Pulse Stability Of The Infectious Disease Model And Branch

Impulsive differential equations have been greatly used to study the dynamics behavior in ecological systems and epidemic models. In this paper, epidemic models with pulses are considered.In chapter 1, we introduce the development and the significance on the epidemic model with pulses.In chapter 2, an SIR epidemic model with logistic population dynamics and nonlinear birth pulses is considered. The basic reproduction number R₀ is defined. We obtain the exact infection-free periodic solution of the impulsive epidemic system. By using the discrete dynamical system generated by a monotone, concave map for the population, we prove the infection-free periodic solution is globally asymptotically stable if R₀＜1. We use standard bifurcation theory to show the existence of positive periodic solution if R₀＞1.In chapter 3, an SIR epidemic model with pulse vaccination and pulse elimination is considered. The basic reproduction number R₀ is defined. By using the impulsive differential inequality and comparison principle, we have proved the infection-free periodic solution is globally asymptotically stable if R₀＜1. Numerical simulation is given in the final part.In chapter 4, an SIS epidemic model with pulse vaccination is considered. By comparison principle, we have proved the infection-free periodic solution is globally asymptotically stable and by bifurcation theory, we obtain a supercritical bifurcation at the threshold for the period of pulsing.