Study on Dynamical Behavior of a Type of Diabetes Mellitus Model With Nonlinear Impulsive Injection of Insulin

ABSTRACT Based on the dynamic management of insulin, we consider the effects of insulin, glucose, and glucocorticoid concentration on the dynamics of the diabetes model with nonlinear impulsive control to simulate the physiological process of exogenous insulin therapy. Firstly, if , in type 1 diabetes mellitus (T1DM), the existence of positive periodic solution of the system is proved by using fixed point theory for difference equations, whereas the global asymptotic stability of positive periodic solution is proved by using the Floquet multiplier theory and comparison theorem. Next, if , in type 2 diabetes mellitus (T2DM), the permanence of the solution of the system is proved by using the comparison theorem. Furthermore, simulation results are performed to verify the correctness of the theoretical findings and to study the system's dynamic behavior under various conditions. By simulating the feedback regulation mechanism between insulin and glucose in a closed‐loop system, the model not only elucidates the dynamic equilibrium relationship between insulin and glucose but also offers a novel and feasible strategy for the treatment and control of diabetes, thereby demonstrating substantial theoretical and practical significance.