## 2 Human Glucose-Insulin Regulatory System  
The human glucose-insulin regulatory system (HGIRS) is part of the human endocrine system [6] and comprises two main hormones, insulin and glucagon, produced and released, respectively, by \(\beta\) and \(\alpha\)-cells of the pancreatic islets. These two hormones, which exert opposing effects, act in concert to maintain blood glucose (BG) in a narrow range. During the fasting state, glucagon stimulates hepatic glucose production (HGP) in order to prevent hypoglycemia, while insulin is secreted at levels sufficient to constrain HGP and to maintain BG concentration at approximately 90 mg/dL (basal secretion). After meals, the increase in BG concentration stimulates insulin secretion (meal-related secretion), suppressing HGP and stimulating glucose uptake by insulin-sensitive tissues such as muscle and adipose tissue, eventually restoring normoglycemia. In this research, due to modeling limitations, we do not consider other hormones such as cortisol and growth hormones, which exert direct and indirect effects on glucose metabolism. Insulin secretion is complex and glucose is the most potent stimulant of insulin release. After eight to ten minutes of food ingestion, the insulin concentration increases, reaching a peak in 30 to 45 min, and then rapidly decreases to baseline values in 90 to 120 min [7]. Insulin is a physiological suppressor of glucagon release; thus, glucagon secretion is low in the postprandial period. On the other hand, glucagon is released during fasting, when BG is in the normal range and insulin concentration is low [7]. In the case of T1D, the autoimmune destruction of pancreatic \(\beta\)-cells prevents insulin secretion. Thus, T1D individuals depend on exogenous insulin administration to mimic the physiological secretion of this hormone, i.e., basal and meal bolus insulin. To maintain the BG concentration to as close as possible to the normal range, it is also necessary to measure BG and to count the amount of macronutrients (especially carbohydrates) before every meal to calculate the bolus insulin dose, which must match the total carbohydrate content of that meal and also correct occasional hyperglycemia. Optimal glycemic control is crucial to avoid the complications associated with chronic hyperglycemia. However, the procedures described above are relatively complex and are influenced by numerous other factors, such as the type and intensity of physical activity, as well as stress, among others, which impair glycemic control, contributing to the occurrence of episodes of hyper and hypoglycemia. The literature presents several predictive models for glucose oscillation, such as [8; 9; 10; 11], the last one having a night prediction of up to 6 h. However, we could not find any model that predicts glucose oscillation in a continuous and personalized way. Quantitative methods used to model the metabolic physiology of T1D individuals are usually based on ordinary differential equations (ODE), and are also called compartment models [12]. We propose an extension to the model proposed by [PERSON] et al., 2014 [3] presented in Equations (1) and (2). \[ \begin{array}{l}G^{\prime}(t)=G_{in}+f_{1}(I(t-\tau_{2}))-f_{2}(G(t)\\ -\gamma[1+s.(m-m_{b})].(f_{3}(G(t)).f_{4}(I(t))\\ \end{array}  \tag{1}\]  
\[ \begin{array}{l}I^{\prime}(t)=I_{in}+\beta f_{5}(G(t-\tau_{1}))-\frac{V_{MAX} I(t)}{K_{M}+I(t)}\\ \end{array}  \tag{2}\]  
In the glucose compartment \(G^{\prime}(t),f_{1}\) describes the hepatic glucose production (HGP); \(f_{2}\) describes the central nervous system glucose utilization; \(f_{3}\) describes the muscle/fat glucose utilization; \(f_{4}\) describes the muscle/fat insulin uptake; and \(f_{5}\) describes the pancreatic insulin production.