Fractional optimal control of breast cancer dynamics using Caputo derivatives: modeling, analysis, and numerical investigation

Abstract In this paper, we present a fractional-order mathematical model for breast cancer dynamics that incorporates memory effects through Caputo fractional derivatives. The model describes the interactions among tumor cells, immune cells, and healthy host cells under the influence of two therapeutic interventions representing chemotherapy and immunotherapy. An optimal control problem is formulated to minimize tumor burden while balancing treatment costs. We establish fundamental mathematical properties of the model, including positivity, boundedness, and existence and uniqueness of solutions. Necessary optimality conditions are derived using the fractional Pontryagin Maximum Principle. A numerical solution strategy based on an L1 discretization combined with a forward–backward sweep algorithm is developed to solve the resulting fractional optimal control problem. Numerical simulations are presented to illustrate the effects of fractional order and combined therapies on tumor suppression and control profiles. The results demonstrate that fractional dynamics and simultaneous therapeutic strategies provide enhanced modeling flexibility and improved treatment outcomes compared to classical integer-order approaches.