几类双时滞传染病模型的全局稳定性分析

发布时间：2018-11-05 19:20

【摘要】：生物数学模型能够形象的描述病毒的传染过程,许多数学工作者通过建立各类传染病模型,研究模型的动力学性质,得到了病毒的传染机理,这对于传染病的预防、控制起到了非常重要的作用.本文主要利用李雅普诺夫函数法和LaSalle不变原理,研究了几类新的双时滞传染病模型的全局稳定性,在已有文献的基础上得到了若干新结果,根据内容,全文分为以下四章.第一章介绍了问题的研究背景和全文各章节研究的主要内容.第二章研究了一类具有Beddington-DeAngelis功能反应函数的双时滞HBV传染病模型的全局稳定性.首先考虑了模型的解的非负性和有界性,在得出基本再生数R0的基础上,得到了系统的平衡点,利用李雅普诺夫函数法和LaSalle不变原理,给出了无病平衡点和有病平衡点全局渐近稳定的充分条件,得到了如下结论:当R 1时,系统的无病平衡点E0全局渐近稳定;当R1时,有病平衡点E1是全局渐近稳定的.第三章研究了一类具有CTL免疫反应和双线性感染率的双时滞HBV传染病模型的全局稳定性.首先考虑了模型的解是非负的并且有界,然后分别给出了不具有CTL免疫的基本再生数和具有CTL免疫的基本再生数,得到了不具有CTL免疫及具有CTL免疫平衡点的存在性,最后通过构造适当的李雅普诺夫函数,证明了各平衡点的全局渐近稳定性.第四章研究了一类具有CTL免疫反应和非线性感染率的双时滞HIV传染病模型的全局稳定性.本章主要通过构造适当的李雅普诺夫函数和利用LaSalle不变原理,得到了模型各三类平衡点全局渐近稳定的充分条件.当R0 1,τ1 0,τ2 0时,无病平衡点E0是全局渐近稳定的;当RCTL 1 R0,τ1 0时,无CTL免疫平衡点E1是全局渐近稳定的;当RCT 1,τ1 0,τ2 0时,具有CTL免疫的平衡点E2是全局渐近稳定的.
[Abstract]:The biological mathematical model can describe the transmission process of the virus vividly. Many mathematics workers establish various infectious disease models, study the dynamic properties of the model, and get the transmission mechanism of the virus, which is the prevention of the infectious disease. Control plays a very important role. In this paper, by using Lyapunov function method and LaSalle invariant principle, we study the global stability of several new infectious disease models with two delays. Some new results are obtained on the basis of existing literature. According to the contents, the paper is divided into four chapters. The first chapter introduces the research background and the main contents of each chapter. In chapter 2, the global stability of a class of HBV infectious disease models with Beddington-DeAngelis function is studied. Firstly, the nonnegative and boundedness of the solution of the model are considered. On the basis of the basic reproducing number R _ 0, the equilibrium point of the system is obtained, and the Lyapunov function method and the LaSalle invariant principle are used. The sufficient conditions for the global asymptotic stability of disease-free and disease-free equilibrium points are given. The following conclusions are obtained: when R 1, the disease-free equilibrium point E _ 0 is globally asymptotically stable; When R 1, the diseased equilibrium point E1 is globally asymptotically stable. In chapter 3, the global stability of a class of HBV epidemic models with CTL immune response and bilinear infection rate is studied. Firstly, the solution of the model is considered as nonnegative and bounded, then the basic number of regeneration without CTL immunity and the number of basic regeneration with CTL immunity are given, respectively, and the existence of the equilibrium point without CTL immunity and with CTL immunity is obtained. Finally, the global asymptotic stability of the equilibrium points is proved by constructing appropriate Lyapunov functions. In chapter 4, the global stability of a class of HIV epidemic models with CTL immune response and nonlinear infection rate is studied. In this chapter, by constructing appropriate Lyapunov functions and using the LaSalle invariant principle, sufficient conditions for the global asymptotic stability of each of the three kinds of equilibrium points are obtained. When R01, 蟿 10, 蟿 20:00, the disease-free equilibrium point E _ 0 is globally asymptotically stable, and if RCTL _ 1 R _ 0, 蟿 10:00, the CTL free immune equilibrium point E _ 1 is globally asymptotically stable, the disease free equilibrium point E _ 0 is globally asymptotically stable. When RCT 1, 蟿 10, 蟿 20:00, the equilibrium point E2 with CTL immunity is globally asymptotically stable.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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