一类具有时滞的流行病模型分析

讨论了一类带有时滞的SE IS流行病模型,并讨论了阈值、平衡点和稳定性.模型是一个具有确定潜伏期的时滞微分方程模型,在这里我们得到了各类平衡点存在条件的阈值R0;当R0<1时,只有无病平衡点P0,且是全局渐近稳定的;当R0>1时,除无病平衡点外还存在唯一的地方病平衡点Pe,且该平衡点是绝对稳定的.     

具有染病年龄结构的SEIR流行病模型的稳定性

建立和研究了一类具有染病年龄结构的SEIR流行病模型.得到了该模型的基本再生数R0的表达式.证明了当R0<1时,无病平衡点E0不仅局部渐近稳定,而且全局吸引;当R0>1时,无病平衡点E0不稳定,此时存在稳定的地方病平衡点.     

具有潜伏年龄和隔离的SEIQ流行病模型的稳定性

建立和研究了具潜伏带年龄和隔离的SEIQ流行病模型.运用微分方程和积分方程中的理论和方法,得到基本再生数R0的表达式,证明了当R0<1时,存在全局渐近稳定的无病平衡点,当R0>1时,无病平衡点不稳定,此时存在局部渐近稳定的地方病平衡点.     

潜伏期和染病期均具有康复的年龄结构MSEIS流行病模型的稳定性

本文建立和研究了潜伏期和染病期均具有康复的年龄结构MSEIS流行病模型.在总人口规模不变的假设下,得到了决定疾病消亡与否的基本再生数R0的表达式,证明了当R0＜1时,无病平衡点是局部和全局渐近稳定的,此时疾病消失;当R0＞1时,无病平衡点不稳定,此时系统至少存在一个地方病平衡点,并在一定条件下证明了地方病平衡点的局部渐近稳定性.     

一类具有接种疫苗的年龄结构传染病模型分析

建立和研究了一类具有接种疫苗的年龄结构SVWIR传染病模型.在总人口规模不变的条件下,运用微分方程和积分方程中的理论和方法,得到与接种疫苗策略Ψ有关的基本再生数R(Ψ)的表达式,证明了当R(Ψ)1时,无病平衡点是局部渐近稳定的;当R(0)1时,无病平衡点是全局渐近稳定的,此时疾病消亡;当R(Ψ)1时,无病平衡点是不稳定的,此时系统存在地方病平衡点.     

一类具有Logistic增长和病程的SIR模型

研究具有Logistic增长和病程的SIR流行病模型.运用微分、积分方程理论,得到再生数R0<1时,无病平衡点E0是全局渐近稳定的;而当R0>1时,地方病平衡点E*是局部渐近稳定的.     

具有Logistic增长和年龄结构的SIS模型

讨论具有Logistic增长和年龄结构的SIS流行病模型.运用微分、积分方程理论,得到了当再生数R0<1时,无病平衡点E0是全局渐近稳定的;当R0>1时,地方病平衡点E*是局部渐近稳定的.     

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.     

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.     

年龄结构SIQR传染病模型及稳定性

讨论了年龄结构SIQR传染病模型,得出基本再生数R_0和接种再生数R(ψ)的表达式,证明了当R(ψ)1时,无病平衡点局部渐近稳定;当R_01时,无病平衡点全局渐近稳定;当R(ψ)1时,无病平衡点不稳定,此时存在唯一的地方病平衡点,并给出了地方病平衡点的局部渐近稳定性条件,这些条件对于控制疾病的传播具有重要的理论及实际意义,同时用再生数的表达式进一步解释了接种和隔离治疗在控制消除传染病中的作用.     

The dynamic behavior of deterministic and stochastic delayed SIQS model

In this paper, we present the deterministic and stochastic delayed SIQS epidemic models. For the deterministic model, the basic reproductive number $R_{0}$ is given. Moreover, when $R_{0}<1$, the disease-free equilibrium is globally asymptotical stable. When $R_{0}>1$ and additional conditions hold, the endemic equilibrium is globally asymptotical stable. For the stochastic model, a sharp threshold $\overset{\wedge }{R}_{0}$ which determines the extinction or persistence in the mean of the disease is presented. Sufficient conditions for extinction and persistence in the mean of the epidemic are established. Numerical simulations are also conducted in the analytic results.     

具常数输入及饱和发生率的脉冲接种SIQRS传染病模型

研究了具有常数输入及饱和发生率的脉冲接种SIQRS传染病模型,得到了疾病消除与否的阈值R_0=1.证明了当R_01时,系统存在全局渐近稳定的无病周期解;当R_01时,系统一致持久.     

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.     

Birds movement impact on the transmission of West Nile virus between patches

Spatial heterogeneity plays an important role in the distribution and persistence of many infectious disease. In the paper, a multi-patch model for the spread of West Nile virus among $n$ discrete geographic regions is presented that incorporates a mobility process. In the mobility process, we assume that the birds can move among regions, but not the mosquitoes based on scale-space. We show that the movement of birds between patches is sufficient to maintain disease persistence in patches. We compute the basic reproduction number $R_{0}$. We prove that if $R_{0}<1$, then the disease-free equilibrium of the model is globally asymptotically stable. When $R_{0}>1$, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between two regions.     

Global dynamics in a multi-group epidemic model for disease with latency spreading and nonlinear transmission rate

In this paper, we investigate a class of multi-group epidemic models with general exposed distribution and nonlinear incidence rate. Under biologically motivated assumptions, we show that the global dynamics are completely determined by the basic production number $R_0$. The disease-free equilibrium is globally asymptotically stable if $R_0\leq1$, and there exists a unique endemic equilibrium which is globally asymptotically stable if $R_0>1$. The proofs of the main results exploit the persistence theory in dynamical system and a graph-theoretical approach to the method of Lyapunov functionals. A simpler case that assumes an identical natural death rate for all groups and a gamma distribution for exposed distribution is also considered. In addition, two numerical examples are showed to illustrate the results.     

具有种群Logistic增长及饱和传染率的SIS模型的稳定性和Hopf分支

该文研究一类具有种群Logistic增长及饱和传染率的SIS传染病模型,讨论了平衡点的存在性及全局渐近稳定性,得到疾病消除的阈值就是基本再生数$R_{0}=1$. 证明了,当$R_{0}<1$ 时,无病平衡点全局渐近稳定;当$R_{0}>1$ 且$\alpha K\leq 1$ 时,正平衡点全局渐近稳定;当$R_{0}>1$ 且$\Delta ={0}$ 时,系统在正平衡点附近发生Hopf分支;当$R_{0}>1$ 且$\Delta <{0}$ 时,系统在正平衡点外围附近存在唯一稳定的极限环.     

Modelling and stability of epidemic model with free-living pathogens growing in the environment

To understand the impact of free-living pathogens (FLP) on the epidemics, an epidemic model with FLP is constructed. The global dynamics of our model are determined by the basic reproduction number $R_0$. If $R_0<1$, the disease free equilibrium is globally asymptotically stable, and if $R_0>1$, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.     

Global dynamical analysis of a heroin epidemic model on complex networks

In this paper, a heroin epidemic model on complex networks is proposed. By the next generation matrix, the basic reproduction number $R_0$ is obtained. If $R_0<1$, then the drug-free equilibrium is globally asymptotically stable. If $R_0>1$, there is an unique endemic equilibrium and it is also globally asymptotically stable. Our results show that if the degree of the network is large enough, the drug transmission always spreads. Sensitivity analysis of the basic reproduction number with the various parameters in the model are carried out to verify the important effects for control the drug transmission. Some simulations illustrate our theoretical results     

Dynamical Analysis of a Delayed SIQS Epidemic Model on Scale-free Networks

In this paper, we establish a novel delayed SIQS epidemic model on scale-free networks, where time delay represents the average quarantine period. Through mathematical analysis, we present the basic reproduction number $R_{0}$. Then, we provide the global asymptotical stability of the disease-free equilibrium and the local asymptotical stability of the endemic equilibrium. Finally, we perform numerical simulations to verify the correctness of the main results and analyze the sensitivity of parameters. Our research shows that when $R_0>1$, lengthening the quarantine period can slow the spread of the disease and reduce the number of infected individuals.     

Dynamic behavior of a delay cholera model with constant infectious period

In this paper, a delay cholera model with constant infectious period is investigated. By analyzing the characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium of the model is established. It is proved that if the basic reproductive number $\mathcal{R}_0>1$, the system is permanent. If $\mathcal{R}_0<1$, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the disease-free equilibrium. If $\mathcal{R}_0>1$, also by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.