Periodic solutions of an epidemic model with saturated treatment

Based on the fact that many infectious diseases exhibit periodic fluctuations and there is a saturated phenomenon during disease treatment, we study an SIR model with periodic incidence rate and saturated treatment function. Firstly, we find that the basic reproduction number less than 1 cannot insure the global stability of disease-free equilibrium and it needs to add other conditions. Moreover, we establish sufficient conditions for the multiplicity of positive periodic solutions. We also apply the numerical method to confirm theoretical results and show the stability of the periodic solutions. We observe that there are two periodic solutions in the system where one is stable and the other one is unstable. These results will provide some guidance for control measures of disease.