Kunnisai Muniroh, Ummu Habibah, Wuryansari Muharini Kusumawinahyu, Nur'Izzati Hamdan
This paper develops a mathematical model to investigate breast cancer dynamics by incorporating tumor–immune interactions, ketogenic diet effects, and medical treatment. The model is formulated as a system of nonlinear ordinary differential equations and analyzed within an optimal control framework. Time-dependent control variables are introduced to represent treatment strategies aimed at minimizing tumor progression while reducing therapeutic costs. The model’s well-posedness is established through positivity and boundedness analysis. The necessary conditions for optimality are derived using Pontryagin’s Minimum Principle, resulting in a coupled system of state and adjoint equations. Numerical solutions are obtained using the fourth-order Runge–Kutta method combined with a forward–backward sweep algorithm. Simulation results demonstrate that the proposed control strategy effectively suppresses tumor growth and enhances immune response compared to uncontrolled scenarios. Cost-effectiveness analysis also shows the treatment strategies with the greatest impact at minimum costs. The numerical findings are consistent with the analytical results and highlight the importance of optimized treatment scheduling. The proposed framework provides a reliable and computationally efficient approach for analyzing cancer treatment dynamics and may serve as a basis for further extensions involving parameter estimation or data-driven studies. © 2025 by the authors. This work is licensed under a Creative Commons Attribution 4.0 (CC BY) International License. The authors retain ownership of the copyright for their article, but they allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles in MMNSA, so long as the original authors and source are credited. To see the complete license contents, please visit (http://creativecommons.org/licenses/by/4.0/).
Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia; Faculty of Computer and Mathematical Sciences, University of MARA Technology, Selangor, Malaysia
