【作者】 杨丽娟；

【导师】 刘兴波；

【作者基本信息】 华东师范大学 ， 应用数学， 2010， 硕士

【摘要】 在这篇文章中,我们分别在第二章和第三章中研究了两个不同的流行病模型.第二章中主要研究了一个具有一般非线性发生率函数的SEIQR流行病模型.该模型始终存在无病平衡点,当且仅当基本再生数R0>1时存在地方病平衡点,且是唯一的.无病平衡点与地方病平衡点的稳定性与基本再生数R0的取值有着密切联系,当R0≤1时,无病平衡点是全局渐进稳定的；当R0>1时,无病平衡点是不稳定的,此时疾病具有持久性.当R0>1且发生率函数满足§2.2条件H,地方病平衡点是全局渐近稳定的.本章最后给出了几个数值模拟来证明结论的正确性,同时通过改变隔离项的系数来说明对感染者进行一定比例的隔离对消灭流行病的重要作用.第三章我们主要研究了一个具有非单调发生率函数的时滞SIR流行病模型.得到了系统无病平衡点及地方病平衡点的存在与基本再生数之间的关系.证明了当基本再生数R0≤1时无病平衡点是全局渐近稳定的；当R0>1时,疾病是持久的,此时无病平衡点是不稳定的；给出参数满足的一定条件时,地方病平衡点是全局渐近稳定的.更多还原

【Abstract】 In this paper, we study two epidemic models in chapter two and chapter three, respectively.In chapter two, we study a SEIQR epidemic model with a class of nonlinear incidence rate. The model always exhibits the disease-free equilibrium, and the unique endemic equilibrium turns up if and only if the basic reproduction number R0>1. It is shown that if R0≤1, the disease-free equilibrium is globally asymptotically stable and if R0>1, the disease-free equilibrium is unstable. Moreover, we show that if R0>1,the disease is uniformly persistent and the unique endemic equilibrium is globally asymptotically stable under certain condition H which we will give in§2.2. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the effect of quarantine to eliminate the disease.In chapter three, we consider a SIR epidemic model with non-monotone incidence rate and time delay. Through analysis we get the relative of the existence of the disease free equilibrium and endemic equilibrium between the basic reproduction number R0. We show that the disease-free equilibrium is globally asymptotically stable when R0≤1, and if R0>1, the disease free equilib-rium is unstable but the disease is persistent, and the endemic equilibrium is globally asymptoti-cally stable if the parameters satisfied some conditions.更多还原