Well-Posedness and Stability of a Variable-Order Discrete Cancer Model

This paper develops and analyzes a nine-compartment discrete colorectal cancer model governed by a Caputo variable-order difference operator. The model describes the interactions among epithelial cells, adenomatous polyps, oncogenes, tumor suppressor genes, APC mutations, KRAS mutations, microsatellite instability, inflammatory cells, and myofibroblasts. The variable-order framework is used to represent stage-dependent memory effects in colorectal cancer progression. The existence and uniqueness of solutions are established by using a Banach fixed-point argument, while positivity and uniform boundedness are proved to ensure biological feasibility of the model. Equilibrium points and a basic reproduction number are derived, and local stability is investigated through a variable-order spectral-angle criterion. Numerical simulations are performed to study the influence of memory variation on the model dynamics. A comparative analysis is also presented for the integer-order, constant fractional-order, and variable-order cases. Finally, the normalized model output is compared qualitatively with available colorectal cancer incidence data, showing that the proposed framework can capture the general observed trend while preserving flexibility through time-varying memory.