随机HIV传染病模型的稳定性和灭绝分析

发布时间：2018-03-08 15:19

本文选题：随机HIV传染病模型　切入点：矩渐近稳定　出处：《北方民族大学》2017年硕士论文　论文类型：学位论文

【摘要】：由于当今社会人员的流通性越来越广泛,社会价值也呈现多样性的趋势,各种传染疾病的传播也越来越广泛,导致越来越多学者致力于研究传染病传播过程和疾病感染抑制的机制,尤其是艾滋病.本文将随机性引入传染病模型,研究了在噪声环境影响下的疾病传播过程和疾病在个体中的灭绝条件,以及受Allee效应影响下的体内HIV病毒感染的过程.本文工作如下:首先,对一类随机Human Immunodeficiency Virus(简写为HIV)模型进行矩渐近稳定分析.对随机微分方程取集合平均,得到关于系统一阶矩的微分方程,通过Routh-Hurwitz准则得到一阶矩渐近稳定的充要条件.再利用伊藤公式,得到关于系统二阶矩的微分方程,同理可得二阶矩渐近稳定的充要条件.其次,在Allee效应影响的假设下,利用Ito公式、Lyapunov函数、Borel-Cantelli引理和鞅的不等式、鞅的大数定理进一步证明了随机HIV传染病模型全局正解的存在唯一性,并得出了疾病灭绝的条件.并引入了一个具有混合双线性发生率,划分为五个群体的随机HIV传染病模型中证明了全局正解的存在唯一性,患病人群密度几乎必然指数收敛和解的几乎必然收敛的条件,对控制疫情提供了数据上的参考.最后,本文将方块脉冲函数用于求解非线性随机动力系统.以随机HIV传染病模型为例,借助方块脉冲函数,给出了其近似非线性方程.
[Abstract]:As the circulation of social personnel is becoming more and more widespread, the social values are also showing a trend of diversity, and the spread of various infectious diseases is becoming more and more widespread. This has led more and more scholars to study the process of infectious disease transmission and the mechanism of disease infection inhibition, especially AIDS. The process of disease transmission under the influence of noise environment, the condition of disease extinction in individuals and the process of HIV virus infection in vivo under the influence of Allee effect are studied. The main work of this paper is as follows: first of all, The moment asymptotic stability of a class of stochastic Human Immunodeficiency virus model is analyzed. The first order moment differential equation of the system is obtained by taking the set average for the stochastic differential equation. By using the Routh-Hurwitz criterion, the necessary and sufficient conditions for the asymptotic stability of the first order moments are obtained. By using Ito's formula, the necessary and sufficient conditions for the asymptotic stability of the second order moments of the system are obtained by the same theory. Secondly, under the assumption of the influence of the Allee effect, The existence and uniqueness of the global positive solution of stochastic HIV infectious disease model are further proved by using the Ito formula and the inequality of Borel-Cantelli Lemma and martingale, and the large number theorem of martingale. The condition of disease extinction is obtained, and the existence and uniqueness of global positive solution are proved in a stochastic HIV infectious disease model with mixed bilinear incidence and divided into five populations. The density of the infected population is almost certain to converge exponentially and the condition of convergence is almost inevitable, which provides a data reference for the control of the epidemic situation. Finally, In this paper, the block impulse function is used to solve the nonlinear stochastic dynamical system. Taking the stochastic HIV epidemic model as an example, the approximate nonlinear equation is given by means of the block pulse function.
【学位授予单位】：北方民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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