TRAVELING WAVES OF A NONLOCAL DIFFUSION SIRS EPIDEMIC MODEL WITH A CLASS OF NONLINEAR INCIDENCE RATES AND TIME DELAY

Abstract

In this paper, we study the traveling waves of a delayed SIRS epidemic model with nonlocal diffusion and a class of nonlinear incidence rates. When the basic reproduction ratio $ \mathscr{R}_0>1 $, by using the Schauder's fixed point theorem associated with upper-lower solutions, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of traveling wave solutions connecting the disease-free steady state and the endemic steady state. When $ \mathscr{R}_0<1 $, the nonexistence of traveling waves is obtained by the comparison principle.