传染病动力学模型性态分析

发布时间：2018-03-21 09:36

本文选题：传染病　切入点：非线性传染率　出处：《西南大学》2008年博士论文　论文类型：学位论文

【摘要】：本文主要研究了几个寄生虫宿主数学模型的动力学性态。我们首先在第二部分研究了一类具有一般传染率寄生虫宿主模型的全局性态,此传染率是在双线性与标准传染率之间连续变化的。我们利用微分方程定性理论证明了只有在标准传染率时,宿主和寄生虫可能绝灭;我们还得到了宿主和寄生虫共存即系统正平衡点存在的条件,证明了正平衡点只要存在一定是全局稳定的。本文第三部分对一类具有S形函数非线性传染率的传染病模型进行了分支分析,我们得到了两个正平衡点存在的阈值条件,通过分析平衡点的稳定性我们给出系统存在Allee效应的条件,我们证明了在一定条件下,系统经历超临界Hopf分支和次临界Hopf分支;还证明了在一定条件下系统经历Bogdanov-Takens分支,通过数值模拟分析,我们发现,系统可能出现两个极限环并且两个极限环可能由不同情况引起:一种是一个极限环从Hopf分支出现,而另一个极限环由同宿轨分支引起;第二种是首先从Hopf分支分支出第一个极限环,接着从退化的极限点出现第二个极限环,并且发现从退化的极限点出现极限环有三种方式,通过数值模拟表明S形函数的倾角如果有微小的变化,可能会导致系统完全不同的分支结构。本文第四部分分析了第二部分的模型加空间效应后系统的斑图动力学性态,我们利用线性稳定性分析得到具有双线性传染率的反应扩散系统的均匀定态解是稳定的;具有标准传染率的反应扩散系统在一定条件下是稳定的,在一定条件下会出现图灵斑图,我们通过分析系统的振幅方程,给出图灵斑图出现点状和条状及其稳定的条件。进一步,利用中心流形法确定振幅方程的系数,给出在固定参数值时系统出现图灵斑图及图灵斑图形状变化的数值分析。本文第五部分研究了具有Allee效应的反应扩散系统的斑图动力学性态,首先,我们利用线性稳定性理论给出图灵不稳定和Hopf分支发生的条件且分析了图灵斑图的稳定性,进一步,我们对照不加扩散的系统与加扩散的系统的性态,进行了数值模拟,通过数值模拟,我们发现系统可能出现图灵斑图、缺口状斑图、迷宫斑图、螺旋波斑图、静态斑图及混沌斑图。若寄生虫剂量作为分支参数,发现随着寄生虫剂量的增加,有两种情况发生:一种是当易感者宿主的扩散率超过临界值时系统依次出现图灵斑图、缺口状斑图和迷宫斑图,意味着染病者宿主的分布会由疏到密接着再变疏;另一种是当易感者宿主的扩散率低于临界值时系统依次出现螺旋波斑图、静态斑图和迷宫斑图。若取易感者宿主的扩散率作为分支参数,发现易感者宿主的扩散率大于或小于染病者宿主的扩散率都可能发生稳定的螺旋波,并且随着染病者宿主扩散率的增加,系统螺旋波缺陷数增加导致混沌发生。
[Abstract]:In this paper, we mainly study the dynamic behavior of several parasite host models. In the second part, we study the global behavior of a class of parasite host models with general transmission rate. The transmission rate varies continuously between bilinear and standard transmission rates. We use the qualitative theory of differential equation to prove that the host and parasite may be extinct only when the standard infection rate is the standard rate. We also obtain the condition that the host and parasite coexist that is, the existence of the positive equilibrium of the system, and prove that the existence of the positive equilibrium is globally stable. In the third part of this paper, we analyze a class of infectious disease models with S-shaped function nonlinear infection rate. We obtain the threshold conditions for the existence of two positive equilibrium points. By analyzing the stability of the equilibrium point, we give the conditions for the existence of Allee effect in the system. We prove that the system experiences supercritical Hopf bifurcation and subcritical Hopf bifurcation under certain conditions, and that the system experiences Bogdanov-Takens bifurcation under certain conditions. By numerical simulation, we find that there may be two limit cycles and two limit cycles may be caused by different conditions: one is that one limit cycle appears from Hopf bifurcation, the other is caused by homoclinic orbit bifurcation; The second is to first branch out the first limit cycle from the Hopf bifurcation, and then the second limit cycle from the degenerate limit point, and find that there are three ways of appearing the limit cycle from the degenerate limit point. Numerical simulation shows that a slight change in the inclination of the S-shape function may lead to a completely different branching structure of the system. In part 4th, we analyze the pattern dynamics of the second part of the model with spatial effect. We use linear stability analysis to obtain that the uniform steady state solution of the reaction diffusion system with bilinear contagion rate is stable. The reaction diffusion system with standard infection rate is stable under certain conditions. Under certain conditions, Turing pattern will appear. We analyze the amplitude equation of the system. Furthermore, the coefficients of amplitude equation are determined by the center manifold method, and the numerical analysis of the Turing pattern and the shape change of the Turing pattern are given when the parameters are fixed. In 5th, we study the pattern dynamics of a reaction diffusion system with Allee effect. Firstly, we give the conditions of Turing instability and Hopf bifurcation by using linear stability theory and analyze the stability of Turing pattern. Further, we compare the behavior of the system without diffusion with that of the system with diffusion. By numerical simulation, we find that the system may have Turing pattern, notch pattern, labyrinth pattern, spiral pattern. Static pattern and chaotic pattern. If the parasite dose is taken as the branch parameter, it is found that there are two kinds of cases with the increase of parasite dose: one is that the Turing pattern appears in turn when the diffusion rate of the susceptible host exceeds the critical value. Notched patterns and labyrinth patterns mean that the distribution of infected hosts will change from sparse to dense and then sparse. The other is that when the diffusivity of susceptible host is lower than the critical value, the helical pattern appears in turn. Static pattern and labyrinth pattern. If the diffusivity of susceptible host is taken as branch parameter, it is found that stable helical wave may occur in susceptible host if the diffusivity of susceptible host is greater than or less than that of infected host. With the increase of host diffusivity, the number of helical wave defects in the system increases, which leads to chaos.
【学位授予单位】：西南大学
【学位级别】：博士
【学位授予年份】：2008
【分类号】：R181.3;O175

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