A fractal-fractional chemo-immune model for non-muscle-invasive bladder cancer: Analysis and simulation

This research presents a novel mathematical study for analyzing the chemo-immune dynamics of non-muscle-invasive bladder cancer (NMIBC). We generalize a foundational three-compartment model (Mitomycin-C, Tumor, Effector-cells) by employing the fractal-fractional (FF) differential operator in the Caputo sense, characterized by a fractional order $ \alpha $ and a fractal dimension $ \beta $. This advanced operator is uniquely suited to capture the non-local memory effects inherent in immune system activation and the fractal (non-Euclidean) nature of the tumor microenvironment. We first establish the model's mathematical integrity by rigorously proving the existence, uniqueness, positivity, and boundedness of its solutions. The long-term behavior of the system is then analyzed, centered on the derivation of the basic reproduction number ($ \mathcal{R}_0 $). We demonstrate that the disease-free equilibrium is globally asymptotically stable if $ \mathcal{R}_0 \le 1 $, while a unique endemic (tumor) equilibrium emerges and gains stability if $ \mathcal{R}_0 > 1 $, indicating a transcritical bifurcation. For the numerical solution, we develop a semi-analytical scheme using the fractal-fractional Adomian decomposition method (FF-ADM) and validate its high accuracy against established numerical methods. Extensive numerical simulations are presented, including 2D and 3D plots, which visualize the profound impact of the fractional parameters $ \alpha $ and $ \beta $ on the system's trajectory, revealing that they significantly alter the time to tumor clearance. A comprehensive sensitivity analysis identifies the most critical parameters for controlling the disease, and 3D bifurcation plots illustrate the thresholds between tumor elimination and persistence. This work provides a more realistic and flexible tool for understanding NMIBC, with direct implications for optimizing treatment strategies.