Comparative study of blood sugar–insulin model using fractional derivatives

ABSTRACT

Many years have passed since the model that outlined the connection between sugar and insulin was developed, and a great deal of study and investigation has subsequently been done on it. In this work, we investigated the disparities in the Modified Bergman's glucose–insulin model between Yang–Abdel–Cattani derivative and Caputo–Fabrizio derivative. With our improvements, the prior model now includes the diet, which is a crucial component of the blood glucose model. Based on results, the new model outperforms the previous one in terms of accuracy. To highlight the significance of fractional derivatives, we also established the fractional models. In addition, we verified the existence and uniqueness of the results and offered a graphical and numerical depiction of the findings.

KEYWORDS:

• Bergman's minimal model
• Caputo–Fabrizio derivative
• Yang–Abdel–Cattani derivative
• Sumudu transform
• uniqueness and oneness

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 92B05
• 92C60
• Secondary26A33

1. Introduction and background

Nowadays, modelling is proving out a very useful tool in mathematical field. It is used in all fields to achieve or forecast future prospects. We can forecast the goal by translating the results back into real-world languages after using the necessary procedures. It does this by translating real-world issues into mathematical terms. Modelling has recently been applied to fractional calculus, a main but less communal area of maths  [1–8]. Even though we've read a lot of problems and their results using standard calculus, fractional calculus sometimes proves better than previous one, in terms of characterizing the model. Fractional calculus has many practical applications in the engineering and science field  [9–22]. In this work, mathematical modelling and fractional calculus will be used to study and analyse a diabetic. These days, it appears that “diabetes” is a fatal condition. Fundamentally, an imbalance between the body's insulin and glucose levels leads to diabetes. Additionally, diabetes comes in two varieties. Category I is more severe than category II. As per an assessment, category 1 affects 10 percent of people with diabetes, while other affects the other 90 percent. Therefore, this imbalance required a thorough investigation from a research standpoint. Given this, Bergman put forth the important model known as the Bergman minimal model  [11–16]. We have used both operators and fractional order generalization of the model to carry out our work. Fractional calculus was first proposed by Riemann–Liouville and Caputo. However, Caputo and Fabrozio subsequently found that certain systems could not be well-defined by Riemann or Caputo derivatives. As a result, Caputo and Fabrizio derived new operator with non-singular kernel. After this, the Caputo–Fabrizio operator was widely used by researchers to model various mathematical and engineering issues. This operator is frequently used because of its memory and non-singularity properties. In the fields of science, engineering and biomathematics, the operator is used to analyse a wide range of models, including those related to diffusion, heat transfer, dengue, rabies, HIV and electrical engineering, among others. Overall, we find that the results obtained with this operator are more realistic than with other approaches. See  [23–43] for further details regarding the applications of the Caputo–Fabrizio derivative. We added a factor called “Diet” to the model to represent the impact of a meal on blood glucose levels in addition to converting the system to fractional order. We found that, more accurate outcomes were also attained. The fractional order system provides us more comprehensive forecasting about the issue because it covers a wide range of solutions and outcomes at every point of the domain.

2. Pre-requisites

Here, we will go over some basic presumptions pertaining to our article:

2.1. The Caputo–Fabrizio fractional coefficient 26

Assume $h\in H^1(a_1,b_1),b_1>a_1,\text{β}\in [0,1]$ so Caputo–Fabrizio derivative of fractional order: (1) $\begin{array}{r}{D}_t^{\text{β}}\left(\mathrm{h}\left(\mathrm{t}\right)\right)=\frac{M\left(\text{β}\right)}{\left(1-\text{β}\right)}{\int }_a^t{h}^{\prime }\left(x\right){\mathrm{e}}^{\left[-\text{β}\frac{t-x}{1-\text{β}}\right]}\mathrm{d}x.\end{array}$(1) $M\left(\text{β}\right)$ is the normalization function s.t. $M\left(0\right)=M\left(1\right)=1$. If $h\notin H^1(a_1,b_1),$ then derivative is (2) $\begin{array}{rl}& {\mathrm{D}}_t^{\text{β}}\left(\mathrm{h}\left(\mathrm{t}\right)\right)=\frac{N\left(\sigma \right)}{\sigma }{\int }_a^t{h}^{\prime }\left(x\right){\mathrm{e}}^{\left[-\frac{t-x}{\sigma }\right]}\mathrm{dx},\\ & N\left(0\right)=N\left(\mathrm{\infty }\right)=1,\end{array}$(2) more than that (3) $\begin{array}{r}\underset{\sigma \to 0}{lim}\frac{1}{\sigma }{\mathrm{e}}^{\left[-\frac{t-x}{1-\text{β}}\right]}=\delta (x-t).\end{array}$(3) The anti-derivative of this fractional differential coefficient is discovered by Losada and Nieto [44].

2.2. The Caputo–Fabrizio fractional integral

Fractional integral of function g(t) of exponent $\gamma (0<\gamma <1),$ is (4) $\begin{array}{rl}{I}_{\text{β}}^{\gamma }\left(g\left(t\right)\right)& =\frac{2\left(1-\gamma \right)}{\left(2-\gamma \right)M\left(\gamma \right)}g\left(t\right)\\ & +\frac{2\gamma }{\left(2-\gamma \right)M\left(\gamma \right)}{\int }_0^{t}g\left(s\right)\mathrm{d}s,t\ge 0.\end{array}$(4)

Remark 2.1

From (2.4), we see that integral is average of g and its integral of exponent one. Hence Nieto et al. deduced following constraints: (5) $\begin{array}{r}\frac{2\left(1-\gamma \right)}{\left(2-\gamma \right)M\left(\gamma \right)}+\frac{2\gamma }{\left(2-\gamma \right)M\left(\gamma \right)}=1,\end{array}$(5) so we have an evident expression $\begin{array}{r}M\left(\gamma \right)=\frac{2}{\left(2-\gamma \right)},0<\gamma <1.\end{array}$On basis of defined relationship, Nieto and Losada defined Caputo differential coefficient of exponent $0<\gamma <1$ defined as (6) $\begin{array}{r}{}_0^{\mathrm{CF}}{D}_t^{\gamma }\left(\mathrm{g}\left(\mathrm{t}\right)\right)=\frac{1}{\left(1-\gamma \right)}{\int }_{a_1}^t{g}^{\prime }\left(x\right){\mathrm{e}}^{\left[-\gamma \frac{t-x}{1-\gamma }\right]}\mathrm{d}x.\end{array}$(6)

2.3. Yang–Abdel–Cattani differential coefficient

Assume $\psi \in L(0,\mathrm{\infty })$, where $n\in I^{+}$, then YAC fractional operator [45] of ψ with parameters $(\mu ,\chi ,n),\mu \ge 0,\chi >0$ is given as (7) $\begin{array}{r}{}_0^{\mathrm{YAC}}{D}_t^{\mu }\psi \left(t\right)={\int }_0^t{R}_{\mu }\left[-\chi {(t-\zeta )}^{\mu }\right]{\psi }^{\left(n\right)}\left(\zeta \right)\mathrm{d}\zeta ;t>0.\end{array}$(7)

2.4. Yang–Abdel–Cattani integral operator

YAC integral coefficient of exponent ξ is given as (8) $\begin{array}{r}{}_a^{\mathrm{YAC}}{I}_0^{\xi }h\left(t\right)={\int }_0^t{\varphi }_{\xi }\left[-\chi {(t-\varpi )}^{\xi }\right]h\left(\varpi \right)\mathrm{d}\varpi .\end{array}$(8)

2.5. Sumudu transform of Caputo–Fabrizio coefficient 19

Assume j(t) be any function whose CF derivative exists so Sumudu Transform of CF operator of j(t) is (9) $\begin{array}{r}\mathrm{ST}\left({}_0^{\mathrm{CF}}{D}_t^{\text{β}}\right)\left(j\right(t\left)\right)=M\left(\text{β}\right)\left[\frac{\mathrm{ST}\left(j\right(t\left)\right)-j\left(0\right)}{1-\text{β}+\text{β}u}\right].\end{array}$(9)

2.6. Sumudu transform of YAC differential operator

Consider $\mathrm{{\rm Y}}\in H^1(a,b),b>a,\mu \in [0,1]$ then Sumudu transform of YAC derivative is (10) $\begin{array}{rl}& \mathrm{ST}\left\{{}^{\mathrm{YAC}}{D}_t^{\mathrm{n\mu }}\mathrm{{\rm Y}}(\eta ,t)\right\}=\frac{u^{\mu }}{1+\chi u^{\mu +1}}\\ & \times \left[\mathrm{ST}\left\{\mathrm{{\rm Y}}(\eta ,t)\right\}-\sum _{k=0}^{n-1}{u}^k{\mathrm{{\rm Y}}}^k(\eta ,0)\right].\end{array}$(10) Here one should note that Caputo–Fabrizio fractional coefficient and Yang–Abdel–Cattani derivatives have been used for analysis. This is done because these contain the exponential function and we know that exponential functional is an entire function, so gives better results in the domain, and these operators serve better on account of having better memory vis-a-vis other operators.

2.7. Structure of the article

The whole paper has been partitioned into seven parts. Part 1 tells us about the basics of fractional calculus and modelling along with little background. In Section 2, the pre-requisites related to the article have been discussed. Next, we have details about Bergman system of fractional exponent and modified minimal model. The fourth part deals with existence and oneness of results, while fifth section details result of system by using Sumudu Transform. Section 6 enlists numeric results with graphical solution of modified system and in the final seventh segment, the outcomes are concluded.

3. Computational model

3.1. Bergman model of fractional order

In 2017, Bergman model of fractional exponent was introduced, meriting a perfunctory introduction. A sugar chamber is taken into consideration, and the final glucose intake is controlled by the inherent action of plasma insulin throughout the isolate chamber. According to this model, using the fewest possible parameters, it is good enough to meet some validation criteria. This system works well for describing the interaction between insulin and blood sugar. We assume that $\mathrm{\Delta }\left(t\right)$ represents the variation of plasma sugar concentration and $\mathrm{\Theta }\left(t\right)$ represents free plasma insulin concentration in the model that is being presented, based on their starting values. Old system was (11) $\begin{array}{rl}\frac{{\mathrm{d}}^{\alpha }\mathrm{\Delta }\left(t\right)}{\mathrm{d}{t}^{\alpha }}& =-\left(q_1+\mathrm{\Omega }\left(t\right)\right)\mathrm{\Delta }\left(t\right)+q_1{g}_b,0<\alpha <1\end{array}$(11) (12) $\begin{array}{rl}\frac{{\mathrm{d}}^{{}^{\text{β}}}\mathrm{\Omega }\left(t\right)}{\mathrm{d}{t}^{\text{β}}}& =-q_2\mathrm{\Omega }\left(t\right)+q_3\left(\mathrm{\Theta }\left(t\right)-I_b\right),0<\text{β}<1\end{array}$(12) (13) $\begin{array}{rl}\frac{{\mathrm{d}}^{\gamma }\mathrm{\Theta }\left(t\right)}{\mathrm{d}{t}^{\gamma }}& =q_6{\left(\mathrm{\Delta }\left(t\right)-q_5\right)}^{+}t-q_4\left(\mathrm{\Theta }\left(t\right)-I_b\right),\\ & 0<\gamma <1,\end{array}$(13) with constraints $\mathrm{\Delta }\left(0\right)={\mathrm{\Delta }}_0$, $\mathrm{\Omega }\left(0\right)={\mathrm{\Omega }}_0$, and $\mathrm{\Theta }\left(0\right)={\mathrm{\Theta }}_0$.

3.2. Outline of modified system

Here, we revamped the earlier setup. The model was modified and a few new parameters were included. Considering that fluctuations in blood glucose levels are largely influenced by our food. Now the changed structure of the modified Bergman's system is described by the following equations: (14) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Delta }\left(t\right)& =-(q_1+\mathrm{\Omega }(t\left)\right)\mathrm{\Delta }\left(t\right)+\mathrm{\wp }\left(t\right)-\mathrm{\Omega }\left(t\right)g_b,\end{array}$(14) (15) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Omega }\left(t\right)& =-q_2\mathrm{\Omega }\left(t\right)+q_3\mathrm{\Theta }\left(t\right),\end{array}$(15) (16) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Theta }\left(t\right)& =-n\left[\mathrm{\Theta }\right(t)+I_b]+\frac{u\left(t\right)}{V_1},\end{array}$(16) (17) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\wp }\left(t\right)& =-k\mathrm{\wp }\left(t\right).\end{array}$(17) Note here that $0<\varpi \le 1$ and starting constraints are $\mathrm{\Delta }\left(0\right)={\mathrm{\Delta }}_0,\mathrm{\Omega }\left(0\right)={\mathrm{\Omega }}_0,\mathrm{\Theta }\left(0\right)={\mathrm{\Theta }}_0$, $\mathrm{\wp }\left(0\right)={\mathrm{\wp }}_0$.

This arrangement can be useful in finding artificial pancreas. In this model, we use Caputo–Fabrizio and Yang–Abdel–Cattani operators which are well known in fractional calculus. The parameters are $\mathrm{\Delta }\left(t\right)$ – sugar concentration, $\mathrm{\Omega }\left(t\right)$ – repercussions of potent insulin, $\mathrm{\Theta }\left(t\right)$ – insulin concentration, $\mathrm{\wp }\left(t\right)$ – infusion of extrinsic sugar, $u\left(t\right)$ – insulin dispensation function, $g_b$ – starting sugar cluster, $I_b$ – starting insulin concentration, $V_1$ – insulin issuance volume, n – fractional vanish outlay of insulin, $q_1$ – insulin free sugar dispensation outlay, $q_2$ – active insulin dispensation outlay and $q_3$ – growth in soaking capacity due to insulin.

4. Existence of results of models

Here we will prove some assertions to show the existence of the solution [11–16].

Theorem 4.1

Define $N_1,$ $N_2,$ $N_3$ and $N_4,$ and there relations with variables.

Proof.

The model is (18) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Delta }\left(t\right)& =-(q_1+\mathrm{\Omega }(t\left)\right)\mathrm{\Delta }\left(t\right)+\mathrm{\wp }\left(t\right)-\mathrm{\Omega }\left(t\right)g_b,\end{array}$(18) (19) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Omega }\left(t\right)& =-q_2\mathrm{\Omega }\left(t\right)+q_3\mathrm{\Theta }\left(t\right),\end{array}$(19) (20) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Theta }\left(t\right)& =-n\left[\mathrm{\Theta }\right(t)+I_b]+\frac{u\left(t\right)}{V_1},\end{array}$(20) (21) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\wp }\left(t\right)& =-k\mathrm{\wp }\left(t\right).\end{array}$(21) Here, note that $0<\varpi \le 1$, so writing above model into integral equation (22) $\begin{array}{rl}\mathrm{\Delta }\left(t\right)-\mathrm{\Delta }\left(0\right)& ={}_0^{\mathrm{CF}}{I}_t^{\varpi }[-(q_1+\mathrm{\Omega }\left(t\right)\left)\mathrm{\Delta }\right(t)+\mathrm{\wp }(t)\\ & -\mathrm{\Omega }\left(t\right)g_b],\end{array}$(22) (23) $\begin{array}{rl}\mathrm{\Omega }\left(t\right)-\mathrm{\Omega }\left(0\right)& ={}_0^{\mathrm{CF}}{I}_t^{\varpi }[-q_2\mathrm{\Omega }(t)+q_3\mathrm{\Theta }(t\left)\right],\end{array}$(23) (24) $\begin{array}{rl}\mathrm{\Theta }\left(t\right)-\mathrm{\Theta }\left(0\right)& ={}_0^{\mathrm{CF}}{I}_t^{\varpi }[-n[\mathrm{\Theta }\left(t\right)+I_b]+\frac{u\left(t\right)}{V_1}],\end{array}$(24) (25) $\begin{array}{rl}\mathrm{\wp }\left(t\right)-\mathrm{\wp }\left(0\right)& ={}_0^{\mathrm{CF}}{I}_t^{\varpi }[-k\mathrm{\wp }(t\left)\right],\end{array}$(25) then by Nieto's definition, we obtain (26) $\begin{array}{rl}\mathrm{\Delta }\left(t\right)& =\mathrm{\Delta }\left(0\right)+\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}\\ & \times [-(q_1+\mathrm{\Omega }\left(t\right)\left)\mathrm{\Delta }\right(t)+\mathrm{\wp }(t)-\mathrm{\Omega }(t\left)g_b\right]\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times {\int }_0^t\left[-(q_1+\mathrm{\Omega }(s\left)\right)G\left(s\right)+\mathrm{\wp }\left(s\right)-\mathrm{\Omega }\left(s\right)g_b\right]\mathrm{d}s,\end{array}$(26) (27) $\begin{array}{rl}\mathrm{\Omega }\left(t\right)& =\mathrm{\Omega }\left(0\right)+\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}[-q_2\mathrm{\Omega }(t)+q_3\mathrm{\Theta }(t\left)\right]\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t\left[-q_2\mathrm{\Omega }\left(s\right)+q_3\mathrm{\Theta }\left(s\right)\right]\mathrm{d}s,\end{array}$(27) (28) $\begin{array}{rl}\mathrm{\Theta }\left(t\right)& =\mathrm{\Theta }\left(0\right)\\ & +\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}[-n[\mathrm{\Theta }\left(t\right)+I_b]+\frac{u\left(t\right)}{V_1}]\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t\left[-n\left[\mathrm{\Theta }\right(s)+I_b]+\frac{u\left(s\right)}{V_1}\right]\mathrm{d}s,\end{array}$(28) (29) $\begin{array}{rl}\mathrm{\wp }\left(t\right)& =\mathrm{\wp }\left(0\right)+\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}[-k\mathrm{\wp }(t\left)\right]\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t\left[-k\mathrm{\wp }\left(s\right)\right]\mathrm{d}s.\end{array}$(29) Now assume the kernels are $\begin{array}{rl}{N}_1(t,\mathrm{\Delta })& =-(q_1+\mathrm{\Omega }(t\left)\right)\mathrm{\Delta }\left(t\right)+\mathrm{\wp }\left(t\right)-\mathrm{\Omega }\left(t\right)g_b,\\ N_2(t,\mathrm{\Omega })& =-q_2\mathrm{\Omega }\left(t\right)+q_3\mathrm{\Theta }\left(t\right),\\ N_3(t,\mathrm{\Theta })& =-n\left[\mathrm{\Theta }\right(t)+I_b]+\frac{u\left(t\right)}{V_1},\\ N_4(t,D)& =-k\mathrm{\wp }\left(t\right).\end{array}$

Theorem 4.2

Prove that $N_1,N_2,N_3$ and $N_4$ fulfil Lipchitz status.

Proof.

First, we will prove this for $N_1$. Assume Δ and ${\mathrm{\Delta }}_1$ are any functions, so $\begin{array}{rl}\Vert N_1(t,\mathrm{\Delta })-N_1(t,{\mathrm{\Delta }}_1)\Vert & =\Vert -(q_1+\mathrm{\Omega }(t\left)\right)\mathrm{\Delta }\left(t\right)\\ & +(q_1+\mathrm{\Omega }(t\left)\right){\mathrm{\Delta }}_1\left(t\right)\Vert \\ & =\Vert q_1+\mathrm{\Omega }\left(t\right)\Vert \Vert \mathrm{\Delta }\left(t\right)-{\mathrm{\Delta }}_1\left(t\right)\Vert \\ \Vert N_1(t,\mathrm{\Delta })-N_1(t,{\mathrm{\Delta }}_1)\Vert & \le H\Vert \mathrm{\Delta }\left(t\right)-{\mathrm{\Delta }}_1\left(t\right)\Vert ,\\ & \mathrm{where}\Vert q_1+\mathrm{\Omega }\left(t\right)\Vert \le H<1.\end{array}$Similarly we can have $\begin{array}{rl}\Vert N_2(t,\mathrm{\Omega })-N_2(t,{\mathrm{\Omega }}_1)\Vert & \le H_1\Vert \mathrm{\Omega }\left(t\right)-{\mathrm{\Omega }}_1\left(t\right)\Vert ,\\ & \mathrm{where}\Vert q_2\Vert \le H_1<1,\\ \Vert N_3(t,\mathrm{\Theta })-N_3(t,{\mathrm{\Theta }}_1)\Vert & \le H_2\Vert \mathrm{\Theta }\left(t\right)-{\mathrm{\Theta }}_1\left(t\right)\Vert ,\\ & \mathrm{where}\Vert n\Vert \le H_2<1\end{array}$and $\begin{array}{rl}& \Vert N_4(t,\mathrm{\wp })-N_4(t,{\mathrm{\wp }}_1)\Vert \le H_3\Vert \mathrm{\wp }\left(t\right)-{\mathrm{\wp }}_1\left(t\right)\Vert \\ & \mathrm{where}\Vert k\Vert \le H_3<1.\end{array}$Now take the recursive relation (30) $\begin{array}{rl}{\mathrm{\Delta }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_{n-1})\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,{\mathrm{\Delta }}_{n-1})\mathrm{d}s,\end{array}$(30) (31) $\begin{array}{rl}{\mathrm{\Omega }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_2(t,{\mathrm{\Omega }}_{n-1})\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_2(s,{\mathrm{\Omega }}_{n-1})\mathrm{d}s,\end{array}$(31) (32) $\begin{array}{rl}{\mathrm{\Theta }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_3(t,{\mathrm{\Theta }}_{n-1})\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_3(s,{\mathrm{\Theta }}_{n-1})\mathrm{d}s,\end{array}$(32) (33) $\begin{array}{rl}{\mathrm{\wp }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_4(t,{\mathrm{\wp }}_{n-1})\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_4(s,{\mathrm{\wp }}_{n-1})\mathrm{d}s.\end{array}$(33) Again consider deviation between two successive terms is (34) $\begin{array}{rl}{U}_n\left(t\right)& ={\mathrm{\Delta }}_n\left(t\right)-{\mathrm{\Delta }}_{n-1}\left(t\right)\\ & =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_{n-1})\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,{\mathrm{\Delta }}_{n-1})\mathrm{d}s\\ & -\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_{n-2})\\ & -\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,{\mathrm{\Delta }}_{n-2})\mathrm{d}s,\end{array}$(34) (35) $\begin{array}{rl}{U}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_{n-1})\\ & -\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_{n-2})\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times {\int }_0^t\left\{N_1(s,{\mathrm{\Delta }}_{n-1})-N_1(s,{\mathrm{\Delta }}_{n-2})\right\}\mathrm{d}s.\end{array}$(35) Now (36) $\begin{array}{rl}\Vert U_n\left(t\right)\Vert & =\Vert {\mathrm{\Delta }}_n\left(t\right)-{\mathrm{\Delta }}_{n-1}\left(t\right)\Vert ,\\ & =\Vert \frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_{n-1})\\ & -\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_{n-2})\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times {\int }_0^t\left\{N_1(s,{\mathrm{\Delta }}_{n-1})-N_1(s,{\mathrm{\Delta }}_{n-2})\right\}\mathrm{d}s\Vert ,\\ & \le \frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}\Vert N_1(t,{\mathrm{\Delta }}_{n-1})-N_1(t,{\mathrm{\Delta }}_{n-2})\Vert \\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times \Vert {\int }_0^t\left\{N_1(s,{\mathrm{\Delta }}_{n-1})-N_1(s,{\mathrm{\Delta }}_{n-2})\right\}\mathrm{d}s\Vert .\end{array}$(36) But $K_1$ satisfies Lipchitz condition so (37) $\begin{array}{rl}& \Vert U_n\left(t\right)\Vert \le \frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}H\Vert {\mathrm{\Delta }}_{n-1}-{\mathrm{\Delta }}_{n-2}\Vert \\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}K{\int }_0^t\Vert {\mathrm{\Delta }}_{n-1}-{\mathrm{\Delta }}_{n-2}\Vert \mathrm{d}s.\end{array}$(37) In the same way, we obtain (38) $\begin{array}{rl}\Vert V_n\left(t\right)\Vert & \le \frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{H}_1\Vert {\mathrm{\Omega }}_{n-1}-{\mathrm{\Omega }}_{n-2}\Vert \\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{J}_1{\int }_0^t\Vert {\mathrm{\Omega }}_{n-1}-{\mathrm{\Omega }}_{n-2}\Vert \mathrm{d}s,\end{array}$(38) (39) $\begin{array}{rl}\Vert W_n\left(t\right)\Vert & \le \frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{H}_2\Vert {\mathrm{\Theta }}_{n-1}-{\mathrm{\Theta }}_{n-2}\Vert \\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{J}_2{\int }_0^t\Vert {\mathrm{\Theta }}_{n-1}-{\mathrm{\Theta }}_{n-2}\Vert \mathrm{d}s,\end{array}$(39) (40) $\begin{array}{rl}\Vert T_n\left(t\right)\Vert & \le \frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{H}_3\Vert {\mathrm{\wp }}_{n-1}-{\mathrm{\wp }}_{n-2}\Vert \\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{J}_3{\int }_0^t\Vert {\mathrm{\wp }}_{n-1}-{\mathrm{\wp }}_{n-2}\Vert \mathrm{d}s.\end{array}$(40)

Theorem 4.3

Demonstrate that minimum framework for sugar-insulin kinetics is Bergman's system of fractional order.

Proof.

By recursive technique, we get (41) $\begin{array}{rl}\Vert U_n\left(t\right)\Vert & \le \Vert \mathrm{\Delta }\left(0\right)\Vert +{\left\{\frac{2(1-\varpi )H}{(2-\varpi )M\left(\varpi \right)}\right\}}^n\\ & +{\left\{\frac{2\mathrm{\varpi Kt}}{(2-\varpi )M\left(\varpi \right)}\right\}}^n,\end{array}$(41) (42) $\begin{array}{rl}\Vert V_n\left(t\right)\Vert & \le \Vert \mathrm{\Omega }\left(0\right)\Vert +{\left\{\frac{2(1-\varpi )H_1}{(2-\varpi )M\left(\varpi \right)}\right\}}^n\\ & +{\left\{\frac{2\varpi J_{1}t}{(2-\varpi )M\left(\varpi \right)}\right\}}^n,\end{array}$(42) (43) $\begin{array}{rl}\Vert W_n\left(t\right)\Vert & \le \Vert \mathrm{\Theta }\left(0\right)\Vert +{\left\{\frac{2(1-\varpi )H_2}{(2-\varpi )M\left(\varpi \right)}\right\}}^n\\ & +{\left\{\frac{2\varpi J_{2}t}{(2-\varpi )M\left(\varpi \right)}\right\}}^n,\end{array}$(43) (44) $\begin{array}{rl}\Vert T_n\left(t\right)\Vert & \le \Vert \mathrm{\wp }\left(0\right)\Vert +{\left\{\frac{2(1-\varpi )H_3}{(2-\varpi )M\left(\varpi \right)}\right\}}^n\\ & +{\left\{\frac{2\varpi J_{3}t}{(2-\varpi )M\left(\varpi \right)}\right\}}^n.\end{array}$(44) So, existence is confirmed and is continuous too. Hence we have $\begin{array}{rl}\mathrm{\Delta }\left(t\right)& ={\mathrm{\Delta }}_n\left(t\right)+E_n\left(t\right),\\ \mathrm{\Omega }\left(t\right)& ={\mathrm{\Omega }}_n\left(t\right)+F_n\left(t\right),\\ \mathrm{\Theta }\left(t\right)& ={\mathrm{\Theta }}_n\left(t\right)+R_n\left(t\right),\\ \mathrm{\wp }\left(t\right)& ={\mathrm{\wp }}_n\left(t\right)+S_n\left(t\right).\end{array}$where $E_n,F_n,R_n$ and $S_n$ are reminders of series result. So, $\begin{array}{rl}\mathrm{\Delta }\left(t\right)-{\mathrm{\Delta }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,{\mathrm{\Delta }}_n)\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,{\mathrm{\Delta }}_n)\mathrm{d}s,\\ \mathrm{\Delta }\left(t\right)-{\mathrm{\Delta }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,\mathrm{\Delta }-E_n(t\left)\right)\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times {\int }_0^t{N}_1(s,\mathrm{\Delta }-E_n(s\left)\right)\mathrm{d}s.\end{array}$Similarly, we have $\begin{array}{rl}\mathrm{\Omega }\left(t\right)-{\mathrm{\Omega }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_2(t,\mathrm{\Omega }-F_n(t\left)\right)\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times {\int }_0^t{N}_2(s,\mathrm{\Omega }-F_n(s\left)\right)\mathrm{d}s,\\ \mathrm{\Theta }\left(t\right)-{\mathrm{\Theta }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_3(t,\mathrm{\Theta }-R_n(t\left)\right)\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times {\int }_0^t{N}_3(s,\mathrm{\Theta }-R_n(s\left)\right)\mathrm{d}s,\\ \mathrm{\wp }\left(t\right)-{\mathrm{\wp }}_n\left(t\right)& =\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_4(t,\mathrm{\wp }-S_n(t\left)\right)\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\\ & \times {\int }_0^t{N}_4(s,\mathrm{\wp }-S_n(s\left)\right)\mathrm{d}s.\end{array}$From the above, we have $\begin{array}{rl}& \mathrm{\Delta }\left(t\right)-{\mathrm{\Delta }}_n\left(t\right)=\frac{2(1-\varpi )}{(2-\varpi )M\left(\varpi \right)}{N}_1(t,\mathrm{\Delta }-E_n(t\left)\right)\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,\mathrm{\Delta }-E_n(s\left)\right)\mathrm{d}s,\\ & \mathrm{\Delta }\left(t\right)-\mathrm{\Delta }\left(0\right)-\frac{2(1-\varpi )N_1(t,\mathrm{\Delta })}{(2-\varpi )M\left(\varpi \right)}\\ & -\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,\mathrm{\Delta })\mathrm{d}s=E_n\left(t\right)\\ & +\frac{2(1-\varpi )N_1(t,\mathrm{\Delta }-E_n(t\left)\right)}{(2-\varpi )M\left(\varpi \right)}\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,\mathrm{\Delta }-E_n(s\left)\right)\mathrm{d}s.\end{array}$Now, (45) $\begin{array}{rl}& \Vert \mathrm{\Delta }\left(t\right)-\frac{2(1-\varpi )N_1(t,\mathrm{\Delta })}{(2-\varpi )M\left(\varpi \right)}-\mathrm{\Delta }\left(0\right)\\ & -\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,\mathrm{\Delta })\mathrm{d}s\Vert \le \Vert E_n\left(t\right)\Vert \\ & +\left\{\frac{2(1-\varpi )H}{(2-\varpi )M\left(\varpi \right)}+\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}\mathrm{Kt}\right\}\Vert E_n\left(t\right)\Vert ,\end{array}$(45) (46) $\begin{array}{rl}& \Vert \mathrm{\Omega }\left(t\right)-\frac{2(1-\varpi )N_2(t,\mathrm{\Omega })}{(2-\varpi )M\left(\varpi \right)}-\mathrm{\Omega }\left(0\right)\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_2(s,\mathrm{\Omega })\mathrm{d}s\Vert \\ & \le \Vert F_n\left(t\right)\Vert +\left\{\frac{2(1-\varpi )H_1}{(2-\varpi )M\left(\varpi \right)}+\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{J}_{1}t\right\}\\ & \times \Vert F_n\left(t\right)\Vert ,\end{array}$(46) (47) $\begin{array}{rl}& \Vert \mathrm{\Theta }\left(t\right)-\frac{2(1-\varpi )N_3(t,\mathrm{\Theta })}{(2-\varpi )M\left(\varpi \right)}-\mathrm{\Theta }\left(0\right)\\ & -\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_3(s,\mathrm{\Theta })\mathrm{d}s\Vert \\ & \le \Vert R_n\left(t\right)\Vert +\left\{\frac{2(1-\varpi )H_2}{(2-\varpi )M\left(\varpi \right)}+\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{J}_{2}t\right\}\\ & \times \Vert R_n\left(t\right)\Vert ,\end{array}$(47) (48) $\begin{array}{rl}& \Vert \mathrm{\wp }\left(t\right)-\frac{2(1-\varpi )N_4(t,\mathrm{\wp })}{(2-\varpi )M\left(\varpi \right)}-\mathrm{\wp }\left(0\right)\\ & -\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_4(s,\mathrm{\wp })\mathrm{d}s\Vert \\ & \le \Vert S_n\left(t\right)\Vert +\left\{\frac{2(1-\varpi )H_3}{(2-\varpi )M\left(\varpi \right)}+\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{J}_{3}t\right\}\\ & \times \Vert S_n\left(t\right)\Vert .\end{array}$(48) Taking $n\to \mathrm{\infty }$ (49) $\begin{array}{rl}\mathrm{\Delta }\left(t\right)& =\mathrm{\Delta }\left(0\right)+\frac{2(1-\varpi )N_1(t,\mathrm{\Delta })}{(2-\varpi )M\left(\varpi \right)}\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_1(s,\mathrm{\Delta })\mathrm{d}s,\end{array}$(49) (50) $\begin{array}{rl}\mathrm{\Omega }\left(t\right)& =\mathrm{\Omega }\left(0\right)+\frac{2(1-\varpi )N_2(t,\mathrm{\Omega })}{(2-\varpi )M\left(\varpi \right)}\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_2(s,\mathrm{\Omega })\mathrm{d}s,\end{array}$(50) (51) $\begin{array}{rl}\mathrm{\Theta }\left(t\right)& =\mathrm{\Theta }\left(0\right)+\frac{2(1-\varpi )N_3(t,\mathrm{\Theta })}{(2-\varpi )M\left(\varpi \right)}\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_3(s,\mathrm{\Theta })\mathrm{d}s,\end{array}$(51) (52) $\begin{array}{rl}\mathrm{\wp }\left(t\right)& =\mathrm{\wp }\left(0\right)+\frac{2(1-\varpi )N_4(t,\mathrm{\wp })}{(2-\varpi )M\left(\varpi \right)}\\ & +\frac{2\varpi }{(2-\varpi )M\left(\varpi \right)}{\int }_0^t{N}_4(s,\mathrm{\wp })\mathrm{d}s.\end{array}$(52) Hence, we can say that the result of model exists.

4.1. Oneness of result

Here, we are going to prove that the results mentioned in above section are totally solitary. For this, consider another set of solutions for the set up given by Equations (14(14) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Delta }\left(t\right)& =-(q_1+\mathrm{\Omega }(t\left)\right)\mathrm{\Delta }\left(t\right)+\mathrm{\wp }\left(t\right)-\mathrm{\Omega }\left(t\right)g_b,\end{array}$(14) ), (15(15) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Omega }\left(t\right)& =-q_2\mathrm{\Omega }\left(t\right)+q_3\mathrm{\Theta }\left(t\right),\end{array}$(15) ), (16(16) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\Theta }\left(t\right)& =-n\left[\mathrm{\Theta }\right(t)+I_b]+\frac{u\left(t\right)}{V_1},\end{array}$(16) ) and (17(17) $\begin{array}{rl}{}_0^{\mathrm{CF}}{D}_t^{\varpi }\mathrm{\wp }\left(t\right)& =-k\mathrm{\wp }\left(t\right).\end{array}$(17) ) say $\mathrm{\Delta }\left(t\right)$, $\mathrm{\Theta }\left(t\right)$, $\mathrm{\Omega }\left(t\right)$ and $D\left(t\right)$, then (53) $\begin{array}{rl}\mathrm{\Delta }\left(t\right)-{\mathrm{\Delta }}_1\left(t\right)& =\frac{2\left(1-\varpi \right)}{M\left(\varpi \right)\left(2-\varpi \right)}\\ & \times \left[N_1(t,\mathrm{\Delta })-N_1(t,{\mathrm{\Delta }}_1)\right]\\ & +\frac{2\left(\varpi \right)}{M\left(\varpi \right)\left(2-\varpi \right)}\\ & \times {\int }_0^t\left[N_1(s,\mathrm{\Delta })-N_1(s,{\mathrm{\Delta }}_1)\right]\mathrm{d}s,\end{array}$(53) on both side taking norm, we get (54) $\begin{array}{rl}\Vert \mathrm{\Delta }-{\mathrm{\Delta }}_1\Vert & =\frac{2\left(1-\varpi \right)}{M\left(\varpi \right)\left(2-\varpi \right)}\left[\Vert N_1(t,\mathrm{\Delta })-N_1(t,{\mathrm{\Delta }}_1)\Vert \right]\\ & +\frac{2\left(\varpi \right)}{M\left(\varpi \right)\left(2-\varpi \right)}\\ & \times {\int }_0^t\left[\Vert N_1(s,\mathrm{\Delta })-N_1(s,{\mathrm{\Delta }}_1)\Vert \right]\mathrm{d}s,\end{array}$(54) by Lipchitz constraints, we get (55) $\begin{array}{rl}\Vert \mathrm{\Delta }-{\mathrm{\Delta }}_1\Vert & <\frac{2\left(1-\varpi \right)}{M\left(\varpi \right)\left(2-\varpi \right)}\mathrm{HE}\\ & +{\left(\frac{2\left(\varpi \right)}{M\left(\varpi \right)\left(2-\varpi \right)}{J}_1\mathrm{Et}\right)}^n,\end{array}$(55) which is true for all n, hence (56) $\begin{array}{r}\mathrm{\Delta }={\mathrm{\Delta }}_1,\end{array}$(56) similarly (57) $\begin{array}{r}\mathrm{\Omega }={\mathrm{\Omega }}_1,\mathrm{\Theta }={\mathrm{\Theta }}_1\mathrm{and}\mathrm{\wp }={\mathrm{\wp }}_1.\end{array}$(57) So, it proves oneness of model.

In the same way, we can establish the existence and oneness of results for Yang–Abdel–Cattani fractional operator.

5. Solution of model by Sumudu transform

Precise result-finding is challenging due to multiple equations being inherent to the system. Iterative method in conjunction with the Sumudu Transform shall be used to solve the model. Initially, the output of the Yang–Abdel–Cattani derivative system shall be determined, followed by the application of Sumudu transform on each side, we shall have $\begin{array}{rl}& \mathrm{ST}{(}_0^{\mathrm{YAC}}{D}_t^{\varpi }\left)\right(\mathrm{\Delta }\left(t\right))\\ & =\mathrm{ST}\left\{-(q_1+\mathrm{\Omega }(t\left)\right)\mathrm{\Delta }\left(t\right)+\mathrm{\wp }\left(t\right)-\mathrm{\Omega }\left(t\right)g_b\right\}\end{array}$or $\begin{array}{rl}& \frac{s^{\alpha }}{\left(1+s^{\alpha +1}\right)}.\left[\mathrm{ST}\left\{\mathrm{\Delta }\left(t\right)\right\}-\mathrm{\Delta }\left(0\right)\right]\\ & =\mathrm{ST}\left\{-\left(q_1+\mathrm{\Omega }\left(t\right)\right)\mathrm{\Delta }\left(t\right)+\mathrm{\wp }\left(t\right)-\mathrm{\Omega }\left(t\right)g_b\right\}\end{array}$or $\begin{array}{rl}\mathrm{ST}\left\{\mathrm{\Delta }\left(t\right)\right\}& =\mathrm{\Delta }\left(0\right)+\frac{\left(1+s^{\alpha +1}\right)}{s^{\alpha }}.\mathrm{ST}\{-\left(q_1+\mathrm{\Omega }\left(t\right)\right)\mathrm{\Delta }(t)\\ & +\mathrm{\wp }\left(t\right)-\mathrm{\Omega }\left(t\right)g_b\}.\end{array}$Now, take inverse ST on both sides, we get (58) $\begin{array}{rl}\mathrm{\Delta }\left(t\right)& =\mathrm{\Delta }\left(0\right)+S{T}^{-1}[\left(\frac{1}{s^{\alpha }}+s\right)\\ & \times .\mathrm{ST}\{-\left(q_1+\mathrm{\Omega }\left(t\right)\right)\mathrm{\Delta }(t)+\mathrm{\wp }(t)-\mathrm{\Omega }(t\left)g_b\right\}\phantom{\left(\frac{1}{s^{\alpha }}+s\right)}].\end{array}$(58) Similarly we obtain (59) $\begin{array}{rl}\mathrm{\Omega }\left(t\right)& =\mathrm{\Omega }\left(0\right)\\ & +S{T}^{-1}\left[\left(\frac{1}{s^{\alpha }}+s\right).\mathrm{ST}\left\{-q_2\mathrm{\Omega }\left(t\right)+q_3\mathrm{\Theta }\left(t\right)\right\}\right],\end{array}$(59) (60) $\begin{array}{rl}\mathrm{\Theta }\left(t\right)& =\mathrm{\Theta }\left(0\right)+S{T}^{-1}[\left(\frac{1}{s^{\alpha }}+s\right)\\ & \times .\mathrm{ST}\left\{-n\left(\mathrm{\Theta }\left(t\right)+I_b\right)+\frac{u\left(t\right)}{V_1}\right\}\phantom{\left(\frac{1}{s^{\alpha }}+s\right)}],\end{array}$(60) (61) $\begin{array}{rl}\mathrm{\wp }\left(t\right)& =\mathrm{\wp }\left(0\right)+S{T}^{-1}\left[\left(\frac{1}{s^{\alpha }}+s\right).\mathrm{ST}\left\{-k\mathrm{\wp }\left(t\right)\right\}\right].\end{array}$(61) Hence we have following recurrent form from above: (62) $\begin{array}{rl}{\mathrm{\Delta }}_{n+1}\left(t\right)& =\mathrm{\Delta }\left(0\right)+S{T}^{-1}[\left(\frac{1}{s^{\alpha }}+s\right).\mathrm{ST}\{-\left(q_1+{\mathrm{\Omega }}_n\left(t\right)\right)\\ & \times {\mathrm{\Delta }}_n\left(t\right)+{\mathrm{\wp }}_n\left(t\right)-{\mathrm{\Omega }}_n\left(t\right)g_b\phantom{\left(\frac{1}{s^{\alpha }}+s\right)}\}],\end{array}$(62) (63) $\begin{array}{rl}{\mathrm{\Omega }}_{n+1}\left(t\right)& =\mathrm{\Omega }\left(0\right)+S{T}^{-1}[\left(\frac{1}{s^{\alpha }}+s\right)\\ & +.\mathrm{ST}\left\{-q_2{\mathrm{\Omega }}_n\left(t\right)+q_3{\mathrm{\Theta }}_n\left(t\right)\right\}],\end{array}$(63) (64) $\begin{array}{rl}{\mathrm{\Theta }}_{n+1}\left(t\right)& =\mathrm{\Theta }\left(0\right)+S{T}^{-1}[\left(\frac{1}{s^{\alpha }}+s\right).\\ & \times \mathrm{ST}\left\{-n\left({\mathrm{\Theta }}_n\left(t\right)+I_b\right)+\frac{u\left(t\right)}{V_1}\right\}],\end{array}$(64) (65) $\begin{array}{rl}{\mathrm{\wp }}_{n+1}\left(t\right)& =\mathrm{\wp }\left(0\right)+S{T}^{-1}\left[\left(\frac{1}{s^{\alpha }}+s\right).\mathrm{ST}\left\{-k{\mathrm{\wp }}_n\left(t\right)\right\}\right].\end{array}$(65) Similarly, we obtain the following mathematical outcomes for the system with a CF derivative: (66) $\begin{array}{rl}{\mathrm{\Delta }}_{n+1}\left(t\right)& ={\mathrm{\Delta }}_n\left(0\right)+S{T}^{-1}[\frac{(1-\varpi +\mathrm{\varpi u})}{M\left(\varpi \right)}\\ & \times \mathrm{ST}\{-\left(q_1+{\mathrm{\Omega }}_n\left(t\right)\right){\mathrm{\Delta }}_n(t)\\ & \times +{\mathrm{\wp }}_n\left(t\right)-{\mathrm{\Omega }}_n\left(t\right)g_b\}\phantom{\frac{(1-\varpi +\mathrm{\varpi u})}{M\left(\varpi \right)}}],\end{array}$(66) (67) $\begin{array}{rl}{\mathrm{\Omega }}_{n+1}\left(t\right)& ={\mathrm{\Omega }}_n\left(0\right)+S{T}^{-1}[\frac{(1-\varpi +\mathrm{\varpi u})}{M\left(\varpi \right)}\\ & +\mathrm{ST}\left\{-q_2{\mathrm{\Omega }}_n\left(t\right)+q_3{\mathrm{\Theta }}_n\left(t\right)\right\}\phantom{\frac{(1-\varpi +\mathrm{\varpi u})}{M\left(\varpi \right)}}],\end{array}$(67) (68) $\begin{array}{rl}{\mathrm{\Theta }}_{n+1}\left(t\right)& ={\mathrm{\Theta }}_n\left(0\right)+S{T}^{-1}[\frac{(1-\varpi +\mathrm{\varpi u})}{M\left(\varpi \right)}\\ & +\mathrm{ST}\left\{-n\left(I_n\left(t\right)+I_b\right)+\frac{u\left(t\right)}{V_1}\right\}],\end{array}$(68) (69) $\begin{array}{rl}{\mathrm{\wp }}_{n+1}\left(t\right)& ={\mathrm{\wp }}_n\left(0\right)+S{T}^{-1}\\ & \times \left[\frac{(1-\varpi +\mathrm{\varpi u})}{M\left(\varpi \right)}\mathrm{ST}\left\{-k{\mathrm{\wp }}_n\left(t\right)\right\}\right].\end{array}$(69) The result is obtained by $\begin{array}{rl}\mathrm{\Delta }\left(t\right)& =\underset{n\to \mathrm{\infty }}{lim}{\mathrm{\Delta }}_n\left(t\right),\\ \mathrm{\Omega }\left(t\right)& =\underset{n\to \mathrm{\infty }}{lim}{\mathrm{\Omega }}_n\left(t\right),\\ \mathrm{\Theta }\left(t\right)& =\underset{n\to \mathrm{\infty }}{lim}{\mathrm{\Theta }}_n\left(t\right)\end{array}$and $\begin{array}{r}\mathrm{\wp }\left(t\right)=\underset{n\to \mathrm{\infty }}{lim}{\mathrm{\wp }}_n\left(t\right).\end{array}$

6. Numerical result

For numerical results, we use following parametric values [36].

The numeric solutions can now be ascertained quickly for predefined system by using defined values mentioned above. The Sumudu transform was employed to derive the numerical result in the aforementioned analysis. It is evident from the numerical results that diet plays a significant role in glucose levels (Table 1).

Therefore, we can conclude that, when compared to the linear model, the fraction model with the Caputo–Fabrizio and Yang–Abdel–Cattani fractional derivatives produces better results. Furthermore, our model provided a better definition of this real-world issue.

Table 1. Initial value and parameters.

Parameter	Value
${\mathrm{\Delta }}_0$	287 mg/dL
${\mathrm{\Omega }}_0$	0
${\mathrm{\Theta }}_0$	241mu/L
${\mathrm{\wp }}_0$	9 mg/dL
$g_b$	92 mg/dL
$I_b$	7.3 mu/L
$q_1$	0.01572
$q_2$	0.01301
$q_3$	$0.4031\times {10}^{-5}$
n	0.3606/min
k	0.05

7. Conclusion

The work that is being presented aims to provide an explanation for the existence and unity of modified Bergman model, which is extended by the CF fractional coefficient, within the context of blood sugar and insulin levels. From the graphs (Figures 1 and 2), it can be seen that for various value of ϖ the error difference reducing as compared to $\varpi =1$. So, we get approx. results of set-up which shows the aftermath of time on cluster $\mathrm{\Delta }\left(t\right),\mathrm{\Omega }\left(t\right),\mathrm{\Theta }\left(t\right)$ and $\mathrm{\wp }\left(t\right)$. Further, if we consider comparative result of previous papers with the new model and shows that it is more reliable and useful. From the graphical results obtained by the models, one can easily see that the results obtained in this paper are far improved than the results obtained in the paper “The solution of modified fractional Bergman's minimal blood glucose insulin model” in the year 2017 and even if we talk to compare both the results obtained in the current paper then again, we can see that Yang operator is giving more precise results than Caputo–Fabrizio operator. In future work, we can study this model or its extension with help of various fractional operators (Figures 3–8).

Figure 1. Graph of sugar concentration (Δ) w.r.t t for different ϖ in CF case.

Figure 2. Graph of insulin concentration (Ω) w.r.t t for different ϖ in CF case.

Figure 3. Graph of aftermath of effective insulin (Θ) w.r.t t for different ϖ in CF case.

Figure 4. Graph of infusion of exogenous glucose (℘) w.r.t t for different ϖ in CF case.

Figure 5. Graph of sugar concentration (Δ) w.r.t t for different ϖ in YAC case.

Figure 6. Graph of insulin concentration (Ω) w.r.t t for different ϖ in YAC case.

Figure 7. Graph of aftermath of effective insulin (Θ) w.r.t t for different ϖ in YAC case.

Figure 8. Graph of infusion of exogenous glucose (℘) w.r.t t for different ϖ in YAC case.

Author's contribution

The study was led by Manvendra Narayan Mishra, who also carried out all the mathematical computations. Mounirah Areshi drew the figures/graphs. Pranay Goswami created the study path and arranged the related literature. The draught was read, corrected and polished by both the authors.

Availability of statistics and materials

Availability of statistics is already cited in the article.

Disclosure statement

No potential conflict of interest was reported by the author(s).