Global Dynamic Analysis On Discrete Epidemic Models

To study discrete epidemic model is a hot topic in the field of biomathematics. In some cases, we know that it shows much more behavior than continues model and press close to reality. Hence, more and more people pay attention to the discrete epidemic model and then achieve plenty of conclusions. In this paper, we study the the dynamical behavior of the discrete epidemic model by the difference equation theory.This paper is split into four chapters, the main ideas can be summarized as follows:In the first chapter, we show the biological background and meaning of discrete epidemic model, some lately research results and the main work in this paper.In the second chapter, we study global dynamics for a discrete viral infection model with general incidence rate and CTL immune response. In this paper, we firstly obtain discrete model by applying the Micken's non-standard finite difference method. Then we study global positivity and ultimate boundness of solution and the existence of equi-librium points. Based on the assumption of function f(x,y,v), we study the dynamical behavior of equilibrium points of model by constructing the Lyapunov function and linearization method. Finally, we give numerical simulations to show that no-immune e-quilibrium points and infected equilibrium points may globally asymptotic stability even though assumption A4 does not hold.In the third chapter, we study a discrete time analogue for coupling within-host and between-host dynamics in environmentally-driven infectious disease. Firstly, we prove the existence of positive solution, ultimate boundness of solution and the existence of equilibrium points in the quick system. Then we study the dynamical behavior of equilibrium points of model by constructing the Lyapunov function and linearization method. Following, we also obtain the existence of positive solution, ultimate boundness of solution and the existence of equilibrium points in the environmentally-driven slow system, Then we study the dynamical behavior of equilibrium points of model by using linearization method. However, we do not give the theoretical proof to local stability of the two positive equilibrium points. we only show stability of the two positive equilibrium points by numerical simulations. Last, we study the coupling system, we obtain the existence of equilibrium points and the local stability of disease-free and infection-free equilibrium and disease-free equilibrium. However, we do not give the theoretical proof to local stability of endemic equilibrium and the two positive equilibrium points, we only show stability of the endemic equilibrium and two positive equilibrium points by numerical simulations.In the four chapter, we do some discussions and conclusions which we have study on this paper.