Existence of entire solutions for delayed monostable epidemic models

Wu, Shi-Liang; Hsu, Cheng-Hsiung

doi:10.1090/tran/6526

Existence of entire solutions for delayed monostable epidemic models
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by Shi-Liang Wu and Cheng-Hsiung Hsu PDF

Trans. Amer. Math. Soc. 368 (2016), 6033-6062 Request permission

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.

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