Existence of traveling waves for a nonlocal dispersal SIR epidemic model with treatment

Abstract

This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal epidemic model with treatment. The existence of traveling wave solutions is established by Schauder's fixed point theorem, while the nonexistence of traveling wave solutions is proved by two-sided Laplace transform. From the results, we conclude the minimal wave speed, which is an important threshold to predict how fast the disease invades. Compared with the work in [35], we obtain more accurate results about the existence and nonexistence of nontrivial traveling wave solutions. We prove that when the basic reproduction number R0>1, there exists a critical number c1⁎>0 such that for each c>c1⁎, the system has a nontrivial traveling wave solution with speed c, while for 0<c<c1⁎ the system admits no nontrivial traveling wave solution. When R0<1, we show that there exists no nontrivial traveling wave solution. In addition, based on [24], we obtain the existence of traveling waves with the critical speed c=c1⁎ under certain conditions.