Scaling behavior of threshold epidemics

We study the classic Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease. In this stochastic process, there are two competing mechanism: infection and recovery. Susceptible individuals may contract the disease from infected individuals, while infected ones recover from the disease at a constant rate and are never infected again. Our focus is the behavior at the epidemic threshold where the rates of the infection and recovery processes balance. In the infinite population limit, we establish analytically scaling rules for the time-dependent distribution functions that characterize the sizes of the infected and the recovered sub-populations. Using heuristic arguments, we also obtain scaling laws for the size and duration of the epidemic outbreaks as a function of the total population. We perform numerical simulations to verify the scaling predictions and discuss the consequences of these scaling laws for near-threshold epidemic outbreaks.     

Variations in epidemic distribution with some characteristic parameters

Considering the spread of an epidemic among a population of mobile agents that can get infected and maintain the infection for a period, we investigate the variation in the homogeneity of the distribution of the epidemic with the remaining time of infection τ, the velocity modulus of the agent v, and the infection rate α. We find that the distribution of the infected cluster size is always exponential. By analyzing the variation of the characteristic infected cluster size coefficient, we show that the inhomogeneity of epidemic distribution increases with an increase in τ for very low v, while it decreases with an increase in τ for moderate v. The epidemic distribution also tends to a homogeneous state as both v and α increase.     

Variations of the epidemic distribution with some characteristic parameters

Considering the epidemic spread among a population of mobile agents which can get infected and maintain the infection for a period, we investigate the variation of the homogeneity of the epidemic distribution with the remaining time of infection τ, the velocity modulus of the agent v, and the infection rate α. We find that the distribution of the infected cluster size is always exponential. By analyzing the variation of the characteristic infected cluster size coefficient, we show that, the inhomogeneity of the epidemic distribution increases with the increase of τ for very low v, while decreases with the increase of τ for moderate v. And the epidemic distribution tends to a homogeneous state as both v and α increase.     

Epidemic dynamics behavior in some bus transport networks

We abstract bus transport networks (BTNs) to complex networks using the Space P approach. First, we select three actual BTNs in three major cities in China, namely, Beijing, Shanghai and Hangzhou. Using the SIS model, we simulate and study the epidemic spreading in the three BTNs. We obtain the density of infected vertices varying with time and the stationary density of infected vertices varying with infection rate. Second, we simulate and study the epidemic spreading in a recently introduced BTN evolution model, the network properties of which correspond well with those of actual BTNs. Third, we use mean-field theory to analyze the epidemic dynamics behavior of the BTN evolution model and obtain the theoretical epidemic threshold of this model. The theoretical value agrees well with the simulation results. Based on the work in this paper, we provide the following possible forecasts for epidemic dynamics in actual BTNs. An actual BTN should have a finite positive epidemic threshold. If the effective infection rate is above this threshold, the epidemic spread in the network and the density of infected vertices finally stabilizes in a balanced state. Below this threshold, the number of infected vertices decays exponentially fast and the epidemic cannot spread on a large scale.     

Steady States in SIRS Epidemical Model of Mobile Individuals

We consider an epidemical model within socially interacting mobile individuals to study the behaviors of steady states of epidemic propagation in 2D networks. Using mean-field approximation and large scale simulations, we recover the usual epidemic behavior with critical thresholds δ_c and p_c below which infectious disease dies out. For the population density δ far above δ_c, it is found that there is linear relationship between contact rate λ and the population density δ in the main. At the same time, the result obtained from mean-field approximation is compared with our numerical result, and it is found that these two results are similar by and large but not completely the same.     

Effects of City-Size Heterogeneity on Epidemic Spreading in a Metapopulation: A Reaction-Diffusion Approach

We review and introduce a generalized reaction-diffusion approach to epidemic spreading in a metapopulation modeled as a complex network. The metapopulation consists of susceptible and infected individuals that are grouped in subpopulations symbolizing cities and villages that are coupled by human travel in a transportation network. By analytic methods and numerical simulations we calculate the fraction of infected people in the metapopulation in the long time limit, as well as the relevant parameters characterizing the epidemic threshold that separates an epidemic from a non-epidemic phase. Within this model, we investigate the effect of a heterogeneous network topology and a heterogeneous subpopulation size distribution. Such a system is suited for epidemic modeling where small villages and big cities exist simultaneously in the metapopulation. We find that the heterogeneous conditions cause the epidemic threshold to be a non-trivial function of the reaction rates (local parameters), the network’s topology (global parameters) and the cross-over population size that separates “village dynamics” from “city dynamics”.     

Effects of disease characteristics and population distribution on dynamics of epidemic spreading among residential sites

We investigate the spreading processes of epidemic diseases among many residential sites for different disease characteristics and different population distributions by constructing and solving a set of integrodifferential equations for the evolutions of position-dependent infected and infective rates, taking into account the infection processes both within a single site and among different sites. In a spreading process the states of an individual include susceptible (S), incubative (I), active (A) and recovered (R) states. Although the transition from S to I mainly depends on the active rate, the susceptible rate and the connectivity among individuals, the transitions from I to A and from A to R are determined by intrinsic characteristics of disease development in individuals. We adopt incubation and infection periods to describe the intrinsic features of the disease. By numerically solving the equations we find that the asymptotic behavior of the spreading crucially depends on the infection period and the population under affection of an active individual. Other factors, such as the structure of network and the popular distribution, play less important roles. The study may provide useful information for analyzing the key parameters affecting the dynamics and the asymptotic behavior.     

Epidemic outbreaks in complex heterogeneous networks

We present a detailed analytical and numerical study for the spreading of infections with acquired immunity in complex population networks. We show that the large connectivity fluctuations usually found in these networks strengthen considerably the incidence of epidemic outbreaks. Scale-free networks, which are characterized by diverging connectivity fluctuations in the limit of a very large number of nodes, exhibit the lack of an epidemic threshold and always show a finite fraction of infected individuals. This particular weakness, observed also in models without immunity, defines a new epidemiological framework characterized by a highly heterogeneous response of the system to the introduction of infected individuals with different connectivity. The understanding of epidemics in complex networks might deliver new insights in the spread of information and diseases in biological and technological networks that often appear to be characterized by complex heterogeneous architectures. Received 20 September 2001 and Received in final form 4 February 2002     

一类SEIRS型流行病的细胞自动机模型

本文通过分析SEIRS类流行病，建立了该类疾病的二维概率细胞自动机模型。在二维中，每个细胞的状态代表易感者，潜伏者，患者，恢复者（或免疫者）和死亡者五个部分个体之一。我们研究了两种情况下，即对潜伏者和患者隔离与不隔离将对疾病转播的影响。经研究我们发现，如果不隔离疾病将持续流行，而及时的隔离则将会减缓疾病的流行。本模型给出了对具体疾病利用细胞自动进行仿真的算法。我们发现当恢复者的失去免疫力大于时，疾病潜伏者和患者的密度序列将在正平衡点附近振荡。最后，我们用计算机对模型进行了仿真。     

Sexually Transmitted Diseases on Bipartite Graph

We study the susceptible infected-susceptible （SIS） epidemic model on bipartite graph. According to the difference of sex conception in western and oriental nations, we construct the Barabasi Albert Barabasi Albert （BA-BA） model and Barabasi-Albert Homogeneity （BA-HO） model for sexually transmitted diseases （STDs）. Applying the rate equation approach, the positive equilibria of both models are given analytically. We find that the ratio between infected females and infected males is distinctly different in both models and the infected density in the BA-HO model is much less than that in the BA-BA model. These results explain that the countries with small ratio have less infected density than those with large ratio. Our numerical simulations verify these theoretical results.     

Epidemic Propagation and Microscopic Structure of Complex Networks

How the microscopic structure of complex network takes influence on the epidemic propagation is investigated. Special attention is paid to the growing network where its average degree changes with time. A formula for the final density of infected individuals is given and is confirmed by numerical simulations. Our results show that the final density of refractory increases nonlinearly with both the average degree of nodes and the adjustable random parameter of network structure.     

Comparison of immunization strategies in geographical networks

The epidemic spread and immunizations in geographically embedded scale-free (SF) and Watts-Strogatz (WS) networks are numerically investigated. We make a realistic assumption that it takes time which we call the detection time, for a vertex to be identified as infected, and implement two different immunization strategies: one is based on connection neighbors (CN) of the infected vertex with the exact information of the network structure utilized and the other is based on spatial neighbors (SN) with only geographical distances taken into account. We find that the decrease of the detection time is crucial for a successful immunization in general. Simulation results show that for both SF networks and WS networks, the SN strategy always performs better than the CN strategy, especially for more heterogeneous SF networks at long detection time. The observation is verified by checking the number of the infected nodes being immunized. We found that in geographical space, the distance preferences in the network construction process and the geographically decaying infection rate are key factors that make the SN immunization strategy outperforms the CN strategy. It indicates that even in the absence of the full knowledge of network connectivity we can still stop the epidemic spread efficiently only by using geographical information as in the SN strategy, which may have potential applications for preventing the real epidemic spread.     

The Role of Caretakers in Disease Dynamics

One of the key challenges in modeling the dynamics of contagion phenomena is to understand how the structure of social interactions shapes the time course of a disease. Complex network theory has provided significant advances in this context. However, awareness of an epidemic in a population typically yields behavioral changes that correspond to changes in the network structure on which the disease evolves. This feedback mechanism has not been investigated in depth. For example, one would intuitively expect susceptible individuals to avoid other infecteds. However, doctors treating patients or parents tending sick children may also increase the amount of contact made with an infecteds, in an effort to speed up recovery but also exposing themselves to higher risks of infection. We study the role of these caretaker links in an adaptive network models where individuals react to a disease by increasing or decreasing the amount of contact they make with infected individuals. We find that, for both homogeneous networks and networks possessing large topological variability, disease prevalence is decreased for low concentrations of caretakers whereas a high prevalence emerges if caretaker concentration passes a well defined critical value.     

Disease spreading in structured scale-free networks

We study the spreading of a disease on top of structured scale-free networks recently introduced. By means of numerical simulations we analyze the SIS and the SIR models. Our results show that when the connectivity fluctuations of the network are unbounded whether the epidemic threshold exists strongly depends on the initial density of infected individuals and the type of epidemiological model considered. Analytical arguments are provided in order to account for the observed behavior. We conclude that the peculiar topological features of this network and the absence of small-world properties determine the dynamics of epidemic spreading. Received 16 October 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: yamir@ictp.trieste.it     

Two different kinds of time delays in a stochastic system

The steady state properties of a noise-driven bistable system are investigated when there are two different kinds of time delays existed in the deterministic and fluctuating forces respectively. Using the approximation of the probability density approach, the delayed Fokker-Planck equation is obtained. The stationary probability distribution (SPD) and the variance of the system are derived. It is found that the time delay τ in the deterministic force can reduce the fluctuations while the time delay β in the fluctuating force can enhance the fluctuations. Numerical simulations are presented and are in good agreement with the approximate theoretical results.     

Scattering of light in an inhomogeneous fluctuating plasma

Scattering of light in an inhomogeneous fluctuating plasma is treated whose mean density distribution is given by a solution of the Korteweg-de Vries equation. The amplitude and phase correlation functions are computed for given mean density and spectral function of the density fluctuations.The non-linear effect of the mean inhomogeneous density distribution on the scattered light leads to beatings in the spectral functions of the amplitude and phase correlations. It is shown that the covariance function for the log-amplitude (which is to first order proportional to the intensity of the scattered light) can be approximated in a large L-region by a power law whose exponent is smaller than in the homogeneous case.     

Impact of heterogeneous human activities on epidemic spreading

Recent empirical observations suggest a heterogeneous nature of human activities. The heavy-tailed inter-event time distribution at the population level is well accepted, while whether the individual acts in a heterogeneous way is still under debate. Motivated by the impact of temporal heterogeneity of human activities on epidemic spreading, this paper studies the susceptible-infected model on a fully mixed population, where each individual acts in a completely homogeneous way but different individuals have different mean activities. Extensive simulations show that the heterogeneity of activities at the population level remarkably affects the speed of spreading, even though each individual behaves regularly. Furthermore, the spreading speed of this model is more sensitive to the change of system heterogeneity compared with the model consisted of individuals acting with heavy-tailed inter-event time distributions. This work refines our understanding of the impact of heterogeneous human activities on epidemic spreading.     

Dynamical response of multi-patch, flux-based models to the input of infected people: Epidemic response to initiated events

The time course of an epidemic can be modeled using the differential equations that describe the spread of disease and by dividing people into “patches” of different sizes with the migration of people between these patches. We used these multi-patch, flux-based models to determine how the time course of infected and susceptible populations depends on the disease parameters, the geometry of the migrations between the patches, and the addition of infected people into a patch. We found that there are significantly longer lived transients and additional “ancillary” epidemics when the reproductive rate R is closer to 1, as would be typical of SARS (Severe Acute Respiratory Syndrome) and bird flu, than when R is closer to 10, as would be typical of measles. In addition we show, both analytical and numerical, how the time delay between the injection of infected people into a patch and the corresponding initial epidemic that it produces depends on R.     

A simple stochastic lattice gas model for H1N1 pandemic. Application to the Italian epidemiological data

We construct a very simple epidemic model for influenza spreading in an age-class-distributed population, by coupling a lattice gas model for the population dynamics with a SIR stochastic model for susceptible, infected and removed/immune individuals. We use as a test case the age-distributed Italian epidemiological data for the novel influenza A(H1N1). The most valuable features of this model are its country-independent and virus-independent structure (few demographic, social and virological data are used to fix some parameters), its large statistic due to a very short run-time machine, and its easy generalizability to include mitigation strategies. In spite of its simplicity, the model presented reproduces the epidemiological Italian data, with sensible predictions for the reproduction number and theoretically interesting results for the generation time distribution.