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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. [ABSTRACT FROM AUTHOR]

This paper is concerned with a time-periodic and nonlocal system arising from the spread of a deterministic epidemic in multi-types of population by incorporating a seasonal variation. The existence of the critical wave speed of the periodic traveling wavefronts and its coincidence with the spreading speed were proved in Wu et al. (J Math Anal Appl 463:111–133, 2018). In this paper, we prove the uniqueness and stability of all non-critical periodic wavefronts. Of particular interest is the influences of time-periodicity on the spreading speed in one-dimensional case. It turns out that, in comparison with the autonomous case, the periodicity of the infection rate increases the spreading speed, while the periodicity of the combined death/emigration/recovery rate for infectious individuals decreases the spreading speed. We also find that the contact distribution increases the spreading speed. [ABSTRACT FROM AUTHOR]