Global properties of a two-scale network stochastic delayed human epidemic dynamic model

Complex population structure and the large-scale inter-patch connection human transportation underlie the recent rapid spread of infectious diseases of humans. Furthermore, the fluctuations in the endemicity of the diseases within patch dwelling populations are closely related with the hereditary features of the infectious agent. We present an SIR delayed stochastic dynamic epidemic process in a two-scale dynamic structured population. The disease confers temporary natural or infection-acquired immunity to recovered individuals. The time delay accounts for the time-lag during which naturally immune individuals become susceptible. We investigate the stochastic asymptotic stability of the disease free equilibrium of the scale structured mobile population, under environmental fluctuations and the impact on the emergence, propagation and resurgence of the disease. The presented results are demonstrated by numerical simulation results.     

A network modelling approach for assessing the efficiency of prevention strategies

Survey data and a simulation model based on a stochastic pair formation process are used to construct networks of sexual contacts. We model heterosexual partnerships which can be steady or casual depending on their average duration. Transmission of an infectious disease can take place in pairs of a susceptible and an infected individual. We study networks of sexual contacts accumulated during 1 year for different types of mixing patterns. The networks are constructed on the basis of data from a survey in The Netherlands. We analyze the network structure for different mixing patterns and investigate the relationship between network structure and disease spread; furthermore we study the effect of prevention measures on the structure of the network.     

Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases

In this paper, a non-linear mathematical model for the effects of awareness programs on the spread of infectious diseases such as flu has been proposed and analyzed. In the modeling process it is assumed that disease spreads due to the contact between susceptibles and infectives only. The growth rate of awareness programs impacting the population is assumed to be proportional to the number of infective individuals. It is further assumed that due to the effect of media, susceptible individuals form a separate class and avoid contact with the infectives. The model is analyzed by using stability theory of differential equations. The model analysis shows that the spread of an infectious disease can be controlled by using awareness programs but the disease remains endemic due to immigration. The simulation analysis of the model confirms the analytical results.     

基于Kolmogorov前向方程评估甲型H1N1流感疫情的动态变化北大核心CSCD

基于个体水平的传染病模型可以揭示随机性在传染病疫情防控中的重要作用.研究此类模型的普遍方法是通过事件驱动的、大量重复的随机模拟来确定预测变量的范围.而基于Kolmogorov前向方程(KFE)研究个体水平的传染病模型,不仅不需要大量的重复模拟来确定预测变量的范围,而且可以同时考虑每种状态发生的概率.因此,基于2009年西安市第八医院甲型H1N1流感数据,建立了基于社交网络的个体决策心理模型,以确定行为改变率;进一步地,为得到传染病传播过程中各状态的概率分布,基于改进的个体SIR模型,通过Markov过程推导出KFE.结果表明:通过数值求解KFE可以得到整个爆发过程中每种状态发生的概率分布、最严重的时间段及相应的概率,从而能更快、更准确地了解甲型H1N1疫情的传播过程,因此有助于高效地进行甲型H1N1疫情防控.     

Applications of the Variance of Final Outbreak Size for Disease Spreading in Networks

The assumption that all susceptible individuals are equally likely to acquire the disease during an outbreak (by direct contact with an infective individual) can be relaxed by bringing into the disease spread model a contact structure between individuals in the population. The structure is a random network or random graph that describes the kind of contacts that can result in transmission. In this paper we use an approach similar to the approaches of Andersson (Ann Appl Probab 8(4):1331–1349, 1998) and Newman (Phys Rev E 66:16128, 2002) to study not only the expected values of final sizes of small outbreaks, but also their variability. Using these first two moments, a probability interval for the outbreak size is suggested based on Chebyshev’s inequality. We examine its utility in results from simulated small outbreaks evolving in simulated random networks. We also revisit and modify two related results from Newman (Phys Rev E 66:16128, 2002) to take into account the important fact that the infectious period of an infected individual is the same from the perspective of all the individual’s contacts. The theory developed in this area can be extended to describe other “infectious” processes such as the spread of rumors, ideas, information, and habits.     

Threshold of disease transmission in a patch environment

An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that reproduction number is a threshold of the uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence the persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when the population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when the population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen. It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.     

A nonlinear HIV/AIDS model with contact tracing

A nonlinear mathematical model is proposed and analyzed to study the effect of contact tracing on reducing the spread of HIV/AIDS in a homogeneous population with constant immigration of susceptibles. In modeling the dynamics, the population is divided into four subclasses of HIV negatives but susceptibles, HIV positives or infectives that do not know they are infected, HIV positives that know they are infected and that of AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives and all infectives move with constant rates to develop AIDS. The model is analyzed using the stability theory of differential equations and numerical simulation. The model analysis shows that contact tracing may be of immense help in reducing the spread of AIDS epidemic in a population. It is also found that the endemicity of infection is reduced when infectives after becoming aware of their infection do not take part in sexual interaction.     

具有交通流量异质性的网络传染病模型研究

主要研究了异质集合种群网络上的移动和扩散行为对疾病传播的影响.针对现实社会中的网络所具有的异质性,分析了影响城市疾病传播的主要因素为网络拓扑结构以及城市交通流量异质性,建立了依赖于交通流量移出率的传染病动力学模型.通过分析模型的无病平衡点以及正平衡点的存在及其稳定性,发现人口流动会使交通较发达的城市拥有更多的染病者,更容易促使疾病的爆发.     

Epidemic outbreaks on networks with effective contacts

In real-world networks of disease transmission, the incidence of infection among individuals conforms to a certain fixed probability of effective contact between them, which must meet some necessary conditions for the disease to continue to spread. Based on susceptible/infective/removed (SIR) models in homogeneous or heterogeneous networks, we find that these models evolve dynamically just like in networks without connectivity fluctuations if all the susceptible individuals are supposed to have the same effective contact. This means that effectively heterogeneous contacts play a striking role in epidemic dynamics. To go a step further, we introduce the effective contact function (ECF) into models and present an analytical and numerical study for the threshold and dynamical behaviors of epidemic incidence. The power-law and proportional ECFs are considered, and, we demonstrate analytically that the epidemic incidence is generally a monotone decreasing function of the epidemic threshold and increasing function of the number of effective contacts. Certain exceptional cases are also discussed. This tells us that we cannot always focus on the threshold to evaluate the extent of epidemic outbreaks.     

Modeling the spread of an infectious disease with bacteria and carriers in the environment

An SIS model with immigration for the spread of an infectious disease with bacteria and carriers in the environment is proposed and analyzed. It is assumed that susceptibles get infected directly by infectives as well as by their contacts with bacteria discharged by infectives in the environment. The growth rate of density of bacteria is assumed to be proportional to the density of infectives and decreases naturally as well as by bacterial interactions with susceptibles and carriers. The carrier population density is considered to follow the logistic model and grows due to conducive human population density related factors. It is assumed further that the number of bacteria transported by carriers to susceptibles is proportional to densities of both bacteria and carriers. The model study shows that the spread of the infectious disease increases due to growth of bacteria and carriers in the environment and disease becomes more endemic due to immigration.     

ANALYSIS OF A SUSCEPTIBLE-EXPOSED-INFECTED EPIDEMIC MODEL WITH RANDOM PERTURBATION AND VARYING POPULATION SIZE

In this paper, we study a type of susceptible-exposed-infected (SEI) epidemic model with varying population size and introduce the random perturbation of the constant contact rate into the SEI epidemic model due to the universal existence of fluctuations. Under some moderate conditions, the density of the exposed and the infected individuals exponentially approaches zero almost surely are derived. Furthermore, the stochastic SEI epidemic model admits a stationary distribution around the endemic equilibrium, and the solution is ergodic. Some numerical simulations are carried out to demonstrate the efficiency of the main results.     

Complicated endemics of an SIRS model with a generalized incidence under preventive vaccination and treatment controls

In this paper, we propose and study an SIRS epidemic model that incorporates: a generalized incidence rate function describing mechanisms of the disease transmission; a preventive vaccination in the susceptible individuals; and different treatment control strategies depending on the infective population. We provide rigorous mathematical results combined with numerical simulations of the proposed model including: treatment control strategies can determine whether there is an endemic outbreak or not and the number of endemic equilibrium during endemic outbreaks, in addition to the effects of the basic reproduction number; the large value of the preventive vaccination rate can reduce or control the spread of disease; and the large value of the psychological or inhibitory effects in the incidence rate function can decrease the infective population. Some of our interesting findings are that the treatment strategies incorporated in our SIRS model are responsible for backward or forward bifurcations and multiple endemic equilibria; and the infective population decreases with respect to the maximal capacity of treatment. Our results may provide us useful biological insights on population managements for disease that can be modeled through SIRS compartments.     

Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogenous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school. Supplementary material for this article is available online.     

SIR epidemic models with general infectious period distribution

We show how epidemics in which individuals’ infectious periods are not necessarily exponentially distributed may be naturally modelled as piecewise deterministic Markov processes. For the standard susceptible–infective–removed (SIR) model, we exhibit a family of martingales which may be used to derive the joint distribution of the number of survivors of the epidemic and the area under the trajectory of infectives. We also show how these results may be extended to a model in which the rate at which an infective generates infectious contacts may be an arbitrary function of the number of susceptible individuals present.     

Simulating influenza pandemic dynamics with public risk communication and individual responsive behavior

Individual responsive behavior to an influenza pandemic has significant impacts on the spread dynamics of this epidemic. Current influenza modeling efforts considering responsive behavior either oversimplify the process and may underestimate pandemic impacts, or make other problematic assumptions and are therefore constrained in utility. This study develops an agent-based model for pandemic simulation, and incorporates individual responsive behavior in the model based on public risk communication literature. The resultant model captures the stochastic nature of epidemic spread process, and constructs a realistic picture of individual reaction process and responsive behavior to pandemic situations. The model is then applied to simulate the spread dynamics of 2009 H1N1 influenza in a medium-size community in Arizona. Simulation results illustrate and compare the spread timeline and scale of this pandemic influenza, without and with the presence of pubic risk communication and individual responsive behavior. Sensitivity analysis sheds some lights on the influence of different communication strategies on pandemic impacts. Those findings contribute to effective pandemic planning and containment, particularly at the beginning of an outbreak.     

An epidemic model to evaluate the homogeneous mixing assumption

Many epidemic models are written in terms of ordinary differential equations (ODE). This approach relies on the homogeneous mixing assumption; that is, the topological structure of the contact network established by the individuals of the host population is not relevant to predict the spread of a pathogen in this population. Here, we propose an epidemic model based on ODE to study the propagation of contagious diseases conferring no immunity. The state variables of this model are the percentages of susceptible individuals, infectious individuals and empty space. We show that this dynamical system can experience transcritical and Hopf bifurcations. Then, we employ this model to evaluate the validity of the homogeneous mixing assumption by using real data related to the transmission of gonorrhea, hepatitis C virus, human immunodeficiency virus, and obesity.     

Impacts of TV and radio advertisements on the dynamics of an infectious disease: A modeling study

TV and radio advertisements are widely acknowledged as important interventions in raising issues of public health care and play promising role to control the infection through propagating awareness among the individuals. In this paper, a nonlinear susceptible‐infected‐susceptible (SIS) model is proposed and analyzed to see the impacts of TV and radio advertisements on the spread of influenza epidemic. In the model formulation, it is assumed that the susceptible individuals contract infection through the direct contact with infected individuals. The information regarding the protection against the disease is propagated via TV and radio advertisements, and their growth rates are assumed to be proportional to the fraction of infected individuals. However, the growth rate of TV advertisements decreases with the increase in number of aware individuals. The information broadcasted through TV and radio advertisements induces behavioral changes among the susceptible individuals, and they form an isolated aware class. The epidemiological feasible equilibria, their stability properties, and direction of bifurcation are discussed. The expression for modified basic reproduction number is obtained. The model analysis shows that the dissemination rate of awareness among susceptible individuals due to TV and radio advertisements and baseline number of TV and radio advertisements have potential to reduce the epidemic peak and, thus, control the spread of infection. Further, the analytical findings are well supported through numerical simulation.