Epidemic Spreading In Complex Networks Research Articles

Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks.

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási-Albert (BA), Erdös-Rényi (ER), and Watts-Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM<R0. For RM,R0>1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.

Impact of contact rate on epidemic spreading in complex networks.

Contact reduction is an effective strategy to mitigate the spreading of epidemic. However, the existing reaction-diffusion equations for infectious disease are unable to characterize this effect. Thus, we here propose an extended susceptible-infected-recovered model by incorporating contact rate into the standard SIR model, and concentrate on investigating its impact on epidemic transmission. We analytically derive the epidemic thresholds on homogeneous and heterogeneous networks, respectively. The effects of contact rate on spreading speed, scale and outbreak threshold are explored on ER and SF networks. Simulations results show that epidemic dissemination is significantly mitigated when contact rate is reduced. Importantly, epidemic spreads faster on heterogeneous networks while broader on homogeneous networks, and the outbreak thresholds of the former are smaller.

Impact of human contact patterns on epidemic spreading in time-varying networks.

Human contact behaviors involve both dormant and active processes. The dormant (active) process goes from the disappearance (creation) to the creation (disappearance) of an edge. The dormant (active) time is the elapsed time since the edge became dormant (active). Many empirical studies have revealed that dormant and active times in human contact behaviors tend to show a long-tailed distribution. Previous researches focused on the impact of the dormant process on spreading dynamics. However, the epidemic spreading happens on the active process. This raises the question of how the active process affects epidemic spreading in complex networks. Here, we propose a novel time-varying network model in which the distributions of both the dormant time and active time of edges are adjustable. We develop a pairwise approximation method to describe the spreading dynamical processes in the time-varying networks. Through extensive numerical simulations, we find that the epidemic threshold is proportional to the mean dormant time and inversely proportional to the mean active time. The attack rate decreases with the increase of mean dormant time and increases with the increase of mean active time. It is worth noting that the epidemic threshold and the attack rate (e.g., the infected density in the steady state) are independent of the heterogeneities of the dormant time distribution and the active time distribution. Increasing the heterogeneity of the dormant time distribution accelerates epidemic spreading while increasing the heterogeneity of the active time distribution slows it down.

Efficient network immunization strategy based on generalized Herfindahl–Hirschman index

The topic of finding effective strategies to restrain epidemic spreading in complex networks is of current interest. A widely used approach for epidemic containment is the fragmentation of the contact networks through immunization. However, due to the limitation of immune resources, we cannot always fragment the contact network completely. In this study, based on the size distribution of connected components for the network, we designed a risk indicator of epidemic outbreaks, the generalized Herfindahl–Hirschman index (GHI), which measures the upper bound of the expected infection’s prevalence (the fraction of infected nodes) in random outbreaks. An immunization approach based on minimizing GHI is developed to reduce the infection risk for individuals in the network. Experimental results show that our immunization strategy could effectively decrease the infection’s prevalence as compared to other existing strategies, especially against infectious diseases with higher infection rates or lower recovery rates. The findings provide an efficient and practicable strategy for immunization against epidemic diseases.

Epidemic spreading in complex networks

<p indent=0mm>Human history is accompanied by all kinds of epidemic diseases. Yet, the research progress in epidemic spreading is unexpectedly slow. In the last few decades, by applying the theory of complex networks, the study on epidemic spreading has obtained great success. In this review, we briefly introduce the important progresses on epidemic spreading in complex networks. We mainly focus on the following four aspects: (1) epidemic spreading based on contact network model; (2) epidemic spreading based on meta-population model; (3) the prediction of epidemic spreading; (4) the control of epidemic spreading. These achievements make us deeply understand the epidemic spreading in the real world, and make it possible for us to predict and control the real epidemics.

Epidemic Spreading in Complex Networks with Resilient Nodes: Applications to FMD

At the outbreak of the animal epidemic disease, farms that recover quickly from partially infected state can delay or even suppress the wide spreading of the infection over farm networks. In this work, we focus on how the spatial transmission of the infection is affected by both factors, the topology of networks and the internal resilience mechanism of nodes. We first develop an individual farm model to examine the influence of initial number of infected individuals and vaccination rate on the transmission in a single farm. Based on such intrafarm model, the farm network is constructed which reflects disease transmission between farms at various stages. We explore the impact of the farms vaccinated at low rates on the disease transmission into entire farm network and investigate the effect of the control on hub farms on the transmission over the farm network. It is shown that intensive control on the farms vaccinated at low rates and hub farms effectively reduces the potential risk of foot-and-mouth disease (FMD) outbreak on the farm network.

Estimating the state probability distribution for epidemic spreading in complex networks

The problem of state estimation of spreading phenomena in complex networks is considered on the basis of a detectability-based approach. Using a simple, reduced model based state distribution estimator, where the monitored nodes are driven directly by the measured data, asymptotic convergence conditions are provided in terms of the number and location of the required sensors on the basis of the network topology. The convergence of the estimator is established in terms of the largest eigenvalue of a reduced connectivity matrix which stems from removing the monitored nodes and their connections from the original graph. In the case of unit weights, this condition corresponds to measuring the nodes with highest degree. Numerical simulations for a complete and a scale-free network each of 500 nodes and randomly distributed and unit weights, respectively, illustrate the estimator functioning with 20 sensors for the complete, and 38 sensors for the scale-free network.

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, SF, is introduced to the classical susceptible-infected-recovered model. In the model, SF 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 SF 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 SF state has no impact on the epidemic threshold.

Global and local targeted immunization in networks with community structure

.Immunization plays an important role in the field of epidemic spreading in complex networks. In previous studies, targeted immunization has been proved to be an effective strategy. However, when extended to networks with community structure, it is unknown whether the superior strategy is to vaccinate the nodes who have the most connections in the entire network (global strategy), or those in the original community where epidemic starts to spread (local strategy). In this work, by using both analytic approaches and simulations, we observe that the answer depends on the closeness between communities. If communities are tied closely, the global strategy is superior to the local strategy. Otherwise, the local targeted immunization is advantageous. The existence of a transitional value of closeness implies that we should adopt different strategies. Furthermore, we extend our investigation from two-community networks to multi-community networks. We consider the mode of community connection and the location of community where epidemic starts to spread. Both simulation results and theoretical predictions show that local strategy is a better option for immunization in most cases. But if the epidemic begins from a core community, global strategy is superior in some cases.

Traffic-driven epidemic spreading in correlated networks.

In spite of the extensive previous efforts on traffic dynamics and epidemic spreading in complex networks, the problem of traffic-driven epidemic spreading on correlated networks has not been addressed. Interestingly, we find that the epidemic threshold, a fundamental quantity underlying the spreading dynamics, exhibits a nonmonotonic behavior in that it can be minimized for some critical value of the assortativity coefficient, a parameter characterizing the network correlation. To understand this phenomenon, we use the degree-based mean-field theory to calculate the traffic-driven epidemic threshold for correlated networks. The theory predicts that the threshold is inversely proportional to the packet-generation rate and the largest eigenvalue of the betweenness matrix. We obtain consistency between theory and numerics. Our results may provide insights into the important problem of controlling and/or harnessing real-world epidemic spreading dynamics driven by traffic flows.

An environment aware epidemic spreading model and immune strategy in complex networks

In the standard SIS model, each node has the same probability to be infected by its neighbors regardless of its surrounding environment. In the real world, the probability of a node to be infected is varying with the network environment; the prior model is not suitable for this scenario. In this paper, we consider an actual epidemic spreading model in which the probability of a node to be infected is related with the number of the infected nodes among its neighbors. We develop an analytical model for this epidemic spreading, named environment aware SIS model (EA-SIS) considering the heterogeneous infection rates, and analytically investigate the epidemic spreading in complex networks. We find that the threshold of EA-SIS is smaller than SIS which means the virus is easier to spread out in the EA-SIS model. In addition, we study several existing immune strategies on the EA-SIS model and propose a novel immune strategy which is based on expected infection time, ETB, of the nodes around the infected nodes for EA-SIS. The simulation results show that the EA-SIS model is more efficient that the SIS model, also, the proposed immune strategy, ETB, is more effective than the local information method and is close to the target immune strategy.

Epidemic spreading in time-varying community networks.

The spreading processes of many infectious diseases have comparable time scale as the network evolution. Here, we present a simple networks model with time-varying community structure, and investigate susceptible-infected-susceptible epidemic spreading processes in this model. By both theoretic analysis and numerical simulations, we show that the efficiency of epidemic spreading in this model depends intensively on the mobility rate q of the individuals among communities. We also find that there exists a mobility rate threshold qc. The epidemic will survive when q > qc and die when q < qc. These results can help understanding the impacts of human travel on the epidemic spreading in complex networks with community structure.

Impact of media coverage on epidemic spreading in complex networks

An SIS network model incorporating the influence of media coverage on transmission rate is formulated and analyzed. We calculate the basic reproduction number R0 by utilizing the local stability of the disease-free equilibrium. Our results show that the disease-free equilibrium is globally asymptotically stable and that the disease dies out if R0 is below 1; otherwise, the disease will persist and converge to a unique positive stationary state. This result may suggest effective control strategies to prevent disease through media coverage and education activities in finite-size scale-free networks. Numerical simulations are also performed to illustrate our results and to give more insights into the dynamical process.

Nonperturbative heterogeneous mean-field approach to epidemic spreading in complex networks

Since roughly a decade ago, network science has focused among others on the problem of how the spreading of diseases depends on structural patterns. Here, we contribute to further advance our understanding of epidemic spreading processes by proposing a nonperturbative formulation of the heterogeneous mean-field approach that has been commonly used in the physics literature to deal with this kind of spreading phenomena. The nonperturbative equations we propose have no assumption about the proximity of the system to the epidemic threshold, nor any linear approximation of the dynamics. In particular, we first develop a probabilistic description at the node level of the epidemic propagation for the so-called susceptible-infected-susceptible family of models, and after we derive the corresponding heterogeneous mean-field approach. We propose to use the full extension of the approach instead of pruning the expansion to first order, which leads to a nonperturbative formulation that can be solved by fixed-point iteration, and used with reliability far away from the epidemic threshold to assess the prevalence of the epidemics. Our results are in close agreement with Monte Carlo simulations, thus enhancing the predictive power of the classical heterogeneous mean-field approach, while providing a more effective framework in terms of computational time.

Modeling the effects of social impact on epidemic spreading in complex networks

We investigate by mean-field analysis and extensive simulations the effects of social impact on epidemic spreading in various typical networks with two types of nodes: active nodes and passive nodes, of which the behavior patterns are modeled according to the social impact theory. In this study, nodes are not only the media to spread the virus, but also disseminate their opinions on the virus—whether there is a need for certain self-protection measures to be taken to reduce the risk of being infected. Our results indicate that the interaction between epidemic spreading and opinion dynamics can have significant influences on the spreading of infectious diseases and related applications, such as the implementation of prevention and control measures against the infectious diseases.

Epidemic spreading in complex networks with spreading delay based on cellular automata

In this paper, based on the cellular automata, we propose a new susceptible-infected-susceptible (SIS) model to study epidemic spreading in the networks with spreading delay. Theoretical analysis and simulation results show that the existence of spreading delay can significantly reduce the epidemic threshold and enhance the risk of outbreak of epidemics. It is also found that both the epidemic prevalence and the propagation velocity increase obviously with the spreading delay increasing. Moreover, the SIS model proposed in this paper describes not only the average propagation tendency of epidemics, but also the dynamic evolution process over the time of epidemics and the probability events such as outbreak and extinction of epidemics, and thus can overcome the limitations of the differential equation model based on mean-field method that describes only the average transmitting tendency of epidemics. Meanwhile, some suggestions of how to effectively control the propagation of epidemics are presented.

Effect of incubation period on epidemic spreading in complex networks

We study the effect of incubation period on epidemic spreading in the Barabasi–Albert scale-free network and the Watts–Strogatz small world network by using a Suspectable-Incubated-Infected-Suspectable model. Our analytical investigations show that the epidemic threshold is independent of incubation period in both networks, which is verified by our large-scale simulation results. We also investigate the effect of incubation period on the epidemic dynamics in a supercritical regime. It is found that with the increase of incubation period Ω, a damped oscillation evolution of ρT (the ratio of persons in incubated state) appears and the time needed to reach a saturation value increases. Moreover, the steady value of ρT increases and approaches to an asymptotic constant with the value of Ω increasing. As a result, the infected ratio ρI decreases with the increase of Ω according to a power law.

Epidemic spreading in complex networks.

The study of epidemic spreading in complex networks is currently a hot topic and a large body of results have been achieved. In this paper, we briefly review our contributions to this field, which includes the underlying mechanism of rumor propagation, the epidemic spreading in community networks, the influence of varying topology, and the influence of mobility of agents. Also, some future directions are pointed out.