Wave propagation of a discrete SIR epidemic model with a saturated incidence rate

Abstract

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number [Formula: see text] of the corresponding ordinary differential system and the minimal wave speed [Formula: see text]. More specifically, we first prove the existence of the traveling wave solutions for [Formula: see text] and [Formula: see text] via considering a truncated initial value problem and using the Schauder’s fixed point theorem. The existence of the traveling wave solutions with speed [Formula: see text] is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] is proved.