Dynamic stability analysis of a fractional calculus-based colon cancer model with therapeutic interventions.

Colon cancer is a high-risk malignant tumor worldwide. To address the limited efficacy and overlapping toxicity of current chemoimmunotherapy, this study establishes a fractional-order colon cancer model with a core architecture capturing the probiotics-[Formula: see text]T cells-tumor cells axis. The primary theoretical contributions include: rigorous proof of solution existence and uniqueness through fixed-point theory. Analytical derivation of equilibrium stability conditions using the Routh-Hurwitz criterion. Numerical simulations systematically quantify how the fractional parameter α governs system dynamics and memory effects, while identifying critical drug dosage thresholds for tumor clearance. These findings provide a mathematical foundation for optimizing probiotic-based combination therapies.