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Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals

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Abstract

This paper deals with the qualitative behavior of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals. Firstly, the positivity and boundedness of solutions for integer-order model are proved. The basic reproduction number \({\mathcal {R}}_0\) is driven and it is shown that the disease-free equilibrium is globally asymptotically stable if \({\mathcal {R}}_0<1\) in integer-order model. Using the methods of bifurcations theory, it is proved that the integer-order model exhibits forward bifurcation and Hopf bifurcation. Next, with the aim of the stability theory of fractional-order systems, some conditions, which can guarantee the local stability of the fractional-order model, are developed and occurrence of forward and Hopf bifurcations in this model are studied. Lastly, numerical simulations are illustrated to support the theoretical results and a comparison between the integer and fractional-order systems is presented.

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Authors and Affiliations

  1. Department of Mathematics, Yazd University, Yazd, Iran

    M. H. Akrami

  2. Department of Mathematics, Razi University, Kermanshah, Iran

    A. Atabaigi

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  1. M. H. Akrami
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  2. A. Atabaigi
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Correspondence to M. H. Akrami.

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Akrami, M.H., Atabaigi, A. Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals. J. Appl. Math. Comput. 64, 615–633 (2020). https://doi.org/10.1007/s12190-020-01371-2

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  • DOI: https://doi.org/10.1007/s12190-020-01371-2

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