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Global dynamics of a class of SEIRS epidemic models in a periodic environment. (English) Zbl 1184.34056
Summary: We study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number $$R_{0}$$ which is defined through the spectral radius of a linear integral operator. If $$R_{0}<1$$, then the disease free periodic solution is globally asymptotically stable and if $$R_{0}>1$$, then the disease persists. Our results improve the results in [T. Zhang and Z. Teng, Bull. Math. Biol. 69, No. 8, 2537–2559 (2007; Zbl 1245.34040)] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D30 Epidemiology 34D05 Asymptotic properties of solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations
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