具有重复感染和染病年龄结构的两菌株SIJR流行病模型分析

建立和研究了具有染病年龄结构和重复感染的两菌株SIJR流行病模型,得到了与两菌株相对应的基本再生数的表达式,给出了无病平衡点,各菌株占优平衡点以及共存平衡点的存在性和稳定性条件.最后详细讨论了该模型的特殊情形-重复感染率为常数的情形.     

一类具有垂直传染率的SIS模型的稳定性分析

研究了一类具有垂直传染率的SIS模型,首先计算出该模型的基本再生数和平衡点,其次分析了该模型在无病平衡点处的局部渐近稳定性和全局稳定性;然后构造Lyapunov函数证明了地方病平衡点的全局稳定性;最后得到当基本再生数小于1时,传染病会逐渐消失;基本再生数大于1时,传染病将会流行并最终形成一种地方病.     

带有免疫反应的病毒动力学模型的全局稳定性

利用Lyapunov函数研究了带有免疫反应的病毒动力学模型的全局稳定性.当基本再生数R0≤1时.病毒在体内清除;当R0>1时,病毒在体内持续生存.并且模型的正解当免疫再生数R1≤1时,趋于无免疫平衡点,当R1>1.趋于地方病平衡点.     

具有潜伏期和免疫应答的HIV感染模型的全局稳定性

本文考虑具有CTL免疫应答和细胞内部潜伏阶段的HIV感染数学模型,得到其基本再生数,通过构造适用的Lyapunov函数,研究该模型的健康平衡点和感染平衡点的稳定性.当基本再生数不大于1时,健康平衡点在可行域上是全局稳定的,即HIV在个体体内最终灭绝;当基本再生数大于1时,模型存在惟一的感染平衡点在可行域上是全局稳定的,即HIV在个体体内呈现持续存在状态,且其浓度最终趋于一个常数.     

具有饱和发生率的血吸虫病动力学分析

血吸虫病是我国一种严重的寄生虫病,并且在湖北、安徽、湖南、江苏、四川和云南成地方病.结合中国血吸虫病的现状及特点,考虑人群、牛群以及水环境中的钉螺、尾蚴和毛蚴之间的相互传染,建立了具有饱和发生率的血吸虫病动力学模型,给出了模型的基本再生数.通过构造Lyapunov函数证明了当基本再生数小于1时,模型的无病平衡点全局渐近稳定;当基本再生数大于1时,模型的地方病平衡点也是全局渐近稳定的.最后,利用数值模拟验证了理论结果.     

流感模型的数据模拟和动力学分析

利用常微分方程定性和稳定性理论、计算机工具建立并研究了没有疫苗和带有疫苗的流感模型.根据中国疾控中心的数据,利用MATLAB进行参数模拟,得到了流感基本再生数的取值范围,并对疫苗的年生产量做出了估计;同时,求出了模型的无病平衡点和地方病平衡点,证明了无病平衡点当基本再生数小于1时是全局渐进稳定的、地方病平衡点存在时是局部稳定的.     

具有饱和传染率、免疫接种和垂直传染的SIR传染病模型的稳定性北大核心

研究了一类具有饱和传染率、免疫接种和垂直传染的SIR传染病模型,确定了疾病的基本再生数,得出当疾病的基本再生数小于1时,无病平衡点是全局指数渐近稳定的,当疾病基本再生数大于1时.地方病平衡点是全局渐近稳定的,讨论了其生物意义.     

具有饱和传染率、免疫接种和垂直传染的SIR传染病模型的稳定性

研究了一类具有饱和传染率、免疫接种和垂直传染的SIR传染病模型,确定了疾病的基本再生数,得出当疾病的基本再生数小于1时,无病平衡点是全局指数渐近稳定的,当疾病基本再生数大于1时.地方病平衡点是全局渐近稳定的,讨论了其生物意义.     

基于两斑块和迁移的SIRS传染病模型的稳定性分析

根据传染病动力学原理,考虑人口在两斑块上流动且具有非线性传染率,建立一类基于两斑块和迁移的SIRS传染病模型.利用常微分方程定性与稳定性方法,分析非负平衡点的存在性,通过构造适当的Lyapunov函数,获得无病平衡点和地方病平衡点全局渐近稳定的充分条件.研究结果表明:基本再生数是决定疾病流行与否的阀值,当基本再生数小于等于1时,疾病逐渐消失;当基本再生数大于1且疾病主导再生数大于1时,疾病持续流行并将成为一种地方病.     

一类具有双线性传染率的HIV/AIDS病毒动力学改进模型

针对HIV/AIDS传播的具有常数移民和指数出生的SI型模型,为了更加符合实际意义,对具有双线性传染率的模型进行局部改进,并对改进后的动力学模型进行了简化.对于改进后的模型,证明了平衡点的存在与局部稳定性,并证明了传染病毒的灭绝与持续性,得到了传染病毒的基本再生数.结果表明:当单位时间内从外界迁入人口中染病者的比例系数c近似等于零时,基本再生数小于1时,传染病毒最终灭绝;当基本再生数大于1时,模型存在唯一的正平衡点,且是局部渐近稳定的,说明传染病毒一致持续存在.     

Dynamics analysis of a quarantine model in two patches

A new deterministic model for assessing the impact of quarantine on the transmission dynamics of a communicable disease in a two‐patch community is designed. Rigorous analysis of the model shows that the imperfect nature of quarantine (in the two patches) could induce the phenomenon of backward bifurcation when the associated reproduction number of the model is less than unity. For the case when quarantined susceptible individuals do not acquire infection during quarantine, the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model has a unique Patch i‐only boundary equilibrium (i = 1,2) whenever the associated reproduction number for Patch i is greater than unity. The unique Patch i‐only boundary equilibrium is locally asymptotically stable whenever the invasion reproduction number of Patch 3 ? i is less than unity (and the associated reproduction number for Patch i exceeds unity). The model has at least one endemic equilibrium when its reproduction number exceeds unity (and the disease persists in both patches in this case). It is shown that adding multi‐patch dynamics to a single‐patch quarantine model (which allow the quarantine of susceptible individuals) in a single patch does not alter its quantitative dynamics (with respect to the existence and asymptotic stability of its associated equilibria as well as its backward bifurcation property). Copyright © 2014 John Wiley & Sons, Ltd.     

A predator–prey–disease model with immune response in infected prey

In this paper, a predator–prey–disease model with immune response in the infected prey is formulated. The basic reproduction number of the within-host model is defined and it is found that there are three equilibria: extinction equilibrium, infection-free equilibrium and infection-persistent equilibrium. The stabilities of these equilibria are completely determined by the reproduction number of the within-host model. Furthermore, we define a basic reproduction number of the between-host model and two predator invasion numbers: predator invasion number in the absence of disease and predator invasion number in the presence of disease. We have predator and infection-free equilibrium, infection-free equilibrium, predator-free equilibrium and a co-existence equilibrium. We determine the local stabilities of these equilibria with conditions on the reproduction and invasion reproduction numbers. Finally, we show that the predator-free equilibrium is globally stable.     

Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics

A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease‐free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh‐Volterra type, to be globally asymptotically stable for a special case.     

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.     

GLOBAL STABILITY OF AN SVIR EPIDEMIC MODEL WITH VACCINATION

In this paper, an SIR epidemic model with vaccination for both the newborns and susceptibles is investigated, where it is assumed that the vaccinated individuals have the temporary immunity. The basic reproduction number determining the extinction or persistence of the infection is found. By constructing a Lyapunov function, it is proved that the disease free equilibrium is globally stable when the basic reproduction number is less than or equal to one, and that the endemic equilibrium is globally stable wh...     

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.     

Stability analysis for an epidemic model with stage structure

An epidemic model with stage structure is formulated. The period of infection is partitioned into the early and later stages according to the developing process of infection, and the infectious individuals in the different stages have the different ability of transmitting disease. The constant recruitment rate and exponential natural death, as well as the disease-related death, are incorporated into the model. The basic reproduction number of this model is determined by the method of next generation matrix. The global stability of the disease-free equilibrium and the local stability of the endemic equilibrium are obtained; the global stability of the endemic equilibrium is got under the case that the infection is not fatal.     

基于两斑块和人口流动的SIR传染病模型的稳定性

根据传染病动力学原理,考虑人口在两斑块上流动且具有非线性传染率,建立了一类基于两斑块和人口流动的SIR传染病模型.利用常微分方程定性与稳定性方法,分析了模型永久持续性和非负平衡点的存在性,通过构造适当的Lyapunov函数和极限系统理论,获得无病平衡点和地方病平衡点全局渐近稳定的充分条件.研究结果表明:基本再生数是决定疾病流行与否的阈值,当基本再生数小于等于1时,感染者逐渐消失,病毒趋于灭绝;当基本再生数大于1并满足永久持续条件时,感染者持续存在且病毒持续流行并将成为一种地方病.     

Mathematical model for the dynamics of visceral leishmaniasis–malaria co‐infection

A mathematical model to understand the dynamics of malaria–visceral leishmaniasis co‐infection is proposed and analyzed. Results show that both diseases can be eliminated if R₀, the basic reproduction number of the co‐infection, is less than unity, and the system undergoes a backward bifurcation where an endemic equilibrium co‐exists with the disease‐free equilibrium when one of R_m or R_l, the basic reproduction numbers of malaria‐only and visceral leishmaniasis‐only, is precisely less than unity. Results also show that in the case of maximum protection against visceral leishmaniasis (VL), the disease‐free equilibrium is globally asymptotically stable if malaria patients are protected from VL infection; similarly, in the case of maximum protection against malaria, the disease‐free equilibrium is globally asymptotically stable if VL and post‐kala‐azar dermal leishmaniasis patients and the recovered humans after VL are protected from malaria infection. Numerical results show that if R_m and R_l are greater than unity, then we have co‐existence of both disease at an endemic equilibrium, and malaria incidence is higher than visceral leishmaniasis incidence at steady state. Copyright © 2016 John Wiley & Sons, Ltd.     

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.