几类随机传染病模型阈值的研究

发布时间：2018-03-07 12:33

  本文选题：阈值　切入点：非线性发生率　出处：《新疆大学》2017年硕士论文　论文类型：学位论文

【摘要】：传染病一直危害着人类的健康和生命,所以用数学模型研究传染病的发病规律意义非常大.通过在确定模型上添加随机扰动,从而建立了随机传染病模型.最近几年许多的学者研究了随机传染病模型,他们主要讨论了模型的持久性、灭绝性、解的正性以及平稳分布.主要内容可以概述如下:第一部分,我们首先介绍了随机传染病模型的生物背景及意义,随后介绍了随机传染病模型的研究现状,最后简述了本文的研究内容.第二部分,介绍了一些相关定义,给出了文中证明所用到的定义、记号、引理、定理等内容.第三部分,在这一部分我们研究了一类具有非线性发生率的随机SIVS传染病模型,得到了阈值R0,并且建立了疾病的灭绝性和在均值意义下持久性的判别条件,即(?)1,则疾病依概率1是灭绝的,若(?)1,疾病依概率1在均值意义下是持久的.第四部分,我们讨论了疾病的持久性和灭绝性.我们对传输率系数和因病死亡率进行扰动,在先前的研究中,主要讨论了疾病的灭绝性,但对于疾病的持久性很少进行研究.我们给出了阈值R0s,若R0s1,则疾病依概率1是灭绝的,若R0s1,疾病依概率1在均值意义下是持续的.最后讨论了在白噪声不大时,系统存在一个平稳分布.第五部分,我们研究了一类具有非线性发生率和暂时免疫的随机SIR传染病模型.我们证明了,对任意的初始值,存在唯一的全局正解.并且建立了疾病灭绝和在均值意义下的持久性的条件:若(?)1,则疾病依概率1是灭绝的,若(?)1,疾病依概率1在均值意义下是持久的.第六部分,我们对本文的研究结果进行了讨论和总结.
[Abstract]:Infectious diseases have been harmful to human health and life, so it is of great significance to use mathematical models to study the pathogenesis of infectious diseases. In recent years, many scholars have studied the stochastic infectious disease model, and they have mainly discussed the persistence and extinction of the model. The main contents can be summarized as follows: in the first part, we introduce the biological background and significance of stochastic infectious disease model, and then introduce the research status of stochastic infectious disease model. In the second part, some related definitions are introduced, and the definitions, notation, Lemma, theorems and so on used in the proof are given. In this part, we study a class of stochastic SIVS infectious disease models with nonlinear incidence, obtain threshold R0, and establish the criteria for disease extinction and persistence in the mean sense, I. e. If the disease is extinct according to the probability of 1? In part 4th, we discussed the persistence and extinction of disease. We perturbed the transmission rate coefficient and the disease mortality rate, in previous studies, This paper mainly discusses the extinction of disease, but seldom studies the persistence of disease. We give the threshold R0s, if R0s1, the disease is extinct according to probability 1. If R0s1, the disease depends on probability 1 is persistent in the mean value. Finally, it is discussed that the system has a stationary distribution. 5th, when white noise is small, In this paper, we study a class of stochastic SIR infectious disease models with nonlinear incidence and transient immunity. We prove that for arbitrary initial values, There is a unique global positive solution, and the condition of disease extinction and persistence in the mean sense is established. If the disease is extinct according to the probability of 1? Disease probability 1 is persistent in the sense of mean value. Part 6th, we discuss and summarize the research results of this paper.
【学位授予单位】：新疆大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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本文编号：1579314