Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase

Mostafa Adimy; Abdennasser Chekroun; Claudia Pio Ferreira; Mostafa Adimy; Abdennasser Chekroun; Claudia Pio Ferreira

doi:10.3934/mbe.2020067

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1329-1354. doi: 10.3934/mbe.2020067

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Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase

1.
Inria, CNRS UMR 5208, Institut Camille Jordan, Université Lyon 1, F-69200 - Villeurbanne, France
2.
Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquees, Université de Tlemcen-Tlemcen 13000, Algeria
3.
Department of Biostatistics, São Paulo State University (UNESP), 18618-689-Botucatu, SP, Brazil

Received: 13 July 2019 Accepted: 30 October 2019 Published: 20 November 2019

In this paper, we are concerned with an epidemic model of susceptible, infected and recovered (SIR) population dynamic by considering an age-structured phase of protection with limited duration, for instance due to vaccination or drugs with temporary immunity. The model is reduced to a delay differential-difference system, where the delay is the duration of the protection phase. We investigate the local asymptotic stability of the two steady states: disease-free and endemic. We also establish when the endemic steady state exists, the uniform persistence of the disease. We construct quadratic and logarithmic Lyapunov functions to establish the global asymptotic stability of the two steady states. We prove that the global stability is completely determined by the basic reproduction number.
- SIR epidemic model,
- age-structured PDE,
- delay differential-difference system,
- basic reproduction number,
- Lyapunov functional,
- local and global stability
Citation: Mostafa Adimy, Abdennasser Chekroun, Claudia Pio Ferreira. Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1329-1354. doi: 10.3934/mbe.2020067

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Abstract

In this paper, we are concerned with an epidemic model of susceptible, infected and recovered (SIR) population dynamic by considering an age-structured phase of protection with limited duration, for instance due to vaccination or drugs with temporary immunity. The model is reduced to a delay differential-difference system, where the delay is the duration of the protection phase. We investigate the local asymptotic stability of the two steady states: disease-free and endemic. We also establish when the endemic steady state exists, the uniform persistence of the disease. We construct quadratic and logarithmic Lyapunov functions to establish the global asymptotic stability of the two steady states. We prove that the global stability is completely determined by the basic reproduction number.

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