Fractional optimal control and numerical analysis of a Zika virus transmission model

Abstract Zika virus is a mosquito-borne infectious disease that poses a significant public health threat in many regions of the world. Mathematical modeling plays a crucial role in understanding its transmission dynamics and in designing effective control strategies. In this paper, a fractional-order optimal control model describing the transmission of Zika virus between human and mosquito populations is proposed and analyzed. The model is formulated using Caputo fractional derivatives in order to capture memory and hereditary effects that are often neglected in classical integer-order models. Three time-dependent control measures, representing prevention, treatment of infected individuals, and insecticide-based vector control, are incorporated into the model. Fundamental qualitative properties of the system, including positivity and boundedness of solutions, are established. A threshold parameter governing disease persistence is derived using the next-generation matrix approach. An optimal control problem is then formulated, and the necessary optimality conditions are obtained via the fractional Pontryagin Maximum Principle. Numerical simulations, implemented using a forward–backward sweep algorithm combined with a fractional predictor–corrector scheme, illustrate the effects of fractional order and control strategies on the disease dynamics. The results demonstrate that fractional-order dynamics significantly influence epidemic behavior and that the combined control strategy is the most effective in reducing both human and mosquito infections. These findings highlight the importance of memory effects and coordinated interventions in controlling Zika virus outbreaks.