# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021145
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## Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China 2 School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou, Gansu 730020, China 3 School of Mathematical and Statistical Sciences, University of Texas, Edinburg, Texas 78539, USA

* Corresponding author

Received  January 2021 Revised  June 2021 Early access August 2021

Fund Project: The first author was supported by the innovation fund project for colleges and universities of Gansu Province of China (2020A-062) and NSF of Gansu Province of China (21JR7RA549), the second author was supported by NSF (DMS-1204497), the third author was supported by NSF of China (12071193 and 11731005), and the forth author was supported by NSF of China (12171214, 11701242)

We study the asymptotic spreading properties and periodic traveling wave solutions of a time periodic and diffusive SI epidemic model with demographic structure (follows the logistic growth). Since the comparison principle is not applicable to the full system, we analyze the asymptotic spreading phenomena for susceptible class and infectious class by comparing with respective relevant periodic equations with KPP-type. By applying fixed point theorem to a truncated problem on a finite interval, combining with limit idea, the existence of periodic traveling wave solutions are derived. The results show that the minimal wave speed exactly equals to the spreading speed of infectious class when susceptible class is abundant.

Citation: Shuang-Ming Wang, Zhaosheng Feng, Zhi-Cheng Wang, Liang Zhang. Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021145
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