Mathematical and Numerical Analysis of a Fractional Diabetes Model with Singular Operator

Diabetes mellitus is a chronic disease with complex progression dynamics. This study introduces a fractional order compartmental model based on the Caputo derivative, a singular-kernel derivative, to describe disease progression across four compartments: susceptible, insulin-resistant, diabetic without complications, and diabetic with complications. The model is novel for integrating memory effects into disease-stage transitions while maintaining dimensional consistency. Key mathematical properties, including existence, uniqueness, positivity, boundedness, equilibrium analysis, and both local and global stability, are established. Ulam–Hyers stability is also examined to evaluate the robustness of the model solutions. Numerical approximations are obtained using the Adomian Decomposition Method and its Laplace variant. Simulations indicate that lower fractional orders enhance memory effects, slow disease progression, and influence long-term dynamics. These results demonstrate that the proposed approach provides a flexible and robust framework for studying chronic disease progression and makes a meaningful contribution to the literature on fractional diabetes models.