A stochastic delay differential model for glucose-insulin interactions

To restore normal glucose levels in the body, high glucose levels stimulate β-cells in the pancreas to produce insulin. This paper presents a stochastic differential model for glucose-insulin interactions with two types of time delays. Using a Lyapunov function, we show that the solution to the model exists and is unique. Further, we proved that the stochastic system's solution is bounded and permanent by using Ito's formula and Chebyshev inequality. When noise intensities are high, glucose and insulin levels oscillate around the equilibrium point, mimicking physiological variability. Time delays and noise increase oscillation amplitude and predictability, demonstrating compounding effects on system variability. Numerical simulations are used to illustrate the theoretical conclusions.