Optimal control of a fractional two-strain epidemic model with general incidence rates

Abstract

We study a fractional-order two-strain $\mathcal{S}\mathcal{I}_1\mathcal{I}_2\mathcal{R}$ epidemic model with a general incidence function, incorporating optimal control strategies for each infectious class. The system has four fractional differential equations that describe the evolution of susceptible, two infected populations, and recovered individuals. The model is developed using Caputo fractional derivatives. The existence, uniqueness, positivity and boundedness of solutions are established. The basic reproduction numbers are derived and the stability properties of the equilibria are investigated.An optimal control problem is then formulated by introducing two time-dependent control functions $u_1(t)$ and $u_2(t)$ aimed at reducing the infection levels while minimizing the associated implementation costs. Using Pontryagin's Maximum Principle, the necessary optimality conditions are derived and the optimal controls are characterized.Numerical simulations demonstrate the effectiveness of the proposed control strategies. The results show that optimal interventions significantly reduce infection peaks and improve disease mitigation. Moreover, the fractional-order dynamics reveal the important role of memory effects in shaping the epidemic behavior.