一类SARS传染病自治动力系统的稳定性分析

在K-M传染病模型的基础上,进一步考虑易感人群的密度制约以及患病者类的死亡与治愈率等因素,建立了描述SARS传染病的一个新的动力学模型,分析了该模型平衡点的稳定性态．证明了疾病消除平衡点在一定条件下是全局渐进稳定的,而地方病平衡点不是渐近稳定的．得到了该传染病系统在适当条件下为永久持续生存的结果．     

Stability analysis for differential infectivity epidemic models

We present several differential infectivity (DI) epidemic models under different assumptions. As the number of contacts is assumed to be constant or a linear function of the total population size, either standard or bilinear incidence of infection is resulted. We establish global stability of the infection-free equilibrium and the endemic equilibrium for DI models of SIR (susceptible/infected/removed) type with bilinear incidence and standard incidence but no disease-induced death, respectively. We also obtain global stability of the two equilibria for a DI SIS (susceptible/infected/susceptible) model with population-density-dependent birth and death functions. For completeness, we extend the stability of the infection-free equilibrium for the standard DI SIR model previously proposed.     

A QUALITATIVE ANALYSIS OF A VACCINATION MODEL OF HIV/AIDS WITH STAGED PROGRESSION AND AMELIORATION

An epidemic vaccination model with multiple stages of infection is presented and analyzed. The model allows infected individuals to move from advanced stages of infection back to less advanced stages of infection. A threshold parameter which determines the local stability of the disease-free equilibrium is found. The existence and stability of endemic equilibrium for 2-dimensional phase space are analyzed. At the same time, we put forward an optimal vaccine efficacy.     

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.     

一类具有病毒变异的Logistic死亡率SEIR传染病模型研究分析

本文建立了一类具有病毒变异的Logistic死亡率SEIR传染病模型,借助Lyapunov函数和LaSalle''s不变原理,证明了无病平衡点全局稳定性.利用代数方法构造Lyapunov函数,证明了地方病平衡点全局稳定性.另外,通过数值模拟分析了参数对疾病传播的影响.     

一类具有非线性传染率的阶段结构传染病模型

讨论了一类带有非线性传染率的阶段结构传染病模型,得到了各类平衡点存在的阈值条件.借助Hurwitz判据、Lasalle不变集原理和Bendixson法则,找到了疾病消除平衡点,及在无因病死亡时,地方病平衡点全局渐近稳定的充要条件.     

具有接种的确定性和随机SIS流行病模型的全局稳定性

We study the stability of endemic equilibriums of the deterministic and stochastic SIS epidemic models with vaccination. The deterministic SIS epidemic model with vaccination was proposed by Li and Ma(2004), for which some sufficient conditions for the global stability of the endemic equilibrium were given in some earlier works. In this paper, we first prove by Lyapunov function method that the endemic equilibrium of the deterministic model is globally asymptotically stable whenever the basic reproduction number is larger than one. For the stochastic version, we obtain some sufficient conditions for the global stability of the endemic equilibrium by constructing a class of different Lyapunov functions.     

具有免疫接种且总人口规模变化的SIR传染病模型的稳定性

讨论一类具有预防免疫接种且有效接触率依赖于总人口的SIR传染病模型,给出了决定疾病灭绝和持续生存的基本再生数σ的表达式,在一定条件下证明了疾病消除平衡点的全局稳定性,得到了唯一地方病平衡点的存在性和局部渐近稳定性条件.最后研究了具有双线性传染率和标准传染率的两个具体模型,并证明了当σ>1时该模型地方病平衡点的全局渐近稳定性.     

Asymptotic profiles of the endemic equilibrium of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection

A susceptible‐infected‐susceptible (SIS) epidemic reaction‐diffusion model with saturated incidence rate and spontaneous infection is considered. We establish the existence of endemic equilibrium by using a fixed‐point theorem. The global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. We mainly investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium. When the saturated incidence rate tends to infinity, the susceptible and infective distributes in the habitat in a nonhomogeneous way; this result is in strong contrast with the case of no spontaneous infection, where the endemic equilibrium tends to the disease free equilibrium. Our analysis shows that the spontaneous infection can enhance the persistence of an infectious disease and may provide some useful implications on disease control.     

一类带有一般接触率和常数输入的流行病模型的全局分析

借助极限系统理论和构造适当的Liapunov函数,对带有一般接触率和常数输入的SIR型和SIRS型传染病模型进行讨论．当无染病者输入时,地方病平衡点存在的阈值被找到A·D2对相应的SIR模型,关于无病平衡点和地方病平衡点的全局渐近稳定性均得到充要条件;对相应的SIRS模型,得到无病平衡点和地方病平衡点全局渐近稳定的充分条件．当有染病者输入时,模型不存在无病平衡点．对相应的SIR模型,地方病平衡点是全局渐近稳定的;对相应的SIRS模型,得到地方病平衡点全局渐近稳定的充分条件．     

一类具有时滞传染病模型的稳定性

讨论了具有双时滞的SIS传染病模型.研究了一个边界平衡点的全局稳定性和正平衡点的局部稳定性,得到了传染病最终消失和成为地方病的阈值.     

Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment

In this paper, we consider an SIS epidemic reaction–diffusion model with spontaneous infection and logistic source in a heterogeneous environment. The uniform bounds of solutions are established, and the global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. This paper aims to analyze the asymptotic profile of endemic equilibria (when it exists) as the diffusion rate of the susceptible or infected population is small or large. Our results on this new model reveal that varying total population and spontaneous infection can enhance persistence of infectious disease, which may provide some implications on disease control and prediction.     

A Mathematical Model with Delays for Schistosomiasis Japonicum Transmission

A dynamic model of schistosoma japonicum transmission is presented that incorporates effects of the prepatent periods of the different stages of schistosoma into Baxbour＇s model. The model consists of four delay differential equations. Stability of the disease free equilibrium and the existence of an endemic equilibrium for this model are stated in terms of a key threshold parameter. The study of dynamics for the model shows that the endemic equilibrium is globally stable in an open region if it exists and there is no delays, and for some nonzero delays the endemic equilibrium undergoes Hopf bifurcation and a periodic orbit emerges. Some numerical results are provided to support the theoretic results in this paper. These results suggest that prepatent periods in infection affect the prevalence of schistosomiasis, and it is an effective strategy on schistosomiasis control to lengthen in prepatent period on infected definitive hosts by drug treatment （or lengthen in prepatent period on infected intermediate snails by lower water temperature）.     

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.     

Mathematical analysis of an eco-epidemiological predator–prey model with stage-structure and latency

In this paper, an eco-epidemiological predator–prey model with stage structure for the prey and a time delay describing the latent period of the disease is investigated. By analyzing corresponding characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, the disease-free equilibrium and the endemic equilibrium is addressed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are obtained for the global asymptotic stability of the trivial equilibrium, the predator-extinction equilibrium, the disease-free equilibrium and the endemic equilibrium of the model.     

Global asymptotic properties of an SEIRS model with multiple infectious stages

The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.     

Global analysis of a multi-group animal epidemic model with indirect infection and time delay

The transmission mechanism of some animal diseases is complex because of the multiple transmission pathways and multiple-group interactions, which lead to the limited understanding of the dynamics of these diseases transmission. In this paper, a delay multi-group dynamic model is proposed in which time delay is caused by the latency of infection. Under the biologically motivated assumptions, the basic reproduction number $R_0$ is derived and then the global stability of the disease-free equilibrium and the endemic equilibrium is analyzed by Lyapunov functionals and a graph-theoretic approach as for time delay. The results show the global properties of equilibria only depend on the basic reproductive number $R_0$: the disease-free equilibrium is globally asymptotically stable if $R_0\leq 1$; if $R_0>1$, the endemic equilibrium exists and is globally asymptotically stable, which implies time delay span has no effect on the stability of equilibria. Finally, some specific examples are taken to illustrate the utilization of the results and then numerical simulations are used for further discussion. The numerical results show time delay model may experience periodic oscillation behaviors, implying that the spread of animal diseases depends largely on the prevention and control strategies of all sub-populations.     

GLOBAL DYNAMICS OF AN SEIR EPIDEMIC MODEL WITH IMMIGRATION OF DIFFERENT COMPARTMENTS

The SEIR epidemic model studied here includes constant inflows of new susceptibles, exposeds, infectives, and recovereds. This model also incorporates a population size dependent contact rate and a disease-related death. As the infected fraction cannot be eliminated from the population, this kind of model has only the unique endemic equilibrium that is globally asymptotically stable. Under the special case where the new members of immigration are all susceptible, the model considered here shows a threshold phenomenon and a sharp threshold has been obtained. In order to prove the global asymptotical stability of the endemic equilibrium, the authors introduce the change of variable, which can reduce our four-dimensional system to a three-dimensional asymptotical autonomous system with limit equation.     

Optimal control and stability analysis of an online game addiction model with two stages

In this paper, we establish a mathematical model of online game addiction with two stages to research the dynamic properties of it. The existence of all equilibria is obtained, and the basic reproduction number is calculated by the method of next-generation matrix. The global asymptotic stability of disease-free equilibrium (DFE) is proved by comparison principle, and the global asymptotic stability of endemic equilibrium (EE) is proved by constructing an appropriate Lyapunov function. Then we use the Pontryagin's maximum principle to find the optimal solution of the model, so that the number of infected people can be minimized. In numerical simulation, firstly, we validate the global stability of DFE and EE. Secondly, we consider three kind of control measures (treatment, isolation, and education) and divide them into four cases. The models with control and without control are solved numerically by using forward and backward sweep Runge-Kutta method. In order to achieve the best control effect, we suggest that three kind of measures should be used simultaneously according to the optimal control strategy.     

Global stability for a multi-group SIRS epidemic model with varying population sizes

In this paper, by extending well-known Lyapunov function techniques to SIRS epidemic models, we establish sufficient conditions for the global stability of an endemic equilibrium of a multi-group SIRS epidemic model with varying population sizes which has cross patch infection between different groups. Our proof no longer needs such a grouping technique by graph theory commonly used to analyze the multi-group SIR models.