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Global Asymptotic Stability of a Hybrid Differential–Difference System Describing SIR and SIS Epidemic Models with a Protection Phase and a Nonlinear Force of Infection

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Abstract

We study the local and global asymptotic stability of the two steady-states, disease-free and endemic, of hybrid differential–difference SIR and SIS epidemic models with a nonlinear force of infection and a temporary phase of protection against the disease, e.g. by vaccination or medication. The initial model is an age-structured system that is reduced using the method of characteristic lines to a hybrid system, coupled between differential equations and a time continuous difference equation. We first prove that the solutions of the original system can be obtained from the reduced one. We then focus on the reduced system to obtain new results on the asymptotic stability of the two steady-states. We determine the local asymptotic stability of the two steady-states by studying the associated characteristic equation. We then discuss their global asymptotic stability in various situations (SIR, SIS, mass action, nonlinear force of infection), by constructing appropriate Lyapunov functions.

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Acknowledgements

A. Chekroun thanks Grant, PRFU: C00L03UN29012022002, from DGRSDT of Algeria.

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Author notes

  1. Mostafa Adimy, Abdennasser Chekroun, Charlotte Dugourd-Camus and Hanene Meghelli have contributed equally to this work.

Authors and Affiliations

  1. Inria, Univ Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, 69200, Villeurbanne, France

    Mostafa Adimy & Charlotte Dugourd-Camus

  2. Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, University of Tlemcen, 13000, Tlemcen, Algeria

    Abdennasser Chekroun & Hanene Meghelli

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  1. Mostafa Adimy
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  2. Abdennasser Chekroun
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  3. Charlotte Dugourd-Camus
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  4. Hanene Meghelli
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Correspondence to Mostafa Adimy.

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Adimy, M., Chekroun, A., Dugourd-Camus, C. et al. Global Asymptotic Stability of a Hybrid Differential–Difference System Describing SIR and SIS Epidemic Models with a Protection Phase and a Nonlinear Force of Infection. Qual. Theory Dyn. Syst. 23, 34 (2024). https://doi.org/10.1007/s12346-023-00891-z

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