具有饱和发生率和免疫响应的病毒感染模型的稳定性分析改进

通过构造Lyapunov函数,改进了具有饱和发生率和免疫响应的病毒感染数学模型的稳定性分析,得到了当病毒感染的基本再生数R_01时,无病平衡点全局渐进稳定;当R_01时,在一定条件下,免疫耗竭平衡点和持续带毒平衡点全局渐近稳定的结论.     

具有Crowley-Martin功能反应和CTL免疫反应的病毒动力学模型的全局稳定性（英文）

本文讨论了一类具有Growley-Martin功能反应和CTL免疫反应的病毒动力学模型的全局稳定性.利用Lyapunov函数和LaSalle不变原理证明:当基本再生数R_0≤1时,无病平衡点全局渐近稳定;当基本再生数R_01且免疫基本再生数R_0≤1时,免疫平衡点全局渐近稳定;当R_01时,地方病平衡点全局渐近稳定.     

一类具有免疫反应的HollingⅡ型发生率病毒动力学模型

建立并讨论了一类具有潜伏期、抗体免疫反应和CTL免疫反应的Holling II型发生率病毒动力学模型.定义了决定这个模型动力学性质的五个阈值,借助适当的Lyapunov函数得到:当R_(01)≤1时,无病平衡点全局渐近稳定,病毒被清除;当R_(01)1,R_(02)≤1,R_(03)≤1时,无免疫平衡点全局渐近稳定;当R_(02)1,R_(04)≤1时,CTL免疫主导平衡点全局渐近稳定;当R_(03)〉1,R_(04)≤1时,抗体免疫主导平衡点全局渐近稳定;当R_(04)1,R′_(04)1时,正平衡点全局渐近稳定.     

具有非线性发生率的SEIS模型的定性分析

研究了一类具有非线性发生率的SEIS传染病模型,给出了其基本再生数R_0.当R_01时,无病平衡点是全局渐近稳定的;当R_0〉1时,得到了唯一的地方病平衡点是全局渐近稳定的条件.     

具有非单调传染率的SIQR传染病模型的稳定性分析

该文研究了一类具有非单调传染率的SIQR传染病模型,讨论了平衡点的存在性,运用特征值法、Hurwit判据和极限方程理论证明了当阈值R_01时无病平衡点是全局渐近稳定的;当R_01时,无病平衡点是不稳定的.并采用Lipunov函数法和Lasalle不变性原理证明了地方性平衡点是全局渐近稳定的.最后进行了数值模拟,验证了理论结果的有效性.     

霍乱的动力学行为分析

与通常的SIR类传染病模型有所不同,本文中所研究的模型考虑霍乱菌受环境和时滞的影响.在文中,当基本再生数R_01时,利用Lyapunov泛函,证明了无病平衡点的全局渐近稳定性.当R_01时,证明正平衡点是局部渐近稳定的和持久的.     

改进的病毒感染动力学模型的稳定性分析

提出一个改进的乙肝病毒感染动力学模型.本模型有三个平衡点.对于HBV感染人群,三个平衡点分别对应于三类人群：感染病毒后自愈人群、健康带毒人群、慢性乙肝患者人群.证明了当模型导出的基本复制数R_0〈1时病毒清除平衡点具有局部稳定性和全局渐近稳定性,当1〈R_0〈k_3d/（k_2λ-k_3a）＋1时持续带毒平衡点具有局部稳定性.     

一类具有非线性传染力的传染病模型的稳定性分析

研究了一类具有非线性发生率的急慢性阶段传染病模型,得到了确定模型全局动力性的阀值参数-基本再生数R_0,证明了R_01时,无病平衡点是全局渐近稳定的,疾病消失;若R_01,则存在地方病平衡点且是稳定结点,并证明了一定条件下地方病平衡点是全局渐近稳定的,疾病将蔓延.     

一类具有治愈率的HBV病毒感染模型的全局稳定性

本文研究了一类具有治愈率的HBV病毒感染模型的动力学性质.通过分析,证明了当基本再生数小于1时,未感染病毒平衡点全局渐近稳定,病毒在宿主体内被清除.当基本再生数大于1时,病毒在宿主体内持续生存,同时给出了病毒感染平衡点全局渐近稳定和存在轨道稳定周期解的充分条件.     

一类具有潜伏时滞和非线性疾病发生率的SEIRS传染病模型

本文提出一类具有潜伏时滞和非线性疾病发生率的SEIRS传染病模型,通过分析对应的特征方程,运用时滞微分方程的稳定性理论得出:当基本再生数R_01时无病平衡点处的局部渐近稳定性,R_0 1时地方病平衡点处的局部渐近稳定性.通过构造Lyapunov泛函,运用LaSalle's不变集原理得到:当基本再生数R_0≤1时无病平衡点处的全局渐近稳定性;通过比较方法得到R_01时系统的一致持久性     

具有非线性传染率的SEIQR流行病模型的全局稳定性

研究了一类具有非线性传染率的SEIQR流行病数学模型,得到了疾病灭绝与否的基本再生数R_O,当R_O≤1时,无病平衡点全局渐近稳定,且疾病最终消亡;当R_O>1时,惟一地方病平衡点全局渐近稳定.     

Global Properties of <Emphasis Type="Italic">SIR</Emphasis> and <Emphasis Type="Italic">SEIR</Emphasis> Epidemic Models with Multiple Parallel Infectious Stages

We consider global properties of compartment SIR and SEIR models of infectious diseases, where there are several parallel infective stages. For instance, such a situation may arise if a fraction of the infected are detected and treated, while the rest of the infected remains undetected and untreated. We assume that the horizontal transmission is governed by the standard bilinear incidence rate. The direct Lyapunov method enables us to prove that the considered models are globally stable: There is always a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number R ₀, this state can be either endemic (R ₀>1), or infection-free (R ₀≤1).     

Global dynamics of a staged-progression model with amelioration for infectious diseases

We analyze the global dynamics of a mathematical model for infectious diseases that progress through distinct stages within infected hosts with possibility of amelioration. An example of such diseases is HIV/AIDS that progresses through several stages with varying degrees of infectivity; amelioration can result from a host's immune action or more commonly from antiretroviral therapies, such as highly active antiretroviral therapy. For a general n-stage model with constant recruitment and bilinear incidence that incorporates amelioration, we prove that the global dynamics are completely determined by the basic reproduction number R(0). If R(0)≤1, then the disease-free equilibrium P(0) is globally asymptotically stable, and the disease always dies out. If R(0)>1, P(0) is unstable, a unique endemic equilibrium P* is globally asymptotically stable, and the disease persists at the endemic equilibrium. Impacts of amelioration on the basic reproduction number are also investigated.     

Global stability of an SEIS epidemic model with recruitment and a varying total population size

This paper considers an SEIS epidemic model that incorporates constant recruitment, disease-caused death and disease latency. The incidence term is of the bilinear mass-action form. It is shown that the global dynamics is completely determined by the basic reproduction number R(0). If R(0)1, a unique endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium.     

An age-structured within-host HIV-1 infection model with virus-to-cell and cell-to-cell transmissions

In this paper, a within-host HIV-1 infection model with virus-to-cell and direct cell-to-cell transmission and explicit age-since-infection structure for infected cells is investigated. It is shown that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number. By analysing the corresponding characteristic equations, the local stability of an infection-free steady state and a chronic-infection steady state of the model is established. By using the persistence theory in infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if the basic reproduction number is less than unity, the infection-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the chronic-infection steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.     

Global dynamics of an SEIR epidemic model with saturating contact rate

Heesterbeek and Metz [J. Math. Biol. 31 (1993) 529] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number R0 is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If R0 < or =1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium is also analyzed.     

The Ross-Macdonald model in a patchy environment

We generalize to n patches the Ross-Macdonald model which describes the dynamics of malaria. We incorporate in our model the fact that some patches can be vector free. We assume that the hosts can migrate between patches, but not the vectors. The susceptible and infectious individuals have the same dispersal rate. We compute the basic reproduction ratio R(0). We prove that if R(0)1, then the disease-free equilibrium 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 minus the disease-free equilibrium.     

Dynamics of an HIV-1 therapy model of fighting a virus with another virus

In this paper, we rigorously analyse an ordinary differential equation system that models fighting the HIV-1 virus with a genetically modified virus. We show that when the basic reproduction ratio ?(0)<1, then the infection-free equilibrium E (0) is globally asymptotically stable; when ?(0)>1, E (0) loses its stability and there is the single-infection equilibrium E (s). If ?(0)∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium E (s) that is globally asymptotically stable, while when ?(0)>1+δ, E (s) becomes unstable and the double-infection equilibrium E (d) comes into existence. When ?(0) is slightly larger than 1+δ, E (d) is stable and it loses its stability via Hopf bifurcation when ?(0) is further increased in some ways. Through a numerical example and by applying a normal form theory, we demonstrate how to determine the bifurcation direction and stability, as well as the estimates of the amplitudes and the periods of the bifurcated periodic solutions. We also perform numerical simulations which agree with the theoretical results. The approaches we use here are a combination of analysis of characteristic equations, fluctuation lemma, Lyapunov function and normal form theory.     

Global Stability of Equilibria in a Two-Sex HPV Vaccination Model

Human papillomavirus (HPV) is the primary cause of cervical carcinoma and its precursor lesions, and is associated with a variety of other cancers and diseases. A prophylactic quadrivalent vaccine against oncogenic HPV 16/18 and warts-causing genital HPV 6/11 types is currently available in several countries. Licensure of a bivalent vaccine against oncogenic HPV 16/18 is expected in the near future. This paper presents a two-sex, deterministic model for assessing the potential impact of a prophylactic HPV vaccine with several properties. The model is based on the susceptible-infective-removed (SIR) compartmental structure. Important epidemiological thresholds such as the basic and effective reproduction numbers and a measure of vaccine impact are derived. We find that if the effective reproduction number is greater than unity, there is a locally unstable infection-free equilibrium and a unique, globally asymptotically stable endemic equilibrium. If the effective reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable, and HPV will be eliminated.     

一个具非单调传染率的SEIR传染病模型的全局稳定性

本文研究一类描述某种严重疾病的传染数目变大时在心理上产生影响的非单调传染率的SEIR传染病模型.研究表明模型的动力行为和疾病的爆发完全由基本再生数R0决定.当R0≤1时,无病平衡点是全局稳定的,疾病消亡;当R0〉1时,地方病平衡点是全局稳定的,疾病持续且发展成地方病.