Study On Asymptotic Properties Of Several Kinds Of Stochastic Biological Mathematical Models

The epidemic model can be established according to the transmission mechanism of diseases,and be analyzed to explore the law of disease transmission,predict the development trend of disease.Analyzing the models and comparing the control effects of different prevention measures can provide reference for eliminating diseases and optimizing the control measures.Life is filled with all kinds of stochastic factors,which will inevitably interfere with the spread of disease.The development and perfection of stochastic differential equations and stochastic processes provide theoretical guidance for the study of stochastic systems.This thesis will study the asymptotic properties of several types of stochastic epidemic models and the structure of the full text is as follows:First,it is the introduction,which mainly summarizes the research history and development status of infectious disease model,and briefly introduces the main research content of this thesis.Secondly,the theoretical knowledge of stochastic differential equation is introduced,and some definitions and lemmas involved in this paper are given.Then the asymptotic behavior of a multi-group SEIR model with stochastic perturbations and nonlinear incidence rate is studied and the existence and uniqueness of the solution to the model we discuss are given.The global asymptotical stability in probability of the model with R0<1 is established by constructing Lyapunov functions.Next it proves that the disease can die out exponentially under certain stochastic perturbation while it is persistent in the deterministic case when R0>1.Several examples and numerical simulations are provided to illustrate the dynamic behavior of the model and verify our analytical results.Subsequently,the SIRS models with two different incidence rates and Markovian switching is considered.First,consider the perturbation of parameters due to random environments modulated by Markovian switching.The segment method is used to prove that the model has a unique solution and the estimate of the solution is provided.The threshold values for determining extinction or persistence in mean of diseases are presented by theoretical analysis and some inequalities techniques.Furthermore,some results reveal that stochastic disturbances can suppress the disease outbreak.Because of regime switching,the diseases will be extinct(or persistent)although they might be persistent(or extinct)in some certain environments.Then,the model in which incidence rate functions are perturbed by random environment is also discussed and the values to judge the disease extinction are obtained.At last,a few examples are set to illustrate these interesting phenomena,and their simulations have been carried out to verify our theoretical outcomes.Chapter 5 discusses a stochastic SIQS model via isolation with regimeswitching.The range of positive solution of the model is presented.Threshold to determine extinction and invariant measure is obtained by a new technique,which can be seen as the sufficient and almost necessary condition.Meantime,a value to judge the existence of stationary distribution is acquired by constructing the suitable hybrid Lyapunov function.Two values are proved to be consistent.Several examples are enumerated to test the theoretical results.Chapter 6 concerns the dynamic behavior of a stochastic epidemic model with information intervention and vertical transmission.The threshold to judge the extinction and persistence of the disease is obtained.When Δ<0,the three classes It,Mt and Rt appearing in the model go to extinct at exponential rate and the susceptible class St almost surely converges to the solution of the boundary equation exponentially.When Δ>0,the disease in the model is persistent in the mean.Besides,the existence of invariant probability measure under this condition is proved by constructing proper Lyapunov functions.Some examples are listed to check the reached results.Finally,the conclusion and outlook briefly summarize the research results of this thesis,and provide prospects for future research content and directions.