## 3 Models  
The idea of creating an artificial model for a glucose-insulin system for several purposes has been widely explored. In [39], a minimal glucose-insulin system is proposed; the model starts with the Bergman model [17] and presents modifications to improve, considering the glucose level's description in the subcutaneous layer and a meal disturbance term. The developed application uses four mathematical models related to them, following the different dynamics that they describe:  
1. Glucose-Insulin Model: [PERSON] in [40] proposed it. This model comprises 19 nonlinear differential equations, which include significant metabolic effects related to glucose regulation. The model was obtained through a compartmental technique, where the pancreas compartment is omitted to consider a type 1 diabetic patient. Recently, this glucose-insulin model was extended to consider exercise periods [41]. 2. Subcutaneous Insulin model: [PERSON] and [PERSON] [42] obtained it. This model describes the subcutaneous insulin uptake pattern after an insulin injection. The model considers interactions between components after subcutaneous injections such as insulin absorption and elimination in plasma insulin, active insulin, glucose utilization, plasma glucose, and glucose input. The model is based on a set of differential equations. Variations in plasma insulin concentration (\(I\)) can be estimated by Equation (1). \[\frac{dI}{dt}=\frac{I_{\text{abs}}}{V_{i}}-k_{e}\cdot I\] (1) where \(I_{\text{abs}}\) represents the rate of insulin absorption, \(V_{i}\) is the volume of insulin distribution, and \(k_{e}\) is the first-order rate constant of insulin elimination. The absorption of insulin can be calculated as follows: \[I_{\text{abs}}=\frac{st^{2}T_{50}^{\text{s}}D}{t\left(T_{50}^{\text{s}}+t^{ \text{s}}\right)^{2}}\] (2) where \(s\) is a parameter associated with the insulin absorption pattern, \(t\) represents the time of the injection, \(T_{50}\) is when the dose (D) is absorbed in an amount of 50%. To estimate steady-state conditions, insulin profile (\(I_{ss}\)) using the superposition principle is computed assuming three days as follows: \[I_{ss}(t)=I(t)+I(t+24)+I(t+48)\] (3) \[I_{a,ss}(t)=I_{a}(t)+I_{a}(t+24)+I_{a}(t+48)\] (4) where \(I_{a}\) is the build-up and the deactivation of the insulin pump. 4. Meal Intake model: this model was proposed in 1992 by [PERSON] and [PERSON] [43]. It describes the gastric glucose absorption into the bloodstream by a meal intake represented by its carbohydrates content. According to this model, glucose utilization \(G_{out}\) (\(G\)) at plasma insulin concentration is: \[G_{out}(G)=\frac{V_{max}G}{k_{m}+G}\] (5) \(G\) represents the glucose concentration, \(V_{max}\) maximal glucose utilization for reference insulin concentration, and \(K_{m}\) is glucose concentration for the half-maximal response. \(V_{max}\) depends on insulin; mathematically, this is \[V_{max}(I_{ss})=\frac{G_{out}(I_{ss})(K_{m}+G_{x})}{G_{x}}\] (6) where \(I_{ss}\) is plasma insulin concentration equivalent to steady-state insulin, and \(G_{x}\) is the reference glucose value. Glucose utilization for any combination of plasma glucose and insulin levels is given by \[G_{out}(G,I_{ss})=\frac{G(c\cdot I_{ss}+d)(K_{m}+G_{x})}{G_{x}(K_{m}+G)}\] (7) where \(c\) is insulin sensitivity, and \(d\) represents insulin-independent glucose utilization. The insulin absorption dynamics for regular, NPH, lente, ultralente formulations were characterized in [42] just by changing some specific parameters in the general model. Recently, in [44], the Lispro insulin formulation was also characterized by the model in [42]. 5. The exercise model was considered from [41]. The liver's glycogen reservoir (\(GLY\)) can be calculated as shown in Equation (8). \[\frac{dGLY(t)}{dt}=-\varnothing\Big{(}PVO_{2}^{max}(t),GLY(t)\Big{)}+\Psi(G_{abs }(t)+GLY(t))\] (8) \[\varnothing\Big{(}PVO_{2}^{max},GLY\Big{)}= \begin{cases}&\zeta\Big{(}PVO_{2}^{ max}\Big{)}\quad\text{if}\quad 0<GLY\leq GLY_{max}\\ &0\quad\text{if}\quad GLY\geq 0\end{cases}