A Caputo (discretization) fractional-order model of glucose-insulin interaction: numerical solution and comparisons with experimental data

In this paper, we investigate a (discretization) Caputo fractional glucose-insulin model qualitatively with incommensurate orders that appear in Bergman's minimal model. After intravenous tolerance testing, the model is used to characterize the blood insulin and glucose metabolism. We also prove that the presented model possesses existence, uniqueness, non-negative, and boundedness solution. We also proceed a systematical studies on the stability of the (discretization) Caputo fractional. Comparisons between the results of the fractional-order, the integer order and the measured real data obtained from patients are presented. These comparisons is shown that the presented Caputo fractional order model is better representative of the system than its integer order form. Numerical solutions of the Caputo fractional model are obtained by using the method of Adams-Bashforth-Moulton type to handle the fractional derivatives. Also, numerical simulations of the discretization fractional derivative order model are used to support the analytical results.

In [6], De Gaetano and Arino has intended a model called the dynamical model which couples the two different parts of the "Minimal Model" into one part given byẋ 6 , and c 7 are parameters. In [7], Derouich et al. have been used a version of the minimal model in modified form to introduce parameters related to physical exercise: In [8], Li et al. had reinvestigated the dynamical analysis of the "Minimal Model" in both modelling and physiological aspect to understanding blood glucose regulatory system: The concept of fractional calculus has great importance in many branches and is also important for modelling real world problems . In this paper, we concerned on the discrete version Caputo fractional order of the minimal model (1): with This paper concerned on a analytical studies of a Caputo fractional-order glucose-insulin model (2) and its discretization. The fractional calculus has great importance for modelling real world problems and is also important in many branches. After intravenous tolerance testing, this model used to characterize the metabolism of blood insulin and glucose. We show that the model (2) possesses existence, uniqueness, nonnegative, and boundedness solution. We also prove that the presented model possesses existence, uniqueness, non-negative, and boundedness solution. We also proceed a systematical studies on the stability of the (discretization) Caputo fractional. Comparisons between the results of the fractional-order, the integer order and the measured real data obtained from patients are presented. These comparisons is shown that the presented Caputo fractional order model is better representative of the system than its integer order form. Numerical solutions of the Caputo fractional model are obtained by using the method of Adams-Bashforth-Moulton type to handle the fractional derivatives. Also, numerical simulations of the discretization fractional derivative order model are used to support the analytical results.

Notation and definitions
For ν ∈ R, the fractional derivative D ν t , can represented by Define the Euler-Gamma function as In [12], the Riemann-Liouville definition first introduced in 1847 and is given by In [13], the Caputo definition first introduced in 1967, and is given by Anton Karl Grunwald [37] and Aleksey Vasilievich Letnikov [38], introduced the Grünwald-Letnikov definition over the interval [a, t] with n ∈ N, is the step size, and a binomial .

Definition 2.2:
For β > 0, the function is called the Mittag-Leffler function of β.

Proof: One has
Then, the solutions of the model (2)  Proof: As in [21], one consider the function Thus, for all γ > 0, Thus, by choosing γ < min{q 1 − q 4 |t|, q 2 , q 6 − q 3 }, one obtains Following to Lemma 2.3, one obtains Thus, as starting in R 3 + , the model (2) has uniformly bounded solution lies in the region , with

Stability analysis
For the model (2), we assume that Then, the model (2) has only one equilibrium point E • = (x b , 0, z b ) and its Jacobian matrix J(E • ) at E • is given by Also, its characteristic equation 1 (λ) is given by Then Thus, by using [32], E • is asymptotically stable.

Numerical results
In this subsection, the numerical solutions for Caputo fractional order system (2) are simulated by using the method of Adams-Bashforth-Moulton existed in [40]. Values of the parameters, given in Table 1, are taken from [11], in which these values are obtained using a computer program, named "MINMOD". Consider the following: with
As t −→ (n + 1)m, one obtains the corresponding equation of the model (2) with a piecewise constant argument is given as:

Stability analysis
Here, the dynamical behaviours and stability analysis of the Caputo fractional discretized Glucose-Insulin model (4) is investigated here at the equilibrium point . First, we compute the Jacobian matrix J(E • ) of (4) as follows and its characteristic equation is given by (5) Its discriminant is given by From the Jury's criterion [41], From the Jury test, E • is asymptotically stable if 3 − a 2 1 , and d 2 = a 3 a 2 − a 1 a 2 .

Numerical simulations
Taking the parameter values as shown in Table 1 and consider the following discretized fractional order: By calculation, the corresponding eigenvalue is D = −5.8328e−05. Then, system (4) has a free equilibrium point E • = (287, 0, 403.4). By (6) and Proposition 3.1, the solution of (4) converges to E • (see Figures 4-7). Consequently, the insulin and the activity of insulin excitable tissue glucose uptake are increased and the glucose decreased. For these parameter the corresponding eigenvalues are D = −5.8328e−05. Furthermore, glucose, insulin excitable tissue glucose uptake, and insulin concentration versus time for different cases of ν.Then, (6) and Proposition 3.1 are satisfied and then E • is asymptomatically stable. Behaviour of x(t), y(t), and z(t), for different values of ν, showing glucose, activity of insulin excitable tissue glucose uptake and insulin dynamics are shown in Figures 4-6. Also, the behaviour of Glucose, Insulin excitable tissue glucose uptake and Insulin concentration versus time for different cases ν = 1, ν = 0.95 and ν = 0.90 are shown in Figures 7-9. Now, we list some numerical results for the discretized fractional order (7) of IVGTT glucose-insulin interaction

Comparison results
In this section, Adams-Bashforth-Moulton method was employed as a reasonable basis for studying the solution of a fractional-order model of glucose-insulin system (2). We have tuned for the order of fractional derivative which ensures better fit. We compared the fractional-order model to the experimental data obtained based upon the experimental data used in [11], given in Table 2, during primary glucose-insulin interaction. Furthermore, based upon this experimental data, we demonstrate that, fractional order Bergman's minimal model is better representative of the system of glucose and insulin in blood as compared to its integer order version. As in Figure 1, the numerical results of the fractional-order model are closer to the real measured data of the patients more than the results of the integer-order. For ν = 0.95, this fractional order model gives better fit on the experimental data. It is worthy to note that the provision of changing fractions in different ways as well as changing parameters is still there, and by availing this provision, it is possible to get a very close fit. In comparison with its integer order version, the proposed model is superior. The reason is that the increase in the glucose level is less that of the integer order version. The Plasma insulin concentration in (mU/L) is illustrated in Figure 2. As shown in this figure, the proposed model outperforms the integer order version. The initial increase of Plasma insulin concentration for the proposed model is much less than that its integer order version. Simulation results verify the satisfactory performance of the proposed model in comparison with  Time  Glucose  Insulin   0  9 2  1 1  2  350  26  4  287  130  6  251  85  8  240  51  10  216  49  12  211  45  14  205  41  16  196  35  18  192  30  20  172  30  22  163  27  32  142  30   a previous related work. Comparison of average and rms values of absolute difference from the experimental data is given in the Table 3. On the other hand, discretization method was employed as a reasonable basis for studying the solution of a fractional-order model of glucose-insulin system (2). We have utilized the above mentioned model, and tuned for the order of fractional derivative which ensures better fit.

Conclusions
The Caputo fractional-order glucose-insulin model (2) and its discretization system (4) are investigated. We showed that the fractional system (2) possesses existence, uniqueness, non-negative, boundedness solution. We also deduced a detailed analysis on the stability of the model (2) and its discretization system (4). Comparisons between the results of the Caputo fractionalorder (2), the model of integer one and the measured real data obtained from patients are presented. These comparisons is concluded that the presented fractional order model is better representative of the system than its integer order one. Numerical solutions of the model (2) are obtained by using the method of Adams-Bashforth-Moulton type to handle the fractional derivatives. We also obtained the solution of the discretization model (4) and a numerical solution of the system which shows that effect of time on the concentrations x(t), y(t) and z(t).