Global Dynamic Analysis On Several Kinds Of Stochastic Epidemic Models

With the development of human society and the progress of science and technology, population ecology and epidemic spread increasingly become a problem that is so intertwined with human life. Researchers have set up many biological population and infectious disease model which described the relationship between dierent species and the external environment.A mathematical model can be divided into continuous system and discrete system or deterministic and stochastic systems according to the dierent properties. The ?rst two have been extensively studied. And compared with the deterministic systems, people ?nd the real world inevitably aected by random or uncertain factors, so stochastic dierential equations can more accurately and really describe the real world. Therefore, stochastic model has become one of the hot research subjects in the biological mathematics theory.In this paper, we study the dynamic behavior of infectious diseases in stochastic model by using the theory of stochastic dierential equation.This paper is divided into ?ve chapters, the main contents can be summarized as follows:In the ?rst chapter, we introduce the biological background and signi?cance of stochastic epidemic model,and then introduce basic theory of stochastic dierential equations and research results in stochastic infectious disease as well as the main work in this paper.In the second chapter, some main concepts, inequality, lemma, theorem are introduced in order to prove our results in the paper.In the third chapter, we studied the SIRS epidemic model with Brown motion and jump. In this paper, we ?rst obtain stochastic model by interfering the deterministic model with white noise. Then global positivity of solution is studied. Based on global positive solution, the dynamic behavior of solution around the equilibrium points of the deterministic model is invested. Numerical simulations are given in the end to con?rm our results.In the fourth chapter, the dynamical behavior of a stochastic SIS epidemic model with nonlinear incidence rate is investigated. in this section, we not only prove that the existence of global positive solution, extinction and persistence of the disease, but also through the previous proof of the theorem we discuss the existence of the stationary distribution and ergodicity of the model. And the mean and variance are estimated in the system. Finally, numerical simulation is carried out to verify our results.In the ?fth chapter, we study the probability distribution of SIS infectious disease model. That is the s(t), I(t) are regarded the birth and death process. We compute corresponding s(t), I(t) probability distribution by calculating the Karl kolmogoro equation.We know that using Karl kolmogoro equations to compute the probability distribution is a more dicult process. For the pure birth or death process, we can obtain the probability distribution by Laplase transformation. For the process of the mixture of the two, here we use the idea of discussing ?nally to obtain the probability distribution of each state. In the process, we ?nd some special phenomena and make a summary for the special phenomena in the end of the paper.In the sixth chapter, we do some discussions and conclusions about we have study on this paper.