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

## TL;DR

This paper develops a fractional calculus model to study colon cancer dynamics and optimize treatment strategies involving probiotics and immune cells.

## Contribution

The paper introduces a fractional-order model with rigorous proofs of solution existence and stability analysis for colon cancer therapy.

## Key findings

- Solution existence and uniqueness are rigorously proven using fixed-point theory.
- Equilibrium stability conditions are derived analytically using the Routh-Hurwitz criterion.
- Fractional parameter α influences system dynamics and identifies drug dosage thresholds for tumor clearance.

## Abstract

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-\documentclass[12pt]{minimal}
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\begin{document}$$\hbox {CD4}^{+}$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\alpha$$\end{document} 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.