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Global Dynamics of an SEIR Model with Two Age Structures and a Nonlinear Incidence

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Abstract

In this paper, we study an SEIR model with both infection and latency ages and also a very general class of nonlinear incidence. We first present some preliminary results on the existence of solutions and on bounds of solutions. Then we study the global dynamics in detail. After proving the existence of a global attractor \(\mathcal{A}\), we characterize it in two cases distinguished by the basic reproduction number \(R_{0}\). When \(R_{0}<1\), we apply the Fluctuation Lemma to show that the disease-free equilibrium \(E_{0}\) is globally asymptotically stable, which means \(\mathcal{A}=\{E_{0}\}\). When \(R_{0}>1\), we show the uniform persistence and get \(\mathcal{A}=\{E_{0}\}\cup C \cup \mathcal{A}_{1}\), where \(C\) consists of points with connecting orbits from \(E_{0}\) to \(\mathcal{A}_{1}\) and \(\mathcal{A}_{1}\) attracts all points with initial infection force. Under an additional condition, we employ the approach of Lyapunov functional to find that \(\mathcal{A}_{1}\) just consists of an endemic equilibrium.

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Funding

The research of Chen was supported partially by NSERC of Canada.

S. Bentout, S. Djilali are partially supported by DGESTR of Algeria No. C00L03UN130120200004.

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Authors and Affiliations

  1. Laboratoire d’Analyse Non Linéaire et Mathématiques Appliquées, University of Tlemcen, Tlemcen, Algeria

    Soufiane Bentout & Salih Djilali

  2. Department of Mathematics and Informatics, Center of Belhadj Bouchaib, Ain Temouchent, BP 284 RP, 46000, Algeria

    Soufiane Bentout

  3. Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

    Yuming Chen

  4. Faculty of Exact sciences and Informatics, Mathematic Department, Hassiba Benbouali University, Chlef, Algeria

    Salih Djilali

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  1. Soufiane Bentout
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  2. Yuming Chen
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Correspondence to Salih Djilali.

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Bentout, S., Chen, Y. & Djilali, S. Global Dynamics of an SEIR Model with Two Age Structures and a Nonlinear Incidence. Acta Appl Math 171, 7 (2021). https://doi.org/10.1007/s10440-020-00369-z

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