Stability analysis for Nabla discrete fractional-order of Glucose–Insulin Regulatory System on diabetes mellitus with Mittag-Leffler kernel

In this paper, we study an existence theory as well as stability results to the new fractional Nabla difference biological model of glucose–insulin regulatory system (GIRS) on diabetes mellitus involving the Atangana Baleanu–Caputo (ABC) derivative. We consider the proposed model under ABC derivative, and it has the nonsingular Mittag-Leffler function as its kernel. We utilize the fixed point technique for the existence and uniqueness analysis. The stability of the concerned solution in Hyers–Ulam sense is also investigated. Further to derive the approximate solution in the form of series to the considered model, we use iterative technique method. Numerical simulations are given to support the theoretical results. The results show that order of the fractional derivative has a significant effect on the dynamic process. Many informations on the dynamics of GIRS in diabetes mellitus were obtained using this model. It is recognized as a deterministic fractional Nabla difference model for diabetes mellitus that provides a better control approach at fractional values for the development of an artificial pancreas.