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Existence of periodic solutions of a periodic SEIRS model with general incidence. (English) Zbl 1354.34085
Summary: For a family of periodic SEIRS models with general incidence, we prove the existence of at least one endemic periodic orbit when some condition related to \(\mathcal{R}_0\) holds. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when \(\mathcal{R}_0 < 1\). In particular, our main result generalizes the one in [T. Zhang et al., Appl. Math., Praha 57, No. 6, 601–616 (2012; Zbl 1274.34150)]. We also discuss some examples where our results apply and show that, in some particular situations, we have a sharp threshold between existence and non existence of an endemic periodic orbit.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C25 Periodic solutions to ordinary differential equations
92D30 Epidemiology
37C60 Nonautonomous smooth dynamical systems
34D20 Stability of solutions to ordinary differential equations
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