Studies On Asymptotic Behavior Of Some SIRS Models

In this dissertation, we propose some SIRS models. We obtain sufcient conditions for exis-tence and local stability of equilibria in those models and study the boundedness of solutions ofthose models. Moreover, we give some correspond numerical simulations to illustrate the validityof our results.The dissertation has six chapters.In the frst chapter, we introduce the background and research of epidemic models, especiallySIRS models, and give the main results of the thesis.In the second chapter, we propose a new stochastic SIRS model, and get sufcient conditionsfor the existence of the positive solution in the model. Moreover, we give numerical simulations toillustrate our results.In the third chapter, taking both white and color environmental noises into account, we in-troduce a new stochastic SIRS model under regime switching. We show the equilibria of the modelare stochastically asymptotically stable under certain conditions, and the model has a stationarydistribution under specifed conditions. Moreover, we show that our model is stochastically ulti-mately bounded and the moment average in time of the solution of the model is bounded. Finally,we give an example to illustrate the boundedness of the moment average in time of the solution inthe model.In the fourth chapter, we set up two new kinds of high-dimensional SIRS models: high-dimensional continuous SIRS model and high-dimensional discrete SIRS model, and obtain suf-cient conditions for existence and local stability of the equilibria in the two models. Moreover, weuse numerical simulations to illustrate the validity of our results.In the ffth chapter, we establish a new stochastic high-dimensional SIRS model, and demon-strate that the model is stochastically ultimately bounded and the moment average in time of thepositive solution of the model is bounded. Furthermore, we give numerical simulations to illustrateour main results.In the sixth chapter, we give a summary and the outlook of the dissertation.