Suppression of epidemic spreading in complex networks by local information based behavioral responses

Epidemic spreading under game-based self-quarantine behaviors: The different effects of local and global information

During the outbreak of an epidemic, individuals may modify their behaviors in response to external (including local and global) infection-related information. However, the difference between local and global information in influencing the spread of diseases remains inadequately explored. Here, we study a simple epidemic model that incorporates the game-based self-quarantine behavior of individuals, taking into account the influence of local infection status, global disease prevalence, and node heterogeneity (non-identical degree distribution). Our findings reveal that local information can effectively contain an epidemic, even with only a small proportion of individuals opting for self-quarantine. On the other hand, global information can cause infection evolution curves shaking during the declining phase of an epidemic, owing to the synchronous release of nodes with the same degree from the quarantined state. In contrast, the releasing pattern under the local information appears to be more random. This shaking phenomenon can be observed in various types of networks associated with different characteristics. Moreover, it is found that under the proposed game-epidemic framework, a disease is more difficult to spread in heterogeneous networks than in homogeneous networks, which differs from conventional epidemic models.

Effect of decay behavior of information on disease dissemination in multiplex network

The diseases dissemination always brings serious problems in the economy and livelihood issues. It is necessary to study the law of disease dissemination from multiple dimensions. Information quality about disease prevention has a great impact on the dissemination of disease, that is because only the real information can inhibit the dissemination of disease. In fact, the dissemination of information involves the decay of the amount of real information and the information quality becomes poor gradually, which will affect the individual's attitude and behavior towards disease. In order to study the influence of the decay behavior of information on disease dissemination, in the paper, an interaction model between information and disease dissemination is established to describe the effect of the decay behavior of information on the coupled dynamics of process in multiplex network. According to the mean-field theory, the threshold condition of disease dissemination is derived. Finally, through theoretical analysis and numerical simulation, some results can be obtained. The results show that decay behavior is a factor that greatly affects the disease dissemination and can change the final size of disease dissemination. The larger the decay constant, the smaller final size of disease dissemination. In the process of information dissemination, emphasizing key information can reduce the impact of decay behavior.

Fractional dynamic analysis and optimal control problem for an SEIQR model on complex networks

A fractional order susceptible-exposed-infected-quarantined-recovered model is established on the complex networks. We calculate a specific expression for the basic reproduction number R₀, prove the existence and uniqueness with respect to the solution, and prove the Ulam-Hyers stability of the model. Using the Latin hypercube sampling-partial rank correlation coefficient method, the influence of parameters on the R₀ is analyzed. Based on the results of the analysis, the optimal control of the model is investigated as the control variables with vaccination rate and quarantine rate applying Pontryagin's minimum principle. The effects of α, degree of nodes, and network size on the model dynamics are simulated separately by the prediction correction method.

Risk perception and subsidy policy-based voluntary vaccination driven by multiple information sources

Exploring vaccination behavior is fundamental to understand the role of vaccine in suppressing the epidemic. Motivated by the efficient role of the risk perception and the subsidy policy in promoting vaccination, we propose the Risk Perception and the Risk Perception with Subsidy Policy voluntary vaccination strategies with imperfect vaccine. The risk perception is driven by multiple information sources based on global information (released by Public Health Bureau) and local information (from first-order neighbors). In time-varying networks, we use the mean-field approach and the Monte Carlo simulations to analyze the epidemic dynamics under vaccination behavior with imperfect vaccine. We find that vaccination with the incorporation of risk perception and subsidy policy can effectively control the epidemic. Moreover, information from different sources plays different roles. Global information is more helpful in promoting vaccination than local information. In addition, to further understand the influence of vaccination strategies, we calculate the social cost as the cost for the vaccine and treatment, and find that excess vaccination cost results in a higher social cost after the herd immunity. Thus, for balancing the epidemic control and social cost, providing individuals with more global information as well as local information would be helpful in vaccination. These results are expected to provide insightful guidance for designing the policy to promote vaccination.

Effects of official information and rumor on resource-epidemic coevolution dynamics

Epidemic-related information and resources have proven to have a significant impact on the spread of the epidemic during the Corona Virus Disease 2019 (COVID-19) pandemic. The various orientation role of information has different effects on the epidemic spreading process, which will affect the individual’ awareness of resources allocation and epidemic spreading scale. Based on this, a three-layer network is established to describe the dynamic coevolution process among information dissemination, resource allocation, and epidemic spreading. In order to analyze dynamic coevolution process, the microscopic Markov chain (MMC) theory is used. Then, the threshold of epidemic spreading is deduced. Our results indicated that the official information orientation intensity inhibits the epidemics spreading, while rumor orientation intensity promotes epidemic spreading. At the same time, the efficiency of resource utilization restrains the expansion of the infection scale. The two kinds of information are combined with resources respectively. Official information will enhance the inhibitory effect of resources epidemics spreading, while rumor will do the opposite.

The impact of the global and local awareness diffusion on epidemic transmission considering the heterogeneity of individual influences

In this paper, we propose a coupled awareness-epidemic spreading model considering the heterogeneity of individual influences, which aims to explore the interaction between awareness diffusion and epidemic transmission. The considered heterogeneities of individual influences are threefold: the heterogeneity of individual influences in the information layer, the heterogeneity of individual influences in the epidemic layer and the heterogeneity of individual behavioral responses to epidemics. In addition, the individuals' receptive preference for information and the impacts of individuals' perceived local awareness ratio and individuals' perceived epidemic severity on self-protective behavior are included. The epidemic threshold is theoretically established by the microscopic Markov chain approach and the mean-field approach. Results indicate that the critical local and global awareness ratios have two-stage effects on the epidemic threshold. Besides, either the heterogeneity of individual influences in the information layer or the strength of individuals' responses to epidemics can influence the epidemic threshold with a nonlinear way. However, the heterogeneity of individual influences in the epidemic layer has few effect on the epidemic threshold, but can affects the magnitude of the final infected density.

Microscopic edge-based compartmental modeling method for analyzing the susceptible-infected-recovered epidemic spreading on networks

The edge-based compartmental modeling (EBCM) approach has been used widely to characterize the nonrecurrent epidemic spreading dynamics (e.g., the susceptible-infected-recovered model) in complex networks. By using the probability theory, we derived an individual-based formulation for this approach, which we herein refer to as the microscopic EBCM method. We found that both for small and large initial infection numbers, the epidemic evolution agreed well with the ensemble averages of our stochastic simulations on different complex networks. Moreover, we showed that the dynamical message passing model, the standard EBCM system, and the pair quenched mean-field equations can be deduced by our microscopic EBCM method. In addition, the microscopic EBCM method was used to analyze the effect of epidemic awareness on networks. Importantly, the simple EBCM model for exponential awareness was developed. Our method provides a way for handling nontrivial disease transmission processes with irreversible dynamics.

Asymmetrical dynamics of epidemic propagation and awareness diffusion in multiplex networks

To better explore asymmetrical interaction between epidemic spreading and awareness diffusion in multiplex networks, we distinguish susceptibility and infectivity between aware and unaware individuals, relax the degree of immunization, and take into account three types of generation mechanisms of individual awareness. We use the probability trees to depict the transitions between distinct states for nodes and then write the evolution equation of each state by means of the microscopic Markovian chain approach (MMCA). Based on the MMCA, we theoretically analyze the possible steady states and calculate the critical threshold of epidemics, related to the structure of epidemic networks, the awareness diffusion, and their coupling configuration. The achieved analytical results of the mean-field approach are consistent with those of the numerical Monte Carlo simulations. Through the theoretical analysis and numerical simulations, we find that global awareness can reduce the final scale of infection when the regulatory factor of the global awareness ratio is less than the average degree of the epidemic network but it cannot alter the onset of epidemics. Furthermore, the introduction of self-awareness originating from infected individuals not only reduces the epidemic prevalence but also raises the epidemic threshold, which tells us that it is crucial to enhance the early warning of symptomatic individuals during pandemic outbreaks. These results give us a more comprehensive and deep understanding of the complicated interaction between epidemic transmission and awareness diffusion and also provide some practical and effective recommendations for the prevention and control of epidemics.

Simplicial SIRS epidemic models with nonlinear incidence rates

Mathematical epidemiology that describes the complex dynamics on social networks has become increasingly popular. However, a few methods have tackled the problem of coupling network topology with complex incidence mechanisms. Here, we propose a simplicial susceptible-infected-recovered-susceptible (SIRS) model to investigate the epidemic spreading via combining the network higher-order structure with a nonlinear incidence rate. A network-based social system is reshaped to a simplicial complex, in which the spreading or infection occurs with nonlinear reinforcement characterized by the simplex dimensions. Compared with the previous simplicial susceptible-infected-susceptible (SIS) models, the proposed SIRS model can not only capture the discontinuous transition and the bistability of a complex system but also capture the periodic phenomenon of epidemic outbreaks. More significantly, the two thresholds associated with the bistable region and the critical value of the reinforcement factor are derived. We further analyze the stability of equilibrium points of the proposed model and obtain the condition of existence of the bistable states and limit cycles. This work expands the simplicial SIS models to SIRS models and sheds light on a novel perspective of combining the higher-order structure of complex systems with nonlinear incidence rates.

Epidemic Propagation With Positive and Negative Preventive Information in Multiplex Networks

We propose a novel epidemic model based on two-layered multiplex networks to explore the influence of positive and negative preventive information on epidemic propagation. In the model, one layer represents a social network with positive and negative preventive information spreading competitively, while the other one denotes the physical contact network with epidemic propagation. The individuals who are aware of positive prevention will take more effective measures to avoid being infected than those who are aware of negative prevention. Taking the microscopic Markov chain (MMC) approach, we analytically derive the expression of the epidemic threshold for the proposed epidemic model, which indicates that the diffusion of positive and negative prevention information, as well as the topology of the physical contact network have a significant impact on the epidemic threshold. By comparing the results obtained with MMC and those with the Monte Carlo (MC) simulations, it is found that they are in good agreement, but MMC can well describe the dynamics of the proposed model. Meanwhile, through extensive simulations, we demonstrate the impact of positive and negative preventive information on the epidemic threshold, as well as the prevalence of infectious diseases. We also find that the epidemic prevalence and the epidemic outbreaks can be suppressed by the diffusion of positive preventive information and be promoted by the diffusion of negative preventive information.

Compact models for influential nodes identification problem in directed networks

Influential nodes identification problem (INIP) is one of the most important problems in complex networks. Existing methods mainly deal with this problem in undirected networks, while few studies focus on it in directed networks. Moreover, the methods designed for identifying influential nodes in undirected networks do not work for directed networks. Therefore, in this paper, we investigate INIP in directed networks. We first propose a novel metric to assess the influence effect of nodes in directed networks. Then, we formulate a compact model for INIP and prove it to be NP-Complete. Furthermore, we design a novel heuristic algorithm for the proposed model by integrating a 2-opt local search into a greedy framework. The experimental results show that, in most cases, the proposed methods outperform traditional measure-based heuristic methods in terms of accuracy and discrimination.

Influential Nodes Identification in Complex Networks via Information Entropy

Identifying a set of influential nodes is an important topic in complex networks which plays a crucial role in many applications, such as market advertising, rumor controlling, and predicting valuable scientific publications. In regard to this, researchers have developed algorithms from simple degree methods to all kinds of sophisticated approaches. However, a more robust and practical algorithm is required for the task. In this paper, we propose the EnRenew algorithm aimed to identify a set of influential nodes via information entropy. Firstly, the information entropy of each node is calculated as initial spreading ability. Then, select the node with the largest information entropy and renovate its l-length reachable nodes’ spreading ability by an attenuation factor, repeat this process until specific number of influential nodes are selected. Compared with the best state-of-the-art benchmark methods, the performance of proposed algorithm improved by 21.1%, 7.0%, 30.0%, 5.0%, 2.5%, and 9.0% in final affected scale on CEnew, Email, Hamster, Router, Condmat, and Amazon network, respectively, under the Susceptible-Infected-Recovered (SIR) simulation model. The proposed algorithm measures the importance of nodes based on information entropy and selects a group of important nodes through dynamic update strategy. The impressive results on the SIR simulation model shed light on new method of node mining in complex networks for information spreading and epidemic prevention.

Co-evolution spreading of multiple information and epidemics on two-layered networks under the influence of mass media

During epidemic outbreaks, there are various types of information about epidemic prevention disseminated simultaneously among the population. Meanwhile, the mass media also scrambles to report the information related to the epidemic. Inspired by these phenomena, we devise a model to discuss the dynamical characteristics of the co-evolution spreading of multiple information and epidemic under the influence of mass media. We construct the co-evolution model under the framework of two-layered networks and gain the dynamical equations and epidemic critical point with the help of the micro-Markov chain approach. The expression of epidemic critical point show that the positive and negative information have a direct impact on the epidemic critical point. Moreover, the mass media can indirectly affect the epidemic size and epidemic critical point through their interference with the dissemination of epidemic-relevant information. Though extensive numerical experiments, we examine the accuracy of the dynamical equations and expression of the epidemic critical point, showing that the dynamical characteristics of co-evolution spreading can be well described by the dynamic equations and the epidemic critical point is able to be accurately calculated by the derived expression. The experimental results demonstrate that accelerating positive information dissemination and enhancing the propaganda intensity of mass media can efficaciously restrain the epidemic spreading. Interestingly, the way to accelerate the dissemination of negative information can also alleviate the epidemic to a certain extent when the positive information hardly spreads. Current results can provide some useful clues for epidemic prevention and control on the basis of epidemic-relevant information dissemination.

Effects of asymptomatic infection on the dynamical interplay between behavior and disease transmission in multiplex networks

Multiplex network theory is widely introduced to deepen the understanding of the dynamical interplay between self-protective behavior and epidemic spreading. Most of the existing studies assumed that all infected individuals can transmit disease- related information or can be perceived by their neighbors. However, owing to lack of distinct symptoms for patients in the initial stage of infection, the disease information cannot be transmitted in the population, which may lead to the wrong perception of infection risk and inappropriate behavior response. In this work, we divide infected individuals into Exposed-state (without obvious clinical symptoms) individuals and Infected-state (with evident clinical symptoms) individuals, both of whom can spread disease, but only Infected-state individuals can diffuse disease information. Then, in this work we establish UAU-SEIS (Unaware–Aware–Unaware–Susceptible–Exposed–Infected–Susceptible) model in multiplex networks and analyze the effect of asymptomatic infection and the isolation of Infected-state individuals on the density of infection and the epidemic threshold. Furthermore, we extend the UAU-SEIS model by taking the individual heterogeneity into consideration. Combined Markov chain approach and Monte-Carlo Simulations, we find that asymptomatic infection has an effect on the density of infected individuals and the epidemic threshold, and the extent of the effect is influenced by whether Infected-state individuals are isolated or treated. In addition, results show that the individual heterogeneity can lower the density of infected individuals, but cannot enhance the epidemic threshold.

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We study the impact of asymptomatic infection on the epidemic spread dynamics in multiplex networks.

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We assume infected can be isolation and non isolation, then compare the research results of these two cases.

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We take the individual heterogeneity into consideration and study whether it affect research results.

The coevolution of contagion and behavior with increasing and decreasing awareness

Understanding the effects of individual awareness on epidemic phenomena is important to comprehend the coevolving system dynamic, to improve forecasting, and to better evaluate the outcome of possible interventions. In previous models of epidemics on social networks, individual awareness has often been approximated as a generic personal trait that depends on social reinforcement, and used to introduce variability in state transition probabilities. A novelty of this work is to assume that individual awareness is a function of several contributing factors pooled together, different by nature and dynamics, and to study it for different epidemic categories. This way, our model still has awareness as the core attribute that may change state transition probabilities. Another contribution is to study positive and negative variations of awareness, in a contagion-behavior model. Imitation is the key mechanism that we model for manipulating awareness, under different network settings and assumptions, in particular regarding the degree of intentionality that individuals may exhibit in spreading an epidemic. Three epidemic categories are considered-disease, addiction, and rumor-to discuss different imitation mechanisms and degree of intentionality. We assume a population with a heterogeneous distribution of awareness and different response mechanisms to information gathered from the network. With simulations, we show the interplay between population and awareness factors producing a distribution of state transition probabilities and analyze how different network and epidemic configurations modify transmission patterns.

Coevolution spreading in complex networks

The propagations of diseases, behaviors and information in real systems are rarely independent of each other, but they are coevolving with strong interactions. To uncover the dynamical mechanisms, the evolving spatiotemporal patterns and critical phenomena of networked coevolution spreading are extremely important, which provide theoretical foundations for us to control epidemic spreading, predict collective behaviors in social systems, and so on. The coevolution spreading dynamics in complex networks has thus attracted much attention in many disciplines. In this review, we introduce recent progress in the study of coevolution spreading dynamics, emphasizing the contributions from the perspectives of statistical mechanics and network science. The theoretical methods, critical phenomena, phase transitions, interacting mechanisms, and effects of network topology for four representative types of coevolution spreading mechanisms, including the coevolution of biological contagions, social contagions, epidemic-awareness, and epidemic-resources, are presented in detail, and the challenges in this field as well as open issues for future studies are also discussed.

Suppression of epidemic spreading process on multiplex networks via active immunization

Spatial epidemic spreading, a fundamental dynamical process upon complex networks, attracts huge research interest during the past few decades. To suppress the spreading of epidemic, a couple of effective methods have been proposed, including node vaccination. Under such a scenario, nodes are immunized passively and fail to reveal the mechanisms of active activity. Here, we suggest one novel model of an observer node, which can identify infection through interacting with infected neighbors and inform the other neighbors for vaccination, on multiplex networks, consisting of epidemic spreading layer and information spreading layer. In detail, the epidemic spreading layer supports susceptible-infected-recovered process, while observer nodes will be selected according to several algorithms derived from percolation theory. Numerical simulation results show that the algorithm based on large degree performs better than random placement, while the algorithm based on nodes' degree in the information spreading layer performs the best (i.e., the best suppression efficacy is guaranteed when placing observer nodes based on nodes' degree in the information spreading layer). With the help of state probability transition equation, the above phenomena can be validated accurately. Our work thus may shed new light into understanding control of empirical epidemic control.

Self-awareness control effect of cooperative epidemics on complex networks

Coinfection mechanism is a common interacting mode between multiple diseases in real spreading processes, where the diseases mutually increase their susceptibility, and has aroused widespread studies in network science. We use the bond percolation theory to characterize the coinfection model under two self-awareness control strategies, including immunization strategy and quarantine strategy, and to study the impacts of the synergy effect and control strategies on cooperative epidemics. We find that strengthening the synergy effect can reduce the epidemic threshold and enhance the outbreak size of coinfected networks. On Erdős-Rényi networks, the synergy effect will induce a crossover phenomenon of phase transition, i.e., make the type of phase transition from being continuous to discontinuous. Self-awareness control strategies play a non-negligible role in suppressing cooperative epidemics. In particular, increasing immunization or the quarantine rate can enhance the epidemic threshold and reduce the outbreak size of cooperative epidemics, and lead to a crossover phenomenon of transition from being discontinuous to continuous. The impact of quarantine strategy on cooperative epidemics is more significant than the immunization strategy, which is verified on scale-free networks.

A stochastic SIR network epidemic model with preventive dropping of edges

A Markovian Susceptible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ Infectious \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ Recovered (SIR) model is considered for the spread of an epidemic on a configuration model network, in which susceptible individuals may take preventive measures by dropping edges to infectious neighbours. An effective degree formulation of the model is used in conjunction with the theory of density dependent population processes to obtain a law of large numbers and a functional central limit theorem for the epidemic as the population size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document}N→∞, assuming that the degrees of individuals are bounded. A central limit theorem is conjectured for the final size of the epidemic. The results are obtained for both the Molloy–Reed (in which the degrees of individuals are deterministic) and Newman–Strogatz–Watts (in which the degrees of individuals are independent and identically distributed) versions of the configuration model. The two versions yield the same limiting deterministic model but the asymptotic variances in the central limit theorems are greater in the Newman–Strogatz–Watts version. The basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0 and the process of susceptible individuals in the limiting deterministic model, for the model with dropping of edges, are the same as for a corresponding SIR model without dropping of edges but an increased recovery rate, though, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0>1$$\end{document}R0>1, the probability of a major outbreak is greater in the model with dropping of edges. The results are specialised to the model without dropping of edges to yield conjectured central limit theorems for the final size of Markovian SIR epidemics on configuration-model networks, and for the size of the giant components of those networks. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations are good, even for moderate N.

Identifying influential spreaders in complex networks by propagation probability dynamics

Numerous well-known processes of complex systems such as spreading and cascading are mainly affected by a small number of critical nodes. Identifying influential nodes that lead to broad spreading in complex networks is of great theoretical and practical importance. Since the identification of vital nodes is closely related to propagation dynamics, a novel method DynamicRank that employs the probability model to measure the ranking scores of nodes is suggested. The influence of a node can be denoted by the sum of probability scores of its i order neighboring nodes. This simple yet effective method provides a new idea to understand the identification of vital nodes in propagation dynamics. Experimental studies on both Susceptible-Infected-Recovered and Susceptible-Infected-Susceptible models in real networks demonstrate that it outperforms existing methods such as Coreness, H-index, LocalRank, Betweenness, and Spreading Probability in terms of the Kendall τ coefficient. The linear time complexity enables it to be applied to real large-scale networks with tens of thousands of nodes and edges in a short time.

Bifurcation analysis of a network-based SIR epidemic model with saturated treatment function

In this paper, we present a study on a network-based susceptible-infected-recovered (SIR) epidemic model with a saturated treatment function. It is well known that treatment can have a specific effect on the spread of epidemics, and due to the limited resources of treatment, the number of patients during severe disease outbreaks who need to be treated may exceed the treatment capacity. Consequently, the number of patients who receive treatment will reach a saturation level. Thus, we incorporated a saturated treatment function into the model to characterize such a phenomenon. The dynamics of the present model is discussed in this paper. We first obtained a threshold value R₀, which determines the stability of a disease-free equilibrium. Furthermore, we investigated the bifurcation behavior at R₀=1. More specifically, we derived a condition that determines the direction of bifurcation at R₀=1. If the direction is backward, then a stable disease-free equilibrium concurrently exists with a stable endemic equilibrium even though R₀<1. Therefore, in this case, R₀<1 is not sufficient to eradicate the disease from the population. However, if the direction is forward, we find that for a range of parameters, multiple equilibria could exist to the left and right of R₀=1. In this case, the initial infectious invasion must be controlled to a lower level so that the disease dies out or approaches a lower endemic steady state.

Interaction between epidemic spread and collective behavior in scale-free networks with community structure

Many real-world networks exhibit community structure: the connections within each community are dense, while connections between communities are sparser. Moreover, there is a common but non-negligible phenomenon, collective behaviors, during the outbreak of epidemics, are induced by the emergence of epidemics and in turn influence the process of epidemic spread. In this paper, we explore the interaction between epidemic spread and collective behavior in scale-free networks with community structure, by constructing a mathematical model that embeds community structure, behavioral evolution and epidemic transmission. In view of the differences among individuals' responses in different communities to epidemics, we use nonidentical functions to describe the inherent dynamics of individuals. In practice, with the progress of epidemics, individual behaviors in different communities may tend to cluster synchronization, which is indicated by the analysis of our model. By using comparison principle and Gers˘gorin theorem, we investigate the epidemic threshold of the model. By constructing an appropriate Lyapunov function, we present the stability analysis of behavioral evolution and epidemic dynamics. Some numerical simulations are performed to illustrate and complement our theoretical results. It is expected that our work can deepen the understanding of interaction between cluster synchronization and epidemic dynamics in scale-free community networks.

Effect of time varying transmission rates on the coupled dynamics of epidemic and awareness over a multiplex network

A non-linear stochastic model is presented to study the effect of time variation of transmission rates on the co-evolution of epidemics and its corresponding awareness over a two layered multiplex network. In the model, the infection transmission rate of a given node in the epidemic layer depends upon its awareness probability in the awareness layer. Similarly, the infection information transmission rate of a node in the awareness layer depends upon its infection probability in the epidemic layer. The spread of disease resulting from physical contacts is described in terms of a Susceptible Infected Susceptible process over the epidemic layer and the spread of information about the disease outbreak is described in terms of an Unaware Aware Unaware process over the virtual interaction mediated awareness layer. The time variation of the transmission rates and the resulting co-evolution of these mutually competing processes are studied in terms of a network topology dependent parameter ( α ). Using a second order linear theory, it is shown that in the continuous time limit, the co-evolution of these processes can be described in terms of damped and driven harmonic oscillator equations. From the results of a Monte-Carlo simulation, it is shown that for a suitable choice of the parameter ( α ) , the two processes can either exhibit sustained oscillatory or damped dynamics. The damped dynamics corresponds to the endemic state. Furthermore, for the case of an endemic state, it is shown that the inclusion of the awareness layer significantly lowers the disease transmission rate and reduces the size of the epidemic. The infection probability of the nodes in the endemic state is found to have a dependence on both the transmission rates and on their absolute degrees in each of the network layers and on the relative differences between their degrees in the respective layers.

A colored mean-field model for analyzing the effects of awareness on epidemic spreading in multiplex networks

We study the impact of susceptible nodes' awareness on epidemic spreading in social systems, where the systems are modeled as multiplex networks coupled with an information layer and a contact layer. We develop a colored heterogeneous mean-field model taking into account the portion of the overlapping neighbors in the two layers. With theoretical analysis and numerical simulations, we derive the epidemic threshold which determines whether the epidemic can prevail in the population and find that the impacts of awareness on threshold value only depend on epidemic information being available in network nodes' overlapping neighborhood. When there is no link overlap between the two network layers, the awareness cannot help one to raise the epidemic threshold. Such an observation is different from that in a single-layer network, where the existence of awareness almost always helps.

Information filtering by smart nodes in random networks

Diffusion of information in social networks has drawn extensive attention from various scientific communities, with many contagion models proposed to explain related phenomena. In this paper, we present a simple contagion mechanism, in which a node will change its state immediately if it is exposed to the diffusive information. By considering two types of nodes (smart and normal) and two kinds of information (true and false), we study analytically and numerically how smart nodes influence the spreading of information, which leads to information filtering. We find that for randomly distributed smart nodes, the spreading dynamics over random networks with Poisson degree distribution and power-law degree distribution (with relatively small cutoffs) can both be described by the same approximate mean-field equation. Increasing the heterogeneity of the network may elicit more deviations, but not much. Moreover, we demonstrate that more smart nodes make the filtering effect on a random network better. Finally, we study the efficacy of different strategies of selecting smart nodes for information filtering.

Structural calculations and propagation modeling of growing networks based on continuous degree

When a network reaches a certain size, its node degree can be considered as a continuous variable, which we will call continuous degree. Using continuous degree method (CDM), we analytically calculate certain structure of the network and study the spread of epidemics on a growing network. Firstly, using CDM we calculate the degree distributions of three different growing models, which are the BA growing model, the preferential attachment accelerating growing model and the random attachment growing model. We obtain the evolution equation for the cumulative distribution function F(k,t), and then obtain analytical results about F(k,t) and the degree distribution p(k,t). Secondly, we calculate the joint degree distribution p(k1,k2,t) of the BA model by using the same method, thereby obtain the conditional degree distribution p(k1|k2). We find that the BA model has no degree correlations. Finally, we consider the different states, susceptible and infected, according to the node health status. We establish the continuous degree SIS model on a static network and a growing network, respectively. We find that, in the case of growth, the new added health nodes can slightly reduce the ratio of infected nodes, but the final infected ratio will gradually tend to the final infected ratio of SIS model on static networks.

The impact of individual heterogeneity on the coupled awareness-epidemic dynamics in multiplex networks

Awareness of disease outbreaks can trigger changes in human behavior and has a significant impact on the spread of epidemics. Previous studies usually considered the coupled awareness-epidemic dynamics to be two competing processes that interact in the information and epidemic layers. However, these studies mostly assumed that all aware individuals have the same reduced infectivity and that different neighbors have the same influence on one's perception, ignoring the heterogeneity of individuals. In this paper, we propose a coupled awareness-epidemic spreading model in multiplex networks incorporating three types of heterogeneity: (1) the heterogeneity of individual responses to disease outbreaks, (2) the influence heterogeneity in the epidemic layer, and (3) the influence heterogeneity in the information layer. The theoretical analysis shows that the influence heterogeneity in the information layer has two-stage effects on the epidemic threshold. Moreover, we find that the epidemic threshold in the higher stage depends on the heterogeneity of individual responses and the influence heterogeneity in the epidemic layer, while the epidemic threshold in the lower stage is independent of awareness spreading and individual behaviors. The results give us a better understanding of how individual heterogeneity affects epidemic spreading and provide some practical implications for the control of epidemics.

The impact of individual heterogeneity on the coupled awareness-epidemic dynamics in multiplex networks

Awareness of disease outbreaks can trigger changes in human behavior and has a significant impact on the spread of epidemics. Previous studies usually considered the coupled awareness-epidemic dynamics to be two competing processes that interact in the information and epidemic layers. However, these studies mostly assumed that all aware individuals have the same reduced infectivity and that different neighbors have the same influence on one's perception, ignoring the heterogeneity of individuals. In this paper, we propose a coupled awareness-epidemic spreading model in multiplex networks incorporating three types of heterogeneity: (1) the heterogeneity of individual responses to disease outbreaks, (2) the influence heterogeneity in the epidemic layer, and (3) the influence heterogeneity in the information layer. The theoretical analysis shows that the influence heterogeneity in the information layer has two-stage effects on the epidemic threshold. Moreover, we find that the epidemic threshold in the higher stage depends on the heterogeneity of individual responses and the influence heterogeneity in the epidemic layer, while the epidemic threshold in the lower stage is independent of awareness spreading and individual behaviors. The results give us a better understanding of how individual heterogeneity affects epidemic spreading and provide some practical implications for the control of epidemics.

Social Synchrony on Complex Networks

Social synchrony (SS) is an emergent phenomenon in human society. People often mimic others which, over time, can result in large groups behaving similarly. Drawing from prior empirical studies of SS in online communities, here we propose a discrete network model of SS based on four attributes: 1) depth of action; 2) breadth of impact, i.e., a large number of actions are performed with a large group of people involved; 3) heterogeneity of role, i.e., people of higher degree play more important roles; and 4) lastly, emergence of phenomenon, i.e., it is far from random. We analyze our model both analytically and with simulations, and find good agreement between the two. We find this model can well explain the four characters of SS, and thus hope it can help researchers better understand human collective behavior.

Use of social network analysis and global sensitivity and uncertainty analyses to better understand an influenza outbreak

In the summer of 2014, an influenza A(H3N2) outbreak occurred in Yichang city, Hubei province, China. A retrospective study was conducted to collect and interpret hospital and epidemiological data on it using social network analysis and global sensitivity and uncertainty analyses. Results for degree (χ²=17.6619, P<0.0001) and betweenness(χ²=21.4186, P<0.0001) centrality suggested that the selection of sampling objects were different between traditional epidemiological methods and newer statistical approaches. Clique and network diagrams demonstrated that the outbreak actually consisted of two independent transmission networks. Sensitivity analysis showed that the contact coefficient (k) was the most important factor in the dynamic model. Using uncertainty analysis, we were able to better understand the properties and variations over space and time on the outbreak. We concluded that use of newer approaches were significantly more efficient for managing and controlling infectious diseases outbreaks, as well as saving time and public health resources, and could be widely applied on similar local outbreaks.

Susceptible-infected-recovered epidemics in random networks with population awareness

The influence of epidemic information-based awareness on the spread of infectious diseases on networks cannot be ignored. Within the effective degree modeling framework, we discuss the susceptible-infected-recovered model in complex networks with general awareness and general degree distribution. By performing the linear stability analysis, the conditions of epidemic outbreak can be deduced and the results of the previous research can be further expanded. Results show that the local awareness can suppress significantly the epidemic spreading on complex networks via raising the epidemic threshold and such effects are closely related to the formulation of awareness functions. In addition, our results suggest that the recovered information-based awareness has no effect on the critical condition of epidemic outbreak.

Epidemic spreading between two coupled subpopulations with inner structures

The structure of underlying contact network and the mobility of agents are two decisive factors for epidemic spreading in reality. Here, we study a model consisting of two coupled subpopulations with intra-structures that emphasizes both the contact structure and the recurrent mobility pattern of individuals simultaneously. We show that the coupling of the two subpopulations (via interconnections between them and round trips of individuals) makes the epidemic threshold in each subnetwork to be the same. Moreover, we find that the interconnection probability between two subpopulations and the travel rate are important factors for spreading dynamics. In particular, as a function of interconnection probability, the epidemic threshold in each subpopulation decreases monotonously, which enhances the risks of an epidemic. While the epidemic threshold displays a non-monotonic variation as travel rate increases. Moreover, the asymptotic infected density as a function of travel rate in each subpopulation behaves differently depending on the interconnection probability.

Efficiency of prompt quarantine measures on a susceptible-infected-removed model in networks

This study focuses on investigating the manner in which a prompt quarantine measure suppresses epidemics in networks. A simple and ideal quarantine measure is considered in which an individual is detected with a probability immediately after it becomes infected and the detected one and its neighbors are promptly isolated. The efficiency of this quarantine in suppressing a susceptible-infected-removed (SIR) model is tested in random graphs and uncorrelated scale-free networks. Monte Carlo simulations are used to show that the prompt quarantine measure outperforms random and acquaintance preventive vaccination schemes in terms of reducing the number of infected individuals. The epidemic threshold for the SIR model is analytically derived under the quarantine measure, and the theoretical findings indicate that prompt executions of quarantines are highly effective in containing epidemics. Even if infected individuals are detected with a very low probability, the SIR model under a prompt quarantine measure has finite epidemic thresholds in fat-tailed scale-free networks in which an infected individual can always cause an outbreak of a finite relative size without any measure. The numerical simulations also demonstrate that the present quarantine measure is effective in suppressing epidemics in real networks.

Accurate ranking of influential spreaders in networks based on dynamically asymmetric link weights

We propose an efficient and accurate measure for ranking spreaders and identifying the influential ones in spreading processes in networks. While the edges determine the connections among the nodes, their specific role in spreading should be considered explicitly. An edge connecting nodes i and j may differ in its importance for spreading from i to j and from j to i. The key issue is whether node j, after infected by i through the edge, would reach out to other nodes that i itself could not reach directly. It becomes necessary to invoke two unequal weights w_{ij} and w_{ji} characterizing the importance of an edge according to the neighborhoods of nodes i and j. The total asymmetric directional weights originating from a node leads to a novel measure s_{i}, which quantifies the impact of the node in spreading processes. An s-shell decomposition scheme further assigns an s-shell index or weighted coreness to the nodes. The effectiveness and accuracy of rankings based on s_{i} and the weighted coreness are demonstrated by applying them to nine real-world networks. Results show that they generally outperform rankings based on the nodes' degree and k-shell index while maintaining a low computational complexity. Our work represents a crucial step towards understanding and controlling the spread of diseases, rumors, information, trends, and innovations in networks.

How heterogeneous susceptibility and recovery rates affect the spread of epidemics on networks

In this paper, an extended heterogeneous SIR model is proposed, which generalizes the heterogeneous mean-field theory. Different from the traditional heterogeneous mean-field model only taking into account the heterogeneity of degree, our model considers not only the heterogeneity of degree but also the heterogeneity of susceptibility and recovery rates. Then, we analytically study the basic reproductive number and the final epidemic size. Combining with numerical simulations, it is found that the basic reproductive number depends on the mean of distributions of susceptibility and disease course when both of them are independent. If the mean of these two distributions is identical, increasing the variance of susceptibility may block the spread of epidemics, while the corresponding increase in the variance of disease course has little effect on the final epidemic size. It is also shown that positive correlations between individual susceptibility, course of disease and the square of degree make the population more vulnerable to epidemic and avail to the epidemic prevalence, whereas the negative correlations make the population less vulnerable and impede the epidemic prevalence.

Behavioral synchronization induced by epidemic spread in complex networks

During the spread of an epidemic, individuals in realistic networks may exhibit collective behaviors. In order to characterize this kind of phenomenon and explore the correlation between collective behaviors and epidemic spread, in this paper, we construct several mathematical models (including without delay, with a coupling delay, and with double delays) of epidemic synchronization by applying the adaptive feedback motivated by real observations. By using Lyapunov function methods, we obtain the conditions for local and global stability of these epidemic synchronization models. Then, we illustrate that quenched mean-field theory is more accurate than heterogeneous mean-field theory in the prediction of epidemic synchronization. Finally, some numerical simulations are performed to complement our theoretical results, which also reveal some unexpected phenomena, for example, the coupling delay and epidemic delay influence the speed of epidemic synchronization. This work makes further exploration on the relationship between epidemic dynamics and synchronization dynamics, in the hope of being helpful to the study of other dynamical phenomena in the process of epidemic spread.

Interplay between epidemic spread and information propagation on metapopulation networks

The spread of an infectious disease has been widely found to evolve with the propagation of information. Many seminal works have demonstrated the impact of information propagation on the epidemic spreading, assuming that individuals are static and no mobility is involved. Inspired by the recent observation of diverse mobility patterns, we incorporate the information propagation into a metapopulation model based on the mobility patterns and contagion process, which significantly alters the epidemic threshold. In more details, we find that both the information efficiency and the mobility patterns have essential impacts on the epidemic spread. We obtain different scenarios leading to the mitigation of the outbreak by appropriately integrating the mobility patterns and the information efficiency as well. The inclusion of the impacts of the information propagation into the epidemiological model is expected to provide an support to public health implications for the suppression of epidemics.

Effects of awareness diffusion and self-initiated awareness behavior on epidemic spreading - An approach based on multiplex networks

In this paper, we study the interplay between the epidemic spreading and the diffusion of awareness in multiplex networks. In the model, an infectious disease can spread in one network representing the paths of epidemic spreading (contact network), leading to the diffusion of awareness in the other network (information network), and then the diffusion of awareness will cause individuals to take social distances, which in turn affects the epidemic spreading. As for the diffusion of awareness, we assume that, on the one hand, individuals can be informed by other aware neighbors in information network, on the other hand, the susceptible individuals can be self-awareness induced by the infected neighbors in the contact networks (local information) or mass media (global information). Through Markov chain approach and numerical computations, we find that the density of infected individuals and the epidemic threshold can be affected by the structures of the two networks and the effective transmission rate of the awareness. However, we prove that though the introduction of the self-awareness can lower the density of infection, which cannot increase the epidemic threshold no matter of the local information or global information. Our finding is remarkably different to many previous results on single-layer network: local information based behavioral response can alter the epidemic threshold. Furthermore, our results indicate that the nodes with more neighbors (hub nodes) in information networks are easier to be informed, as a result, their risk of infection in contact networks can be effectively reduced.

Dynamics of epidemic diseases on a growing adaptive network

The study of epidemics on static networks has revealed important effects on disease prevalence of network topological features such as the variance of the degree distribution, i.e. the distribution of the number of neighbors of nodes, and the maximum degree. Here, we analyze an adaptive network where the degree distribution is not independent of epidemics but is shaped through disease-induced dynamics and mortality in a complex interplay. We study the dynamics of a network that grows according to a preferential attachment rule, while nodes are simultaneously removed from the network due to disease-induced mortality. We investigate the prevalence of the disease using individual-based simulations and a heterogeneous node approximation. Our results suggest that in this system in the thermodynamic limit no epidemic thresholds exist, while the interplay between network growth and epidemic spreading leads to exponential networks for any finite rate of infectiousness when the disease persists.

Collective effect of personal behavior induced preventive measures and differential rate of transmission on spread of epidemics

In the present work, the effect of personal behavior induced preventive measures is studied on the spread of epidemics over scale free networks that are characterized by the differential rate of disease transmission. The role of personal behavior induced preventive measures is parameterized in terms of variable λ, which modulates the number of concurrent contacts a node makes with the fraction of its neighboring nodes. The dynamics of the disease is described by a non-linear Susceptible Infected Susceptible model based upon the discrete time Markov Chain method. The network mean field approach is generalized to account for the effect of non-linear coupling between the aforementioned factors on the collective dynamics of nodes. The upper bound estimates of the disease outbreak threshold obtained from the mean field theory are found to be in good agreement with the corresponding non-linear stochastic model. From the results of parametric study, it is shown that the epidemic size has inverse dependence on the preventive measures (λ). It has also been shown that the increase in the average degree of the nodes lowers the time of spread and enhances the size of epidemics.

Behavioural change models for infectious disease transmission: a systematic review (2010-2015)

We review behavioural change models (BCMs) for infectious disease transmission in humans. Following the Cochrane collaboration guidelines and the PRISMA statement, our systematic search and selection yielded 178 papers covering the period 2010-2015. We observe an increasing trend in published BCMs, frequently coupled to (re)emergence events, and propose a categorization by distinguishing how information translates into preventive actions. Behaviour is usually captured by introducing information as a dynamic parameter (76/178) or by introducing an economic objective function, either with (26/178) or without (37/178) imitation. Approaches using information thresholds (29/178) and exogenous behaviour formation (16/178) are also popular. We further classify according to disease, prevention measure, transmission model (with 81/178 population, 6/178 metapopulation and 91/178 individual-level models) and the way prevention impacts transmission. We highlight the minority (15%) of studies that use any real-life data for parametrization or validation and note that BCMs increasingly use social media data and generally incorporate multiple sources of information (16/178), multiple types of information (17/178) or both (9/178). We conclude that individual-level models are increasingly used and useful to model behaviour changes. Despite recent advancements, we remain concerned that most models are purely theoretical and lack representative data and a validation process.

The Role of Node Heterogeneity in the Coupled Spreading of Epidemics and Awareness

Exploring the interplay between information spreading and epidemic spreading is a topic that has been receiving increasing attention. As an efficient means of depicting the spreading of information, which manifests as a cascade phenomenon, awareness cascading is utilized to investigate this coupled transmission. Because in reality, different individuals facing the same epidemic will exhibit distinct behaviors according to their own experiences and attributes, it is important for us to consider the heterogeneity of individuals. Consequently, we propose a heterogeneous spreading model. To describe the heterogeneity, two of the most important but radically different methods for this purpose, the degree and k-core measures, are studied in this paper through three models based on different assumptions. Adopting a Markov chain approach, we succeed in predicting the epidemic threshold trend. Furthermore, we find that when the k-core measure is used to classify individuals, the spreading process is robust to these models, meaning that regardless of the model used, the spreading process is nearly identical at the macroscopic level. In addition, the k-core measure leads to a much larger final epidemic size than the degree measure. These results are cross-checked through numerous simulations, not only of a synthetic network but also of a real multiplex network. The presented findings provide a better understanding of k-core individuals and reveal the importance of considering network structure when investigating various dynamic processes.

Impacts of complex behavioral responses on asymmetric interacting spreading dynamics in multiplex networks

Information diffusion and disease spreading in communication-contact layered network are typically asymmetrically coupled with each other, in which disease spreading can be significantly affected by the way an individual being aware of disease responds to the disease. Many recent studies have demonstrated that human behavioral adoption is a complex and non-Markovian process, where the probability of behavior adoption is dependent on the cumulative times of information received and the social reinforcement effect of the cumulative information. In this paper, the impacts of such a non-Markovian vaccination adoption behavior on the epidemic dynamics and the control effects are explored. It is found that this complex adoption behavior in the communication layer can significantly enhance the epidemic threshold and reduce the final infection rate. By defining the social cost as the total cost of vaccination and treatment, it can be seen that there exists an optimal social reinforcement effect and optimal information transmission rate allowing the minimal social cost. Moreover, a mean-field theory is developed to verify the correctness of simulation results.

Local immunization program for susceptible-infected-recovered network epidemic model

The immunization strategies through contact tracing on the susceptible-infected-recovered framework in social networks are modelled to evaluate the cost-effectiveness of information-based vaccination programs with particular focus on the scenario where individuals belonging to a specific set can get vaccinated due to the vaccine shortages and other economic or humanity constraints. By using the block heterogeneous mean-field approach, a series of discrete-time dynamical models is formulated and the condition for epidemic outbreaks can be established which is shown to be not only dependent on the network structure but also closely related to the immunization control parameters. Results show that increasing the immunization strength can effectively raise the epidemic threshold, which is different from the predictions obtained through the susceptible-infected-susceptible network framework, where epidemic threshold is independent of the vaccination strength. Furthermore, a significant decrease of vaccine use to control the infectious disease is observed for the local vaccination strategy, which shows the promising applications of the local immunization programs to disease control while calls for accurate local information during the process of disease outbreak.

Explosive Contagion in Networks

The spread of social phenomena such as behaviors, ideas or products is an ubiquitous but remarkably complex phenomenon. A successful avenue to study the spread of social phenomena relies on epidemic models by establishing analogies between the transmission of social phenomena and infectious diseases. Such models typically assume simple social interactions restricted to pairs of individuals; effects of the context are often neglected. Here we show that local synergistic effects associated with acquaintances of pairs of individuals can have striking consequences on the spread of social phenomena at large scales. The most interesting predictions are found for a scenario in which the contagion ability of a spreader decreases with the number of ignorant individuals surrounding the target ignorant. This mechanism mimics ubiquitous situations in which the willingness of individuals to adopt a new product depends not only on the intrinsic value of the product but also on whether his acquaintances will adopt this product or not. In these situations, we show that the typically smooth (second order) transitions towards large social contagion become explosive (first order). The proposed synergistic mechanisms therefore explain why ideas, rumours or products can suddenly and sometimes unexpectedly catch on.

Coupled disease-behavior dynamics on complex networks: A review

It is increasingly recognized that a key component of successful infection control efforts is understanding the complex, two-way interaction between disease dynamics and human behavioral and social dynamics. Human behavior such as contact precautions and social distancing clearly influence disease prevalence, but disease prevalence can in turn alter human behavior, forming a coupled, nonlinear system. Moreover, in many cases, the spatial structure of the population cannot be ignored, such that social and behavioral processes and/or transmission of infection must be represented with complex networks. Research on studying coupled disease-behavior dynamics in complex networks in particular is growing rapidly, and frequently makes use of analysis methods and concepts from statistical physics. Here, we review some of the growing literature in this area. We contrast network-based approaches to homogeneous-mixing approaches, point out how their predictions differ, and describe the rich and often surprising behavior of disease-behavior dynamics on complex networks, and compare them to processes in statistical physics. We discuss how these models can capture the dynamics that characterize many real-world scenarios, thereby suggesting ways that policy makers can better design effective prevention strategies. We also describe the growing sources of digital data that are facilitating research in this area. Finally, we suggest pitfalls which might be faced by researchers in the field, and we suggest several ways in which the field could move forward in the coming years.

Effects of local and global network connectivity on synergistic epidemics

Epidemics in networks can be affected by cooperation in transmission of infection and also connectivity between nodes. An interplay between these two properties and their influence on epidemic spread are addressed in the paper. A particular type of cooperative effects (called synergy effects) is considered, where the transmission rate between a pair of nodes depends on the number of infected neighbors. The connectivity effects are studied by constructing networks of different topology, starting with lattices with only local connectivity and then with networks that have both local and global connectivity obtained by random bond-rewiring to nodes within a certain distance. The susceptible-infected-removed epidemics were found to exhibit several interesting effects: (i) for epidemics with strong constructive synergy spreading in networks with high local connectivity, the bond rewiring has a negative role in epidemic spread, i.e., it reduces invasion probability; (ii) in contrast, for epidemics with destructive or weak constructive synergy spreading on networks of arbitrary local connectivity, rewiring helps epidemics to spread; (iii) and, finally, rewiring always enhances the spread of epidemics, independent of synergy, if the local connectivity is low.

Interplay between the local information based behavioral responses and the epidemic spreading in complex networks

The spreading of an infectious disease can trigger human behavior responses to the disease, which in turn plays a crucial role on the spreading of epidemic. In this study, to illustrate the impacts of the human behavioral responses, a new class of individuals, S(F), is introduced to the classical susceptible-infected-recovered model. In the model, S(F) state represents that susceptible individuals who take self-initiate protective measures to lower the probability of being infected, and a susceptible individual may go to S(F) state with a response rate when contacting an infectious neighbor. Via the percolation method, the theoretical formulas for the epidemic threshold as well as the prevalence of epidemic are derived. Our finding indicates that, with the increasing of the response rate, the epidemic threshold is enhanced and the prevalence of epidemic is reduced. The analytical results are also verified by the numerical simulations. In addition, we demonstrate that, because the mean field method neglects the dynamic correlations, a wrong result based on the mean field method is obtained-the epidemic threshold is not related to the response rate, i.e., the additional S(F) state has no impact on the epidemic threshold.