Stability analysis of a three-dimensional system of Topp model with diabetes

TL;DR

This paper analyzes the stability of a three-dimensional Topp model for diabetes, providing insights into its fixed points and stability properties through theoretical and numerical methods.

Contribution

It offers a comprehensive stability analysis of the Topp model's dynamics, including local and global stability, with biological interpretations and numerical validation.

Findings

Identification of invariant positive trajectories

Existence and stability of fixed points

Numerical validation of theoretical results

Abstract

Mathematical models of glucose, insulin, and pancreatic $\beta$-cell mass dynamics are essential for understanding the physiological basis of type 2 diabetes. This paper investigates the Topp model's discrete-time dynamics to represent these interactions. We perform a comprehensive analysis of the system's trajectory, examining both local and global behavior. First, we establish the invariance of the positive trajectory and analyze the existence of fixed points. Then, we conduct a complete stability analysis, determining the local and global asymptotic stability of these fixed points. Finally, numerical examples validate the effectiveness and applicability of our theoretical findings. Additionally, we provide biological interpretations of our results.