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Global dynamics of a time-fractional spatio-temporal SIR model with a generalized incidence rate

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Abstract

This paper delves into the study of a diffusive SIR epidemic model characterized by reaction–diffusion equations enriched with a fractional derivative within the Caputo framework. Within this model, we incorporate a general incidence function and meticulously analyze how the adoption of mask-wearing and adherence to physical distancing protocols intricately shape the dynamics of susceptible and infected individuals. Our exploration commences by establishing the existence and uniqueness of a positively bounded solution for the model, employing powerful Banach’s fixed point theorem. Moreover, we showcase that this solution demonstrates distinctive global mild attributes. Subsequently, we elucidate the two equilibrium points inherent in the system: the disease-free and endemic points. Employing the LaSalle–Lyapunov theorem, we establish that the global stability of these equilibrium points is predominantly contingent upon the basic reproduction number of the system. This stability assertion holds true across various values of the non-integer order derivative. Lastly, we substantiate our findings with a series of numerical simulations that provide tangible support for the preceding analytical results.

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Acknowledgements

The authors wish to extend their deepest appreciation to the editor and the anonymous reviewers for their remarkable dedication and invaluable feedback, which greatly improved the paper’s quality.

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The authors assert that no funding was received for the preparation and execution of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, LAMAO Laboratory, Science Faculty, Mohammed First University of Oujda, 60000, Oujda, Morocco

    Ayoub Bouissa & Najib Tsouli

  2. Department of Mathematics, MAIS Laboratory, AMNEA Group, FST Errachidia, Moulay Ismail University of Meknes, P.O. Box. 509, 52000, Boutalamine, Errachidia, Morocco

    Mostafa Tahiri & Moulay Rchid Sidi Ammi

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  1. Ayoub Bouissa
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  2. Mostafa Tahiri
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  3. Najib Tsouli
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Bouissa, A., Tahiri, M., Tsouli, N. et al. Global dynamics of a time-fractional spatio-temporal SIR model with a generalized incidence rate. J. Appl. Math. Comput. 69, 4779–4804 (2023). https://doi.org/10.1007/s12190-023-01932-1

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