Analysis of a delayed SIR epidemic model with pulse vaccination

In this paper, a delayed SIR epidemic model with pulse vaccination is investigated. By the comparison theorem for impulsive differential equations, we obtain that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is permanent if the vaccination proportion is less than some critical value under appropriate condition. By Brouwer’s fixed-point theorem, we establish sufficient condition for the existence of positive periodic solution. Our results indicate that a large vaccination rate or a short period of pulsing is a sufficient condition for the eradication of the disease.