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Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate. (English) Zbl 1239.34038

Summary: I. A. Moneim and D. Greenhalgh [Math. Biosci. Eng. 2, No. 3, 591–611 (2005; Zbl 1079.92058)] proposed an SEIRS epidemic model with general periodic vaccination strategy and seasonally varying contact rate. Their investigation shows that when \(R_0^{\sup}<1\), there exists a globally asymptotically stable disease-free periodic state, and when \(R_0^{\inf}>1\), 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.

MSC:

34C25 Periodic solutions to ordinary differential equations
92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D20 Stability of solutions to ordinary differential equations

Citations:

Zbl 1079.92058
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References:

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