UNIDIRECTIONAL WAVE PROPAGATION IN A NONLOCAL DISPERSAL ENDEMIC MODEL WITH NONLINEAR INCIDENCE

This paper is concerned with existence and non-existence of traveling wave solutions in a nonlocal dispersal endemic model with nonlinear incidence. With the aid of upper-lower solutions method and Schauder's fixed point theorem together with Lyapunov functional technique, we derive the existence of super-critical and critical traveling wave solutions connecting disease-free equilibrium to endemic equilibrium. In a combination with the theory of two-sided Laplace transform and local skilled analysis, we obtain the non-existence of sub-critical traveling wave solutions. Our results illustrate that: (ⅰ) the existence and non-existence of traveling waves are determined by the basic reproduction number and the wave speed; (ⅱ) the critical wave speed is equal to the minimal wave speed; (ⅲ) the traveling waves only propagate along one direction.