How to maintain long-term euglycemia in a noisy environment: Insight from a stochastic glucose–insulin metabolism model with correlated Gaussian colored noise

Abstract

Understanding the response of glucose homeostasis regulation mechanism to stochastic environmental fluctuation in vivo and in vitro may benefit to gain quantitative insight into the physiological processes of progression of hyperglycemia. In this paper, we propose a stochastic glucose insulin regulation model, which takes into account the correlated Gaussian color noises to describe the environmental internal and external variability. First, the noise-induced state transition from physiological steady state $N^{\ast }$ to pathological steady state $N_0$ is observed. Then, by analyzing stochastic stability and the stationary probability density (SDP) of the limiting Itô stochastic system, we theoretically explore the factors that influence the individuals to deviate from the physiological steady state. Additionally, the mean first passage time (MFPT) of the attraction domain of potential well corresponding to the physiological steady state is also calculated to validate the observations. These results demonstrate that the correlation direction of two noises presents different influence pathways of the state transition from $N^{\ast }$ to $N_0$. More precisely, (i) for positive correlation degree, the higher intensity of noises may be more likely to induce the state transitions from $N^{\ast }$ to $N_0$. (ii) Multiplicative and additive noises with negative correlation present a non-monotonic U-shape trend of the probability of state transition, which indicates an antagonistic interaction effect. These findings may provide the theoretical and numerical explanations of the impact of correlated Gaussian color noises on the state transition in the glucose metabolism system and an insight into the mechanism of maintaining long-term euglycemia in a noisy environment.