Master–slave synchronization for glucose–insulin metabolism of type-1 diabetic Mellitus model based on new fractal–fractional order derivative

Type-1 diabetes mellitus (T1DM) is one of the most extensive diseases in the world. In this disorder, a malfunction occurs in the glucose-insulin regulatory system when one intake food. Therefore, the synchronization of T1DM and perturbed T1DM models are necessary to transform the disturbance behavior of T1DM into a typical T1DM model. In this paper, a type-1 diabetic mellitus glucose-insulin metabolism (T1DMGIM) model is concerned with a new fractal-fractional operator β ∈ ( 0 , 1 ] , η ∈ ( 0 , 1 ] by adding food intake as a disturbance to the plant system. A new generalized differentiation operator is introduced as the convolution of the power law. This new operators is based on the fractal-fractional differentiation and integration operators in the Riemann–Liouville (R-L) sense. Sufficient conditions are achieved for the existence, uniqueness, and stability of an error T1DMGIM model by employing the Mönch contraction principle. Finally, numerical simulations are provided to verify the theoretical results.