Dynamic analysis and optimal control for a fractional-order delayed SIR epidemic model with saturated treatment

This paper studies a fractional-order delayed susceptible–infected–recovered epidemic model with saturated incidence rate and saturated treatment function. In particular, the positivity and boundedness of all solutions are determined. The existence and stability of all possible equilibria are also discussed. The study reveals that the model has multiple equilibria and the possible existence of backward bifurcation whose implications for the spread of disease are discussed. Then, a fractional-order delayed optimal control problem related to vaccination and treatment enhancement is proposed and analyzed. The existence of an optimal pair is shown and the optimal control solution is characterized. Three control strategies (viz., vaccination only; treatment enhancement only; the combination of vaccination and treatment enhancement) are discussed and compared through numerical simulations. In terms of the control effect and cost, the combination of vaccination and treatment enhancement is significantly superior to vaccination, and vaccination is better than treatment enhancement.