【作者】 周锐；

【导师】 王稳地；

【作者基本信息】 西南大学 ， 应用数学， 2007， 硕士

【摘要】 本文仍然采用建立数学模型的方法来研究流行病的传播机理。第一部分研究了非线性发生率的传染病模型，给出了对其施加防治策略以控制传染病的流行。前两节考虑具有非线性发生率（κIS）／（1+α（S+I））的SIRS模型以及具有（κIS）／（1+α₁S+α₂I）型非线性发生率的SIRS传染病模型。我们建立了使疾病消除和流行的阈值，讨论了平衡状态的存在性和局部渐近稳定性，得到了一些使无病平衡状态或地方病平衡状态全局渐近稳定的充分条件。在第四节中讨论了一类具有非线性发生率的种群力学流行病模型，分析了模型平衡点的存在性和稳定性，得到了疾病消除平衡点和疾病传染平衡点全局渐近稳定的充分条件。在第五节中我们研究了文[41]中的具有饱和发生率的传染病模型的全局性态，通过定性分析，模型具有疾病平衡点的鞍结分支，Hopf分支及Bogdanov-Takens分支，并且还出现了双稳定现象的丰富的动力学性态。本文第二部分讨论了一类捕食系统的传染病模型，主要研究具有染病食饵的捕食与被捕食种群的SIS传染病模型。也就是说，在生态学意义上，两种群为捕食与被捕食类型；在疾病的传染机理上，两种群为SIS的类型。从两方面对模型（3．1．2）进行了改进。我们考虑的模型采用双线性发生率代替标准发生率，并且考虑捕食者易捕获易感食饵的情况，通过改进得到一些有趣的生物现象。我们通过分析模型研究了在食饵种群中疾病传播的规律，得到了捕食者是如何控制在食饵种群中的疾病的流行，还在一定条件下通过数值模拟给出了控制策略，通过对平衡状态的稳定性的分析给出了疾病持久和灭绝的条件，并且分析了捕食者对食饵优先选择的影响。更多还原

【Abstract】 The aim of this work is to construct several mathematical models in epidemiology and to analyze the asymptotic behavior of these models. In the first part, some epidemic models with nonlinear incidence rate are studied to consider effects of protection measures and intervention policies. In the first two sections, we consider SIRS epidemiological models with nonlinear incidence rates （kIS）/（1+α（S+I）） and （kIS）/（1+α₁S+α₂I）. A threshold for the disease to be epidemic or to be eradicated is established. The behavior of solution and the threshold conditions of existence and locally asymptotic stability are obtained. The sufficient conditions on the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium. In the third section, a species mechanics epidemical model with nonlinear rate was considered. The sufficient conditions on the global asymptotic stability of the extinct equilibrium and the survival equilibrium are obtained. In the fourth section, we study the global dynamics of an epidemic model with a complete form of the nonmonotonic incidence rate of [41]. by carrying out the qualitative analysis of the model, it is shown that the model admits rich dynamical behaviors. It has saddle node bifurcations of endemic equilibria, Hopf bifurcations, the saddle node bifurcation of nonhyperbolic periodic orbit, which implies the existence of two limit cycles and homoclinic bifurcations. We have also found that the model admits bistable steady states such that the outcome of disease evolution depends upon initial positions.In the second part of this paper, we consider a kind of epidemic model with disease transmission in the predator-prey system. A predator-prey model modified to include an SIS parasitic infection in the prey with infected prey being more vulnerable to predation is studied. This means that in ecology it is predator-prey system, in the mechanism of disease transmission it is an SIS epidemiological model. we improve （3.1.2） in two-folds. First, we replace the standard incidence by a bilinear incidence. Secondly, we also consider the case whereα₂ <α₁, which essentially means that predators prefer to healthy prey. This induces interesting biological phenomena. An epidemic model with disease transmission in prey population is studied to find how predators regulate the evolution of disease in the prey population. To have further insights into the effect of biological control of predators to the spread of epidemic disease, we present the example. Conditions for the extinction of disease and conditions for the persistence of disease are established, and effects of the preference of predators to prey individuals are analyzed.更多还原