Pulse vaccination in the periodic infection rate sir epidemic model

In this paper a pulse vaccination SIR model with periodic infection rate β(t) is studied. The basic reproductive number R ₀ is defined. The dynamical behavior of the model is analyzed. It is proved that the infection-free periodic solution is globally stable if R ₀ < 1. The infection-free periodic solution is unstable and the disease will uniform persistence when R ₀ > 1. We use standard bifurcation theory to show the existence of the positive periodic solution when R ₀ → 1 ⁺. Numerical simulation can give suggestion, the system has a unique positive periodic, and it is globally stable when R ₀ > 1.