一类具有双线性发生率的传染病模型--SIR传染病毕业设计论文.doc

一类具有双线性发生率的传染病模型 摘 要：本文研究了一类具有因病死亡因素的SIR传染病模型和一类具有接种免疫的SIR传染病模型. 具有因病死亡因素的SIR传染病模型中，讨论了模型的无病平衡点和正平衡点的稳定性，通过线性化方法和构造合适的Lyapunov函数, 得到如下结论：当时，无病平衡点 QUOTE E0 是全局渐近稳定的；当时， QUOTE E0 不稳定；当时，正平衡点 QUOTE E+ 是全局渐近稳定的.同时，在经典SIR模型的基础上构建了一种具有接种免疫的SIR传染病模型，利用数值分析的方法对其传播过程进行研究，通过理论分析证明其渐近稳定性．与传统的统计方法相比，利用该模型能够更好地了解流行过程中的一些全局性态． 关键词：SIR传染病；无病平衡点；全局渐近稳定性；接种免疫 An Sis Epidemic Model With Bilinear Incidence Rate Abstract: We investigate an SIR epidemic model with disease-reduced death rate. Then the stability of the disease-free equilibrium and the endemic equilibrium is discussed. By the linearization method and using Lyapunov functions, we obtain the following results: when the basic reproduction number, the disease-free equilibrium QUOTE E0 is globally asymptotically stable; while， QUOTE E0 is unstable, and the endemic equilibrium QUOTE E+ is globally asymptotically stable. At the same time, based on the traditional SIR model to construct a SIR epidemic model with vaccination, using the numerical analysis method to study the propagation process, through theoretical analysis proves the asymptotic stability. Compared with the traditional statistical methods, using the model to better understand the epidemic process in some global behavior. Key words: SIR model； Disease-free equilibrium；Global stability；Vaccination 目 录 TOC \o 1-3 \h \z \u HYPERLINK \l _Toc326857551 1 绪论 PAGEREF _Toc326857551 \h 1 HYPERLINK \l _Toc326857552 1.1 论文研究的背景及意义 PAGEREF _Toc326857552 \h 1 HYPERLINK \l _Toc326857553 1.1.1 研究的背景 PAGEREF _Toc326857553 \h 1 HYPERLINK \l _Toc326857554 1.1.2 国内外研究现状 PAGEREF _Toc326857554 \h 4 HYPERLINK \l _Toc326857555 1.2 研究的主要内容 PAGEREF _Toc326857555 \h 4 HYPERLINK \l _Toc326857556 1.3 采用的方法、手段以及步骤等 PAGEREF _Toc326857556 \h 4 HYPERLINK \l _Toc326857557 2 SIR模型简介 PAGEREF _Toc326857557 \h 6 HYPERLINK \l _Toc326857558 2.1 传染病动力学的基本概念 PAGEREF _Toc326857558 \h 6 HYPERLINK \l _Toc326857559 2.1.1 传染病动力学模型的提出 PAGEREF _Toc326857559 \h 6 HYPERLINK