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

TL;DR

This paper establishes the conditions under which periodic solutions exist in a class of periodic SEIRS epidemic models, providing a clear threshold based on the basic reproduction number R_0.

Contribution

It proves the existence of endemic periodic orbits when R_0>1 and the stability of disease-free states when R_0<1, clarifying the threshold for disease persistence.

Findings

Existence of endemic periodic orbits when R_0>1

Global stability of disease-free orbit when R_0<1

Sharp threshold between endemic and disease-free states

Abstract

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.