Mathematical Investigation of Cancer-Immune-Angiogenesis Model Using Fuzzy Piecewise Fractional Derivatives

This work develops a fuzzy piecewise fractional derivative (FPFD) model for cancer-immune-angiogenesis dynamics under uncertainty. Five fuzzy state variables track tumor cells, immune effectors, vessel density, oxygen, and drug concentration. We employ fuzzy triangular numbers with α-cut interval arithmetic using constrained fuzzy arithmetic model parametric uncertainty, with numerical values. Oxygen-dependent carrying capacity follows a Hill-type function; hypoxia-induced angiogenesis follows a decreasing Michaelis–Menten function. The model transitions at t1=50 days from memoryless fuzzy classical derivative to fuzzy ABC fractional derivative of order ψ. The transition time t1=50 days is biologically justified based on experimental observations of the angiogenic switch in solid tumors, which typically occurs within 4–8 weeks post-inoculation. Positivity, boundedness, Lipschitz continuity, existence, and uniqueness of fuzzy solutions are proved via Banach fixed-point theorem in a weighted norm. A basic reproduction number interval R0=R̲0,R¯0 is derived; local and global stability conditions are established for disease-free and endemic equilibria using fuzzy differential inclusions. Global sensitivity analysis using latin hypercube sampling with N=500 samples explores the range of possible outcomes across the fuzzy parameter support. In the numerical implementation, we use a fourth-order fuzzy Runge–Kutta method (Phase I), and a fractional Adams–Bashforth–Moulton predictor-corrector method (Phase II), ensuring preservation of fuzzy number characteristics.