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Periodic Behaviour of an Epidemic in a Seasonal Environment with Vaccination

Author

Listed:
  • Miled El Hajji

    (Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
    ENIT-LAMSIN, Tunis El Manar University, BP. 37, Tunis-Belvédère, Tunis 1002, Tunisia)

  • Dalal M. Alshaikh

    (Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia)

  • Nada A. Almuallem

    (Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia)

Abstract

Infectious diseases include all diseases caused by the transmission of a pathogenic agent such as bacteria, viruses, parasites, prions, and fungi. They, therefore, cover a wide spectrum of benign pathologies such as colds or angina but also very serious ones such as AIDS, hepatitis, malaria, or tuberculosis. Many epidemic diseases exhibit seasonal peak periods. Studying the population behaviours due to seasonal environment becomes a necessity for predicting the risk of disease transmission and trying to control it. In this work, we considered a five-dimensional system for a fatal disease in a seasonal environment. We studied, in the first step, the autonomous system by investigating the global stability of the steady states. In a second step, we established the existence, uniqueness, positivity, and boundedness of a periodic orbit. We showed that the global dynamics are determined using the basic reproduction number denoted by R 0 and calculated using the spectral radius of an integral operator. The global stability of the disease-free periodic solution was satisfied if R 0 < 1 , and we show also the persistence of the disease once R 0 > 1 . Finally, we displayed some numerical investigations supporting the theoretical findings, where the trajectories converge to a limit cycle if R 0 > 1 .

Suggested Citation

  • Miled El Hajji & Dalal M. Alshaikh & Nada A. Almuallem, 2023. "Periodic Behaviour of an Epidemic in a Seasonal Environment with Vaccination," Mathematics, MDPI, vol. 11(10), pages 1-20, May.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2350-:d:1149878
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    References listed on IDEAS

    as
    1. Abdulrahman Ali Alsolami & Miled El Hajji, 2023. "Mathematical Analysis of a Bacterial Competition in a Continuous Reactor in the Presence of a Virus," Mathematics, MDPI, vol. 11(4), pages 1-18, February.
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    Cited by:

    1. Amer Hassan Albargi & Miled El Hajji, 2023. "Bacterial Competition in the Presence of a Virus in a Chemostat," Mathematics, MDPI, vol. 11(16), pages 1-17, August.
    2. Miled El Hajji & Rahmah Mohammed Alnjrani, 2023. "Periodic Behaviour of HIV Dynamics with Three Infection Routes," Mathematics, MDPI, vol. 12(1), pages 1-23, December.

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    1. Amer Hassan Albargi & Miled El Hajji, 2023. "Bacterial Competition in the Presence of a Virus in a Chemostat," Mathematics, MDPI, vol. 11(16), pages 1-17, August.

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