Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay

Abstract

In this paper, we investigate the traveling wave solutions of a discrete diffusive epidemic model with nonlinear incidence and time delay. Employing the method of upper and lower solutions, Schauder's fixed point theorem and a limiting approach, we prove the existence of bounded super-critical and critical traveling wave solutions. Moreover, we obtain the positiveness and asymptotic boundary of the traveling wave solutions, which guarantee that the traveling wave solutions are non-trivial. The existence results show that the traveling waves are mixed of front type and pulse type. By way of contradiction and two-sided Laplace transform, we derive the non-existence of non-trivial, positive and bounded traveling wave solutions. It is the first time to apply the method of upper and lower solutions together with Schauder's fixed point theorem and two-sided Laplace transform to investigate the existence and non-existence of traveling wave solutions for discrete diffusive epidemic models, respectively.