两类带有确定潜伏期的SEIS传染病模型的分析

通过研究两类带有确定潜伏期的SEIS传染病模型,发现对种群的常数输入和指数输入会使疾病的传播过程产生本质的差异．对于带有常数输入的情形,找到了地方病平衡点存在及局部渐近稳定的阈值,证明了地方病平衡点存在时一定局部渐近稳定,并且疾病一致持续存在．对于带有指数输入的情形,发现地方病平衡点当潜伏期充分小时是局部渐近稳定的,当潜伏期充分大时是不稳定的．     

Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination

This paper deals with global dynamics of an SIRS epidemic model for infections with non permanent acquired immunity. The SIRS model studied here incorporates a preventive vaccination and generalized non-linear incidence rate as well as the disease-related death. Lyapunov functions are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one, and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one.     

The Influence of Human Behavior on Heroin Dynamics

In this paper, considering the influence of human behavior on heroin abuse, we establish a mathematical model to describe the spread of heroin. When the basic reproduction number is less than one, the heroin-free abuse equilibrium point is globally asymptotically stable. When the basic re- production number is greater than one, the model has a unique heroin abuse equilibrium point which is globally asymptotically stable, and the heroin-free abuse equilibrium point is unstable. Finally, based on the partial rank corre- lation coefficients (PRCCs) and numerical simulations, the dynamic behavior of the model is further revealed. Our results show that human behavior can reduce the heroin abuse level.     

Global stability of the virus dynamics model with intracellular delay and Crowley–Martin functional response

In this paper, a virus dynamics model with intracellular delay and Crowley–Martin functional response is discussed. By constructing suitable Lyapunov functions and using LaSalles invariance principle for delay differential equations, we established the global stability of uninfected equilibrium and infected equilibrium; it is proved that if the basic reproductive number is less than or equal to one, the uninfected equilibrium is globally asymptotically stable; if the basic reproductive number is more than one, the infected equilibrium is globally asymptotically stable. We also discuss the effects of intracellular delay on global dynamical properties by comparing the results with the stability conditions for the model without delay. Copyright © 2013 John Wiley & Sons, Ltd.     

一类具有垂直传染和预防接种的SEIR传染病模型的定性分析

建立并分析了一类对出生时没有被染病母体垂直传染的染病者的新生儿进行免疫接种的SEIR传染病模型.得到了疾病是否灭绝的阈值R0,当R0<1时,无病平衡点全局渐近稳定的.当R0>1时,地方病平衡点局部渐近稳定的,且疾病一致持续生存.     

一类采取隔离措施的非线性传染率传染病模型的全局稳定性

讨论一类采取隔离措施的非线性传染率传染病的数学模型,得到了基本再生数Rθ的表达式,当Rθ<1时,仅存在无病平衡点,是全局渐近稳定的;当Rθ>1时,存在两个平衡点,其中无病平衡点不稳定,地方病平衡点全局渐近稳定.     

Modeling and analysis of the spread of computer virus

Based on a set of reasonable assumptions, we propose a novel dynamical model describing the spread of computer virus. Through qualitative analysis, we give a threshold and prove that (1) the infection-free equilibrium is globally asymptotically stable if the threshold is less than one, implying that the virus would eventually die out, and (2) the infection equilibrium is globally asymptotically stable if the threshold is greater than one. Two numerical examples are presented to demonstrate the analytical results.     

Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with a general contact rate. The model exhibits the traditional threshold behavior. We prove that when the basic reproduction ratio is less than unity, then the disease-free equilibrium is globally asymptotically stable and when the basic reproduction ratio is great than unity, a unique endemic equilibrium exists and is globally asymptotically stable under certain conditions. The stability of equilibria is derived through the use of Lyapunov stability theory and LaSalle’s invariant set theorem. Numerical simulations are provided to illustrate the theoretical results.     

Global stability of a DS–DI epidemic model with age of infection

A model with differential susceptibility, differential infectivity (DS–DI), and age of infection is formulated in this paper. The susceptibles are divided into n groups according to their susceptibilities. The infectives are divided into m groups according to their infectivities. The total population size is assumed constant. Formula for the reproductive number is derived so that if the reproduction number is less than one, the infection-free equilibrium is locally stable, and unstable otherwise. Furthermore, if the reproductive number is less than one, the infection-free equilibrium is globally asymptotically stable. If the reproductive number is greater than one, it is shown that there exists a unique endemic equilibrium which is globally asymptotically stable. This result is obtained through a Lyapunov function.     

一类具有饱和发生率及免疫的时滞SEIR传染病模型的全局渐近稳定性

研究了一类具有饱和发生率及免疫的SEIR,传染病模型、构造适当的Lyapunov泛函并运用时滞微分方程的LaSalle型定理,证明了当基本再生数小于1时,无病平衡点是全局渐进稳定的,当基本再生数大于1时,地方病平衡点存在并且是全局渐近稳定的.     

Threshold dynamics for a cholera epidemic model with periodic transmission rate

A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results.     

Analysis of a viral infection model with immune impairment,intracellular delay and general non-linear incidence rate

In this article we study the dynamical behaviour of a intracellular delayed viral infection with immune impairment model and general non-linear incidence rate. Several techniques, including a non-linear stability analysis by means of the Lyapunov theory and sensitivity analysis, have been used to reveal features of the model dynamics. The classical threshold for the basic reproductive number is obtained: if the basic reproductive number of the virus is less than one, the infection-free equilibrium is globally asymptotically stable and the infected equilibrium is globally asymptotically stable if the basic reproductive number is higher than one.     

Global stability of a virus dynamics model with intracellular delay and CTL immune response

In this paper, the global stability of a virus dynamics model with intracellular delay, Crowley–Martin functional response of the infection rate, and CTL immune response is studied. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global dynamics is established; it is proved that if the basic reproductive number, R₀, is less than or equal to one, the infection‐free equilibrium is globally asymptotically stable; if R₀ is more than one, and if immune response reproductive number, R⁰, is less than one, the immune‐free equilibrium is globally asymptotically stable, and if R⁰ is more than one, the endemic equilibrium is globally asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence

In this paper, the asymptotic behavior of solutions of an autonomous SEIRS epidemic model with the saturation incidence is studied. Using the method of Liapunov–LaSalle invariance principle, we obtain the disease-free equilibrium is globally stable if the basic reproduction number is not greater than one. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions of locally and globally asymptotically stable convergence to an endemic equilibrium are obtained base on the permanence.     

一个有快慢进展的TB模型的全局稳定性分析

建立了一个有快慢进展、接种和治疗的TB模型,定义了模型的基本再生数R0,通过构造Lyapunov函数来研究解的渐近性态.证明了当R01时,无病平衡点是全局渐近稳定的;也证明了当R0>1时,惟一的地方病平衡点是全局渐近稳定的.     

Susceptible-infected-recovered models with natural birth and death on complex networks

This paper proposes two modified susceptible-infected-recovered (SIRS) models on homogenous and heterogeneous networks, respectively. In the study of the homogenous network model, Lyapunov functions are used to study the globally asymptotically stable of the equilibria of the model. It is proved that if the basic reproduction number of the model is less than one, then the disease-free equilibrium is globally asymptotically stable, otherwise, the endemic equilibrium is globally asymptotically stable. In the study of the heterogeneous network model, the existences of the disease-free equilibrium and epidemic equilibrium of the model are discussed. A threshold value is given. It is proved that if the threshold value of the model is less than one, then the disease-free equilibrium is globally asymptotically stable. The simulation examples on the two SIRS models are given. Copyright © 2013 John Wiley & Sons, Ltd.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.     

Cross-diffusion induced Turing instability for a three species food chain model

In this paper, we study a strongly coupled reaction–diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. We first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ODE system. The positive equilibrium solution remains linearly stable for the reaction–diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. We further proved that the positive equilibrium solution is globally asymptotically stable for the reaction–diffusion system without cross-diffusion by constructing a Lyapunov function. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction–diffusion system, hence the instability is driven solely from the effect of cross-diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions.     

带有非线性传染率的传染病模型

对一类带有非线性传染率的SEIS传染病模型,找到了其基本再生数.借助动力系统极限理论,得到当基本再生数小于1时,无病平衡点是全局渐近稳定的,且疾病最终灭绝.当基本再生数大于1时,无病平衡点是不稳定的,而唯一的地方病平衡点是局部渐近稳定的.应用Fonda定理,得到当基本再生数大于1时疾病一致持续存在.