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 , an endemic equilibrium , and the basic reproduction number of the model are obtained. If is locally asymptotically stable and if 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 approaches 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.