Global analysis for an epidemic model with nonlinear vaccination rate

An epidemic model with the nonlinear vaccination rate is investigated. The threshold determining the disease dies out or not is found. Below the threshold, the disease-free equilibrium is globally stable. Above the threshold, the disease-free equilibrium is unstable, and the endemic equilibrium exists and is locally asymptotically stable. The global stability of the endemic equilibrium is proved for some special cases by means of limiting theory and Liapunov function. At last, some numerical simulations are given for two special cases of vaccination rate.