Impulsive vaccination of sir epidemic models with nonlinear incidence rates

The impulsive vaccination strategies of the epidemic SIR models with nonlinear incidence rates $\beta I^{p}S^{q}$ are considered in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic infection-free solution of the impulsive epidemic system and prove that the periodic infection-free solution is globally asymptotically stable. In order to apply vaccination pulses frequently enough so as to eradicate the disease, the threshold for the period of pulsing, i.e. $\tau _{max}$ is shown, further, by bifurcation theory, we obtain a supercritical bifurcation at this threshold, i.e. when $\tau>\tau_{max}$ and is closing to $\tau_{max}$, there is a stable positive periodic solution. Throughout the paper, we find impulsive epidemiological models with nonlinear incidence rates $\beta I^{p}S^{q}$ show a much wider range of dynamical behaviors than do those with bilinear incidence rate $\beta SI$ and our paper extends the previous results, at the same time, theoretical results show that pulse vaccination strategy is distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination, therefore impulsive vaccination strategy provides a more natural, more effective vaccination strategy.