Fractional optimal control analysis of a melanoma tumor–immune–drug interaction model with numerical simulations

Abstract Melanoma is one of the most aggressive forms of skin cancer, characterized by rapid progression, high metastatic potential, and complex interactions with the immune system. Fractional-order mathematical models provide an effective framework for describing such biological processes because they incorporate memory effects and nonlocal dynamics that cannot be captured by classical integer-order models. In this work, we develop a fractional-order melanoma model that describes the interactions among tumor cells, immune effector cells, and drug pharmacokinetics. The dynamics are formulated using the Caputo fractional derivative in order to incorporate biological memory while preserving classical initial conditions. Fundamental properties of the model, including existence, uniqueness, positivity, and boundedness of solutions, are established. An optimal control problem is then formulated to determine an effective drug infusion strategy that minimizes tumor burden and treatment toxicity while preserving immune activity. Necessary optimality conditions are derived using a Pontryagin-type Maximum Principle for fractional systems, leading to an explicit characterization of the optimal control. Numerical simulations based on a predictor–corrector scheme combined with a forward–backward sweep algorithm illustrate the effectiveness of the proposed control strategy. The results show that fractional dynamics significantly influence tumor evolution and optimal dosing profiles, highlighting the importance of memory effects in melanoma treatment modeling.