Stochastic dynamic of intravenous glucose-insulin model

This paper examines a stochastic glucose-insulin regulatory system, introducing random noise to model environmental and biological fluctuations. To account for the time delay between glucose intake and insulin response, we present a delay differential equation model. We first prove the existence and uniqueness of the solutions. Subsequently, we establish key properties: the existence of solutions within a positively invariant set, stochastic ultimate boundedness, and stochastic permanence. These properties demonstrate that the model’s solutions remain within a feasible, biologically plausible state space, denoted Γ, which represents the dynamics of glucose and insulin concentrations. Furthermore, we prove that Γ is almost surely positively invariant. Numerical simulations illustrate the impact of noise on blood glucose and insulin dynamics, supporting the theoretical findings and highlighting the model’s practical relevance.