具有非线性传染率和时滞的传染病模型的稳定性分析
发布时间：2018-11-25 10:28
【摘要】：近年来,许多学者利用时滞微分方程的理论研究传染病模型,并得到了一些重要的结论。本文在前人研究的基础上,研究几类具有非线性传染率和时滞的传染病模型的稳定性。主要内容如下:第一部分,建立一类具有单时滞,传染率为βS~n的SIR传染病模型;然后借助基本再生数R_0,利用微分方程线性化理论、Hurwitz判断,LaSalle不变原理,给出模型局部渐近稳定和全局渐近稳定的一些充分条件。第二部分,建立一类具有单时滞,传染率为的SIRS传染病模型;然后借助基本再生数R_1,利用线性化矩阵、Hurwitz判断,迭代技巧,比较原理,获得模型局部渐近稳定和全局渐近稳定的一些充分条件。第三部分,在第二部分的基础上,我们建立了一类具有非线线性传染和双时滞的SIRS模型,利用第二部分的研究方法,分析了模型的稳定性。
[Abstract]:In recent years, many scholars have made use of the theory of delay differential equation to study the infectious disease model, and got some important conclusions. On the basis of previous studies, this paper studies the stability of several infectious disease models with nonlinear infection rate and time delay. The main contents are as follows: in the first part, a class of SIR infectious disease model with single delay and infection rate of 尾 -Sn is established. Then some sufficient conditions for the local asymptotic stability and global asymptotic stability of the model are given by means of the basic reproducing number R _ 0, using the linearization theory of differential equations, the Hurwitz judgment and the LaSalle invariant principle. In the second part, a class of SIRS infectious disease model with single time delay and infection rate is established. Then some sufficient conditions for the local asymptotic stability and global asymptotic stability of the model are obtained by means of the basic reproducing number R _ 1, using linearization matrix, Hurwitz judgment, iterative technique and comparison principle. In the third part, on the basis of the second part, we establish a class of SIRS model with nonlinear linear contagion and double delay. The stability of the model is analyzed by using the method of the second part.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175
[Abstract]:In recent years, many scholars have made use of the theory of delay differential equation to study the infectious disease model, and got some important conclusions. On the basis of previous studies, this paper studies the stability of several infectious disease models with nonlinear infection rate and time delay. The main contents are as follows: in the first part, a class of SIR infectious disease model with single delay and infection rate of 尾 -Sn is established. Then some sufficient conditions for the local asymptotic stability and global asymptotic stability of the model are given by means of the basic reproducing number R _ 0, using the linearization theory of differential equations, the Hurwitz judgment and the LaSalle invariant principle. In the second part, a class of SIRS infectious disease model with single time delay and infection rate is established. Then some sufficient conditions for the local asymptotic stability and global asymptotic stability of the model are obtained by means of the basic reproducing number R _ 1, using linearization matrix, Hurwitz judgment, iterative technique and comparison principle. In the third part, on the basis of the second part, we establish a class of SIRS model with nonlinear linear contagion and double delay. The stability of the model is analyzed by using the method of the second part.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175
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