Existence of entire solutions for delayed monostable epidemic models

Abstract

The purpose of this work is to study the existence of entire solutions for delayed monostable epidemic models with and without the quasi-monotone condition. In the quasi-monotone case, we first establish the comparison principle and construct appropriate sub-solutions and upper estimates. Then the existence and qualitative features of entire solutions are proved by mixing any finite number of traveling wave fronts with different speeds $c\geq c_{\min }$ and directions and a spatially independent solution, where $c_{\min }>0$ is the critical wave speed. In the non-quasi-monotone case, some new types of entire solutions are constructed by using the traveling wave fronts and spatially independent solutions of two auxiliary quasi-monotone systems and a comparison theorem for the Cauchy problems of the three systems.