexity

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

resources donated to support the recovery of infected nodes+us the recovery rate of infected nodes langrrang drops abruptlywhich in turn promotes the increases of the effective in-fection rate lang1113957λranglangrrang further and then more and number ofinfected nodes appear Consequently the cascading failureof the entire system occurs

Based on the above analysis for a small value of α ieα 01 we can reasonably explain why people are morewilling to contribute a resource while the disease is morelikely to break out

Secondly we study the case when α is significant forexample α 09 As a larger value of α means more sensitiveof the individuals to the disease and a lower willingness toallocate resources +us we observe from Figure 4(a) thatinitially there is a smallest value of langqrang (see the green stars intop pane of Figure 4(a)) and the infection rate lang1113957λrang Conse-quently the infected nodes receive the lowest value of theresource to recover which leads to the smallest value of therecovery rate langrrang as shown by the green stars in Figure 4(c)+en the recovery of infected nodes is delayed leading to ahigh effective infection rate We can observe in Figure 4(d)that when α 09 there is a highest value of lang1113957λranglangrrang +ehigh effective infection rate leads to a rapid increase in thenumber of infected nodes We can observe in the bottom paneof Figure 4(a) that in the early stage there is a second largestvalue of langmrang for α 09 as denoted by the green stars +elarge value of langmrang can further reduce the willingness of re-source donation for the healthy individuals thus we canobserve a continuous decline in langqrang and lang1113957λrang +e worse thingis that the recovery rate of infected nodes keeps declining withless and less resource (see the curve in Figure 4(c)) which leadsto a rapid growth of lang1113957λranglangrrang (see the curve in Figure 4(d))

+us we can explain the reason why a higher sense ofself-protection of the population cannot suppress the diseaseeffectively

At last we observe in Figure 4 that when the value ofα is around the optimal value αopt there is a relativelylower value of langqrang comparing to the case of α 01 in theinitial stage which results in a lower value of lang1113957λrang (see theyellow squares in Figures 4(a) and 4(b)) +e lowerwillingness of resource donation induces to a relativelysmaller value of the recovery rate langrrang as shown inFigure 4(c) However we can observe from Figure 4(d)that the effective infection rate lang1113957λranglangrrang keeps the lowestvalue in the early stage which suggests that the diseasepropagates slowly in the population and the number ofinfected nodes increases slowly which is verified by thecurve in the bottom pane of Figure 4(a) Further thesmall value of langmrang promotes the increase of langqrang (see thecurve in the top pane of Figure 4(a)) which results in theincrease of the recovery rate langrrang And finally the effectiveinfection rate lang1113957λranglangrrang decreases further as shown inFigure 4(d) +us the disease can be suppressed to thegreatest extend

+rough the three steps we explain the optimal phe-nomena in the coupled dynamics of resource allocation anddisease spreading

Finally we further verify our explanation by studying thecritical quantities as the function of parameter α at a fixedtime t and basic infection rate λ Figures 5(a) to 5(d) plot thevalues of langqrang langmrang lang1113957λrang langrrang and lang1113957λranglangrrang as a function of αwhen t 200 and λ 004 For the sake of clarity we denotethe local minimum and maximum value as XLmin and Xmaxand the global minimum and maximum value as Xmin andXmax respectively where X isin [langqrang langmrang lang1113957λrang lang1113957λranglangrrang] Weobserve that although when α is around αopt there is a localmaximum of langqrangLmax and lang1113957λrangLmax+e recovery rate reachesmaximum langqrangmax and the effective infection rate reaches thelowest (lang1113957λranglangrrang)min which indicates that the disease can beoptimally suppressed at this point

200 400 6000t

0

005

01

015

02

⟨r⟩

α = 01α = 03α = 05

α = 07α = 09

(c)

100

105

⟨λ~ ⟩⟨r⟩

102 103tlowast

t

α = 01α = 03α = 05

α = 07α = 09

(d)

Figure 4 Plots of the critical parameters versus t for typical values of α (a) Top pane time evolution of the average donation rate langqrangBottom pane the evolution of average number of infected neighbors of all nodes langmrang (b) Time evolution of the average infection rate lang1113957λrang(c) +e complete evolution of the average recovery rate langrrang (d) Log-log plots of the average effective infection rate lang1113957λranglangrrang Basic infectionrate is fixed at λ 004 +e results of the simulations are obtained by averaging over 300 realizations

8 Complexity

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

42 Effects of Network Structure on Spreading DynamicsIn this section we investigate the effects of the networkstructure on the coupled dynamics of resource allocationand disease spreading To avoid the impact of reactionstrength on the result the parameter α is fixed at α 05In addition we adopt the UCM model to generate scale-free networks with different degree distributionsP(k) sim kminus c As the degree heterogeneity decreases withthe increase of the power exponent c [52 53] it ap-proaches to random regular networks (RRNs) whenc⟶infin [18]

Figure 6 plots the prevalence ρ in the stationary state as afunction of the basic infection rate c for networks with fourtypical values of c c 24 (blue circles) c 28 (uppertriangles) c 32 (purple squares) and c⟶infin (redrhombus) We observe that there are only two stationarystates of the system all healthy or completely infected for allnetworks which implies that the network structure does notalter the first-order transition of ρ Besides we find that withan increase of c the outbreak of disease is delayed graduallyIt suggests that the degree heterogeneity enhances the dis-ease spreading which is consistent with the existing researchconclusions [54]

0

⟨m⟩min

⟨m⟩max

⟨q⟩Lmax

⟨q⟩Lmin

02 04 05 06 08 10α

0

2

4

⟨m⟩

05

1

⟨q⟩

(a)

⟨λ~⟩Lmax

⟨λ~⟩Lmin

0

001

002

003

004

⟨λ~ ⟩

02 04 05 06 08 10α

(b)

⟨r⟩max

02 04 05 06 08 10α

0

005

01

015

⟨r⟩

(c)

(⟨λ~⟩⟨r⟩)min

10ndash2

100

102

104

⟨λ~ ⟩⟨r⟩

02 04 06 08 10α

(d)

Figure 5 Plots of the critical parameters versus α at fixed time t 200 and basic infection rate λ 004 (a) Top pane the average donationrate langqrang as a function of α Bottom pane the average number of infected neighbors of all nodes langmrang as a function of α (b) +e averageinfection rate lang1113957λrang as a function of α (c) +e average recovery rate langrrang as a function of α (d) Plots of average effective infection rate lang1113957λranglangrrang

as a function of α +e results of the simulations are obtained by averaging over 300 realizations

0035 004 0045 005003λ

0

02

04

06

08

1

ρ

γ = 24γ = 28

γ = 32RRNs

Figure 6 +e prevalence ρ in the stationary as a function of λ onscale-free networks with degree exponent c 24 (blue circles)c 28 and c 32 (purple squares) And the result on randomregular networks (RRNs) marked by the red rhombus Symbolsrepresent the results obtained from Monte Carlo simulations andlines represent results of the GDMP method +e parameter α isfixed at α 05

Complexity 9

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

In the end we study the effects of behavior response onthe spreading dynamics systematically Figure 7 is the phasediagram in the parameter plane (α minus λ) on RRNs Colorsencode the prevalence in the stationary state ρ We find thatthere is also an optimal value αopt at which the epidemicthreshold reaches the maximum indicated by the bluedotted line in Figure 7 +e results suggest that the networkstructure does not alert the optimal phenomenon in be-havior response

5 Discussion

In this paper we have focused on the problem of how can weprotect ourselves from being infected while helping othersby donating resources during an outbreak of an epidemic Toanswer this question we have proposed a novel resourceallocation model in controlling the epidemic spreading byconsidering the following two facts namely the healthyindividuals are the providers of essential resources and thereis a kind of game between individualrsquos self-protection andresource contribution To quantify the awareness for self-protection a parameter α has been assigned to each indi-viduals in the model Besides to study the coupled dynamicsof resource allocation and disease spreading a resource-based SIS model has been proposed First of all we havetheoretically analyzed the model by using a generated dy-namic message-passing method and then carried out ex-tensive Monte Carlo simulations on both scale-free andrandom regular networks +rough theoretical analysis andsimulations we have found that the coupled dynamicsconverges to two stationary states the whole infected or allhealthy which indicates that a shortage of resource caninduce an abrupt outbreak of the epidemic More impor-tantly we have found that too cautious or too selfless for thepeople during the outbreak of an epidemic are both notsuitable for epidemic containment +ere is an optimal

(balance) point where the epidemic spreading can be con-trolled to the greatest extent It also suggests that one candonate resource appropriately to support the people in needbut at the same time they should reserve the right amount ofresources for self-protection Further we have located theoptimal point At last we have investigated the effects of thenetwork structure on the coupled dynamics and found thatthe degree heterogeneity promotes the outbreak of diseaseand the network structure does not alter the optimal phe-nomenon in behavior response

Our research is of practical significance in the context ofthe global outbreak of COVID-19 It will guide us to makethe most reasonable choice between resource contributionand self-protection when perceiving the threat of disease andalso have a direct application in the development of strat-egies to suppress the outbreaks of epidemics Moreover oursuggestions that in the face of a global pandemic individualsor countries should strengthen mutual support and cooper-ation while doing their own prevention are consistent withthe current measures taken by most individuals andcountries in combating the epidemic At present not onlythe individuals but also the nations are donating resources tosupport each other while ensuring its own prevention andcontrol needs For example when the outbreak in China iseffectively contained it announces assistance to manyCOVID-19 countries by donating medical resources such asrespirator mask nucleic acid testing reagent and sendingthe medical staffs [55]

+ere is still much more work need to be done Forexample the SIS model adopted in this work has its ownlimitations and it cannot fully describe the characteristics ofmost real epidemics As we all know that there is an in-cubation period in COVID-19 so an SEIR model may bemore suitable Moreover in some other epidemics the re-covered individuals would obtain a short acquired immuneand then turn into the susceptible state again so the SIRSmodel is more suitable +erefore the research of the dy-namical properties when the present mechanisms are ap-plied in various epidemic models would be the futuredirections

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities (nos JBK190972JBK171113 and JBK170505) National Natural ScienceFoundation of China (nos 61903266 71671141 71873108and 61703292) the Financial Intelligence amp Financial En-gineering Key Lab of Sichuan Province China Postdoctoral

0

001

002

003

004

005

006

λ

02 04 06 08 10α

0

02

04

06

08

1

Figure 7 +e phase diagram in the parameter plane (α minus λ) onRRNs Colors encode the value of ρ obtained from Monte Carlosimulations Red circles connected by dotted lines represent the-oretical predictions of epidemic threshold λc +e blue dotted lineindicates the location of optimal value αopt Data are obtained byaveraging 50 Monte Carlo simulations for each point in the grid200 times 200

10 Complexity

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

Science Foundation (no 2018M631073) and China Post-doctoral Science Special Foundation (no 2019T120829)

References

[1] K H Chan P H Li S Y Tan Q Chang and J P XieldquoEpidemiology and cause of severe acute respiratory syn-drome (sars) in Guangdong Peoplersquos Republic of China inFebruary 2003rdquo Lancet vol 362 no 9393 pp 1353ndash13582003

[2] M P Girard J S Tam O M Assossou andM P Kieny ldquo+e2009 a (H1N1) influenza virus pandemic a reviewrdquo Vaccinevol 28 no 31 pp 4895ndash4902 2010

[3] WHO Ebola Response Team ldquoEbola virus disease in westafricathe first 9 months of the epidemic and forward pro-jectionsrdquo New England Journal of Medicine vol 371 no 16pp 1481ndash1495 2014

[4] World Health Organization Coronavirus Disease 2019 (Covid-19)Situation Reportndash96 WHO Geneva Switzerland 2020 httpswwwwhointemergenciesdiseasesnovel-coronavirus-2019situation-reports

[5] R Li S Pei B Chen et al ldquoSubstantial undocumented in-fection facilitates the rapid dissemination of novel corona-virus (sars-cov2)rdquo Science vol 368 no 6490 pp 489ndash4932020

[6] Y Wan S Roy and A Saberi ldquoDesigning spatially hetero-geneous strategies for control of virus spreadrdquo IET SystemsBiology vol 2 no 4 pp 184ndash201 2008

[7] E Gourdin J Omic and P Van Mieghem ldquoOptimization ofnetwork protection against virus spreadrdquo in Proceedings of the2011 8th International Workshop on the Design of ReliableCommunication Networks (DRCN) pp 86ndash93 IEEE KrakowPoland 2011

[8] A Y Lokhov and D Saad ldquoOptimal deployment of resourcesfor maximizing impact in spreading processesrdquo Proceedings ofthe National Academy of Sciences vol 114 no 39pp E8138ndashE8146 2017

[9] D Zhao L Wang Z Wang and G Xiao ldquoVirus propagationand patch distribution in multiplex networks modelinganalysis and optimal allocationrdquo IEEE Transactions on In-formation Forensics and Security vol 14 no 7 pp 1755ndash17672019

[10] S Li D Zhao XWu Z Tian A Li and ZWang ldquoFunctionalimmunization of networks based on message passingrdquo Ap-plied Mathematics and Computation vol 366 Article ID124728 2020

[11] V M Preciado M Zargham C Enyioha A Jadbabaie andG Pappas ldquoOptimal vaccine allocation to control epidemicoutbreaks in arbitrary networksrdquo in Proceedings of the 52ndIEEE Conference on Decision and Control IEEE Firenze Italypp 7486ndash7491 December 2013

[12] V M Preciado M Zargham C Enyioha A Jadbabaie andG J Pappas ldquoOptimal resource allocation for network pro-tection against spreading processesrdquo IEEE Transactions onControl of Network Systems vol 1 no 1 pp 99ndash108 2014

[13] H Chen G Li H Zhang and Z Hou ldquoOptimal allocation ofresources for suppressing epidemic spreading on networksrdquoPhysical Review E vol 96 no 1 Article ID 012321 2017

[14] C Granell S Gomez and A Arenas ldquoDynamical interplaybetween awareness and epidemic spreading in multiplexnetworksrdquo Physical Review Letters vol 111 no 12 Article ID128701 2013

[15] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani ldquoEpidemic processes in complex networksrdquoReviews of Modern Physics vol 87 no 3 pp 925ndash979 2015

[16] W Wang Q-H Liu J Liang Y Hu and T Zhou ldquoCo-evolution spreading in complex networksrdquo Physics Reportsvol 820 no 2 pp 1ndash51 2019

[17] L Bottcher O Woolley-Meza N A Araujo H J Herrmannand D Helbing ldquoDisease-induced resource constraints cantrigger explosive epidemicsrdquo Scientific Reports vol 5 ArticleID 16571 2015

[18] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[19] X Chen WWang S Cai H E Stanley and L A BraunsteinldquoOptimal resource diffusion for suppressing disease spreadingin multiplex networksrdquo Journal of Statistical MechanicsGeory and Experiment vol 2018 no 5 Article ID 0535012018

[20] P Hu L Ding and X An ldquoEpidemic spreading withawareness diffusion on activity-driven networksrdquo PhysicalReview E vol 98 no 6 Article ID 062322 2018

[21] P Zhu XWang S Li Y Guo and ZWang ldquoInvestigation ofepidemic spreading process on multiplex networks by in-corporating fatal propertiesrdquo Applied Mathematics andComputation vol 359 pp 512ndash524 2019

[22] Z Wang Q Guo S Sun and C Xia ldquo+e impact of awarenessdiffusion on sir-like epidemics in multiplex networksrdquo AppliedMathematics and Computation vol 349 pp 134ndash147 2019

[23] S Funk E Gilad and V A A Jansen ldquoEndemic diseaseawareness and local behavioural responserdquo Journal of Ge-oretical Biology vol 264 no 2 pp 501ndash509 2010

[24] Q Wu X Fu M Small and X-J Xu ldquo+e impact ofawareness on epidemic spreading in networksrdquo Chaos AnInterdisciplinary Journal of Nonlinear Science vol 22 no 1Article ID 013101 2012

[25] H Yang C Gu M Tang S-M Cai and Y-C Lai ldquoSup-pression of epidemic spreading in time-varying multiplexnetworksrdquo Applied Mathematical Modelling vol 75pp 806ndash818 2019

[26] WWang Q-H Liu S-M Cai M Tang L A Braunstein andH E Stanley ldquoSuppressing disease spreading by using in-formation diffusion on multiplex networksrdquo Scientific Re-ports vol 6 no 1 Article ID 29259 2016

[27] H-F Zhang Z Yang Z-X Wu B-H Wang and T ZhouldquoBraessrsquos paradox in epidemic game better condition resultsin less payoffrdquo Scientific Reports vol 3 no 1 pp 1ndash8 2013

[28] J A Kulik and H I Mahler ldquoSocial support and recoveryfrom surgeryrdquo Health Psychology vol 8 no 2 pp 221ndash2381989

[29] B Nausheen Y Gidron R Peveler and R Moss-MorrisldquoSocial support and cancer progression a systematic reviewrdquoJournal of Psychosomatic Research vol 67 no 5 pp 403ndash4152009

[30] A S Mackie L Pilote R Ionescu-Ittu E Rahme andA J Marelli ldquoHealth care resource utilization in adults withcongenital heart diseaserdquoGeAmerican Journal of Cardiologyvol 99 no 6 pp 839ndash843 2007

[31] T Jaarsma R Halfens H Huijer Abu-Saad et al ldquoEffects ofeducation and support on self-care and resource utilization inpatients with heart failurerdquo European Heart Journal vol 20no 9 pp 673ndash682 1999

[32] M Gul and A F Guneri ldquoA computer simulation model toreduce patient length of stay and to improve resource

Complexity 11

utilization rate in an emergency department service systemrdquoInternational Journal of Industrial Engineering vol 19 no 5pp 221ndash231 2012

[33] S Funk M Salathe and V A A Jansen ldquoModelling theinfluence of human behaviour on the spread of infectiousdiseases a reviewrdquo Journal of the Royal Society Interfacevol 7 no 50 pp 1247ndash1256 2010

[34] W Wang M Tang H-F Zhang and Y-C Lai ldquoDynamics ofsocial contagions with memory of nonredundant informa-tionrdquo Physical Review E vol 92 no 1 Article ID 012820 2015

[35] S Funk E Gilad C Watkins and V A A Jansen ldquo+espread of awareness and its impact on epidemic outbreaksrdquoProceedings of the National Academy of Sciences vol 106no 16 pp 6872ndash6877 2009

[36] B Karrer and M E Newman ldquoMessage passing approach forgeneral epidemic modelsrdquo Physical Review E vol 82 no 1Article ID 016101 2010

[37] M Shrestha S V Scarpino and C Moore ldquoMessage-passingapproach for recurrent-state epidemic models on networksrdquoPhysical Review E vol 92 no 2 Article ID 022821 2015

[38] S Gomez A Arenas J Borge-Holthoefer S Meloni andY Moreno ldquoDiscrete-time Markov chain approach to con-tact-based disease spreading in complex networksrdquo Euro-physics Letters vol 89 no 3 Article ID 38009 2010

[39] F Krzakala C Moore E Mossel et al ldquoSpectral redemptionin clustering sparse networksrdquo Proceedings of the NationalAcademy of Sciences vol 110 no 52 pp 20935ndash20940 2013

[40] W Wang M Tang H Eugene Stanley and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquo Reports on Progress in Physics vol 80no 3 Article ID 036603 2017

[41] B Schonfisch and A de Roos ldquoSynchronous and asynchro-nous updating in cellular automatardquo BioSystems vol 51 no 3pp 123ndash143 1999

[42] W Wang P Shu Y-X Zhu M Tang and Y-C ZhangldquoDynamics of social contagions with limited contact capac-ityrdquo Chaos An Interdisciplinary Journal of Nonlinear Sciencevol 25 no 10 Article ID 103102 2015

[43] M Girvan and M E J Newman ldquoCommunity structure insocial and biological networksrdquo Proceedings of the NationalAcademy of Sciences vol 99 no 12 pp 7821ndash7826 2002

[44] P Holme and B J Kim ldquoGrowing scale-free networks withtunable clusteringrdquo Physical Review E vol 65 no 2 ArticleID 026107 2002

[45] M Small Y Li T Stemler and K Judd ldquoGrowing optimalscale-free networks via likelihoodrdquo Physical Review E vol 91no 4 Article ID 042801 2015

[46] Z Liu Y-C Lai and N Ye ldquoPropagation and immunizationof infection on general networks with both homogeneous andheterogeneous componentsrdquo Physical Review E vol 67 no 3Article ID 031911 2003

[47] M Molloy and B Reed ldquoA critical point for random graphswith a given degree sequencerdquo Random Structures amp Algo-rithms vol 6 no 2-3 pp 161ndash180 1995

[48] M Catanzaro M Boguntildea and R Pastor-Satorras ldquoGener-ation of uncorrelated random scale-free networksrdquo PhysicalReview E vol 71 no 2 Article ID 027103 2005

[49] M Boguna R Pastor-Satorras and A Vespignani ldquoCut-offsand finite size effects in scale-free networksrdquo Ge EuropeanPhysical Journal B-Condensed Matter and Complex Systemsvol 38 no 2 pp 205ndash209 2004

[50] S C Ferreira C Castellano and R Pastor-Satorras ldquoEpi-demic thresholds of the susceptible-infected-susceptiblemodel on networks a comparison of numerical and

theoretical resultsrdquo Physical Review E vol 86 no 4 Article ID041125 2012

[51] X Chen C Yang L Zhong and M Tang ldquoCrossover phe-nomena of percolation transition in evolution networks withhybrid attachmentrdquo Chaos An Interdisciplinary Journal ofNonlinear Science vol 26 no 8 Article ID 083114 2016

[52] S Boccaletti V Latora Y Moreno M Chavez andD-U Hwang ldquoComplex networks structure and dynamicsrdquoPhysics Reports vol 424 no 4-5 pp 175ndash308 2006

[53] M Newman Networks An Introduction Oxford UniversityPress Oxford UK 2010

[54] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14pp 3200ndash3203 2001

[55] httpfinancesinacomcnchinagncj2020-03-26doc-iimxxsth1817404shtml

12 Complexity