一类有迁移的SIS流行病模型

研究一类种群有迁移的流行病模型,得到了这类模型的基本再生数R0,证明了R0＜1无病平衡点是局部渐近稳定的,而当R0＞1时无病平衡点是不稳定的.进一步讨论了疾病持续存在与无病平衡点和地方病平衡点全局稳定的条件.     

一类具有时滞和接种疫苗年龄的SIS模型

研究具有时滞和接种疫苗年龄的SIS流行病模型.运用微分、积分方程理论,得到再生数R(ψ)<1,且γτ1时,地方病平衡点E*的存在性.     

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.     

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.     

具有空间效应及一般感染率的时滞病毒模型分析

考虑到时滞效应及空间扩散的影响,建立了一个具有一般传染率的病毒感染仓室模型,分析了模型的动力学性态.定义了模型的基本再生数R_0,讨论了平衡点的存在性,并通过构造Lyapunov函数分析了平衡点的稳定性.结果表明,当R_01时,无病平衡点全局渐近稳定;当R_0 1时,无病平衡点不稳定且地方病平衡点在一定条件下全局渐近稳定.同时,以Beddington-DeAngelis感染率为例的数值模拟进一步验证和扩展了理论结果.     

具有时滞的媒介传播模型的稳定性分析

研究具有时滞的媒介传播的传染病模型.确定了疾病是否流行的阈值R0.当R0≤1时,通过构造Lyapunov泛函证明了系统无病平衡点的全局渐近稳定性.     

一类具有logistic增长的SIR时滞传染病模型的稳定性分析

研究了一类具有logistic增长的时滞SIR传染病模型,得到了决定疾病爆发和消亡的阈值R_0,证明了当R_01时,对于任意的时滞τ,无病平衡点都是全局渐近稳定的,此时疾病消亡;当R_01时,系统会出现一个临界值τ_0,当ττ_0时,地方病平衡点不稳定;当ττ_0,且满足给定的条件时,地方病平衡点局部渐近稳定;当τ=τ_0时,系统发生Hopf分支.通过数值模拟,验证了上述结论的正确性,且做了参数的敏感度分析.     

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.     

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.     

具有治疗和Beddington-DeAngelis发生率的时滞肺结核传染病模型研究

建立了一类具有疾病治疗和Beddington-DeAngelis发生率的时滞肺结核传染病模型,给出了基本再生数R0的表达式.当R0<1时,无病平衡点是全局渐近稳定的,而当R0> 1时,地方病平衡点是全局渐近稳定的.数值模拟演示了所得的理论结果的有效性,研究发现考虑肺结核快速发展阶段的潜伏期时滞及在此期间的发病率能够更好地模拟肺结核病的动力学行为,提高肺结核病的治愈率可以更好的预防和控制肺结核病的传播.     

Mathematical Modeling and Analysis of an Epidemic Model with Quarantine, Latent and Media Coverage

Epidemic models are very important in today''s analysis of diseases. In this paper, we propose and analyze an epidemic model incorporating quarantine, latent, media coverage and time delay. We analyze the local stability of either the disease-free and endemic equilibrium in terms of the basic reproduction number $\mathcal{R}_{0}$ as a threshold parameter. We prove that if $\mathcal{R}_{0}<1,$ the time delay in media coverage can not affect the stability of the disease-free equilibrium and if $\mathcal{R}_{0}>1$, the model has at least one positive endemic equilibrium, the stability will be affected by the time delay and some conditions for Hopf bifurcation around infected equilibrium to occur are obtained by using the time delay as a bifurcation parameter. We illustrate our results by some numerical simulations such that we show that a proper application of quarantine plays a critical role in the clearance of the disease, and therefore a direct contact between people plays a critical role in the transmission of the disease.     

Stability of a delayed SIRS epidemic model with a nonlinear incidence rate

In this paper, an SIRS epidemic model with a nonlinear incidence rate and a time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproductive number R₀<1, the disease-free equilibrium is globally asymptotically stable. If R₀>1, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the endemic equilibrium.     

An SIS epidemic model with diffusion

The aim of this paper is to study the dynamics of an SIS epidemic model with diffusion. We first study the well-posedness of the model. And then, by using linearization method and constructing suitable Lyapunov function, we establish the local and global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Furthermore, in view of Schauder fixed point theorem, we show that the model admits traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium when R_0 1 and c c~*. And also, by virtue of the two-sided Laplace transform, we prove that the model has no traveling wave solution connecting the two equilibria when R_0 1 and c ∈ [0, c~*).     

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.     

具有媒体饱和效应影响的时滞SIS模型研究

媒体报道对疾病的预防和控制有着重要的作用,其可以减少人们感染疾病的机会.通过建立具有媒体饱和的传染病时滞模型来刻画媒体报道对感染率的影响,首先计算出无病平衡点和当R_01时存在唯一的地方病平衡点;其次,分析了平衡点的稳定性,并得到当参数满足一定条件时,时滞τ超过临界值τ_0,地方病平衡点处会出现Hopf分支;最后,通过数值模拟来验证理论分析.     

具有标准发生率和因病死亡率的离散SIS传染病模型的全局稳定性分析

主要研究了具有标准发生率和因病死亡率的离散SIS传染病模型的动力学性质,利用构造Lyapunov函数,得到模型无病平衡点和地方性平衡点的全局稳定性,即无病平衡点是全局渐近稳定的当且仅当基本再生数R_0≤1,地方病平衡点是全局渐近稳定的当且仅当R_0>1.     

GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA

An SIS epidemic model with a simple vaccination is investigated in this article. The efficiency of vaccine, the disease-related death rate and population dynamics are also considered in this model. The authors find two threshold R0 and Rc (Rc may not exist). There is a unique endemic equilibrium for R0 > 1 or Rc = R0; there are two endemic equilibria for Rc < R0 < 1; and there is no endemic equilibrium for R0 < Rc < 1. When Rc exists, there is a backward bifurcation from the disease-free equilibrium for R0 = 1. They analyze the stability of equilibria and obtain the globally dynamic behaviors of the model. The results acquired in this article show that an accurate estimation of the efficiency of vaccine is necessary to prevent and controll the spread of disease.     

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.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.