Asymptotic Stability Of Solutions For Two Types Of Stochastic Models

A large number of deterministic models and continuous models have been proposed and achieved remarkable results in the study of biological population and epidemic diseases. But most of the time, there are a large discrepancy between the real life case with our models, so it is very necessary to add random factors in the deterministic model.In this paper the stochastic discrete Logistic model and the stochastic continuous SIR model are studied, and the asymptotic stability of the solutions are obtained.The first chapter introduces the background and significance of the thesis and the current research status both in inland and abroad, and at the same time, the research content of this paper is briefly described.The second chapter gives the theoretical knowledges which is needed in the research of this thesis, including the knowledge and methods in probability theory and mathematical statistics, Lyapunov function theory, stochastic differential equation theory, stochastic inequality, and so on. And we focus on the ?Ito formula, the martingale convergence theorem.The third chapter studies the asymptotic stability of a stochastic discrete Logistic equation with time delay. Under the support of related theories and knowledges, we were obtained the conditions for the stability of the deterministic discrete model and the conditions for the stability of the stochastic discrete model.The fourth chapter studies a stochastic SIR epidemic model. First, we construct a suitable Lyapunov function, and obtain the global existence and uniqueness for the positive solutions of the model; then, asymptotic behaviors of the stochastic model of the equilibrium point are discussed and the maximum value of the upper limit for the solution of the stochastic model is obtained.