Well-Posedness and Stability of the Stochastic OGTT Model

Oral Glucose Tolerance Test (OGTT) is one of many way to produce data in the study of the diabetes dynamic. In a recent paper [1.]:$\mathit{\text{Estimating}}\mathit{\text{insulin}}\mathit{\text{sensitivity}}\mathit{\text{and}}\beta \mathit{\text{-cell}}\mathit{\text{function}}\mathit{\text{from}}\mathit{\text{the}}\mathit{\text{oral}}\mathit{\text{glucose}}\mathit{\text{tolerance}}\mathit{\text{test:}}\mathit{\text{validation}}\mathit{\text{of}}\mathit{\text{a}}\mathit{\text{new}}\mathit{\text{insulin}}\mathit{\text{sensitivity}}\mathit{\text{and}}\mathit{\text{secretion}}\mathit{\text{(ISS)}}\mathit{\text{model,}}\mathit{\text{J.}}\mathit{\text{American}}\mathit{\text{Physiological}}\mathit{\text{Society}}\mathit{\text{,(2024)}}$,Ha J., Chung S.T., and al. proposed a comprehensive OGTT model under the form of a dynamic system. But their model was fully deterministic. Yet, our natural environment interacts with noise; thus taking into account that inherent perturbation could potentially improve the model, which in turn could lead to a better estimation of the parameters of the system. The current paper endeavors to explore the OGTT model proposed by Ha and al. but this time with the addition of white noise perturbations to the system.\ The work is organized as follow: we first establish the existence and uniqueness of a global positive solution to the proposed stochastic model. Then follows the study the statistical stability of the model by examining the existence of an invariant measure to the perturbed system. Once done with the well-posedness and the stability of the stochastic model, we then examine a Maximum Likelihood Estimation (MLE) scheme to estimate the parameters involved in the model. The paper ends with a brief discussion on potential future developments follows by an appendix whose goal is to make the paper as self contained as possible.