Dynamics analysis and optimal control strategy for a SIRS epidemic model with two discrete time delays

In this paper, a SIRS epidemic model with nonlinear incidence rate, saturated treatment and two time delays is investigated. Firstly, by using the method of regeneration matrix, we have determined the basic regeneration number R-0 and demonstrated the existence of the positive equilibrium point. The permanence of the SIRS epidemic model is obtained by mathematical analysis. Moreover, by selecting time delay as the bifurcation parameter, we discuss the local asymptotic stability of the positive equilibrium point and the existence of Hopf bifurcation for six different situations. Afterwards, to minimize the spread of infectious diseases, we introduce an optimal control technique by the Pontryagin's maximum principle. Finally, we verify the correctness of theoretical analysis through numerical simulations.