Stability and numerical approximation of fractional breast cancer model with estrogen-driven immune regulation

Abstract

Breast cancer ($\bc$) progression is strongly influenced by immune competence, estrogen signaling and therapeutic intervention. Classical integer-order tumor--immune models often fail to reflect memory-driven cellular behavior, saturation in immune--tumor interactions and immune exhaustion, all of which are fundamental in chronic malignancy dynamics. Motivated by these limitations, we present a newly extended fractional breast-cancer model formulated using Caputo ($\c$) derivatives, incorporating a Holling type-II tumor-immune response and a distinct compartment for exhausted immune cells. This framework advances the four-compartment formulation of Abdelwaj et al. \cite{Abdelwaj2020}, capturing nonlinear immune suppression, hormonal influence and saturation-regulated killing efficiency. We analytically identify tumor-free, dead and coexisting equilibrium, derive feasibility conditions for each steady state and establish local stability criteria through Jacobian spectral properties supported by Matignon’s fractional stability theorem. A Toufik--Atangana based numerical scheme is then constructed for computational simulation, enabling accurate approximation of memory-dependent tumor progression. The model demonstrates that immune exhaustion and hormonal saturation critically govern long-term tumor persistence, while fractional order significantly modulates transient behavior.