  For a family of periodic SEIRS models with general incidence, we prove the
existence of at least one endemic periodic orbit when R_0>1. 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 establishes a
sharp threshold between existence and non-existence of endemic periodic orbits
for this family of models.

Dynamical Systems;Classical Analysis and ODEs