Dynamics of a fractional breast cancer model: Existence, stability, and Ulam–Hyers robustness

Abstract

This study develops and analyzes a fractional-order compartmental model for breast cancer ($\bc$) progression within Jordan's healthcare system using Caputo ($\c$) fractional derivatives. The model partitions the female population into seven epidemiological states: Susceptible, Preclinical Undetected, Preclinical Screen-detected, Clinical, Treatment, Remission, and Death. We establish the mathematical well-posedness of the system by proving positivity, boundedness, and global existence of solutions. The disease-free equilibrium (DFE) and endemic equilibrium (EE) are derived explicitly, with the basic reproduction number $R_0^{(\kappa)}$ computed via the next-generation matrix approach. Local asymptotic stability of the DFE is established using Matignon's stability criterion for fractional systems, demonstrating that $R_0^{(\kappa)} < 1$ ensures disease elimination. We prove global Ulam–Hyers stability to quantify the system's robustness under bounded perturbations. Existence and uniqueness of solutions are rigorously established through Lipschitz continuity of the nonlinear operator, Banach fixed-point theorem, and the Leray–Schauder alternative. Numerical simulations employing the Modified Fractional Euler Method (MFEM) demonstrate the influence of fractional order $\kappa$, screening rates, and treatment parameters on long-term disease dynamics. The results highlight memory effects inherent in fractional modeling and their implications for optimizing $\bc$ screening and treatment strategies in resource-limited healthcare settings.