Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate

Moneim and Greenhalgh [I.A. Moneim, D. Greenhalgh, Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate, Math. Biosci. Eng. 2 (2005) 591–611] proposed an SEIRS epidemic model with general periodic vaccination strategy and seasonally varying contact rate. Their investigation shows that when R0sup<1, there exists a globally asymptotically stable disease-free periodic state, and when R0inf>1, the disease-free solution is unstable and there is at least one positive periodic solution. But they did not find the threshold condition for uniform persistence and extinction of the disease, and left a conjecture—that is, whether the basic reproduction ratio of the time-averaged system can be the threshold parameter or not. The present paper gives a negative answer to this question and provides a thorough global dynamics for this system. Numerical simulations which show our theoretical results are also given.