Epidemic dynamics on information-driven adaptive networks

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Epidemic dynamics on information-driven adaptive networks

Zhan, Xiuxiu; Liu , Chuang ; Sun, Gui-Quan; Zhang , Zi-Ke

DOI

10.1016/j.chaos.2018.02.010

Publication date

2018

Document Version

Final published version

Published in

Chaos, Solitons and Fractals

Citation (APA)

Zhan, X., Liu , C., Sun, G-Q., & Zhang , Z-K. (2018). Epidemic dynamics on information-driven adaptive

networks. Chaos, Solitons and Fractals, 108, 196-204. https://doi.org/10.1016/j.chaos.2018.02.010

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ContentslistsavailableatScienceDirect

Chaos,

Solitons

and

Fractals

Nonlinear

Science,

and

Nonequilibrium

and

Complex

Phenomena

journalhomepage:www.elsevier.com/locate/chaos

Epidemic

dynamics

on

information-driven

adaptive

networks

Xiu-Xiu

Zhan

a,b

_,

_Chuang

_Liu

a,∗

_,

_Gui-Quan

_Sun

c

_,

_Zi-Ke

_Zhang

a,d,e,∗

a Alibaba Research Center for Complexity Sciences, Hangzhou Normal University, Hangzhou 311121, PR China

b Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft 2628 CD, The Netherlands c Complex Sciences Center, Shanxi University, Taiyuan 030 0 06, PR China

d Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, PR China e Alibaba Research Institute, Hangzhou 311121, PR China

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 8 November 2017 Revised 19 January 2018 Accepted 8 February 2018 Available online 16 February 2018

Keywords: Epidemic spreading Information diffusion Adaptive model Bifurcation analysis

a

b

s

t

r

a

c

t

Researchontheinterplaybetweenthedynamicsonthenetworkandthedynamicsofthenetworkhas at-tractedmuchattentioninrecentyears.Inthiswork,weproposeaninformation-drivenadaptivemodel, where diseaseanddisease informationcanevolvesimultaneously.Forthe information-drivenadaptive process,susceptible (infected)individualswhohaveabilitiestorecognizethe diseasewouldbreakthe linksoftheirinfected(susceptible)neighborstoprevent theepidemicfromfurtherspreading. Simula-tionresultsandnumericalanalysesbasedonthepairwiseapproachindicatethattheinformation-driven adaptiveprocess cannot onlyslowdownthe speedofepidemicspreading,butcanalsodiminishthe epidemicprevalenceatthefinalstate significantly.Inaddition, thediseasespreadingand information diffusionpatternonthelatticeaswellasonareal-worldnetworkgivevisualrepresentationsabouthow thediseaseistrappedintoanisolatedfieldwiththeinformation-drivenadaptiveprocess.Furthermore, weperformthelocalbifurcationanalysisonfourtypesofdynamicalregions,includinghealthy,a con-tinuousdynamicbehavior,bistableandendemic,tounderstandtheevolutionoftheobserveddynamical behaviors.Thisworkmayshedsomelightsonunderstandinghowinformationaffectshumanactivities onrespondingtoepidemicspreading.

1. Introduction

The spreading dynamic is one of the core issues in network science[1–3],wheremostoftherelatedresearchesfocuson epi-demicspreading andinformationdiffusion inrecent years.Much of the work to date focuses on the analysis of these two pro-cesses independently, such asthe spread of singlecontagion [4– 6]orconcurrent diseases[7,8],andthediffusionofvarious kinds ofinformation(e.g.,news [9],rumor[10],innovation [11].). How-ever,the epidemic spreading process is closelycoupled withthe correspondingdiseaseinformationdiffusion(orsayingindividuals’ awareness ofthe disease) inthe real world. Forinstance, during thesevereacuterespiratorysyndrome(SARS)outbreakinChinain 2003,overwhelmingnumberofdiseasereportshavebeenposted. ThesekindofinformationaboutSARSmayaffectthe individuals’ behavior inkeepingaway fromSARSand thus help to make the disease under control [12,13]. Therefore, disease information dif-fusionmayplayan importantrole inthecontrolofthe epidemic

∗ _{Corresponding authors.}

E-mail addresses: liuchuang@hznu.edu.cn (C. Liu), zkz@hznu.edu.cn (Z.-K. Zhang).

outbreak,butitisnoteasytoquantitativelymeasurethestrength ofitsimpact[14].

Nowadays,some modelshavebeenproposed tomodelthe in-teraction between epidemic spreading and information diffusion oncomplexnetworks[14–17].Thefundamentalassumptionisthat, whenadiseasestartstospreadinthepopulation,peoplemayget thediseaseinformationfromtheirfriendsormediabeforethe ad-ventofthe epidemicandtake somepreventive measuresto keep awayfrombeinginfected[15,18,19].Bydepictingpreventive mea-sures asthe reduction of transmittingprobability [20,21]or par-ticularstatesofindividuals(immuneorvaccination)[22],previous models showedthat the disease informationdiffusion indeed in-hibits the epidemic spreading signiﬁcantly (reduce the epidemic prevalence as well as enhance the epidemic threshold) [15,23]. Therefore,theemergenceofmutualfeedbackbetweeninformation diffusionandepidemicspreading [14]exhibitsthe intricate inter-playbetweenthesetwotypesofspreadingdynamics.Theinterplay betweenthesetwo typesofspreading dynamicsis similarto the competingepidemics[24,25]tosomeextent,thatistosay,thereis acompetitivemechanismbetweenepidemicspreadingandthe in-formationdiffusion.Mostofaforementionedstudiesofsuch com-plex interacted spreading dynamics are based on static network,

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X.-X. Zhan et al. / Chaos, Solitons and Fractals 108 (2018) 196–204 197

i.e.,thenetworkstructurestays ﬁxedwhenthetwoprocessesare spreadingonthenetwork.However, individualswouldsometimes cutoff theconnectionswiththeinfectedoneswhentheybecome aware ofthedisease, leadingto thechange ofnetworkstructure. Consequently,howtocharacterizethemutualspreadingprocesson theadaptivenetworksisacrucialissuewewanttoaddressinthis work.

Generally, thenetwork dynamicresearches could be classiﬁed intotwolines:(i)oneisthedynamicsofthenetwork,whichfocuses onthetimeevolution ofnetworkstructure[26–28];(ii)theother isconsidered asthedynamicson thenetwork,whichconcernsthe state changeof the nodes(or interactions) on networks, such as theepidemicspreadingandinformationdiffusionprocess[29,30], theevolutionarygame[31]andsoforth.Currently,researchers be-came to study howthe epidemic wouldspread on adaptive net-works,i.e.,consideringoneepidemicspreadingprocesson dynam-icalchangingnetworks[32].In[32],theauthorproposed amodel byconsideringthatthesusceptibleindividualsareallowedto pro-tect themselves by rewiring their links fromthe infected neigh-borstosome othersusceptibleones[33–35].Manyresearches in-dicate that segregating infected (or susceptible) individuals with theadaptivebehaviorisaneﬃcientstrategytoreducethefraction ofsusceptible-infected(SI)interactions,aswellashinderthe out-breakofthewholeepidemicspreading[36–38].Inaddition, abun-dant temporalbehaviorsare presentedto illustrate thespreading dynamicsontheadaptivenetwork,suchasthecoexistenceof mul-tiple stable equilibrium and the appearance of an oscillatory re-gion, which are absent in the spreading dynamics on static net-works[32,39].Besidestheedgerewiringstrategy,thelinkcutting or temporarily deactivating is also a commonly used rule in the adaptivemodels[40,41].

In this work, we consider a more complicated case that two dynamical processes (i.e., epidemic spreading and disease infor-mation diffusion) are spreading on adaptive networks. Therefore, threedynamicalprocessesarecoupledinthiscase,weaimto illus-trate howthe adaptivebehaviorcan affectthe interplaybetween epidemicspreadingandinformationdiffusion.Theadaptive behav-ior isarousedbytheindividualsawarenessofthedisease. Inthis model,thosewhohavebeeninformedoftheemergenceofdisease canbreaktheirneighbouringconnectionstopreventfurther infec-tion.Additionally,epidemicspreadinganddiseaseinformation dif-fusionaredescribedbytheSIandSISmodel,respectively.The dis-ease informationgeneration ofthe infectedindividualsis consid-eredtoformamutualfeedbackloop betweenthesetwo typesof spreading dynamics[20].Therefore,the effectofinformation dif-fusiononepidemicspreadingcouldbeinterpretedbytwoaspects: (i)reducetheepidemicspreadingprobabilitywithprotective mea-sures;and(ii)cutoff SIlinkswiththeinformation-drivenadaptive process. Bothnumericalanalysesbased onthepairwise approach andsimulationresultsindicatethat theinformationdiffusionand the adaptivebehavior ofthenodescan inhibitthe epidemic out-breaksigniﬁcantly.In addition,wepresenta full localbifurcation diagram to show the abundantdynamical behaviors in the pro-posedmodel.

The paperisorganizedasfollows.In Section 2, we givea de-tailed description of the model as well as mathematical expres-sions based onthe mean-ﬁeldmodel andthepairwise model.In

Section3,weﬁrstanalyzethecaseofepidemicanddisease infor-mation spreading on staticnetwork, i.e., the case ofno adaptive behavioristakenintoaccount.Wefurthergivetheresultsofhow the epidemic anddisease information spreading processes inter-act witheach other on adaptivenetwork. Thesensitivityanalysis oftheparameters anddynamicalcharacterizationofthemodelis giveninthe endofSection 3.Weconcludethe paperwithsome futuredirectionsoftheworkinSection4.

Fig. 1. Transmission diagram of epidemic spreading ( SI model in the horizontal di- rection) and disease information diffusion model (SIS model in the vertical direc- tion).

2. Model

2.1. Modeldescription

Wegivea detailedillustrationofourmodelinFig.1.The ver-ticaltransformationdescribesthediffusionofdiseaseinformation byanSISmodel,whereindividualscanbeatoneofthetwostates: (i)₊ :indicates thattheindividualshaveknowntheexistence of the disease,denoted asthe informed ones; (ii) −:indicates that the individuals have not known the existence of the disease. At eachtime step,theinformednodeswilltransmit theinformation totheirunknown(−)neighbourswithprobability

α

,andeach in-formedindividual mayforgetthe informationofthediseasewith aprobability

λ

.Besides,theonewhohasbeeninfectedbythe dis-ease willbecome to knowthe informationof thedisease with a correspondingrate

ω

[14,16].

InthehorizontaltransformationofFig.1,theepidemic spread-ing is described by an SI model. Each node is at one of two states,susceptible(S)orinfected(I).Thediseasecanbe transmit-ted through the SI links, where the S-state individuals could be infectedwiththeprobabilities

β

,

σ

I

β

,

σ

S

β

and

σ

SI

β

respectively throughS₋I₋,S₋I₊,S₊I₋ andS₊I₊links,where

σ

I,

σ

Sand

σ

SI are theimpactfactorsoftheinformationonepidemicspreading. Gen-erally,whenpeopleknowtheoccurrenceofthedisease(informed individuals), they would like to take some measures to protect themselves,leadingtothereduction ininfectivity(0<

σ

S,

σ

I<1). In particular, the inﬂuence coeﬃcient of the epidemic spreading probability through S₊I₊ links could be calculated as

σ

SI=

σ

S

σ

I, with the assumption of the independent effect of the infection probability.

Additionally,we consider an information-driven adaptive pro-cess which the informed individuals wouldreduce physical con-tactstoprotectthemselvesortheir friends.Thatistosay,the in-formedsusceptibleindividuals(S₊) willkeep awayfromtheir in-fected neighbors to protect themselves from being infected, and informedinfectedindividuals (I+) willalso avoidcontactingtheir susceptibleneighborstopreventtheepidemicfromfurther spread-ing. Consequently,the edge-breaking rule ofadaptive behavior is adopted [40]. Thus, at each time step, the S₊ (I₊) state individ-uals will break the linksconnected to their I (S)-state neighbors withraterS(rI)respectively.Specially,thebreakingrateoftheS+I+

pairscouldbeinterpretedas1−

(

1− rS

)(

1− rI

)

withthe indepen-dentassumption.ItisworthnotingthatthedeactivationofSIlinks onlyrepresentstheavoidanceofphysicalcontactsbetweentheS -andI-state individuals. That is to say, the edge-breaking process will not affect the diffusion of disease information for it can be transmittedthroughothertypesofconnectionssuchasphone, in-ternetandsoforth.Thedynamicoftheepidemicspreading

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degen-Fig. 2. The epidemic spreading dynamics of various information diffusion probabil- ities αwithout considering the effect of adaptive process. The horizontal axis ( T ) is the time step for the Monte Carlo simulation, the vertical axis ( I ) is the density of infected (the vertical axis of the inset ﬁgure is the density of informed population in the network). The parameters are set as β= 0 . 3 , σS = 0 . 5 , σI = 0 . 7 , λ= 0 . 2 , ω = 0 . 75 , r S = r I = 0 . The inset shows the information diffusion dynamics ( Info ) of vari- ous βfor α= 0 . 6 .

eratestoaclassicalSImodelwhenwesetrS=rI=0,i.e.,thereis noedge-breakinginthiscase.

Accordingtothemodeldescribedabove,thespreadingprocess canbe summarized asfollows.Atthe beginning, an individual is randomlyselectedastheI₊node,whichisconsideredastheseed ofboth theepidemicspreadingandinformationdiffusion,andall otherindividualsaresetasS₋ ones. Ateachtime step,(i) the in-fectedindividualswouldtransmitthe diseaseto their susceptible neighborswiththe corresponding probabilities; (ii) theinformed individuals would transmit the disease information to their un-informed neighbors;(iii) the informed individuals can forget the information; (iv) the informed individuals would also break the links with their relevant neighbors by considering the adaptive mechanism. Finally, the spreading process would be terminated whenthesizeoftheinfectedindividualsbecomesstable.

2.2.Numericalmathematicalanalysis

Firstly, we develop theoretical analysis to depict the dynamic processes of both information diffusion and epidemic spreading. In particular, mean-ﬁeld analysis and the pairwise analysis are adopted.Let

χ

bethestatevariable,thus[

χ

]denotestheexpected valuesofindividualsofdifferenttypesonthepopulation(e.g.[S₊] and[S+I+] represent the expectednumber of informed suscepti-blenodesand expectednumberof linksconnectingan informed susceptiblenodetoaninformedinfectednoderespectively).

Therefore, withthe classical mean-ﬁeld approach, we can ob-tain:

d[I+]

dt =

k

[S+]

(

σS

β

[I−]+

σS

σI

β

[I+]

)

+

α

[I−]

(

[S+]+[I+]

)

+

ω

[I−]−

λ

[I+] (1)

comparatively,withthepairwiseapproach,wecanobtain:

d[I₊]

dt =

(

σS

β

[S+I−]+

σS

σI

β

[S+I+]

)

+

α

(

[S+I−]+[I−I+]

)

+

ω

[I−]−

λ

[I+] (2)

where,the ﬁrst terms of Eqs. (1) and (2) describe the infection of the S+-state individuals, the second terms describe the infor-mationacceptanceoftheI₋-state individuals,thethirdterms de-scribethe informationgeneration ofthe I₋-state individuals and

Fig. 3. Comparison of simulation results with the mean-ﬁeld model and the pair approximation model without considering the effect of adaptive process. The horizontal axis ( T ) is the time step for the Monte Carlo simulation, the vertical axis ( I ) is the density of infected. The parameters are set as β= 0 . 3 , σS = 0 . 5 , σI = 0 . 7 , λ= 0 . 2 , ω = 0 . 75 , α= 0 . 6 , r S = r I = 0 .

Fig. 4. Dynamical analysis of the spreading model with adaptive process. (a) Com- parison of the pairwise model with the simulation results, the horizontal axis ( T ) is the time step for the Monte Carlo simulation, the vertical axis ( I ) is the density of infected. (b) Degree distribution of the original network and that after the adaptive process, the horizontal axis ( k ) represents degree, the vertical axis ( p ( k )) is degree probability. The parameters are set as β= 0 . 2 , σS = 0 . 5 , σI = 0 . 7 , λ= 0 . 2 , ω = 0 . 2 , α= 0 . 5 , r S = 0 . 15 , r I = 0 . 1 .

thelast termsrepresenttheinformationloss oftheI₊-state indi-viduals.Simultaneously,thefullsetofdifferentialequationsbased onthosetwoapproachescanbeillustratedinAppendixA.Bythe way,theadaptiveprocesscould bedescribedby thelasttermsof

d[S₊I₋] dt ,

d[S₋I₊]

dt and

d[S₊I₊]

dt inthepairwise approachofEq.(4). It shouldbe notedthat the pairwiseanalysis isbasedon a well-knownclosureapproximation givenby[ABC]= [AB][BC]

[B] withthe assumptionthatthedegreeofeachindividualobeysPoisson distri-bution[42,43].Ingeneral,itmightbeveryhardtogetexact solu-tionsofsuchcomplexdifferentialequations,thuswegive numeri-calsolutionsoftheequationsinsteadofthetheoreticalanalysisin thefollowinganalysis.

3. Results

3.1. Simulationandnumericalanalysiswithoutadaptivebehaviour Inthiswork,weperformourmodelontheER networkwitha totalpopulation ofN₌10_,000andaveragedegree

k

₌6unless otherwisestated.Moreover,allthesimulationresultsaregivenby 10,000realizations.Weﬁrstconsiderasimplecaseofnoadaptive behavior whenthe epidemicanddiseaseinformation are spread-ing in the network, i.e., the caseof spreading onstatic network.

Fig.2givesthesimulationresultofthefractionofinfectednodes evolvingwithtime forvarious informationdiffusion probabilities

α

,withtheepidemicspreadingprobability

β

=0.3.FortheSI pro-cess, thewholepopulation wouldbeinfectedwhen

β

>0forthe

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Fig. 5. The fraction of infected individuals in the stationary state (colors in the phase diagram represent the density of infected individuals at the ﬁnal state, the dashed green curve shows that the prevalence value transmits from near 0 to significantly larger than 0) versus αand βfor (a) pairwise analysis and (b) simulation result. The parameters are set as σS = 0 . 5 , σI = 0 . 7 , λ= 0 . 2 , ω = 0 . 2 , r S = r I = r = 0 . 1 . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

connectedsocialnetworks,resultinginthattheﬁnalinfected den-sityequalsto1forallthevaluesof

α

inFig.2.Thatistosay,the diseaseinformationdiffusioncannotavoidtheepidemicspreading to the whole population when we perform our model on static network. However,we ﬁndthat the diseaseinformationdiffusion can slow down the epidemic spreading when we increase the value of

α

. Furthermore,the time cost forthe whole population becomesinfectedwhen

α

=1isaboutthreetimeslongerthanthat

of

α

=0.Inthissense,thediffusionofthediseaseinformationcan slow down the epidemic spreading significantly.In addition, the insetofFig.2indicates that theepidemicspreading canenhance thedisease informationdiffusion.Actually,accordingtomodel il-lustratedinFig.1,ontheone hand,we realizethat theepidemic spreadingcouldbe influencedbyinformationdiffusionwherethe epidemicspreadingprobability oftheinformedindividualswould change;andontheotherhand,theinformationdiffusioncouldbe influencedbytheepidemicspreadingwherethesocialdisease in-formationlevel(namelyInfointheinsetofFig.2)wouldbehigher ifmorepeopleareinfectedfortheinformationgeneration,denoted bytheparameter

ω

.Inthisway,amutualfeedbackbetween dis-ease spreading andinformation diffusion emerges:higher preva-lenceofthe infectedindividualsmakes moredisease information generatedinthepopulation, whichinturngivesriseto more in-formedindividuals,therebyweakeningthespreadofepidemic.

Fig.3showsa comparisonoftheevolution ofinfecteddensity fromthenumericalanalysisaccordingtoEqs.(3)and(4)andthe simulationresultsonERnetwork.Infecteddensitycurvebasedon theclassicalmean-fieldapproachismuchquickerthanthatofthe simulation result, which would be caused by the mean-field as-sumption on the SI model. In the mean-field assumption, the I -and S-state individuals are well-distributed in the system. How-ever,intheSIprocess,theI-stateindividualsareallwellclustered,

Fig. 6. Illustration to dynamic spreading process by considering the adaptive effect on the lattice. The square gridding patterns show the distribution of the infected and informed individuals in some particular time steps. The red area represents the nodes that are infected by the epidemic, while the gray area represents the informed individuals. The red curves (lower panels) describe the fraction of infected individuals over time with corresponding (the horizontal axis ( T ) is the time step for the Monte Carlo simulation, the vertical axis ( I ) is the density of infected). (a) α= 0 . 1 ; (b) α= 0 . 3 ; (c) α= 0 . 5 ; (d) α= 0 . 7 . Other parameters are set as β= 0 . 4 , σS = 0 . 4 , σI = 0 . 8 , λ= 0 . 1 , ω = 0 . 2 , r S = r I = r = 0 . 1 . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

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Fig. 7. Illustration to dynamic spreading process by considering the adaptive effect on haggle network, while the purple, green and red circles represent individuals in S −, S+ and I state respectively. The red curves (lower panels) describe the fraction of infected individuals over time with corresponding parameters. (a) α= 0 . 02 ; (b) α= 0 . 3 ; (c) α= 0 . 5 ; (d) α= 1 . 0 . Other parameters are set as β= 0 . 1 , σS = 0 . 3 , σI = 0 . 5 , λ= 0 . 08 , ω = 0 . 2 , r S = r I = r = 0 . 08 . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

resultinginthat manyI-state individualshaveno chanceto con-tacttheS-stateindividuals.Inthisway,theclassicalmean-field ap-proachcannotexactlydescribetheSImodel.However,such prob-lemisnotsosignificantinthepairwiseapproach,whichconsider thetime evolution ofthelinks aswell.Fig. 3showsthat the in-fecteddensity curve of the pairwise approach finds good agree-mentwiththesimulationresults.

3.2.Spreadingdynamicswiththeadaptiveprocess

In this part, we shall present the spreading dynamics with theinformation-drivenadaptiveprocess, theresultsare shownin

Fig.4.Differentfromthe resultsofFig.2,the saturationvalue of theinfected densityat the ﬁnal state is much smaller than 1 in

Fig.4(a). That is to say, with the adaptive process based on the informationdiffusion,manyindividualscouldavoidbeinginfected viareducingsomecontacts.Inaddition,wealsoplotthenumerical solutionbasedonthepairwiseapproachinFig.4(a).Itcanbeseen that the pairwise solution isnot well consistent withsimulation forthespreadingdynamicontheadaptivenetwork.Thedifference mightbe causedby the networkstructure variation inthe

adap-tiveprocess,wheretheassumptionofthepairwiseapproachisthe Poissondegreedistribution.Thisconjecture isprovedinFig.4(b), wherethe degreedistribution oftheoriginal network is approxi-matetothePoisson-distributionwithmeandegreearound6(pink circlemarkers), whilethedistributionof thenetworkattheﬁnal state (graydiamond markers) deviatesfromtheoriginal distribu-tion.Inaddition,Fig.4(a)showsthatthedifferencebecomeslarger withtheincrease oftime, wherethedegree distributiondeviates more away fromthe original distribution when the process goes on.

The information-driven adaptive process can not only slow downthespeed ofepidemicspreading,butalsocandiminishthe epidemic prevalence at the ﬁnal state signiﬁcantly according to

Figs. 2and4.Forsimplicity, weassume rS=rI=r inthe follow-inganalysis.Inordertoexhibittheinﬂuenceofinformation diffu-sionindetail,weshow thefullphasediagram

α

₋

β

withr₌0_.1 inFig.5,thecolorgivestheinfecteddensityinthe ﬁnalstate for eachcombinationof

α

and

β

.TheFig.5(a)and(b)arethe numeri-calsolutionofthepairwiseapproachandthesimulationresult, re-spectively.Asstatedpreviously,thenumericalsolutionisnotvery

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Fig. 8. Final fraction of infected individuals versus r . Different curves correspond to different α. Other parameters are set as β= 0 . 2 , σS = 0 . 5 , σI = 0 . 7 , λ= 0 . 2 , ω = 0 . 2 .

Table 1

Statistics of haggle network, where N, E, C represent the number of nodes, the number of links, clustering coeﬃcient of each system respectively.

Network N E C

Haggle 274 2124 0.0337

precise,butitcanmatchtheoveralltrendofsimulationresultwell. For a ﬁxed epidemicspreading probability

β

, epidemic outbreak size reduces with the increase of

α

. That is to say, the disease information diffusion can inhibit the epidemic spreading. Analo-gously,thequickerandbroaderoftheinformationdiffusion(larger

α

)is,themoreeﬃcientinhibitionontheepidemicspreadingwill be.Inaddition,thecurveofthecolormutation(thedashedgreen curve)inFig.5could beconsideredasthetransitionpoint,where theepidemiccan’tspreadoutif

α

and

β

locateattheareaonthe leftofthiscurve(thewhiterange).Thethresholdvalueofthe epi-demicspreadprobabilitybecomeslargerwiththeincreaseof

α

.

Inordertointuitivelydemonstratetheepidemicspreadingand the information diffusion process on adaptive network,we show the simulation results of those two typesof spreading processes for various

α

on two different networks, i.e., a 100× 100 lattice with degree k₌4 as well as a real-world network, e.g., Haggle network [44]. The contacts in Haggle network represent connec-tion between people measured by carrying wireless devices. The statistics of the network is givenin Table 1.The visualization of howepidemicanddiseaseinformationinteractwitheachotherfor thesetwonetworksaregiveninFig.6andFig.7,respectively. Tak-ing Latticeasanexample,we presentfourkindsofdifferent lev-elsofinformationspreadingprocesses(correspondingtodifferent

α

),andobservehowtheinformationdiffusionaffectsthe spread-ing of epidemic. In addition, as the adaptive edge-breaking pro-cessismerelyexecuted ontheepidemicspreadingprocess,while theseedges canstilltransmit information,thus thedensityof in-formed people can still maintainat a high levelin the network. Foreach

α

inFig.6,firstlywegivethefractionoftheinfected in-dividuals ateachtime step(the red curve ineachsubfigure). For some particular time steps, we show the states of each individ-ual withthe gridding patterns,where the red dots and thegray dots represent theinfected andinformed individualsrespectively (thecontactnetworksandtheun-informedsusceptibleindividuals are not shown in the figures). We can intuitively see the distri-butionoftheinfectedandinformedindividualsandconcludethat whenthediffusionofinformationisslowerthantheepidemic,we

Fig. 9. Distribution of the infected density in the ﬁnal state versus different values of r and β. Each distribution is obtained by carrying out 10,0 0 0 independent realizations for the ﬁnal fraction of infected. The parameters are set as r S = r I = r = 0 . 7 , 0 . 35 , 0 . 15 , 0 ;β= 0 . 05 , 0 . 35 , 0 . 25 , 0 . 4 for (a), (b), (c) and (d) respectively. Other parameters are σS = 0 . 5 , σI = 0 . 7 , λ= 0 . 2 , ω = 0 . 2 , α= 0 . 6 .

cannotstoptheepidemicfromspreading(Fig.6(a)and(b)), how-ever,whentheinformationisdiffusingfaster,theepidemicwillbe trappedintoanisolatedareaandcannotspreadanymore(Fig.6(c) and(d)).Furthermore,thevisualization ofthesetwoprocesseson HagglenetworkdisplayssimilarresultsastheresultsonLattice. 3.3.Sensitivityanalysisofthemodel

The sensitivity of the edge-breaking probability on epidemic spreading dynamics. The phase diagram in Fig.5 shows the im-pactofinformationdiffusionrate

α

ontheepidemicspreading dy-namics.In general, the adaptiveedge-breaking probability rS and

rIarealsoimportantparametersinaffectingtheepidemic spread-ingprocess. Fig.8illustrates the epidemicprevalenceintheﬁnal state versus theadaptiveedge-breaking rate(r) forvarious infor-mationdiffusionrate

α

.It canbefound thattheepidemic preva-lencediminisheswiththeincreaseofr,i.e.,theepidemiccouldbe controlled if people are very sensitive with the disease informa-tionand subsequentlykeep away fromthe infected. It shouldbe notedthat thereis nodisease informationdiffusion when

α

=0, butwithconsidering theinformation generation,the infected in-dividualscould stop contactingwith thesusceptibleneighbors to impedethefurtherspreadingofepidemic.Withtheincreaseof

α

, theepidemicprevalencereducessharplyversus randthe contin-uoustransitioncouldbeobserved.Bytheway,itwillchangetoa totalisolationofinfectedindividualsforr=1,whichseems tobe themosteffectivewayincontrollingthecontagion[45,46].

Dynamical characterization of the information-driven rewiring.In order todeeply characterize the complexdynamical features of the proposed process, we concentrate on the distri-bution of the infected density in the ﬁnal state (I∗) rather than thesimpleaveragevalue[32,39].Fig.9showsfourdifferenttypes ofdynamical behavior by calculating thedistribution ofthe ﬁnal fraction of infectedfor various

β

and r . Forthe distribution of

Fig. 9(a), we havecarried out 10,000 realizations ofthe infected density, and above 94% of the infected density is 0.0001, and the maximal is 0.0007, i.e., the infected density I∗→0, thus we consider this distribution indicates a healthy state (the disease can’t spreadout)under theparameters settinghere. Similarly, as tothedistributionofFig.9(d),above90%oftheinfecteddensityis higherthan0.8, indicates a caseofendemic state (epidemic out-breaks).Whereasthecaseillustrated inFig.9(c)isverydifferent, wherethe infecteddensityI∗ is around eitherzeroor a nonzero

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Fig. 10. (a) Bifurcation diagram of the density of the infected I as a function of the infection probability βfor different values of the edge-breaking rate r based on the results of simulation of the full network (diamonds). (b) Two parameter bifurcation diagram showing the dependence on the edge-breaking rate r and the infection probability βbased on the results of simulation of the full network. (c) Full phase diagram r −β for the simulation of the adaptive process. The parameters are the same as Fig. 8 .

value. This indicates that a bistable state [32] is located in this model,wherehealthy state andendemic state areboth stable in thiscase. Inaddition,a continuousdynamicbehavior canalsobe observedinparticularparametersettings(Fig.9(b)).

Accordingtothedynamicalbehaviorillustrated inFig.9under differentparametersets,bifurcationdiagram ofthedensityofthe infectedasa function ofinfected probability

β

for different val-uesoftheedge-breakingraterisgiveninFig.10(a).Without the adaptiveedge-breakingmechanism(r=0),thediseasecanspread outonlyif

β

>0fortheSI process.Whenr>0,thedynamical be-haviorsbecomemorecomplicated,wherethediscontinuousphase transitions,bistable,oscillatory areobserved.Afastedge-breaking (larger) leadstoabroadhealthy andbistablestate range(shows bythe rangeinthe arrows) inFig.10(a). InFig. 10(b),we give a fullr−

β

bifurcation diagramaccordingtoour simulationresults, andwe canclearly identifythe areasofhealthy, a continous dy-namicbehavior,bistabilityandendemicstateinthismodel.Atlast, wepresentthedependenceofthe averagevalue ofinfected den-sityover10,000independentrealizationsonrand

β

inFig.10(c), wherethechangingofthedensityisconsistentwiththearea clas-siﬁcationinFig.10(b).

4. Conclusions

In orderto understandtheinterplaybetweenthe dynamicson thenetwork(the spreadof epidemicspreading anddisease infor-mation)andthedynamicsofthenetwork(thetimevaryingof net-work links), we present two types of spreading dynamics with SIandSISprocess respectivelyon aninformation-driven adaptive network,where the individualswho haveknown the disease in-formationwould probably cut off their links withothers. Firstly, weillustratethemutualfeedbackbetweenepidemicspreadingand informationdiffusionwithout considering theedge-breaking pro-cess (rS=rI=0), where the high epidemic prevalence preserves highdiseaseinformationlevel,whichinturnslowsdownthe epi-demicspreading.Inthiscase,thenumericalanalysisbasedonthe

pairwise approach is consistent with the simulation result very well.Secondly,theresultsareverydifferentwhenthe information-driven edge-breaking process is considered (rS, rI>0). The epi-demic cannot spread out if the spreading probability is smaller than the threshold (shown in Fig. 5). In addition, the disease spreadingandinformationdiffusionpatternonthelattice aswell as on a real-world network give visual representations that the diseasemight betrapped intoan isolated ﬁeldwith information-drivenadaptiveprocess.Therefore,theinformation-drivenadaptive processcaninhibittheepidemicspreadingsigniﬁcantlythatitcan not only slow down the epidemic spreading speed, but also re-ducetheepidemicprevalence.Finally,wegivethelocalbifurcation analysisonfourtypesofdynamicalphenomena,includinghealthy, a continuous dynamicbehavior, bistable andendemic, indicating that thestate changesfromhealthy toa continuousdynamic be-havior,bistable,endemicstateas

β

increases.

Insummary,westudythedependenceoftheepidemic spread-ing on the disease information diffusion and the information-driven adaptive process, with considering the simplest spreading model(SI)andadaptiveprocess(edge-breaking).Recentresearches show the differentfeatures betweenthe epidemicand the infor-mationdiffusion[47,48],andthisdifferencewouldalsoimpactthe interplaybetweenepidemicspreadinganddiseaseinformation dif-fusion signiﬁcantly.Anotherarea for futureextension is toadopt networksprediction[49]orotheradaptationrulesratherthanthe simple edge-breakingstrategy, such asthetemporarily deactivat-ing, where the broken links would be active again after a ﬁxed time[41]or,ifthecorrespondinginfectednodebecomesrecovered

[40].

Acknowledgments

This work was partially supported by Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LR18A050001,

LY18A050004 and LQ16F030006), Natural Science Foundation of China (Grant Nos.61673151 and11671241) andtheEU-FP7 Grant 611272(projectGROWTHCOM).

AppendixA

Denote [

χ

] as the expected values of individuals of different typesdescribedinSection2.2,theepidemicspreadingisdepicted by the parameters

β

,

σ

I

β

,

σ

S

β

and

σ

SI

β

, while thediffusion of disease information iscontrolled by theparameters:

α

,

λ

,

ω

. All theseparametershavebeenexplainedinSection2.1.Accordingto themodeldescribedabove,thedifferentialequationsofthe mean-ﬁeld approach(Eq.(3)) andpairwise approach(Eq.(4)) aregiven asfollows.

⎧

⎪

⎨

⎪

⎩

d[S−] dt =−

k

β

[I−][S−]−

k

σI

β

[I+][S−] −

α

(

[S+]+[I+]

)

[S−]+

λ

[S+] d[S+] dt =−

k

σ

S

β

[I−][S+]−

k

σ

S

σ

I

β

[I+][S+] +

α

(

[S₊]+[I₊]

)

[S₋]−

λ

[S₊] d[I−] dt =

k

β

[I−][S−]+

k

σI

β

[I+][S−] −

α

(

[S+]+[I+]

)

[I−]−

ω

[I−]+

λ

[I+] d[I+] dt =

k

σ

S

β

[I−][S+]+

k

σ

S

σ

I

β

[I+][S+] +

α

(

[S₊]+[I₊]

)

[I₋]+

ω

[I₋]−

λ

[I₊] (3)

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⎧

⎪

⎨

⎪

⎩

d[S−] dt =−

β

[S−I−]−

σI

β

[S−I+]−

α

(

[S−S+]+[S−I+]

)

+

λ

[S+] d[S+] dt =−

σS

β

[S+I−]−

σS

σI

β

[S+I+]+

α

(

[S−S+]+[S−I+]

)

−

λ

[S+] d[I−] dt =

β

[S−I−]+

σI

β

[S−I+]−

α

(

[S+I−]+[I−I+]

)

−

ω

[I−]+

λ

[I+] d[I+] dt =

σ

S

β

[S+I−]+

σ

S

σ

I

β

[S+I+]+

α

(

[S+I−]+[I−I+]

)

+

ω

[I−]−

λ

[I+] d[S−I−] dt =−

β

[S−I−]+

λ

(

[S+I−]+[S−I+]

)

−

ω

[S−I−]+

β

[S−I−]([S−S−]−[S−I−]) [S−] +

σ

I

β

[S−I+]([S−S−]−[S−I−]) [S−] −

α

[S−I−]([S−I+]+[S−S+]) [S−] −

α

[S−I−]([I−I+]+[S+I−]) [I−] d[S−I+] dt =−

σI

β

[S−I+]+

ω

[S−I−]+

λ

[S+I+]−

α

[S−I+]−

λ

[S−I+]−

β

[S−I−][S−I+] [S−] −

σI

β

[S−I+]2 [S−] +

σS

β

[S+I−][S−S+] [S+] +

σS

σI

β

[S+I+][S−S+] [S+] +

α

[S−I−]([I−I+]+[S+I−]) [I−] −

α

[S−I+]([S−I+]+[S−S+]) [S−] − rI[S−I+] d[S+I−] dt =−

σS

β

[S+I−]+

λ

[S+I+]−

λ

[S+I−]−

α

[S+I−]−

ω

[S+I−]−

σS

β

[S+I−]2 [S+] −

σS

σI

β

[S+I+][S+I−] [S+] +

β

[S−I−][S−S+] [S−] +

σI

β

[S−I+][S−S+] [S−] +

α

[S−I−]([S−S+]+[S−I+]) [S−] −

α

[S+I−]([I−I+]+[S+I−]) [I−] − rS[S+I−] d[S+I+] dt =−

σ

S

σ

I

β

[S+I+]+

α

[S−I+]+

α

[S+I−]+

ω

[S+I−]− 2

λ

[S+I+]+

σ

S

β

[S+I−]([S+S+]−[S+I+]) [S+] +

σ

S

σ

I

β

[S+I+]([S+_[_SS₊+_]]−[S+I+])+

α

[S−I+]([S−_[_SI+₋_]]+[S−S+])+

α

[S+I−]([S_[+_II₋−_]]+[I−I+])− [1−

(

1− rS

)(

1− rI

)

][S+I+] d[I−I−] dt =2

β

[S−I−]+2

λ

[I−I+]− 2

ω

[I−I−]+2

β

[S−I−]2 [S−] +2

σ

I

β

[S−I_[+_S][₋S_]−I−]− 2

α

[I−I−]([S+_[_II₋−_]]+[I−I+]) d[I−I+] dt =

σ

I

β

[S−I+]+

σ

S

β

[S+I−]+

ω

(

[I−I−]− [I−I+]

)

+

λ

(

[I+I+]− [I−I+]

)

−

α

[I−I+]+

β

[S−I−][S−I+] [S−] +

σ

I

β

[S−I+]2 [S−] +

σS

β

[S+I−]2 [S+] +

σS

σI

β

[S+I+][S+I−] [S+] +

α

[I−I−]([S+I−]+[I−I+]) [I−] −

α

[I−I+]([S+I−]+[I−I+]) [I−] d[I+I+] dt =2

σS

σI

β

[S+I+]+2

α

[I−I+]+2

ω

[I−I+]− 2

λ

[I+I+]+2

σS

β

[S+I−][S+I+] [S+] +2

σS

σI

β

[S+I+]2 [S+] +2

α

[I−I+]([S+I−]+[I−I+]) [I−] d[S−S−] dt =2

λ

[S−S+]− 2

β

[S−I−][S−S−] [S−] − 2

σI

β

[S−I+][S−S−] [S−] − 2

α

[S−S−]([S−S+]+[S−I+]) [S−] d[S−S+] dt =

λ

[S+S+]−

λ

[S−S+]−

α

[S−S+]−

σS

β

[S+I−][S−S+] [S+] −

σS

σI

β

[S+I+][S−S+] [S+] −

β

[S−I−][S−S+] [S−] −

σI

β

[S−I+][S−S+] [S−] +

α

[S−S−]([S−S+]+[S−I+]) [S−] −

α

[S−S+]([S−S+]+[S−I+]) [S−] d[S+S+] dt =2

α

[S−S+]− 2

λ

[S+S+]− 2

σ

S

β

[S+I−][S+S+] [S+] − 2

σ

S

σ

I

β

[S+I+][S+S+] [S+] +2

α

[S−S+]([S−S+]+[S−I+]) [S−] (4) References

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