Existence of periodic solutions of a periodic SEIRS model with general incidence

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 R-0 holds. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when R-0 < 1. In particular, our main result generalizes the one in Zhang et al. (2012). 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. (C) 2016 Elsevier Ltd. All rights reserved.