摘要本文首先在第一章大概介绍了传染病模型的发展及其现状，并且介绍了几种较为简单但是意义重大的建模方法。第三章介绍了一类较为简单的带有隔离项具有常数移民的传染病模型，并且利用相关知识得到了疾病流行与否的阀值，同时对其平衡点的存在性以及稳定性进行了讨论。在第四章中，主要考虑了具有分段形式的治疗函数以及非线性发生率的SIS传染病模型，并且将痊愈者并不能获得长时间的免疫这一情况考虑在模型中。之后在第五章中，主要研究了连续预防接种而且带有隔离项的传染病模型，并且利用极限系统理论将其降为简单三维系统，并且找到了影响传染病灭绝与否的阀值，得到了模型平衡点存在的条件以及平衡点的局部渐进稳定的充分条件，以及证明了平衡点的全局渐进稳定性，最后利用Matlab进行了数值模拟。65257

毕业论文关键词  传染病  阀值 极限系统理论   

毕业设计说明书（论文）外文摘要

Title    The Study based on several Epidemic Models                    

Abstract

In the first chapter I introduced the development and the present situation about the infectious disease models, and then introduced a few kinds of simple but significant modeling methods. In the third chapter I introduced a simple category with epidemic model with constant immigration quarantine, and obtained the epidemic threshold which influences if the epidemic will spread among the crowd or not with some relevant knowledge, discussed the existence of the equilibrium and stability. In the fourth chapter, I considered the SIS epidemic model which has piecewise form function of treatment and nonlinear incidence, and which in the situation which the sufferer can't be immune for a long time. Then in the fifth chapter, I mainly studied the model which has  continuous vaccination and isolation of epidemic, and reduced it to a simple 3D system by using the limit system theory, and found the threshold which  influence if the infectious disease will spread or not, obtained the sufficient conditions for the balance points and prove global asymptotic stability of the balance points，finally got the numerical simulation which is  carried out with Matlab.

Keywords  Epidemic  Threshold  Limit system theory

目 次

1  绪论 1

1.1 传染病动态模型的发展历史及其现状 1

2  预备知识 3

2.1  基本定义 3

2.2  基本定理 4

3  较为简单的一类传染病模型的建立 6

3.1  引言 6

3.2  模型的建立 7

3.3  模型稳定性分析 8

3.3.1  平衡点的存在性 8

3.3.2  无病平衡点的稳定性 8

3.3.3  地方病平衡点的稳定性 …9

3.4  本章小结 10

4  具有分段形式治疗函数和非线性发生率的SIS传染病模型 10

4.1  引言 10

4.2  平衡点及其分支 11

4.3  全局性态 16

4.4  对模型的讨论 18