The Extinction And Ergodicity Of Stochastic Epidemic Models With Markov Conversion

This paper focus on several actual problems about prevention of infectious diseases established two random mathematical models with Markov conversion. It contains a plant disease model with Markov conversion and impulsive toxicant and a nonlinear SIRS model with Markov conversion. According to the related theories and methods in stochastic differential equation, the extinction, persistence in mean and ergodicity of two models are discussed. Employing numerical simulations validates the main conclusions, and the biological phenomenon for models are discussed.In Chapter 1, we mainly introduce the basic knowledge and research status of the stochastic differential equation and the stochastic process. We quote some relative definitions, lemmas and theorems of the persistence in mean and extinction, the ergodicity and stationary distribution. We also introduce the general theory and method of epidemic dynamics.In Chapter 2, we established a class of plant disease model with Markov conversion and impulsive toxicant input. Firstly, by instructing Lyapunov functions and applying Ito formulas, we can obtain that the infected plant tends to be extinction under certain conditions. Secondly,by using the comparison theorem and strong law of large numbers, we can get the conclusion that the infected plant tends to be persistence in mean; Thirdly, by using the relevant knowledge of Markov conversion show that the system has the ergodicity and the unique stationary distribution. Lastly , we give some numerical simulations and discuss the biological significance of the system.In Chapter 3, we established a class of nonlinear SIRS model with Markov conversion. Firstly, by instructing Lyapunov functions,applying Ito formulas, we get the conditions for extinction and persistence in mean; Secondly, by using the relative knowledge of Markov Conversion and the Matrix theory knowledge, one gets the result that the system is ergodicity and has a unique stationary under certain conditions. Finally, simulations are carried out to illustrate our analytical results.In Chapter 4, we summarize the whole thesis, and make some future assumption for the subsequent research work.