Pulse vaccination in the periodic infection rate SIR epidemic model
  A pulse vaccination SIR model with periodic infection rate $\beta (t)$ have been proposed and studied. The basic reproductive number $R_0$ is defined. The dynamical behaviors of the model are analyzed with the help of persistence, bifurcation and global stability. It has been shown that the infection-free periodic solution is globally stable provided $R_0 < 1$ and is unstable if $R_0>1$. Standard bifurcation theory have been used to show the existence of the positive periodic solution for the case of $R_0 \to1^+$. Finally, the numerical simulations have been performed to show the uniqueness and the global stability of the positive periodic solution of the system. 
Zhen Jin, Mainul Haque and Quanxing Liu