Stability And Optimal Control Of Two Kinds Of SIRS Models With Constant Input Rate

People use ordinary differential equation model to study the law of the spread of the disease and predict of disease development mechanism. But most of the results are based on the unchanged hypothesis of the population. In fact, the mobility of population is large, the immigration and emigration rate of population have great influence on the quantity of the population changing. In this paper, we establish two types of SIRS epidemic with constant immigration rate, and study the stability and optimal control. The thesis is divided into three sections according to contents.Chapter 1 This chapter briefly introduces the development back-ground of infectious disease and the status of SIRS model.Chapter 2 In this chapter, we study SIRS model with constant input rate and standard incidence rateFirst of all, we discuss respectively locally asymptotic stability of equi-librium with no infections input or infections input. Secondly, we analyze asymptotic stability of the equilibrium by constructing Lyapunov func-tion. Finally, we get optimal control scheme by the extremum principle early in the disease breaks out.Chapter 3 In this chapter, we study the SIRS model with constant input rate and stage structureFirst of all, using the F-v function we find the critical threshold of eliminated disease namely the basic reproduction number R0. Secondly, we discuss the global asymptotic stability of equilibrium by constructing Lyapunov function. Finally, after the outbreak of the disease, we con-struct the control system and we get the control project in the optimal performance index by constructing of performance index function. This provides a powerful reference for the effective control of disease.