A stage-structured SEIR model with time-dependent delays in an almost periodic environment

Lizhong Qiang; Ren-Hu Wang; Ruofan An; Zhi-Cheng Wang; Lizhong Qiang; Ren-Hu Wang; Ruofan An; Zhi-Cheng Wang

doi:10.3934/mbe.2020393

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7732-7750. doi: 10.3934/mbe.2020393

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A stage-structured SEIR model with time-dependent delays in an almost periodic environment

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, P. R. China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China
3.
School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, P. R. China

Received: 01 September 2020 Accepted: 22 October 2020 Published: 06 November 2020

In this paper, we propose and investigate an almost periodic SEIR model with stage structure and latency, in which time-dependent maturation and incubation periods are incorporated. Two threshold parameters for the persistence and extinction of population and disease are introduced: the basic reproduction ratio $\hat{R}_{0}$ for population and the basic reproduction ratio $R_{0}$ for disease. If $\hat{R}_{0}<1$, the population extinction state is globally attractive. In the case where $\hat{R}_{0}>1$, it is shown that the disease tends to die out if $R_{0}<1$, while remains persistent if $R_{0}>1$. By virtue of numerical simulations, we verify the analytic results and investigate the effects of the fluctuations of maturation and incubation periods on disease transmission.
- almost periodicity,
- SEIR model,
- time-dependent delay,
- basic reproduction ratio,
- threshold dynamics
Citation: Lizhong Qiang, Ren-Hu Wang, Ruofan An, Zhi-Cheng Wang. A stage-structured SEIR model with time-dependent delays in an almost periodic environment[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7732-7750. doi: 10.3934/mbe.2020393

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Abstract

In this paper, we propose and investigate an almost periodic SEIR model with stage structure and latency, in which time-dependent maturation and incubation periods are incorporated. Two threshold parameters for the persistence and extinction of population and disease are introduced: the basic reproduction ratio $\hat{R}_{0}$ for population and the basic reproduction ratio $R_{0}$ for disease. If $\hat{R}_{0}<1$, the population extinction state is globally attractive. In the case where $\hat{R}_{0}>1$, it is shown that the disease tends to die out if $R_{0}<1$, while remains persistent if $R_{0}>1$. By virtue of numerical simulations, we verify the analytic results and investigate the effects of the fluctuations of maturation and incubation periods on disease transmission.

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