Stability and bifurcations of an SIR model with a nonlinear incidence rate

Abstract

In this paper, an SIR model with a nonlinear incidence rate is studied. A disease-free equilibrium (Formula presented.), an endemic equilibrium (Formula presented.), and the basic reproduction number of the model (Formula presented.) are obtained. If (Formula presented.) is locally asymptotically stable and if (Formula presented.) is locally asymptotically stable. By Barbalat's lemma, we study the global stability of the model. Transcritical bifurcation analysis is investigated by using the Sotomayor theorem. As the infection rate increases, the asymptotic behavior of the system near (Formula presented.) approaches (Formula presented.) and the system has a transcritical bifurcation. Also, we check the existence of Hopf bifurcation for the given system. In addition, a sensitivity analysis is provided for the basic reproduction number. Our results are supported with numerical simulations.