Impact of individual behavior adoption heterogeneity on epidemic transmission in multiplex networks

Liang’an Huo(霍良安) and Yue Yu(于跃)

Business School,University of Shanghai for Science and Technology,Shanghai 200093,China

Keywords: multiplex network,epidemic transmission,behavior

1.Introduction

The impact of infectious diseases has posed many serious threats to the security of society throughout human history.The development of human society has often been accompanied by the threat of sudden epidemics, which hinder social development.In recent years, the highly contagious COVID-19 virus has had a deleterious effect on people’s normal productive lives.[1-3]What’s more,the COVID-19 pandemic has seen the emergence of many new transmission characteristics that have never occurred before, posing a major obstacle to epidemic prevention and control.[4-6]For this reason,studying the intrinsic transmission patterns of epidemics and exploring the transmission dynamics of the epidemic has received attention from many scholars.

Some traditional studies of epidemic transmission in monolayer networks have provided an important basis for studying the dynamic mechanisms.[7]Bailey proposed the traditional SIS model to describe the transmission process of epidemics.[8]Grabowski extended the SIR model by considering immune individuals.[9]Based on these models, many scholars have studied the mechanisms underlying the spread of epidemics.[10-14]However, traditional studies of diffusion phenomena in single-layer networks often fail to provide a deeper understanding of the coupled effects on complex worlds.For example, epidemic transmission is often accompanied by epidemic-related information diffusion, and different information often affects the strength of the immunization measures in the face of the epidemic.A single exposure network analysis often does not provide a more complete picture of these intricate relationships.[15,16]

In recent years, with the development and popularity of multilayer networks, individuals have become increasingly aware that epidemic transmission is not just simple transmission at the physical contact layer but is also accompanied by epidemic-related information, hence making the study of multilayer networks a popular topic for many scholars.[17-19]The multi-layer network modelling approach has proved to be a better representation of the intricate coupling of relationships in real situations.[20,21]Therefore,it is significant to study epidemic transmission on multi-layered networks.Epidemic transmission is often accompanied by the diffusion of epidemic-related information,and different information often affects individuals’ levels of self-protection at the physical contact layer.Therefore, research on information-epidemic coupling effect has great significance for the prevention and control of epidemics.Granellet al.[22]proposed a UAU-SIS model that couples a multilayer network with the epidemic transmission process.Later[23]they introduced mass media to further analyze the influence of local versus global information on epidemic transmission.Wang and Xia[24]investigated the coupling effect of multiple messages and epidemics on a two-layer network under the influence of mass media and discussed the dynamic effects of different competing messages.Li and Hui[25]studied epidemic transmission in multirelational networks.Penget al.[26]investigated the effect of messages on a multi-layer network for two competing views on how to influence the scale of epidemic transmission.Guoet al.[27]proposed a threshold model,arguing that individuals’behavior adoption usually depends on the state of their neighbors, which can be simply understood by what we describe as the crowd effect,or the standing ovation effect.For example,when the number of neighborhoods around an individual reaches a certain value,the individual tends to accept the message or adopt the same behavior.Xuet al.[28]used two threshold models to explore the effects of local versus global awareness on epidemic transmission separately and also considered the heterogeneity of individual responses to risk.Some scholars have also worked on human mobility and spatio-temporal interactions, which are crucial for spreading epidemics.Liet al.[29]proved that a local basic reproductive number greater than unity is neither necessary nor sufficient for epidemic outbreaks.Liet al.[30]also studied how low population density hinders epidemics.Liuet al.[31]revealed general collective patterns behind spatiotemporal interactions between residents and proved that physical contact eventually drives epidemic transmission.

In addition,individuals’behavior patterns are usually influenced by information, resulting in different propagation effects.[32-37]Liuet al.[38]argued that individuals’ behavior adoption patterns are a nonlinear Markovian process, influenced by time effects and social reinforcement of the number of times they receive information.Moreover,Gao and Tian[39]discussed the pattern of adoption of green behaviors under the combined effect of self-awareness and policies, where coercive policies only act on specific negative individuals.Zhanget al.[40]proposed a coupled model of rumor and behavior to study the effect of internode attraction on propagation dynamics.In addition, the behavioral model of vaccination has received much attention,[41-43]especially during the COVID-19 outbreak.Wanget al.[44]investigated the impact of multiple vaccination strategies and further discussed the effect of vaccine failure on epidemic transmission.Daiet al.[45]discussed the effect of vaccine availability on epidemic transmission in the presence of information.Yinet al.[46]proposed a threelevel coupled network model to analyze the effect of negative information and vaccination behavior on epidemic transmission dynamics, and considered the combined effect of vaccination cost and flock psychology on the choice of vaccination behavior.Furthermore, higher-order networks provide methods for studying propagation dynamics from the perspective of network heterogeneity.Liet al.[47]proposed the coevolution of epidemic-infodemic dynamics on the simplicial complex.Liet al.[48]also proposed a competing spread model for two simplicial irreversible epidemics on higher-order networks, and later discussed the immunization strategies for a simplicial irreversible epidemic on the simplicial complex.[49]Other scholars have focused on the effect of individual heterogeneity on dynamics.Nieet al.[50]proposed a dynamics model with a homophily effect and heterogeneous populations.

In most cases,these scholars have tended to consider the direct effect of information on the epidemic.In fact,individuals are heterogeneous and usually make different choices when faced with different situations.However, few scholars have discussed the impact of information and an epidemic on behavior adoption patterns under the influence of individual heterogeneity and have assumed that individuals’ responses are homogeneous.However, individual behavior adoption patterns are influenced by a combination of information diffusion and epidemic transmission.When individuals select behavior adoption they are often influenced by both epidemicrelated information and the epidemic transmission environment.In the dynamic process of epidemic transmission, official information posted on social media often guides individuals to adopt the correct immunization behavior to reduce the probability of infection.However, individual behavior adoption is a complex process that is influenced not only by the credibility of social media but also by the risk sensitivity of the population.For example, when individuals learn about immunization-related official information they often make judgments about the information rather than immediately adopt immunization behaviors;this is often influenced by the credibility of social media.If the credibility of social media is low, even if individuals know the official information they will not adopt the immunization behavior.On the other hand,individuals’immunization behavior adoption processes are also influenced by the risk sensitivity of the population.Individuals who are more sensitive to epidemic risk are more likely to adopt immunization behaviors to protect themselves.For example, during the COVID-19 epidemic some social media sources had lower credibility and individuals preferred not to adopt immunization behaviors, which tended to increase the individual epidemic infection rate.Not only that,when an epidemic is not severe individuals may be less willing to get vaccinated, but when an epidemic is severe most individuals tend to choose to be vaccinated.Thus, the immunization behavior adoption process is not determined by a single adoption process, but by multiple aspects.Based on this phenomenon, we proposed a three-layer coupled model of information-behavior-epidemic to discuss the interaction effect of information diffusion, individual behavior adoption and epidemic transmission.In particular, we focus on the effect of credibility of social media and risk sensitivity of the population on heterogeneity of behavioral adoption.Our results can provide a reference for exploring the intrinsic connections between official information,immunization behavior and epidemics in multiple ways.

The framework of this paper is as follows: we describe our three-layer coupled model with related assumptions in Section 2.In Section 3,we apply a microscopic Markov chain approach and derive the threshold of epidemic transmission.In Section 4, extensive simulations are performed to demonstrate our theoretical analysis.Finally, in Section 5, our conclusions and outlook are summarized.

2.The three-layer coupled model

During the COVID-19 outbreak, social media played an important role in epidemic control by releasing official information in a timely manner.However, even though some individuals were aware of the official information, they still refused to adopt immunization behaviors.For example, social media emphasized the importance of vaccines and encouraged people to get vaccinated in a timely manner,but some individuals do not think that social media can be trusted and thus were reluctant to adopt immunization behaviors.[51-54]If individuals are more sensitive to epidemic risk, they are more likely to adopt immunization behaviors.In fact,individuals’behavior adoption is not only influenced by the credibility of social media but also by individual risk sensitivity to the severity of the epidemic.To better describe this phenomenon, shown in Fig.1,in this section,we propose a three-layer coupled model including official information diffusion,immunization behavior adoption and epidemic transmission, in which an individual’s immunization behavior adoption process is determined by both the credibility of social media and risk sensitivity of the population to the epidemic.

Fig.1.Structure of the three-layer UAU-DKD-SIS propagation network framework.In the information transmission layer, the purple circles represent individuals U who do not know the official information and the yellow circles represent individuals A who know the official information.In the immunization behavior adoption layer,the blue circles represent individuals D who do not adopt the immunization behavior and the green circles represent individuals K who adopt the immunization behavior.In the epidemic transmission layer,the pink circles represent susceptible individuals S and the orange circles represent infected individuals I.

The model assumptions in this paper are given below.

Assumption 1: Official information diffusion process In the official information diffusion layer, we consider official information diffusion represented by social media.In this layer, the nodes include two states, U (unaware) and A(aware).When an individual in state U who does not know the official information comes into contact with a neighbor in state A who is aware of the official information,the individual in state U will be informed about the official information with probabilityλ,transform to state A,and start to disseminate the official information.While an individual in state A will forget the official information or lose interest in the official information with probabilityδ.

Assumption 2: Immunization behavior adoption process In the immunization behavior adoption layer, we mainly consider the adoption of immunization behavior.In this layer,nodes include two states, K (not adopted) and D (adopted).We refer to the wearing of masks,getting vaccinated and taking proper protective measures collectively as immunization behaviors.Considering the impact of the credibility of social media and risk sensitivity of the population to the epidemic on immunization behavior adoption,we define the probability that individualiin the state K decides to adopt immunization behavior as

whereαi(t)is the rate of behavior adoption considering social media at momenttandθi(t) is the rate of behavior adoption considering the sensitivity of the epidemic at momentt.We assume joint influence of the credibility of social media and risk sensitivity of the population to the epidemic on the immunization behavior adoption process.Thus,we useζto represent the number of individuals influenced by the credibility of mass media in the immunization behavior adoption process andζ ∈[0,1]and 1-ζto represent the number of individuals influenced by risk sensitivity of the population to the epidemic in the immunization behavior adoption process.A largerζindicates a greater tendency to be influenced by the credibility of social media in changing behavior,while a smallerζindicates a greater tendency to be influenced by risk sensitivity of the population to the epidemic in changing behavior.We introduce the Heaviside step function to characterize the effect ofαi(t)andθi(t)on the immunization behavior adoption process,and the specific explanations forαi(t)andθi(t)are given in Assumption 4.As illustrated above, individuals in state D who have not adopted immunization behavior will adopt immunization behavior with probabilityxi(t).It is worth noting that in the immunization behavior adoption process, we only consider an individual’s decision to adopt,and do not consider behavior transmission between neighbors.In addition, an individual who adopts immunization behaviors will face immunization failure with probabilityη.

Assumption 3: Epidemic transmission process In the epidemic transmission layer,nodes include two states,S(susceptible)and I(infected).The epidemic transmission process follows the classical SIS model where a node in state S comes into contact with a node in state I and will be infected with probabilityβ,while a node in state I will be cured and return to state S with probabilityµ.

Based on Assumptions 1-3, we can conclude that there are eight node states coupled in the three-layer network,namely UDS, UDI, UKS, UKI, ADS, ADI, AKS and AKI.Here, we assume that once a node receives an infection, it will be informed of the official information as a result of the treatment and communications of the hospital and then adopt immunization behaviors.The individuals with three states,UDI,UKI,AKI,will be transformed into ADI individuals with probability 1.As a result,we can obtain five valid states:UDS,UKS,ADS,AKS and ADI.

Assumption 4: Influence of official information diffusion and epidemic transmission on immunization adoption behavior considering the credibility of social media and risk sensitivity of the population to epidemic In this paper,we focus on the heterogeneity of individuals when engaging in adoption behavior based on the coupled influence of information and the epidemic.The credibility of social media and risk sensitivity of the population to the epidemic are mainly considered.Here,we consider a Heaviside step function to describe the immunization behavior adoption process,i.e.,F(x)=0,ifx ≥0,andF(x)=1,ifx ＜0.

We hypothesize that individuals’ behavior adoption is firstly related to the credibility of social media.In real life,many individuals often choose to disbelieve official information because of the lower credibility of social media.For example, in the early stages of the COVID-19 epidemic, many social media sources promoted the importance of immunization,but there were still individuals who refused to adopt immunization behaviors due to the lower credibility of social media.Also,when the credibility of social media is low,individuals tend to worry about unknown negative impacts of immunization behaviors on health, which also affects individuals’adoption of immunization behaviors.In addition,due to herd mentality, individuals are often reluctant to adopt immunization behaviors when they are surrounded by fewer neighbors who choose to adopt.Therefore,considering the credibility of social media,we calculate the behavior adoption rateαi(t)as

In addition, we hypothesize that an individual’s behavior adoption is also related to risk sensitivity of the population to the epidemic.When individuals have a high sensitivity to the severity of the epidemic they tend to stay alert and will adopt immunization behaviors.In contrast,individuals with a low sensitivity to the epidemic’s severity are more likely not to adopt immunization behaviors by ignoring the prevalence of the epidemic.Therefore,considering the risk sensitivity of the population to the epidemic, we surmise that the behavior adoption rateθi(t)is

Assumption 5: Coupling effect of the immunization behavior adoption layer on the epidemic transmission layer We hypothesized that an individual’s infection rate in the epidemic transmission layer is influenced by the joint influence of the official information diffusion layer and the immunization behavior adoption layer.

Individual behavioral patterns often play a major role in infection rates in epidemics.Individuals who do not adopt immunization behaviors tend to have a higher infection rate.In addition,when individuals know the official information,they tend to take certain precautions to prevent themselves from getting infected more often than people who do not know the official information.Therefore,we assumed that the infection rates of UKS and AKS individuals isβUKSandβAKS,respectively,are

whereγUKSandγAKSrepresent the influence of the UKS and AKS individuals on the infection rate andβis the underlying infection rate independent of other factors.

In addition,we assume that UDS individuals have a lower infection rate than UKS and AKS individuals because they adopt immunization behaviors.Therefore,we assume that the infection rate of UDS individuals isβUDSAKS individuals isβUKSandβAKS,respectively,are

whereγUDScharacterizes the influence of the UDS individuals on the infection rate.

Finally,we hypothesized that ADS individuals tend to reinforce some self-protection due to their knowledge of official information, and their infection rate at the epidemic level is relatively lower than that of UDS individuals.Therefore, we hypothesized that the infection rate of ADS individuals would be

whereγADSrepresents the influence of the ADS individuals on the infection rate.

We assume that the infection rate of an individual is mainly influenced by behavior adoption and,to a lesser extent,by the receipt of information,thusγUDS≤γAKS,and that both the adoption of immunization behavior and knowledge of official information reduces the risk of infection at the epidemic transmission layer,thusγAKS≤γUKS.Specifically,an individual has the smallest infection rate if s/he adopts immunization behavior and is aware of official information, while s/he has the highest infection rate if s/he neither adopts immunization behavior nor is aware of official information.Moreover,since behavior has a greater effect on the infection rate than information,we consider that individuals who know the official information but do not adopt immunization behavior have a higher infection rate than those who do not know the information but adopt immunization behavior.In summary,the influencing parameters satisfyγADS≤γUDS≤γAKS≤γUKS.In addition, in this paper,we argue that an individual’s state in the immunization behavior adoption layer has a direct effect on the epidemic transmission process, while an individual’s state in the information diffusion layer influences their immunization behavior adoption process and then affects the epidemic transmission process indirectly.Therefore, to simplify the model, we letγAKS=γUKS=1,γUDS=φandγADS=φ2,and then we haveβA=φβU=φ2βK=φ2β.The simplified parameters are as follows:

whereX ∈{UKS,AKS,UDS,ADS}.φis used to characterize the attenuation factor of the infection rate in individuals with different status,φ ∈[0,1].

Based on the above discussion,we can obtain the propagation model of the three-layer network as follows:

(i)Official information diffusion

(ii)Immunization behavior adoption

(iii)Epidemic transmission

3.The theoretical analysis

3.1.Probability transition tree and equations

According to the scheme in Fig.2, the microscopic Markov chain approach[55,56]for each state can be proposed as follows:

Fig.2.The probability transmission trees for five possible states(including UDS,UKS,ADS,AKS,and ADI).

3.2.Critical epidemic threshold of the system

Using the transformation equations derived from the probability transmission tree above, we can summarize the epidemic thresholdβcof the model.When the system tends to a steady state, we can obtain thatPi(t+1) =Pi(t) =Piholds for any nodeiand all possible states.The number of infected individuals in the system approaches 0 near the epidemic threshold, that is,Accordingly,equations(9),(10)and(11)can be approximated as

where

and equation(16)in the steady state becomes

This is equivalent to

Since,similarly,removingO(εi)terms in the stationary state of Eqs.(12)-(15)we get

which can be written as

so

where the maximum eigenvalue of matrixHis a nonmicroscopic solution of Eq.(28)and the threshold of the epidemic is determined by the maximum eigenvalue of matrixH,

whereΛmax(H)is the maximum eigenvalue of matrixH,and matrixHdepends on the solutions of Eqs.(23)and(25),which are also solved by iteration.

4.The numerical simulations

In this section, we perform a number of simulations to validate the applicability of our proposed model.We mainly describe the role of different parameters by discussing the epidemic outbreak thresholdβcand the final number of infected individualsρI.Here, we assume a three-layer scale-free network including 3000 nodes in each layer,where the number of edges to attach from a new node to existing nodes is 6.Each point in the figures of size 40×40 is obtained by 50 simulations.For simplicity, we assumeθ0=α0.In addition, the initial numbers of nodes A,K,and I were each assumed to be 0.1.In order to make the simulation data more reliable, the parameter settings in this paper are based on the references shown in Table 1.

Table 1.Definition of parameters and related references.

We first discuss the effect of the credibility of social mediamon the final number of infected individualsρIfor different information diffusion ratesλin Fig.3.As we hypothesized in Eq.(2),a largermindicates a lower credibility of social media.Therefore,we can see from the figure that the final number of infected individuals increases asmincreases,while the epidemic thresholdβcexhibits a two-stage change asmincreases.Whenmis small, changingmhas little effect on the epidemic threshold,and whenmis large the epidemic threshold gradually decreases asmincreases.It is worth noting that at this point only two variables,mandβ,are available and all other variables are fixed.In addition, the epidemic thresholdβcincreases and the final number of infected individualsρIdecreases as the information diffusion rateλincreases.In real life,the phenomenon of social governance chaos caused by the public’s loss of trust in social media occurs from time to time.Therefore,in the prevention and control of an epidemic,social media should build a credible social image and at the same time increase efforts to publish and diffuse the official information, which increases the epidemic transmission threshold and effectively reduces the chances of epidemic outbreaks.

Fig.3.The influence of β and m on ρI. λ is set as follows: (a)λ =0.2,(b)λ =0.5,(c)λ =0.8.Other parameters are set to be α0=0.4,δ =0.6,η =0.4,µ =0.8,φ =0.5,π =0.5,ζ =0.5.

Fig.4.The ρI as a function of β for different values of m.Other parameters are set as follows: λ =0.5, α0 =0.4, δ =0.6, η =0.5, µ =0.8,φ =0.5,π =0.5,ζ =0.5.

In Fig.4,we further discuss the effect of the credibility of social media on the epidemic thresholdβc.The trend between the different lines similarly illustrates that that for smallerm(high credibility of social media), the epidemic threshold increases significantly and the final number of infected individuals decreases, which is consistent with our conclusion in Fig.3.It is worth noting that increasing trust in social media can be effective in curbing epidemic transmission.Forβ ＞βc,when epidemic transmission is out of control, asβincreases the credibility of the media has little impact on the final number of infected individuals.At this moment,the government’s control policy should focus more on control of physical contact rather than the diffusion of information via social media.Therefore,in the embryonic stage of disease transmission,increasing the credibility of social media is to a certain extent important for controlling epidemic transmission.

In Fig.5, we analyze the effect of risk sensitivity of the population to the epidemic on the final number of infected individualsρIfor different information diffusion ratesλ.As we can see in Fig.5(a), as the risk sensitivity of the populationπincreases,the epidemic thresholdβcdecreases and the final number of infected individualsρIincreases.It is worth noting that at this point only two variables,πandβ,are available and all other variables are fixed.Similarly, the same can be observed in Figs.5(b)and 5(c).The reason for this phenomenon is that asπincreases, individuals become more sensitive to risk and therefore are more likely to be aware of the threat of the epidemic and thus adopt immunization behaviors.Moreover,by comparing the different subplots,we find that the final number of infected individuals keeps increasing as the information diffusion rate decreases,especially for smallerπ.This shows that increasing the awareness of the population about the importance of taking precautions and always maintaining good immunization behaviors can facilitate the epidemic control process.

To more clearly illustrate the impact of risk sensitivity of the population to the epidemic on the epidemic threshold,we show the changes ofρIunder differentβandπin Fig.6.We can observe a more pronounced two-stage effect: increasingπhas a strong effect on raising the epidemic threshold,while forβ ＞βcthe impact of increasing risk sensitivity of the population to the epidemic on the epidemic threshold gradually diminishes and increasingπstill reduces the final number of infected individuals.For the different colored lines in Fig.6,the variables are also onlyπandβ.This likewise suggests that, within the controllable range of epidemic transmission,increasing risk sensitivity of the population to the epidemic facilitates raising the epidemic threshold.Therefore,the government should strengthen science education and raise the risk sensitivity of the population to the epidemic, which plays an essential role in epidemic control.In addition, it is important for individuals to maintain a state of moderate tension in the face of epidemic transmission so that they can take effective measures to reduce their risk.

Fig.5.The influence of β and π on ρI. λ is set as follows: (a)λ =0.2,(b)λ =0.5,(c)λ =0.8,from left to right.Other parameters are set to be α0=0.4,δ =0.6,η =0.6,µ =0.8,φ =0.5,m=0.5,ζ =0.5.

Fig.6.The ρI as a function of β for different values of π.Other parameters are set as follows: λ =0.5, α0 =0.4, δ =0.6, η =0.6, µ =0.8,φ =0.5,m=0.5,ζ =0.5.

Subsequently,we focus on exploring the effect of different epidemic attenuation factorsφon the final number of infected individuals.By comparing Figs.7(a)-7(c) we can see that the reduction ofφhas a beneficial effect on epidemic control.Asφdecreases,the adoption of immunization measures intensifies, which in turn reduces the infection rate.Therefore, during epidemic transmission people should strengthen their self-protection, raise their awareness of self-prevention and take more effective measures to prevent disease,measures which have a high inhibitory effect on the spread of infectious diseases.

Next,we demonstrate the impact of different epidemic attenuation factorsφon the epidemic threshold in Fig.8.Unsurprisingly,increase inφwill lead to expansion of the epidemic,the same conclusion as in Fig.7.What’s more,the influence ofφon the final number of infected individuals is gradually decreasing.Therefore,the government can control the epidemic by encouraging people to adopt good protective measures.

Fig.7.The influence of β and π on ρI. φ is set as follows: (a)φ =0.2,(b)φ=0.5,(c)φ=0.8.Other parameters are set to be λ =0.5,α0=0.4,δ =0.6,η =0.5,µ =0.8,m=0.5,ζ =0.5.

Fig.8.The ρI as a function of β for different values of φ.Other parameters are set as follows: λ =0.5, α0 =0.4, δ =0.6, η =0.4, µ =0.8,m=0.5,π =0.5,ζ =0.5.

In Fig.9, we investigate the effect of behavior adoption preference on the final number of infected individuals,where a largerζrepresents a greater tendency for individuals to be influenced by the information layer and choose to adopt immunization behavior.As can be seen from Fig.9, the epidemic threshold increases and the final number of infected individuals decreases asζincreases.The reason for this phenomenon is that for largerζindividuals adopt immunization behavior when there are sufficient neighborhoods that know the official information, while for smallerζindividuals only adopt immunization behavior when the epidemic outbreak reaches a certain level.Thus, whenζis small individuals pay more attention to effects originating from the information diffusion layer,and whenζis large individuals pay more attention to effects originating from the epidemic transmission layer.In this regard,relevant managers should aim to diffuse official information in a timely manner at the early stage of the epidemic to raise individuals’ awareness, and also make a great effort after the outbreak to increase people’s perception of danger and encourage them to adopt immunization behaviors, which is a good reference for epidemic control.Furthermore, by comparing Figs.9(a)-9(c),we can see that epidemic transmission is well suppressed with increasingα0.Therefore,when faced with immunization behavior,individuals should also maintain good self-judgment, improve their own hygiene and actively choose to adopt the right behavior,which is also conducive to the control of the epidemic.

Fig.9.The influence of β and ζ on ρI. α0 is set as follows: (a)α0=0.2,(b) α0 = 0.5, (c) α0 = 0.8.Other parameters are set to be λ = 0.2,δ =0.6,η =0.5,µ =0.8,φ =0.5,m=0.5,π =0.5.

Fig.10.The influence of π and m on ρI.Other parameters are set to be λ =0.5,α0=0.4,δ =0.6,η =0.5,µ =0.8,φ =0.5,β =0.5,ζ =0.5.

Finally, to investigate the joint effect of the credibility of social mediamand risk sensitivity of the population to epidemicπon the final number of infected individualsρI,we perform the simulations shown in Fig.10.We explore the effect of individual heterogeneity in the information diffusion layer and individual heterogeneity in the epidemic transmission layer on adoption of immunization behavior.When the effect of the credibility of social media is not considered(m=0), the two-stage effect ofπoccurs at about 0.225.Asmincreases (credibility of social media decreases), the epidemic threshold gradually decreases.In addition, when the risk sensitivity of the population to the epidemic is not considered(π=1)the two-stage effect ofmoccurs at around 0.425.Asπdecreases(an individual’s risk sensitivity decreases),the epidemic threshold gradually decreases.What’s more,the results show that increasingπmakes a more significant contribution to epidemic control than decreasingm.In reality, facing limited resources for fighting epidemics,should put more effort into epidemic control rather than a single propaganda source for official information.

5.Conclusion

In this paper we propose a three-layer coupled model that considers the influence of official information diffusion and adoption of immunization behavior on epidemic transmission.Behavior adoption by individuals is influenced by both the official information diffusion layer and the epidemic transmission layer, and individual heterogeneity in behavior adoption is related to the credibility of social media and the risk sensitivity of the population to epidemic.We investigated the influence of the factors described by the model, focussing on epidemic transmission dynamics and the epidemic threshold,by using the microscopic Markov chain approach.At the same time, we conducted a large number of simulations to demonstrate the feasibility and realism of our model.Through this research we found that increasing the credibility of social media can raise the epidemic transmission threshold, which has an important impact in increasing the epidemic transmission threshold and effectively reducing the chances of epidemic outbreaks.In addition, increasing the risk sensitivity of the population to an epidemic is equally beneficial in epidemic prevention and can reduce the number of infected individuals and delay epidemic outbreak.At this stage of COVID-19 development,due to low risk sensitivity of the population to the epidemic and lack of knowledge about COVID-19,many people may simply regard some symptoms as a common cold and not choose to go for further treatment, which in turn hinders epidemic control.Therefore,increasing the ability of individuals to perceive risks and encouraging them to adopt immunization behaviors can reduce the size of an epidemic.The optimal control of an epidemic can be achieved by understanding the coupling effects of multiple factors.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant Nos.72174121 and 71774111),the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning,and the Natural Science Foundation of Shanghai (Grant No.21ZR1444100).