Definition 1: Suppose h∈H1(a1,b1), b₁ > a₁, β∈[0,1] hence the Caputo-Fabrizio differential coefficient of fractional order:

(1) Dtβ(h(t))=M(β)(1-β)∫ ath′(x)e[-βt-x1-β]dx,

M(β) is function of normalization such as M(0) = M(1) = 1.

If, h∉H1(a1,b1), then the derivative is

(2) Dtβ(h(t))=N(σ)σ∫ ath′(x)e[-t-xσ]dx,N(0)=N(∞)=1,

also,

(3) limσ→01σe[-t-x1-β]=δ(x-t).

One thing is to be noted here that the Caputo-Fabrizio operator has an exponent differential coefficient. We know that the exponential function has no singularity which, means the derivative exists at every point of the territory. So CF operator gives better results in its domain, see [49–56].

Definition 2: The fractional integral of the function h(t) of order β (0<β<1), is

(4) Iβt(h(t))=2(1-β)(2-β)M(β)h(t)+2β(2-β)M(β)∫ 0th(s)ds,t≥0.

Remark 1: From the above equation, we conclude that the fractional integral of order β (0<β<1), is the mean of h and its anti-derivative. So Nieto et al. gave this condition,

(5) 2(1-β)(2-β)M(β)+2β(2-β)M(β)=1.

M(β)=2(2-β),0<β<1.

Based on the above relation, Nieto and Losada defined the following operator;

(6) 0 CF D t β ( h(t) )= 1 ( 1−β ) ∫ a 1 t h ′ ( x ) e [ −β t−x 1−β ] dx.

Suppose f(t) be a function whose Caputo-Fabrizio derivative occurs, hence the Sumudu Transform of Caputo-Fabrizio fractional differential coefficient of f(t) is defined below:

(7) ST( 0 CF D t β )(f(t))=M(β)[ ST(f(t))−f(0) 1−β+βu ]

The whole paper is divided into six sections. Section first introduces fractional calculus and mathematical modeling with its brief history. The second section describes the Bergman model’s fractional exponent and an overview of the fractional modified minimal model. The third section discusses the existence and uniqueness of the modified mathematical model with the help of the Caputo-Fabrizio operator. Section 4 is devoted to the solution of the model by using the Sumudu Transform operator. Section 5 deals with numerical solutions along with graphical representations of the modified Bergman model. In section 6, we have concluded our findings.

Here, we are going to discuss the Bergman model of fractional order. In this model, we consider a glucose chamber and plasma insulin which is inherent to carry out across the isolated chamber to control the final glucose intake. This model, asserts that this is good to satisfy specific validation criteria with minimum parameters. This system is efficient in describing the gesture of interactivity of blood sugar and insulin. In the presented model, we suppose that G(t) is the imbalance of plasma glucose cluster and I(t) is the free plasma insulin cluster, from their initial values. The model was

(8) dαG(t)dtα=-(p1+X(t))G(t)+p1Gb,0<α<1

(9) dβX(t)dtβ=-p2X(t)+p3(I(t)-Ib),0<β<1

with starting conditions G(0) = G₀, X(0) = X₀, and I(0) = I₀.

Here, we redefined the structure of the previous model. We made some changes in the model and added few parameters. As we have, ‘Diet’ having significant effect on the glucose level. Now the changed structured Bergman modified system has been described in the following equations-

(11) 0 CF D t τ G(t)=−( p 1 +X(t))G(t)+D(t)−X(t) G b ,

(12) 0 CF D t τ X(t)=− p 2 X(t)+ p 3 I(t),

Here one thing is to be noted that 0<τ≤1 and initial restrictions are, G(0) = G₀, X(0) = X₀, I(0) = I₀ and D(0) = D₀.

This model can be a tool in search of an artificial pancreas. But, unfortunately, it also adopts the problems with the glucose-minimal model. Here, G(t)–-blood glucose cluster, X(t)–-aftermath of effective insulin, I(t)–-blood insulin cluster, D(t)–-infusion of exogenous glucose, u(t)–-insulin distribution function, G_b–-initial blood glucose cluster, I_b–-initial blood insulin cluster, V₁–-insulin issuance volume, n–-fractional fading rate of insulin, p₁–-insulin free glucose dispensation rate, p₂–-active insulin dispensation rate and p₃–-the increase in absorption capacity caused by insulin.

Define K₁, K₂, K₃, K₄ and their relations with variables.

Proof: Since the system is

(15) 0 CF D t τ G(t)=−( p 1 +X(t))G(t)+D(t)−X(t) G b ,

(16) 0 CF D t τ X(t)=− p 2 X(t)+ p 3 I(t),

Here one thing is to be noted that 0<τ≤1. Then converting the above system into a system of integral equations

(19) G(t)−G(0) = 0 CF I t τ [−( p 1 +X(t))G(t)+D(t)−X(t) G b ],

(20) X(t)−X(0) = 0 CF I t τ [− p 2 X(t)+ p 3 I(t)],

Then by definition stated by Nieto, we get

(23) G(t)=G(0)+2(1-τ)(2-τ)M(τ)[-(p1+X(t))G(t)+D(t)-X(t)Gb]+2τ(2-τ)M(τ)∫0t[-(p1+X(s))G(s)+D(s)-X(s)Gb]ds,

(24) X(t)=X(0)+2(1-τ)(2-τ)M(τ)[-p2X(t)+p3I(t)]+2τ(2-τ)M(τ)∫ 0t[-p2X(s)+p3I(s)]ds,

Now let us suppose the kernels are given as K1(t,G)=-(p1+X(t))G(t)+D(t)-X(t)Gb,K2(t,X)=-p2X(t)+p3I(t),K3(t,I)=-n[I(t)+Ib]+u(t)V1,K4(t,D)=-kD(t).

Show that K₁, K₂, K₃ and K₄ satisfies the Lipchitz condition.

Proof: At first, we shall show this for K₁. Suppose G and G₁ are any functions, so ∥K1(t,G)-K1(t,G1)∥=∥-(p1+X(t))G(t)+(p1+X(t))G1(t)∥=∥p1+X(t)∥∥G(t)-G1(t)∥∥K1(t,G)-K1(t,G1)∥≤H∥G(t)-G1(t)∥,where∥p1+X(t)∥≤H<1.

Similarly, we can have ∥K2(t,X)-K2(t,X1)∥≤H1∥X(t)-X1(t)∥,where∥p2∥≤H1<1.∥K3(t,I)-K3(t,I1)∥≤H2∥I(t)-I1(t)∥,where∥n∥≤H2<1. and ∥K4(t,D)-K4(t,D1)∥≤H3∥D(t)-D1(t)∥where∥k∥≤H3<1.

Now consider the recursive formula

(27) Gn(t)=2(1-τ)(2-τ)M(τ)K1(t,Gn-1)+2τ(2-τ)M(τ)∫ 0tK1(s,Gn-1)ds,

(28) Xn(t)=2(1-τ)(2-τ)M(τ)K2(t,Xn-1)+2τ(2-τ)M(τ)∫ 0tK2(s,Xn-1)ds,

Now suppose that the deviation amid two successive terms is

Un(t)=Gn(t)-Gn-1(t)

(31) =2(1-τ)(2-τ)M(τ)K1(t,Gn-1)+2τ(2-τ)M(τ)∫0tK1(s,Gn-1)ds-2(1-τ)(2-τ)M(τ)K1(t,Gn-2)-2τ(2-τ)M(τ)∫0tK1(s,Gn-2)ds,

Now

(33) ∥Un(t)∥=∥Gn(t)-Gn-1(t)∥=∥2(1-τ)(2-τ)M(τ)K1(t,Gn-1)-2(1-τ)(2-τ)M(τ)K1(t,Gn-2)+2τ(2-τ)M(τ)∫0t{K1(s,Gn-1)-K1(s,Gn-2)}ds∥,≤2(1-τ)(2-τ)M(τ)∥K1(t,Gn-1)-K1(t,Gn-2)∥+2τ(2-τ)M(τ)∥∫0t{K1(s,Gn-1)-K1(s,Gn-2)}ds∥.

But K₁ satisfies Lipchitz condition so,

(34) ∥Un(t)∥≤2(1-τ)(2-τ)M(τ)H∥Gn-1-Gn-2∥+2τ(2-τ)M(τ)K∫ 0t∥Gn-1-Gn-2∥ds.

Similarly, we have

(35) ‖ V n (t) ‖≤ 2(1−τ) (2−τ)M(τ) H 1 ‖ X n−1 − X n−2 ‖+ 2τ (2−τ)M(τ) J 1 ∫ 0 t ‖ X n−1 − X n−2 ‖ds ,

(36) ∥Wn(t)∥≤2(1-τ)(2-τ)M(τ)H2∥In-1-In-2∥+2τ(2-τ)M(τ)J2∫ 0t∥In-1-In-2∥ds,

(37) ∥Tn(t)∥≤2(1-τ)(2-τ)M(τ)H3∥Dn-1-Dn-2∥+2τ(2-τ)M(τ)J3∫ 0t∥Dn-1-Dn-2∥ds.

Establish that Bergman Minimal Model with Fractional-order is a minimum system of sugar insulin dynamics.

Proof: Using the recursive technique, we have

(38) ∥Un(t)∥≤∥G(0)∥+{2(1-τ)H(2-τ)M(τ)}n+{2τKt(2-τ)M(τ)}n

(39) ∥Vn(t)∥≤∥X(0)∥+{2(1-τ)H1(2-τ)M(τ)}n+{2τJ1t(2-τ)M(τ)}n

Hence the existence of results is verified, which is continuous too. Now we get G(t)=Gn(t)+Pn(t),X(t)=Xn(t)+Qn(t),I(t)=In(t)+Rn(t),D(t)=Dn(t)+Sn(t), where P_n, Q_n, R_n and S_n are residues of series solution.

So, G(t)-Gn(t)=2(1-τ)(2-τ)M(τ)K1(t,Gn)+2τ(2-τ)M(τ)∫0tK1(s,Gn)ds,G(t)-Gn(t)=2(1-τ)(2-τ)M(τ)K1(t,G-Pn(t))+2τ(2-τ)M(τ)∫0tK1(s,G-Pn(s))ds.

Similarly, we have X(t)-Xn(t)=2(1-τ)(2-τ)M(τ)K2(t,X-Qn(t))+2τ(2-τ)M(τ)∫0tK2(s,X-Qn(s))ds,I(t)-In(t)=2(1-τ)(2-τ)M(τ)K3(t,I-Rn(t))+2τ(2-τ)M(τ)∫0tK3(s,I-Rn(s))ds, D(t)− D n (t)= 2(1−τ) (2−τ)M(τ) K 4 (t,D− S n (t))+ 2τ (2−τ)M(τ) ∫ 0 t K 4 (s,D− S n (s))ds .

From the above, it is clear that, G(t)-Gn(t)=2(1-τ)(2-τ)M(τ)K1(t,G-Pn(t))+2τ(2-τ)M(τ)∫0tK1(s,G-Pn(s))ds,G(t)-G(0)-2(1-τ)K1(t,G)(2-τ)M(τ)-2τ(2-τ)M(τ)∫0tK1(s,G)ds=Pn(t)+2(1-τ)K1(t,G-Pn(t))(2-τ)M(τ)+2τ(2-τ)M(τ)∫0tK1(s,G-Pn(s))ds.

Now,

(42) ∥G(t)-2(1-τ)K1(t,G)(2-τ)M(τ)-G(0)-2τ(2-τ)M(τ)∫0tK1(s,G)ds∥≤∥Pn(t)∥+{2(1-τ)H(2-τ)M(τ)+2τ(2-τ)M(τ)Kt}∥Pn(t)∥,

(43) ∥X(t)-2(1-τ)K2(t,X)(2-τ)M(τ)-X(0)-2τ(2-τ)M(τ)∫0tK2(s,X)ds∥≤∥Qn(t)∥+{2(1-τ)H1(2-τ)M(τ)+2τ(2-τ)M(τ)J1t}∥Qn(t)∥,

Now taking n→∞ we have

(46) G(t)=G(0)+2(1-τ)K1(t,G)(2-τ)M(τ)+2τ(2-τ)M(τ)∫ 0tK1(s,G)ds,

(47) X(t)=X(0)+2(1-τ)K2(t,X)(2-τ)M(τ)+2τ(2-τ)M(τ)∫ 0tK2(s,X)ds,

(48) I(t)=I(0)+2(1-τ)K3(t,I)(2-τ)M(τ)+2τ(2-τ)M(τ)∫ 0tK3(s,I)ds,

(49) D(t)=D(0)+2(1-τ)K4(t,D)(2-τ)M(τ)+2τ(2-τ)M(τ)∫ 0tK4(s,D)ds.

On behalf of the above equations, we can state that the solution of the system exists.

To show the uniqueness of results, we suppose that other sets of results exist for the system specified from Eqs. (46) to (49) such that

(50) G(t)-G1(t)=2(1-τ)M(τ)(2-τ)[K1(t,G)-K1(t,G1)]+2(τ)M(τ)(2-τ)∫0t[K1(s,G)-K1(s,G1)]ds,

taking norm both sides, we get

(51) ∥G-G1∥=2(1-τ)M(τ)(2-τ)[∥K1(t,G)-K1(t,G1)∥]+2(τ)M(τ)(2-τ)∫0t[∥K1(s,G)-K1(s,G1)∥]ds,

using Lipchitz condition, we obtain

(52) ∥G-G1∥<2(1-τ)M(τ)(2-τ)HD+(2(τ)M(τ)(2-τ)J1Dt)n,

which is valid for all n, so

(53) G=G1,

similarly

(54) X=X1,I=I1,andD=D1.

Hence, it claims uniqueness of the system.