Bifurcations in an epidemic model with constant removal rate of the infectives

An epidemic model with a constant removal rate of infective individuals is proposed to understand the effect of limited resources for treatment of infectives on the disease spread. It is found that it is unnecessary to take such a large treatment capacity that endemic equilibria disappear to eradicate the disease. It is shown that the outcome of disease spread may depend on the position of the initial states for certain range of parameters. It is also shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation, and homoclinic bifurcation.     

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.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

Effects of population density on the spread of disease

The effect of population density on the epidemic outbreak of measles or measles-like infectious diseases was evaluated. Using average-number contacts with susceptible individuals per infectious individual as a measure of population density, an analytical model for the distribution of the nonstationary stochastic process of susceptible contact is presented. A 5-dimensional lattice simulation model of disease spread was used to evaluate the effects of four different population densities. A zero-inflated Poisson probability model was used to quantify the nonstationarity of the contact rate in the stochastic epidemic process. Analysis of the simulation results identified a decrease in a susceptible contact rate from four to three, resulted in a dramatic effect on the distribution of contacts over time, the magnitude of the outbreak, and, ultimately, the spread of disease. © 2001 John Wiley & Sons, Inc.     

Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination

Tuberculosis is a global epidemic disease and almost two billion people across the globe are infected with the tuberculosis bacilli. Many countries like China, Europe and United States has achieved dramatic decrease in TB mortality rate but country like India is still struggling hard to control this epidemic. Jharkhand one of the states of India is highly epidemic toward this disease. We propose a mathematical model to understand the spread of tuberculosis disease in human population for both pulmonary and drug-resistant subjects. A number of new vaccines are currently in development. Keeping in mind, vaccination as one of the treatment for TB patients may be infant or adult in future; an assumption for the transfer of proportion of susceptible population to the vaccination class is considered. Quarantine class is also considered in our epidemic model for multidrug-resistant patients, and it is observed that it may play a vital role for controlling the disease. Threshold and equilibria are obtained and the condition for epidemic under different conditions of threshold is established. Real parametric values of the Jharkhand state are taken into account to simulate the system developed, and the results so obtained validate our analytical results.     

POPULATION EXTINCTION IN DETERMINISTICAND STOCHASTIC DISCRETE‐TIME EPIDEMIC MODELS WITH PERIODIC COEFFICIENTS WITH APPLICATIONS TO AMPHIBIAN POPULATIONS

ABSTRACT. Discrete‐time deterministic and stochastic epidemic models are formulated for the spread of disease in a structured host population. The models have applications to a fungal pathogen affecting amphibian populations. The host population is structured according to two developmental stages, juveniles and adults. The juvenile stage is a post‐metamorphic, nonreproductive stage, whereas the adult stage is reproductive. Each developmental stage is further subdivided according to disease status, either susceptible or infected. There is no recovery from disease. Each year is divided into a fixed number of periods, the first period represents a time of births and the remaining time periods there are no births, only survival within a stage, transition to another stage or transmission of infection. Conditions are derived for population extinction and for local stability of the disease‐free equilibrium and the endemic equilibrium. It is shown that high transmission rates can destabilize the disease‐free equilibrium and low survival probabilities can lead to population extinction. Numerical simulations illustrate the dynamics of the deterministic and stochastic models.     

Spreading and vanishing in a West Nile virus model with expanding fronts

We study a simplified version of a West Nile virus(WNv) model discussed by Lewis et al.(2006),which was considered as a first approximation for the spatial spread of WNv. The basic reproduction number R_0 for the non-spatial epidemic model is defined and a threshold parameter R_0~D for the corresponding problem with null Dirichlet boundary condition is introduced. We consider a free boundary problem with a coupled system, which describes the diffusion of birds by a PDE and the movement of mosquitoes by an ODE. The risk index R_0~F(t) associated with the disease in spatial setting is represented. Sufficient conditions for the WNv to eradicate or to spread are given. The asymptotic behavior of the solution to the system when the spreading occurs is considered. It is shown that the initial number of infected populations, the diffusion rate of birds and the length of initial habitat exhibit important impacts on the vanishing or spreading of the virus. Numerical simulations are presented to illustrate the analytical results.     

Global behavior of an SEIRS epidemic model with time delays

This is a study of dynamic behavior of an SEIRS epidemic model with time delays. It is shown that disease-free equilibrium is globally stable if the reproduction number is not greater than one. When the reproduction number is greater than 1, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Local stability of endemic equilibrium is also discussed.     

Optimal control of a birth and death epidemic process

We employ a birth and death process to describe the spread of an infectious disease through a closed population. Control of the epidemic can be effected at any instant by varying the birth and death rates to represent quarantine and medical care programs. An optimal strategy is one which minimizes the expected discounted losses and costs resulting from the epidemic process and the control programs over an infinite horizon. We formulate the problem as a continuous-time Markov decision model. Then we present conditions ensuring that optimal quarantine and medical care program levels are nonincreasing functions of the number of infectives in the population. We also analyze the dependence of the optimal strategy on the model parameters. Finally, we present an application of the model to the control of a rumor.     

Modelling the Epidemiological Consequences of HIV Infection and AIDS: A Contribution from Operational Research

As part of the worldwide effort involved in modelling the spread of AIDS, nothing has been heard from the operational research community. Yet OR, with its client-oriented modelling philosophy, patently has a role to play in assisting clinicians, epidemiologists and health planners to obtain greater understanding of this staggering threat to public health. Using the methods of system dynamics, a model of the spread of AIDS in the UK homosexual population has been developed. It is implemented on a personal computer, and possesses the capability to capture complex virological and behavioural features of the epidemic whilst portraying the consequences in an easily digested graphical form. Some examples of these results are presented. The model's structure and parameters are also fully described, together with sensitivity tests on the crucial incubation time distribution.     

Relating disease and predation: equilibria of an epidemic model

In this paper, we would like compare the spread of an infectious disease in a population without the influence of a predator and under its influence. We show that it is possible to control an epidemic in a population with the help of predators. Copyright © 2004 John Wiley & Sons, Ltd.     

Global dynamics of SIS models with transport-related infection

To understand the effect of transport-related infection on disease spread, an epidemic model for several regions which are connected by transportation of individuals has been proposed by Cui, Takeuchi and Saito [J. Cui, Y. Takeuchi, Y. Saito, Spreading disease with transport-related infection, J. Theoret. Biol. 239 (2006) 376-390]. Transportation among regions is one of the main factors which affects the outbreak of diseases. The purpose of this paper is the further study of the local asymptotic stability of the endemic equilibrium and the global dynamics of the system. Sufficient conditions are established for global asymptotic stability of the endemic equilibrium. Permanence is also discussed. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. This implies that transport-related infection on disease can make the disease endemic even if all the isolated regions are disease free.     

流行病灭绝时间和费用估计随机模型

本文用具有吸收状态的生灭马氏过程建立了流行病随机模型,研究了灭绝之前生灭过程的分布,发现初始分布是拟平稳分布时,其灭绝时间服从指数分布,并得到了灭绝时间与状态概率的关系式和费用估计的期望值.应用模型给出了一个固定人口为N的流行病灭绝时间和平均费用的数值模拟结果.     

An SIRS epidemic model on a dispersive population

The spatial spread of a disease in an SIRS epidemic model with immunity imparted by subclinical infection on a population has been considered. The incidence rate of infection and the rate of immunization are both of nonlinear type. The dynamics of the infectious disease and its endemicity in local and global sense have been investigated.     

Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate

In this paper, an SVEIS epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated. The role that temporary immunity (natural, disease induced, vaccination induced) plays in the spread of disease, is incorporated in the model. The total host population is bounded and the incidence term is of the Holling-type II form. It is shown that the model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. The global dynamics are completely determined by the basic reproduction number R₀. If R₀<1, the disease-free equilibrium is globally stable which leads to the eradication of disease from population. If R₀>1, a unique endemic equilibrium exists and is globally stable in the feasible region under certain conditions. Further, the transcritical bifurcation at R₀=1 is explored by projecting the flow onto the extended center manifold. We use the geometric approach for ordinary differential equations which is based on the use of higher-order generalization of Bendixson’s criterion. Further, we obtain the threshold vaccination coverage required to eradicate the disease. Finally, taking biologically relevant parametric values, numerical simulations are performed to illustrate and verify the analytical results.     

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.     

一类具有接种和时滞的非自治捕食-食饵的SIR传染病模型(英文)

A non-autonomous SIR epidemic model of prey-predator with vaccination and time delay is investigated in this paper. And an infectious disease is only considered to spread among the prey population. By using comparison principle and Lyapunov functional methods, we obtain the sufficient criteria for the permanence, extinction of infectious disease and the global attractively of the model. Furthermore, some numerical simulations illustrate that the vaccination has a better effect for disease controlling of infective prey.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza

The basic reproduction number and the point of endemic equilibrium are two very important factors in any deterministic compartmental epidemic model as the basic reproduction number and the point of endemic equilibrium represent the nature of disease transmission and disease prevalence respectively. In this article the sensitivity analysis based on mathematical as well as statistical techniques has been performed to determine the importance of the epidemic model parameters. It is observed that 6 out of the 11 input parameters play a prominent role in determining the magnitude of the basic reproduction number. It is shown that the basic reproduction number is the most sensitive to the transmission rate of disease. It is also shown that control of transmission rate and recovery rate of the clinically ill are crucial to stop the spreading of influenza epidemics.     

一类带有种群迁移的SIS传染病模型的全局分析

通过假设同一地区内易感者和染病者具有相同的迁移率系数,建立了一类两地区间种群迁移的SIS传染病模型,得到了地方病平衡点存在的阈值条件,并借助比较定理和极限系统理论证明了无病平衡点和疾病不导致死亡时地方病平衡点的全局稳定性,最后讨论了种群迁移对传染病传播的影响.