Backward bifurcation of an epidemic model with saturated treatment function

An epidemic model with saturated incidence rate and saturated treatment function is studied. Here the treatment function adopts a continuous and differentiable function which can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical condition is limited. The global dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when such effect is weak. However, it is shown that a backward bifurcation will take place when this delayed effect for treatment is strong. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. And a critical value at the turning point is deduced as a new threshold. Some sufficient conditions for the disease-free equilibrium and the endemic equilibrium being globally asymptotically stable are also obtained. Mathematical results in this paper suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.