Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model

A delayed SEIRS epidemic model with pulse vaccination and bilinear incidence rate is investigated. Using Krasnoselskii’s fixed-point theorem, we obtain the existence of disease-free periodic solution (DFPS for short) of the delayed impulsive epidemic system. Further, using the comparison method, we prove that under the condition R∗ < 1, the DFPS is globally attractive, and that R∗ > 1 implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than θ∗ and the disease is uniformly persistent if the vaccination rate is less than θ∗. Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease.