Well‐Posedness and Ulam–Hyers Stability of a Normalized Caputo–Fabrizio Fractional Model for Ischemic Heart Disease Progression

Source abstract

We develop and analyze a fractional‐order cascade model for ischemic heart disease (IHD) governed by the normalized Caputo–Fabrizio (NCF) derivative of order β ∈ (0, 1). The model comprises six clinically meaningful compartments: low‐risk ( S ), high‐risk ( R ), chronic IHD ( I ), acute coronary syndrome ( A ), postevent ( P ), and a cumulative death counter ( D ), and incorporates time‐dependent controls representing population‐level risk reduction ( u σ ), lipid‐lowering therapy ( u η ), and revascularization intensity ( ρ ). The NCF operator introduces a nonsingular exponential memory kernel that captures short‐term hereditary effects while preserving the equilibrium structure of the corresponding integer‐order system. Using the NCF–Volterra integral equivalence, we establish positivity of solutions, derive a sharp positively invariant region for the living population, and prove existence and uniqueness via a Banach fixed‐point argument. The robustness of the fractional dynamics is further quantified using Ulam–Hyers stability estimates. For numerical implementation, we construct an efficient Adams–Bashforth predictor–corrector scheme tailored to the normalized kernel, exploiting recurrence relations for low‐memory computation. Analytical results and numerical simulations consistently show that fractional memory reshapes transient disease progression and recovery pathways without altering steady‐state equilibria or threshold relations. The proposed framework provides a mathematically rigorous and computationally efficient tool for exploring memory‐driven cardiovascular dynamics and assessing the impact of clinical interventions.