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A reaction–diffusion epidemic model with incubation period in almost periodic environments

Part of: Applications - Dynamical systems and ergodic theory Parabolic equations and systems

Published online by Cambridge University Press:  25 September 2020

LIZHONG QIANG ,
BIN-GUO WANG Open the ORCID record for BIN-GUO WANG [Opens in a new window]  and
ZHI-CHENG WANG
LIZHONG QIANG
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China emails: qianglzh13@lzu.edu.cn; wangbinguo@lzu.edu.cn; wangzhch@lzu.edu.cn
BIN-GUO WANG
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China emails: qianglzh13@lzu.edu.cn; wangbinguo@lzu.edu.cn; wangzhch@lzu.edu.cn
ZHI-CHENG WANG
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China emails: qianglzh13@lzu.edu.cn; wangbinguo@lzu.edu.cn; wangzhch@lzu.edu.cn
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Abstract

In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent λ* for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of λ* It is shown that the disease-free almost periodic solution is globally attractive if λ* < 0, while the disease is persistent if λ* > 0. By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

Keywords

Almost periodicityincubation periodspatial heterogeneityupper Lyapunov exponentthreshold dynamics

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 37N25: Dynamical systems in biology 92D30: Epidemiology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 6: Special issue featuring papers on Professor Sam Howison , December 2021 , pp. 1153 - 1176
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Supported by NSF of China (11501269, 11731005, 11371179, 11801241) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2020-13).

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