Malignant melanoma fractional-order mathematical model with stabilized fuzzy sliding mode control

Malignant melanoma, an aggressive form of skin cancer, poses significant challenges due to its rapid progression, metastatic potential, and resistance to therapies. This study aims to develop a fractional-order mathematical model capturing melanoma dynamics (tumor-immune interactions, extracellular matrix remodeling, nutrient dynamics) and introduce a Stabilized Fuzzy Sliding Mode Control (SFSMC) strategy to suppress tumor growth and restore microenvironmental homeostasis.

A fractional-order model was derived using Caputo derivatives to incorporate memory effects and long-range dependencies. The SFSMC combines sliding mode control with fuzzy logic to manage uncertainties. Theoretical analysis included well-posedness, stability (via Lyapunov functions), and computation of the reproduction number R₀. Numerical simulations were performed using a predictor-corrector method with parameters calibrated from clinical data.

The model demonstrated stability when R₀<1, indicating tumor suppression. SFSMC reduced tumor cell populations by 78% and circulating tumor cells by 65% while improving immune response (45% increase in immune cells) and nutrient availability (30% recovery). Sensitivity analysis revealed R₀ is mostly influenced by tumor growth rate, natural degradation rate of extracellular matrix (ECM), rate of ECM degradation by tumor cells, and ECM production rate, suggesting their potential role in suppressing tumor growth.

The fractional-order framework and SFSMC offer a robust approach to modeling and controlling melanoma, with potential clinical implications for adaptive therapy.