一类结核病动力系统模型的稳定性分析

发布时间：2018-06-02 04:00

本文选题：结核病 + 全局渐近稳定性　；参考：《华中师范大学》2013年硕士论文

【摘要】：结核病是一种由结核杆菌引起的传染病.结核病传染性极强,全球每年都会有数百万人死于结核病.一旦感染结核病,自身将长期忍受病痛的折磨,甚至会危及生命.因此研究结核病的发病机理,传播规律,治疗措施等都具有很重要的实际意义. 数学模型为研究结核病的传播以及评估医疗干预带来的结果提供了一种有效的工具.本文讨论的是一类双线性发生率的结核病动力系统模型.我们利用再生矩阵的方法得到了模型的基本再生数R0,并且利用构造李雅普诺夫函数的方法,证明了当基本再生数R0≤1时,系统的无病平衡点是全局渐近稳定的；利用复合系统理论,证明了当基本再生数R0＞1时,系统的唯一的流行病平衡点是全局渐近稳定的. 本文的组织结构如下：第一章引言,简单介绍了结核病相关的医学背景,以及结核病动力系统模型的主要研究成果.第二章数学工具,列举了本文中相关的微分方程动力系统理论知识.第三章,建立了一个典型的结核病传播SEI模型,通过计算求得这个动力系统模型的无病平衡点,以及这个系统的的基本再生数；并且证明了系统的无病平衡点是全局稳定的；然后给出了该模型流行病平衡点的存在和稳定性条件.第四章对上述模型进行了改进,建立了新的模型,运用类似的方法对该模型进行了定性分析.第五章,数值模拟.最后,第六章是文章的结论.
[Abstract]:Tuberculosis is an infectious disease caused by TB bacilli. Tuberculosis is very infectious. Millions of people die from tuberculosis every year in the world. Once it is infected, it will endure the pain and even endanger life for a long time. Therefore, it is very important to study the pathogenesis of tuberculosis, the law of transmission and treatment, and so on. Significance.
The mathematical model provides an effective tool to study the spread of tuberculosis and to evaluate the results of medical intervention. This paper discusses a class of bilinear incidence of the model of the dynamic system of tuberculosis. We use the method of regeneration matrix to obtain the basic regeneration number R0 of the model, and use the method of constructing Lyapunov function. It is proved that the disease-free equilibrium point of the system is globally asymptotically stable when the basic regeneration number is R0 less than 1. By using the composite system theory, it is proved that the only epidemic equilibrium point of the system is globally asymptotically stable when the basic regeneration number is R0 > 1.
The organizational structure of this paper is as follows: the first chapter, introduction, briefly introduces the medical background of tuberculosis and the main research results of the model of the dynamic system of tuberculosis. The second chapter is a mathematical tool, which enumerates the theoretical knowledge of the dynamic system of differential equations in this paper. The third chapter builds a typical SEI model for tuberculosis transmission. The disease-free equilibrium point of the dynamic system model and the basic regeneration number of the system are calculated, and the disease free equilibrium point is proved to be global stable. Then the existence and stability conditions of the epidemic equilibrium point of the model are given. In the fourth chapter, the above model is improved, a new model is established and the application class is used. The qualitative analysis of the model is carried out in a similar way. The fifth chapter is numerical simulation. Finally, the sixth chapter is the conclusion of the article.
【学位授予单位】：华中师范大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：R52;O242.1

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