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Time periodic traveling wave solutions for a Kermack–McKendrick epidemic model with diffusion and seasonality

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Abstract

In this paper, we study the time periodic traveling wave solutions for a Kermack–McKendrick SIR epidemic model with individuals diffusion and environment heterogeneity. In terms of the basic reproduction number \(R_0\) of the corresponding periodic ordinary differential model and the minimal wave speed \(c^*\), we establish the existence of periodic traveling wave solutions by the method of super- and sub-solutions, the fixed-point theorem, as applied to a truncated problem on a large but finite interval, and the limiting arguments. We further obtain the nonexistence of periodic traveling wave solutions for two cases involved with \(R_0\) and \(c^*\).

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Acknowledgements

The authors are grateful to the anonymous referees for their insightful comments and suggestions which contributed to greatly improve the original version of the manuscript. Both Zhang and Wang’s research was supported by NNSF of China (11371179, 11731005, 11701242) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2017-27, lzujbky-2019-79), and Zhao’s research was supported in part by the NSERC of Canada.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Liang Zhang & Zhi-Cheng Wang

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL , A1C 5S7, Canada

    Xiao-Qiang Zhao

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  1. Liang Zhang
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  2. Zhi-Cheng Wang
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Correspondence to Zhi-Cheng Wang.

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Zhang, L., Wang, ZC. & Zhao, XQ. Time periodic traveling wave solutions for a Kermack–McKendrick epidemic model with diffusion and seasonality. J. Evol. Equ. 20, 1029–1059 (2020). https://doi.org/10.1007/s00028-019-00544-2

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