Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment

The dynamical behaviors of an SIR epidemic model with nonlinear incidence and treatment is investigated. It is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low. Theoretical and numerical results suggest that decreasing the basic reproduction number below one is insufficient for disease eradication.