非齐次空间的一种SIRS传染病模型的稳态解
发布时间：2018-08-23 09:36
【摘要】：传染病模型是为了方便研究传染病在个体之间和地区之间的发病机制及扩散规律,通过运用一些合理的假设,建立适当的数学模型,并将可决定传染病扩散的各个因素转化为已建立数学模型中的相关数学变量,利用动力学理论来分析疾病的发展趋势,以达到帮助我们预测和控制日常生活中疾病的目的.基本再生数是传染病模型中的重要参数,而基本再生数决定了疾病是否蔓延或者消退.但是我们逐渐认识到空间扩散和环境的异质性不仅是影响疾病消退和蔓延的重要因素,还决定了疾病传播方式和传播速度.这样来说的话通常的基本再生数不足以描述疾病的传播,也不能反映所研究区域的空间特征.从而很有必要研究扩散对于疾病在区域中的传播和控制所起的作用.伴随这些要求,楼元老师对一类非均质区域下的SIS传染病模型的稳定性进行了分析.先考虑固定区域上的SIS反应扩散问题,通过定义具有齐次Neumann边界条件的反应扩散问题的基本再生数R_0~N,讨论无病平衡点和染病平衡点的稳定性;在此基础上,引入自由边界描述传染病传播的边沿,定义具有齐次Dirichlet边界条件的反应扩散问题的基本再生数R_0~D,从而引入具有自由边界的SIS模型的基本再生数R_0~F(t),并讨论了疾病的消退和蔓延.本文采用了一种新的非齐次的SIRS传染病模型.基本思路是:先是构造本模型在具有Neumann边界条件下的基本再生数R_0,同时讨论传染病者的扩散对基本再生数R_0的影响,即如果R_01,则无病平衡点全局渐近稳定,如果R_01,则无病平衡点不稳定.再是,在低危险区域,我们运用分叉理论研究染病平衡点的存在性和稳定性.最终结果显示,减少染病者的扩散并不有利于传染病的消除,但染病平衡点的不稳定性表明传染病可以得到控制.本文在第一章绪论的第一节中具体介绍了SIRS传染病模型的背景来源,第二节中给出近来研究现状;第二章第一节给出了Lyapunov稳定性的,第二节给出Crandall-Rabinowitz分叉理论的知识,第三节给出局部分叉图像和稳定性变换原则的相关知识;第三章中讨论了所研究的SIRS反应扩散传染病模型的基本再生数的定义和特征、无病平衡点的稳定性、染病平衡点的存在性与稳定性和局部分叉图像的方向.第四章对本文的研究做了相关总结.
[Abstract]:The infectious disease model is to facilitate the study of the pathogenesis and diffusion of infectious diseases between individuals and regions, and to establish appropriate mathematical models by using some reasonable assumptions. The factors that can determine the spread of infectious diseases are transformed into relevant mathematical variables in established mathematical models, and the development trend of diseases is analyzed by using the kinetic theory to help us predict and control diseases in our daily life. The number of basic regeneration is an important parameter in infectious disease model, and the number of basic regeneration determines whether the disease is spreading or fading. However, we have come to realize that spatial diffusion and environmental heterogeneity are not only important factors that affect the extinction and spread of disease, but also determine the mode and speed of disease transmission. In this case, the usual number of basic regeneration is not sufficient to describe the spread of disease, nor can it reflect the spatial characteristics of the region studied. It is therefore necessary to study the role of diffusion in the spread and control of disease in the region. In response to these requirements, the stability of a class of SIS infectious disease models in heterogeneous regions was analyzed. Considering the SIS reaction-diffusion problem in a fixed region, the stability of disease-free equilibrium point and disease-free equilibrium point is discussed by defining the basic regenerative number of the reaction-diffusion problem with homogeneous Neumann boundary condition. The free boundary is introduced to describe the edge of infectious disease propagation, and the basic number of reaction-diffusion problems with homogeneous Dirichlet boundary condition is defined. The basic number of reproduction of SIS model with free boundary is introduced, and the extinction and spread of disease are discussed. In this paper, a new non-homogeneous SIRS infectious disease model is used. The basic ideas are as follows: first, we construct the basic regenerative number R0 with Neumann boundary condition, and at the same time discuss the influence of the diffusion of infectious diseases on the basic regenerative number R0, that is, if R201, the disease-free equilibrium point is globally asymptotically stable. If RW is 01, the disease-free equilibrium is unstable. Then, in the low risk area, we use bifurcation theory to study the existence and stability of the infection equilibrium. The final results show that reducing the spread of infectious diseases is not conducive to the elimination of infectious diseases, but the instability of infection equilibrium points indicates that infectious diseases can be controlled. In the first section of the first chapter, the background of SIRS infectious disease model is introduced in detail. In the second section, the recent research status is given. In the second chapter, the stability of Lyapunov is given, and the knowledge of Crandall-Rabinowitz bifurcation theory is given in the second section. In the third section, we give the knowledge of the local bifurcation image and the principle of stability transformation. In chapter 3, we discuss the definition and characteristics of the basic reproduction number of the SIRS model, and the stability of the disease-free equilibrium. The existence and stability of the equilibrium point and the direction of the local bifurcation image. Chapter four summarizes the research of this paper.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2198648
[Abstract]:The infectious disease model is to facilitate the study of the pathogenesis and diffusion of infectious diseases between individuals and regions, and to establish appropriate mathematical models by using some reasonable assumptions. The factors that can determine the spread of infectious diseases are transformed into relevant mathematical variables in established mathematical models, and the development trend of diseases is analyzed by using the kinetic theory to help us predict and control diseases in our daily life. The number of basic regeneration is an important parameter in infectious disease model, and the number of basic regeneration determines whether the disease is spreading or fading. However, we have come to realize that spatial diffusion and environmental heterogeneity are not only important factors that affect the extinction and spread of disease, but also determine the mode and speed of disease transmission. In this case, the usual number of basic regeneration is not sufficient to describe the spread of disease, nor can it reflect the spatial characteristics of the region studied. It is therefore necessary to study the role of diffusion in the spread and control of disease in the region. In response to these requirements, the stability of a class of SIS infectious disease models in heterogeneous regions was analyzed. Considering the SIS reaction-diffusion problem in a fixed region, the stability of disease-free equilibrium point and disease-free equilibrium point is discussed by defining the basic regenerative number of the reaction-diffusion problem with homogeneous Neumann boundary condition. The free boundary is introduced to describe the edge of infectious disease propagation, and the basic number of reaction-diffusion problems with homogeneous Dirichlet boundary condition is defined. The basic number of reproduction of SIS model with free boundary is introduced, and the extinction and spread of disease are discussed. In this paper, a new non-homogeneous SIRS infectious disease model is used. The basic ideas are as follows: first, we construct the basic regenerative number R0 with Neumann boundary condition, and at the same time discuss the influence of the diffusion of infectious diseases on the basic regenerative number R0, that is, if R201, the disease-free equilibrium point is globally asymptotically stable. If RW is 01, the disease-free equilibrium is unstable. Then, in the low risk area, we use bifurcation theory to study the existence and stability of the infection equilibrium. The final results show that reducing the spread of infectious diseases is not conducive to the elimination of infectious diseases, but the instability of infection equilibrium points indicates that infectious diseases can be controlled. In the first section of the first chapter, the background of SIRS infectious disease model is introduced in detail. In the second section, the recent research status is given. In the second chapter, the stability of Lyapunov is given, and the knowledge of Crandall-Rabinowitz bifurcation theory is given in the second section. In the third section, we give the knowledge of the local bifurcation image and the principle of stability transformation. In chapter 3, we discuss the definition and characteristics of the basic reproduction number of the SIRS model, and the stability of the disease-free equilibrium. The existence and stability of the equilibrium point and the direction of the local bifurcation image. Chapter four summarizes the research of this paper.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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