Dynamic Studies Of The Stochastic SIS And HIV-1 Models

Dynamies of epidemiology is the study of the spread and development of dis-eases with the purpose to trace factors that are contribute to their occurrenee. In recent decades, epidemiology modeling by mathematical research has been received a great attention within the academia. In this paper, since stochastic differential equations can reflect the reality more accurately, we extended some classical deter-ministic model for the epidemics by inducing random perturbations in the models. Using the qualitative theory of stochastic differential equation, we study their dy-namic behaviors for the stochastic epidemic models, such as the persistence of the system, the stability of the equilibrium. Moreover, the parameters’influence on the dynamical behaviors is also explored here. And, by means of numerical simu-lations, we discuss the trend of the epidemiology. To be specific, our research is as follows:In Chapter 1, a brief introduction is given for the background of mathematical modeling on infectious diseases, including some basic compartmental models, some definitions and prerninary theorems which will be used in this paper are introduced exhaustively. Moreover, a survey of the results in this thesis is also presented.In Chapter 2, we investigate a stochastic SIS model with saturated incidence rate and a stochastic SIS model with nonlinear incidence rate, respectively, we analyze the dynamic behaviors of the two stochastic SIS models. First, show the two models both exit the unique global positive solution. Second, we investigate the asymptotic behavior of the positive solution, and obtain the threshold between prevalence and extinction of the disease, that is, if Rs0<1, the disease will die out with probability one, if Rs0>1, the two systems has ergodic property and derive the expression for its invariant density, which means the disease will become endemic. Finally, we illustrate our results with computer simulations.In chapter 3, we introduce stochasticity into a multigroup SIS model. The aim of the chapter is to study asymptotical behaviour of the solutions of the stochastic system. We show that the stochastic system has a unique non-negative solution, we present the sufficient condition for the exponential extinction of the disease. In the case of persistence of the disease, we prove that there exists an invariant distribution for the stochastic system and it is ergodic, the disease becomes endemic.In chapter 4, we introduce the stochasticity into an HIV-1 infection model with cytotoxic T lymphocytes (CTLs) immune response via the technique of pa-rameter perturbation, investigate Dynamics of an HIV-1 Infection Model with Cell-Mediated Immune Response and Stochastic Perturbation, we show there is a unique non-negative solution of the system. Then we analyze the long time behavior of this model, and investigate the dynamics of the system around E0, E1 and E2 respec-tively, and give the conditions for the solution fluctuating around the two infection equilibria. Finally, the simulations are carried out to conform to our analytical results.In the last chapter, we made a summary of full paper...