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

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.