Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals. (English) Zbl 1484.37102
Summary: 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.
MSC:
| 37N25 | Dynamical systems in biology |
| 26A33 | Fractional derivatives and integrals |
| 92D30 | Epidemiology |
| 92D25 | Population dynamics (general) |
Keywords:
SIR epidemic model; logistic growth; fractional-order derivative; forward bifurcation; Hopf bifurcationReferences:
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