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Stability and Threshold Dynamics in a Seasonal Mathematical Model for Measles Outbreaks with Double-Dose Vaccination

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  • Mahmoud A. Ibrahim

    (National Laboratory for Health Security, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., 6720 Szeged, Hungary
    Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

  • Attila Dénes

    (National Laboratory for Health Security, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., 6720 Szeged, Hungary)

Abstract

Measles is a highly contagious viral disease that can lead to serious complications, including death, particularly in young children. In this study, we developed a mathematical model that incorporates a seasonal transmission parameter to examine the measles transmission dynamics. We define the basic reproduction number ( R 0 ) and show its utility as a threshold parameter for global dynamics and the existence of periodic solutions. The model was applied to the measles outbreak that occurred in Pakistan from 2019 to 2021 and provided a good fit to the observed data. Our estimate of the basic reproduction number was found to be greater than one, indicating that the disease will persist in the population. The findings highlight the need to increase vaccination coverage and efficacy to mitigate the impact of the epidemic. The model also shows the long-term behavior of the disease, which becomes endemic and recurs annually. Our simulations demonstrate that a shorter incubation period accelerates the spread of the disease, while a higher vaccination coverage rate reduces its impact. The importance of the second dose of the measles vaccine is emphasized, and a higher vaccine efficacy rate can also help bring R 0 below one. Our study provides valuable information for the development and implementation of effective measles control strategies. To prevent future outbreaks, increasing vaccination coverage among the population is the most effective way to reduce the transmission of measles.

Suggested Citation

  • Mahmoud A. Ibrahim & Attila Dénes, 2023. "Stability and Threshold Dynamics in a Seasonal Mathematical Model for Measles Outbreaks with Double-Dose Vaccination," Mathematics, MDPI, vol. 11(8), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1791-:d:1119230
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    References listed on IDEAS

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    1. Ibrahim, Mahmoud A. & Dénes, Attila, 2021. "Threshold and stability results in a periodic model for malaria transmission with partial immunity in humans," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    2. Yuyi Xue & Xiaoe Ruan & Yanni Xiao, 2020. "Modelling the Periodic Outbreak of Measles in Mainland China," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-13, March.
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    Cited by:

    1. Fawaz K. Alalhareth & Mohammed H. Alharbi & Mahmoud A. Ibrahim, 2023. "Modeling Typhoid Fever Dynamics: Stability Analysis and Periodic Solutions in Epidemic Model with Partial Susceptibility," Mathematics, MDPI, vol. 11(17), pages 1-26, 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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