Backward Bifurcation An SIR Epidemic Model With Saturated Treatment Rate

The asymptotic behavior of epidemic models has been studied by many researchers. It is common that a basic reproduction number is a threshold which determines the outcome of the disease. If it is below 1, the disease-free equilibrium is globally stable and the disease dies out. while if it is greater than 1, a unique endemic equilibrium is globally stable. So the bifurcation leading from a disease free equilibrium to an endemic equilibrium is forward. In recent years, papers found backward bifurcations due to social groups with different susceptibilities, nonlinear incidences, and age structures in epidemic models and so on. In this case, the basic reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. Thus, it is important to identify backward bifurcations to obtain thresholds for the control of diseases. In this paper, we studied an epidemic model with saturated treatment function. Through mathematical analysis and numerical simulation, we obtain the following main results:1 .When the effect of the infected being delayed for treatment is weak, the basic reproduction number being unity is a strict threshold for the control of the disease. However, when this delayed effect is strong, a backward bifurcation will take place. Thus it is not enough for us to drive the basic reproduction number below one to eradicate the disease.2.When the backward bifurcation takes place, there is a critical value R~c at the turning point which can be taken as a new threshold for the control of the disease.3.The disease-free equilibrium is globally stable if the basic reproduction number reduces further to some degree.4. 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.Finally, based on the works of early researchers, we suggest our effort in the future.