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

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 R 0 of the corresponding ordinary differential system and the minimal wave speed c ∗ . More specifically, we first prove the existence of the traveling wave solutions for R 0 > 1 and c > c ∗ via considering a truncated initial value problem and using the Schauder's fixed point theorem. The existence of the traveling wave solutions with speed c = c ∗ 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 R 0 > 1 , 0 < c < c ∗ and R 0 ≤ 1 , c > 0 is proved. [ABSTRACT FROM AUTHOR]