### For REFEREES

 Traveling waves of a nonlocal diffusion SIRS epidemic model with a class of nonlinear incidence rates and time delay Keywords:SIRS, traveling waves, nonlocal diffusion, nonlinear incidence rates 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.