一类具有Allee影响的捕食与被捕食模型

分析并建立了具有Allee影响的捕食与被捕食模型,被捕食者由于自身繁殖或是被捕食而具有了Allee效应,分别讨论了强Allee和弱Allee对被捕食种群的影响,讨论了解的有界性和各平衡点的存在性,并证明了各平衡点的局部渐近稳定性,进一步通过构造适当的Lyapunov函数分析了正平衡点E*的全局渐近稳定性.     

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.     

疾病在食饵中流行的捕食与被捕食模型的分析

分析并建立了疾病在食饵中传播的生态-传染病模型,同时考虑到两种群都受密度制约因素的影响,讨论了模型解的有界性和各平衡点的存在性,利用Routh-Hurwitz判据证明了各平衡点的局部渐进稳定性,通过构造Lyapunov函数分析了各平衡点的全局渐进稳定性,得到了疾病存在与否的充分性条件.     

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.     

一类比率型功能性反应捕食模型的稳定性分析

研究了一类具有比率型功能性反应的捕食模型,对模型进行了定性和稳定性分析,讨论了模型唯一正平衡点的存在条件,以及模型各个平衡点的性态.得到了各个平衡点全局渐近稳定的充分条件.通过绘制模型的相轨线,分析轨线的走向得到了原点全局渐近稳定的条件,并证明了模型不存在非平凡正周期解的条件,通过构造Lyapunov函数得到了模型的唯一正平衡点是全局渐近稳定的结论.     

具接种的随机SIQR流行病模型中平衡点的稳定性分析

基于对具接种的确定性SIQR流行病模型的研究,我们引进了随机扰动,建立随机SIQR流行病模型.通过构造Lyapunov函数,运用Ito公式,得到了随机系统的无病平衡点和地方病平衡点随机渐近稳定的充分条件.进一步猜想当随机扰动的强度较大时,平衡点的稳定性将会被破坏.最后,利用数值仿真验证了所得结论及猜想的正确性.     

一类带有潜伏期和部分免疫的传染病模型的全局分析

通过假设被接种者具有部分免疫,建立了一类具有潜伏期和接种的SEIR传染病模型,借助再生矩阵得到了确定此接种模型动力学行为的基本再生数.当基本再生数小于1时,模型只有无病平衡点;当基本再生数大于1时,除无病平衡点外,模型还有唯一的地方病平衡点.借助Liapunov函数,证明了无病平衡点和地方病平衡点的全局稳定性.     

一类具有非线性发生率的SEIR疾病模型的稳定性和分支分析

考虑医院治疗的因素,给出了一个具有非线性发生率和非线性康复率的SEIR模型,讨论该模型的无病平衡点和地方病平衡点,证明向后分支的出现;进一步通过应用Lyapunov函数给出了它全局稳定性的分析.所得结果改进和扩展了文献中的相应结果.     

一类疾病在食饵中传播的生态-传染病模型分析

分析并建立疾病在食饵中传播的生态-传染病模型,且考虑易感食饵具有常数输入,捕食者种群以Logistic模型增长,讨论了系统解的有界性和各平衡点的存在性,以及局部渐近稳定性,通过构造适当的Lyapunov函数分析了各平衡点的全局渐近稳定性,并运用比较定理证明了系统的持久性.     

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.     

Backward bifurcation of an epidemic model with saturated treatment function

An epidemic model with saturated incidence rate and saturated treatment function is studied. Here the treatment function adopts a continuous and differentiable function which can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical condition is limited. The global dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when such effect is weak. However, it is shown that a backward bifurcation will take place when this delayed effect for treatment is strong. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. And a critical value at the turning point is deduced as a new threshold. Some sufficient conditions for the disease-free equilibrium and the endemic equilibrium being globally asymptotically stable are also obtained. Mathematical results in this paper suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.     

Bifurcation analysis of an SIRS epidemic model with standard incidence rate and saturated treatment function

An epidemic model with standard incidence rate and saturated treatment function of infectious individuals is proposed to understand the effect of the capacity for treatment of infective individuals on the disease spread. The treatment function in this paper is a continuous and differential function which exhibits the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. It is proved that the existence and stability of the disease-free and endemic equilibria for the model are not only related to the basic reproduction number but also to the capacity for treatment of infective individuals. And a backward bifurcation is found when the capacity is not enough. By computing the first Lyapunov coefficient, we can determine the type of Hopf bifurcation, i.e., subcritical Hopf bifurcation or supercritical Hopf bifurcation. We also show that under some conditions the model undergoes Bogdanov-Takens bifurcation. Finally, numerical simulations are given to support some of the theoretical results.     

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.     

An age-structured within-host HIV model with T-cell competition

Recent studies demonstrate that resource competition is an essential component of T-cell proliferation in HIV progression, which can contribute instructively to the disease development. In this paper, we formulate an age-structured within-host HIV model, in the form of a hyperbolic partial differential equation (PDE) for infected target cells coupled with two ordinary differential equations for uninfected T-cells and the virions, to explore the effects of both the T-cell competition and viral shedding variations on the viral dynamics. The basic reproduction number is derived for a general viral production rate which determines the local stability of the infection-free equilibrium. Two special forms of viral production rates, which are extensively investigated in previous literature, the delayed exponential distribution and a step function rate, are further investigated, where the original system can be reduced into systems of delay differential equations. It is confirmed that there exists a unique positive equilibrium for two special viral production rates when the basic reproduction number is greater than one. However, the model exhibits the phenomenon of backward bifurcation, where two positive steady states coexist with the infection-free equilibrium when the basic reproduction number is less than one.     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

Asymptotic stability of positive equilibrium solution for a delayed prey–predator diffusion system

This article is concerned with a delayed Lotka–Volterra two-species prey–predator diffusion system with a single discrete delay and homogeneous Dirichlet boundary conditions. By applying the implicit function theorem, the asymptotic expressions of positive equilibrium solutions are obtained. And then, the asymptotic stability of positive equilibrium solutions is investigated by linearizing the system at the positive equilibrium solutions and analyzing the associated eigenvalue problem. It is demonstrated that the positive equilibrium solutions are asymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than this critical value. In addition, it is also found that the system under consideration can undergo a Hopf bifurcation when the delay crosses through a sequence of critical values. Finally, to verify our theoretical predictions, some numerical simulations are also included.     

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.     

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.     

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.     

一类具有非线性发生率的时滞传染病模型的全局稳定性

充分考虑人口统计效应、疾病的潜伏期与传播规律的复杂性,研究了一类具有非线性发生率的时滞SIRS传染病模型的动力学行为.通过分析对应的线性化近似系统的特征方程,证明了无病平衡点的局部稳定性.利用Lyapunov-LaSalle不变集原理,当基本再生数R₀<1时,证明了无病平衡点是全局渐近稳定的;当R₀>1时,得到了地方病平衡点全局渐近稳定的充分条件.所得结论可为人们有效预防和控制传染病传播提供一定的理论依据.