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    Chaos and proportional integral derivative (PID) control on cancer dynamics with fractal fractional operator
    (Elsevier BV, 2025-06) Muhammad Farman; Kottakkaran Sooppy Nisar; Khadija Jamil; Ali Akgül; Aceng Sambas; Mustafa Bayram; Mustafa Habib; Murad Khan Hassani
    This study presents a cancer dynamics model incorporating a fractal-fractional operator with a Mittag-Leffler kernel to capture complex interactions among cancer cells, tumor suppressor cells, immune cells, and oncolytic viruses. The model aims to enhance understanding of tumor-immune dynamics and improve treatment strategies. The existence and uniqueness of the solution are established using fixed point theory under the Lipschitz condition. Lyapunov stability of the system is also analyzed in the context of the fractal-fractional operator. To address chaotic behavior in cancer progression, chaos and Proportional-Integral-Derivative (PID) control techniques are implemented. These control methods effectively stabilize the system and regulate treatment administration. Numerical simulations illustrate the influence of fractional-order derivatives on tumor suppression and immune response, confirming the model's effectiveness in reflecting real-world cancer dynamics.