基于Markov转换的随机传染病模型的灭绝性与遍历性

发布时间：2018-03-14 01:43

本文选题：Markov转换　切入点：平均意义下持久性　出处：《山东科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文研究传染病防治等实际问题,建立了两类具有Markov转换的随机生物数学模型:一是具有Markov转换和脉冲毒素输入的植物传染病模型,二是具有Markov转换的随机非线性SIRS模型.运用随机微分方程相关理论,研究了模型的灭绝性、平均意义下的持久性以及遍历性.通过数值模拟,验证了结论的正确性,并讨论了模型的生物意义.第一章,主要介绍随机微分方程以及随机过程的基本理论和方法,给出了了平均意义下的持久性与灭绝性、遍历性与平稳分布的相关定义、引理和定理,并对传染病动力学的一般理论进行了简要介绍.第二章,建立一类具有Markov转换和脉冲毒素输入的植物传染病模型.首先,通过构造Lyapunov函数并运用伊藤公式得出染病植株趋于灭绝的条件;其次,通过比较定理、强大数定律,证明在一定条件下,染病植株具有平均意义下的持久性;然后,运用Markov转换相关理论证明该系统是遍历的,且具有平稳分布;最后,通过具体实例,利用数值模拟验证了理论分析的正确性并讨论了系统的生物意义.第三章,建立了一类具有Markov转换的随机非线性SIRS传染病模型.首先,通过构造Lyapunov函数并运用伊藤公式得出疾病灭绝与持久的阈值;其次,运用Markov转换及矩阵理论证明该系统在一定条件下是遍历的并且有唯一平稳分布;最后,通过具体实例,利用数值模拟验证了结论的正确性.第四章,论文总结及未来工作的展望.
[Abstract]:In this paper, two kinds of stochastic biological mathematical models with Markov conversion are established. One is the plant infectious disease model with Markov conversion and pulse toxin input. The second is the stochastic nonlinear SIRS model with Markov transformation. The extinction, persistence and ergodicity of the model are studied by using the theory of stochastic differential equation. The correctness of the conclusion is verified by numerical simulation. In chapter 1, the basic theories and methods of stochastic differential equations and stochastic processes are introduced, and the definitions of persistence and extinction, ergodicity and stationary distribution in the mean sense are given. Lemma and theorem, and a brief introduction to the general theory of infectious disease dynamics. In chapter 2, a class of plant infectious disease models with Markov conversion and pulse toxin input are established. By constructing Lyapunov function and using Ito formula, the conditions of plant extinction are obtained. Secondly, through comparison theorem and law of strong numbers, it is proved that the infected plant has average persistence under certain conditions. The Markov conversion theory is used to prove that the system is ergodic and stable. Finally, the correctness of the theoretical analysis is verified by numerical simulation and the biological significance of the system is discussed. A class of stochastic nonlinear SIRS infectious disease model with Markov transformation is established. Firstly, the threshold of disease extinction and persistence is obtained by constructing Lyapunov function and using Ito formula. The Markov transformation and matrix theory are used to prove that the system is ergodic and has a unique stationary distribution under certain conditions. Finally, numerical simulation is used to verify the correctness of the conclusion in chapter 4th. Summary of the paper and prospects for future work.
【学位授予单位】：山东科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.62

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