Spatial dynamics for a time-periodic epidemic model in discrete media

Abstract

This paper considers the spreading speed and periodic traveling waves for a time-periodic epidemic model in discrete media. There were some known results on spatial dynamics for autonomous discrete epidemic models. However, only little results had been done for the same issue of time-periodic epidemic models in discrete media. The main difficulties arise from the lack of comparison principle and compactness of solution operators for such systems. To overcome these difficulties, we first characterize the spreading speed of the system which can be used to estimate how fast the disease spreads. Then, based on constructing two different pairs of explicit subsolutions and supersolutions, we establish the existence of supercritical and critical periodic traveling waves by using the asymptotic fixed point theorem together with the properties of Kuratowski-measure of non-compactness for solution maps to a related linear equation. We further derive the non-existence of periodic traveling waves when the wave speed is smaller than the spreading speed. To the best of our knowledge, this might be the first time to consider the spatial dynamics of time-periodic epidemic models in discrete media.