几类传染病模型的全局分析
发布时间：2018-10-22 08:53
【摘要】：本文研究了几类生态系统中的传染病模型.全文共分为四章:第一章,绪论,介绍了本文的研究背景和主要工作.以及所用到的预备知识.第二章.考虑了一类具有飞沫和直接接触感染的传染病.建立了具有非线性接触率和非线性治愈率的脉冲时滞SIRS传染病模型.首先,运用脉冲微分方程理论讨论了该系统无病周期解的全局吸引性.其次,通过构造适当的V函数证明了该系统的持久性.最后,通过数值模拟验证本章的相关结论.第三章,在第二章的基础上研究了具有两个时滞和一般非线性发生率的传染病模型.将与飞沫感染的个体接触的发病率和与直接接触感染的个体的发病率分别用非线性发生率函数f(P),f(I)来表示,使得该系统更具有一般性.首先,证明了该系统解的正有界性及平衡点的存在性.其次,构造Lyapunov函数证明了无病平衡点的全局渐近稳定性.本文还利用开关原理证明了地方病平衡点的局部渐近稳定性并分析了 Hopf分支存在的条件.最后通过数值模拟验证本章的主要理论结果.第四章.研究了在社会网络中,具有潜伏期和常数补充的谣言传播模型.首先,本章证明了无谣言平衡点和谣言平衡点的存在性.其次,运用特征值的方法证明了无谣言平衡点的局部渐近稳定性并且利用左右特征向量的方法研究了超临界分支的存在性,利用M矩阵的方法证明了无谣言平衡点的全局渐近稳定性.在一定条件下,该系统存在唯一的谣言平衡点.利用Routh-Hurwitz判据证明了谣言平衡点的局部渐近稳定性,而且该平衡点是全局稳定的.最后.给出简单的结论以及一些理论性结果的数值模拟.
[Abstract]:The infectious disease models in several ecosystems are studied in this paper. The paper is divided into four chapters: the first chapter, introduction, introduces the research background and main work of this paper. And the preparatory knowledge used. Chapter 2 A class of infectious diseases with droplet and direct contact infection was considered. A impulsive time-delay SIRS epidemic model with nonlinear contact rate and nonlinear cure rate is established. Firstly, the global attractivity of the disease-free periodic solution of the system is discussed by using the theory of impulsive differential equations. Secondly, the persistence of the system is proved by constructing appropriate V function. Finally, the relevant conclusions of this chapter are verified by numerical simulation. In chapter 3, based on the second chapter, the infectious disease model with two delays and general nonlinear incidence is studied. The incidence rate of individual contact with droplet infection and the incidence rate with direct contact infection are represented by nonlinear incidence function (f (P), f (I), which makes the system more general. Firstly, the positive boundedness and the existence of equilibrium point are proved. Secondly, Lyapunov function is constructed to prove the global asymptotic stability of disease-free equilibrium. This paper also proves the local asymptotic stability of endemic equilibrium by using the switching principle and analyzes the conditions for the existence of Hopf bifurcation. Finally, the main theoretical results of this chapter are verified by numerical simulation. Chapter IV A rumour propagation model with latent period and constant supplement in social network is studied. Firstly, this chapter proves the existence of non-rumor equilibrium and rumor equilibrium. Secondly, the eigenvalue method is used to prove the local asymptotic stability of the non-rumor equilibrium point, and the existence of supercritical bifurcation is studied by using the left and right eigenvector method. The global asymptotic stability of the rumor free equilibrium is proved by using the method of M matrix. Under certain conditions, the system has a unique equilibrium of rumors. By using Routh-Hurwitz criterion, the local asymptotic stability of the rumor equilibrium is proved, and the equilibrium is globally stable. Last A simple conclusion and numerical simulation of some theoretical results are given.
【学位授予单位】：山西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2286699
[Abstract]:The infectious disease models in several ecosystems are studied in this paper. The paper is divided into four chapters: the first chapter, introduction, introduces the research background and main work of this paper. And the preparatory knowledge used. Chapter 2 A class of infectious diseases with droplet and direct contact infection was considered. A impulsive time-delay SIRS epidemic model with nonlinear contact rate and nonlinear cure rate is established. Firstly, the global attractivity of the disease-free periodic solution of the system is discussed by using the theory of impulsive differential equations. Secondly, the persistence of the system is proved by constructing appropriate V function. Finally, the relevant conclusions of this chapter are verified by numerical simulation. In chapter 3, based on the second chapter, the infectious disease model with two delays and general nonlinear incidence is studied. The incidence rate of individual contact with droplet infection and the incidence rate with direct contact infection are represented by nonlinear incidence function (f (P), f (I), which makes the system more general. Firstly, the positive boundedness and the existence of equilibrium point are proved. Secondly, Lyapunov function is constructed to prove the global asymptotic stability of disease-free equilibrium. This paper also proves the local asymptotic stability of endemic equilibrium by using the switching principle and analyzes the conditions for the existence of Hopf bifurcation. Finally, the main theoretical results of this chapter are verified by numerical simulation. Chapter IV A rumour propagation model with latent period and constant supplement in social network is studied. Firstly, this chapter proves the existence of non-rumor equilibrium and rumor equilibrium. Secondly, the eigenvalue method is used to prove the local asymptotic stability of the non-rumor equilibrium point, and the existence of supercritical bifurcation is studied by using the left and right eigenvector method. The global asymptotic stability of the rumor free equilibrium is proved by using the method of M matrix. Under certain conditions, the system has a unique equilibrium of rumors. By using Routh-Hurwitz criterion, the local asymptotic stability of the rumor equilibrium is proved, and the equilibrium is globally stable. Last A simple conclusion and numerical simulation of some theoretical results are given.
【学位授予单位】：山西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
【参考文献】
相关期刊论文 前1条
1 Yoichi Enatsu;Yukihiko Nakata;Yoshiaki Muroya;;GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS[J];Acta Mathematica Scientia;2012年03期
，本文编号：2286699
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