Fractal-fractional modeling of chronic myelogenous leukemia immune dynamics using Laguerre wavelets method

This investigation proposes and analyzes a novel mathematical framework for understanding the complex tumor-immune system dynamics in Chronic Myelogenous Leukemia (CML). We adapt the established Moore and Li three-population model (naive T-cells, effector T-cells, and CML cells) by reformulating it using fractal-fractional differential operators of the Atangana–Baleanu type. This advanced approach offers a significant enhancement over standard models, as it simultaneously captures the fractal geometry of tumor structures and the non-local memory effects inherent in biological immune responses. We first establish the theoretical foundations, providing rigorous proofs for the existence, uniqueness, positivity, and boundedness of solutions, ensuring biological viability. Stability analysis of the tumor-present equilibrium is then conducted. For numerical simulation, a robust and highly accurate scheme based on the Laguerre Wavelets Method (LWM) is implemented with its efficacy demonstrated via comparison and convergence analysis. Numerical experiments visualize the system’s behavior under various fractal dimensions and fractional orders, revealing their profound impact on tumor progression. A sensitivity analysis identifies key parameters for potential therapeutic intervention. This work provides a more flexible and realistic mathematical tool for predictive oncology.