# Existence and Uniqueness Solution of the Model of Enzyme Kinetics in the Sense of Caputo–Fabrizio Fractional Derivative

Authors: Geremew Kenassa Edessa
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

In this paper, a model of the rates of enzyme-catalyzed chemical reactions in the sense of Caputo–Fabrizio a fractional derivative was investigated. Its existence and uniqueness as a solution of the model was proved by setting different criteria. An iterative numerical scheme was provided to support the findings. In order to verify the applicability of the result, numerical simulations using the MATLAB software package that confirms the analytical result was lucidly shown.


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## Body

## 1. Introduction

Mathematical models involving the known ordinary differentiation play an indispensable role in different fields of disciplines such as physical sciences and biology, as well as in other fields of studies [1, 2]. It is mainly used to describe the real phenomena leading to the design of better prediction, prevention, management, and control techniques [3].Mathematical models with integer-order derivatives do not determine the high degree of accuracy needed to model infectious diseases. However, when the situation is unpredictable, due to uncertainties associated with ordinal derivatives and their associated integral operators, there is a deficiency [4].In order to overcome such restrictions, fractional order differential equation models seem more realistic than the integer order models [5]. As a result, fractional differential equations were introduced to handle such problems, which have many applications in applied fields like production problems, optimization problems, artificial intelligence, medical diagnoses, robotics, cosmology, and many more.In the last few decades, the fractional differential has been used in the mathematical modeling of biological phenomena [6–11]. This is because fractional calculus can explain and process the maintenance and legacy properties of various materials more accurately than integer-order models. Researchers, therefore, expanded the classical calculus to the fractional-order via fractional-order modeling using different mathematical techniques [12–14]. Some authors have considered mathematical models of biological systems under fractional order derivatives and produced very good results [8, 9]. In spite of all the above investigations, prediction, and control via a mathematical model, there is still room for improvement.Accordingly, the subject of fractional calculus has gained popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields of science. For instance, fractional calculus has been successfully applied to system biology, physics, chemistry, biochemistry, hydrology, medicine, and finance [12, 14–20]. Several fractional differential operators like Riemann–Liouville, Hilfer, and Caputo, are mostly used in the modeling of physical problems. However, these fractional derivatives possess a power law kernel, have their own limitations, and reduce the field of application of fractional derivative.To deal with this type of difficulty, Caputo and Fabrizio have developed an alternate fractional differential operator having a nonsingular kernel with exponential decay [11, 17, 21]. The Caputo–Fabrizio operator has attracted many research scholars due to the fact that it has a nonsingular kernel and to be found most appropriate for modeling some class of real-world problem. Some researchers [21–23] used a mathematical model in the sense of fractional order derivative and received remarkable results. For example, Ali et al. modeled fractional order-based mathematical model for the Ebola Virus spreading in certain parts of countries [24, 25]. They provided a numerical solution for the generalized model by using the Atangana and Owolabi numerical methods. They showed that model has nonnegative solutions, as is preferred in any population dynamics. Clouthier and Pelletier also investigated a fractional order derivative model and obtained numerical simulation for immunogenic tumors [26]. They studied the model based on a fractional derivative growing tumor cell population and also observed that growth rate in the death of immune cells has a significant role in the tumor dynamical system.Instantly, it is evident that dealing with a dynamical system with memory effects is one of the biggest challenges for researchers. Fractional calculus has a direct link to dynamical systems (with the memory effect). Therefore, fractional differential equations (FDEs) present a novel technique developed to model phenomena related to the dynamics of the aforesaid fields of science [27].From the biological phenomenon (system), the importance of bio-catalytic processes and reactions for organic synthesis and the pharmaceutical food and cosmetics industry has been constantly growing during the last few years [10, 28]. Enzymes of one type, but from different origins, are specialized for substrates, positions in substrates, and products [11].Enzyme reactions do not follow the law of mass action directly. The rate of the reaction only increases to a certain extent as the concentration of substrate increases. The maximum reaction rate is reached at high substrate concentration due to enzyme saturation. This is in contrast to the law of mass action that states that the reaction rate increases as the concentration of substrate increases [11]. Various simplified analytical models have been developed over the last 20 years. In brief, the analysis involves the construction and solution of reaction or diffusion differential equations, resulting in the development of approximate analytical expressions for nonlinear enzyme catalyzed reaction processes.The simplest model that explains the kinetic behavior of enzyme reactions is the classic 1913 model of Michaelis and Menten which is widely used in biochemistry for many types of enzymes. The Michaelis–Menten model is based on the assumption that the enzyme binds the substrate to form an intermediate complex which then dissociates to form the final product and release the enzyme in its original form [28].Despite all the above studies, as per the author’s knowledge, there is still a paucity of knowledge with regard to the mathematical model of enzyme kinetics in the sense of fractional derivatives. Consequently, the main objective of this paper is to develop a new mathematical model for enzyme kinetics in the sense of a Caputo–Fabrizio fractional derivative based on a compartmental approach to investigate some rigorous mathematical analysis and numerical simulations.The rest of the paper is organized as follow: In Section2, assumptions and model formulation was stated. In Section 3 some preliminary results and iterative numerical schemes were lucidly investigated. In Section 4 numerical simulations was illustrated. In Section 5 discussions were given. In the last section concluding remark was forwarded.


## 2. Assumptions and Mathematical Model Formulation

### 2.1. Assumptions

Based on the biological phenomenon, the following biological assumptions have been considered in developing the model that represents the enzyme reaction. These assumptions are stated as follows. The substrate of the metabolic is represented byS, the enzyme for the reaction to be proceed by E, the enzyme substrate complex is denoted by C, whereas the enzyme product complex and product would be represented by D and P, respectively.The aforementioned biocatalyst assumption was indicated in the following block diagram, as shown in Figure1. Based on these assumptions, the model would be stated using the system of differential calculus in the following section.Figure 1
Flow diagram of an enzyme reaction.From the above schematic diagram, we have the following system of nonlinear ordinary differential equation that governs us to express the system using the fractional derivative as follows.


### 2.2. Mathematical Model Formulation

Here is the compartmental model with the stated assumptions defined by the system of differential equations, and the description of each parameter and variable involved in the system is given in Table1.(1)dsdt=−α1SE+α2CdEdt=−α1SE+α2−α3C+α4−α5D+α6PEdCdt=−α1SE−α2+α3C+α4DdDdt=α3C−α4+α5D+α6PEdPdt=α5D−α6PE.Table 1
Description of parameters and variables of the model.
ParametersDescription of parametersα1The rate at which enzyme substrate associate to form enzyme substrate complexα2The rate at which enzyme substrate complex dissociate to enzyme substrateα3The rate at which enzyme substrate complex catalyzed to enzyme product complexα4The rate at which enzyme product complex catalyzed to enzyme substrate complexα5The rate at which enzyme product complex dissociate to enzyme productα6The rate at which enzyme product associate to enzyme product complexηOrder of Caputo–Fabrizio fractional derivativeVariablesDescription of variablesSSubstrateEEnzymeCEnzyme substrate complexDEnzyme product complexPProductSubjected to initial conditions;S0= S0 > 0, E0= E0 > 0, C(0) = C0 ≥ 0, D(0) = D0 ≥ 0, and P(0) = P0 ≥ 0.Now extending the mathematical model proposed in the following equation to Caputo–Fabrizio fractional derivative yields as follows:(2)CFDtηSt=−α1SE+α2C,CFDtηEt=−α1SE+α2−α3C+α4−α5D+α6PE.CFDtηCt=−α1SE−α2+α3C+α4D,CFDtηDt=α3C−α4+α5D+α6PE,CFDtηPt=α5D−α6PE.With initial conditionsS(0) = S0 > 0, E(0) = E0 > 0, C(0) = C0 ≥ 0, D(0) = D0 ≥ 0, and P(0) = P0 ≥ 0.


## 2.1. Assumptions

Based on the biological phenomenon, the following biological assumptions have been considered in developing the model that represents the enzyme reaction. These assumptions are stated as follows. The substrate of the metabolic is represented byS, the enzyme for the reaction to be proceed by E, the enzyme substrate complex is denoted by C, whereas the enzyme product complex and product would be represented by D and P, respectively.The aforementioned biocatalyst assumption was indicated in the following block diagram, as shown in Figure1. Based on these assumptions, the model would be stated using the system of differential calculus in the following section.Figure 1
Flow diagram of an enzyme reaction.From the above schematic diagram, we have the following system of nonlinear ordinary differential equation that governs us to express the system using the fractional derivative as follows.


## 2.2. Mathematical Model Formulation

Here is the compartmental model with the stated assumptions defined by the system of differential equations, and the description of each parameter and variable involved in the system is given in Table1.(1)dsdt=−α1SE+α2CdEdt=−α1SE+α2−α3C+α4−α5D+α6PEdCdt=−α1SE−α2+α3C+α4DdDdt=α3C−α4+α5D+α6PEdPdt=α5D−α6PE.Table 1
Description of parameters and variables of the model.
ParametersDescription of parametersα1The rate at which enzyme substrate associate to form enzyme substrate complexα2The rate at which enzyme substrate complex dissociate to enzyme substrateα3The rate at which enzyme substrate complex catalyzed to enzyme product complexα4The rate at which enzyme product complex catalyzed to enzyme substrate complexα5The rate at which enzyme product complex dissociate to enzyme productα6The rate at which enzyme product associate to enzyme product complexηOrder of Caputo–Fabrizio fractional derivativeVariablesDescription of variablesSSubstrateEEnzymeCEnzyme substrate complexDEnzyme product complexPProductSubjected to initial conditions;S0= S0 > 0, E0= E0 > 0, C(0) = C0 ≥ 0, D(0) = D0 ≥ 0, and P(0) = P0 ≥ 0.Now extending the mathematical model proposed in the following equation to Caputo–Fabrizio fractional derivative yields as follows:(2)CFDtηSt=−α1SE+α2C,CFDtηEt=−α1SE+α2−α3C+α4−α5D+α6PE.CFDtηCt=−α1SE−α2+α3C+α4D,CFDtηDt=α3C−α4+α5D+α6PE,CFDtηPt=α5D−α6PE.With initial conditionsS(0) = S0 > 0, E(0) = E0 > 0, C(0) = C0 ≥ 0, D(0) = D0 ≥ 0, and P(0) = P0 ≥ 0.


## 3. Main Result and Discussion

### 3.1. Existence and Uniqueness of Solutions

In this section the main result of the paper was investigated applying Caputo–Fabrizio fractional integral to both sides of equation (2) gives the following equations:(3)St−S0=CFItη−α1SE+α2C,Et−I0=CFItη−α1SE+α2−α3C+α4−α5D+α6PE,Et−I0=CFItη−α1SE+α2−α3C+α4−α5D+α6PE,Ct−I0=CFItη−α1SE−α2+α3C+α4D,Dt−I0=CFItηα3C−α4+α5D+α6PE,Pt−I0=CFItηα5D−α6PE.Using the definition of Caputo–Fabrizio fractional integral, we obtain(4)St−S0=21−η2−ηMη−α1SE+α2C+2η2−ηMη∫0t−α1SxEx+α2Cxdx,Et−E0=21−η2−ηMη−α1SE+α2−α3C+α4−α5D+α6PE+2η2−ηMη∫0t−α1SxEx+α2−α3Cx+α4−α5Dx+α6PxEx−α1SxEx+α2−α3Cx+α4−α5Dx+α6PxExdx,Ct−I0=21−η2−ηMη−α1SE−α2+α3C+α4D+2η2−ηMη∫0t−α1SxEx−α2+α3Cx+α4Dxdx,Dt−I0=21−η2−ηMηα3C−α4+α5D+α6PE+2η2−ηMη∫0tα3Cx−α4+α5Dx+α6PxExdx,Pt−S0=21−η2−ηMηα5D−α6PE+2η2−ηMη∫0tα5Dx−α6PxExdx.For the sake of convenience, we consider the following equations:(5)k1t,S−α1SE+α2C,k2t,E=−α1SE+α2−α3C+α4−α5D+α6PE,k3t,C=−α1SE−α2+α3C+α4D,k4t,D=α3C−α4+α5D+α6PE,k5t,P=α5D−α6PE.Theorem 1.
The kernelk1t,S satisfies the Lipschitz condition and a contraction if the following inequality holds: 0<α1l1≤1.Proof.
Consider functions andSt and S1(t);(6)k1t,St−k1t,S1t=−α1SxEx+α2Cx−−α1S1xEx+α2Cx=−α1ExSx−S1x=−α1ExSx−S1x=α1ExSx−S1x≤α1l1Sx−S1x=λ1Sx−S1x,where λ1=α1l1 and l1=Ex and this is abounded function, then we have(7)k1t,St−k1t,S1t≤λ1Sx−S1x.
Thus, the Lipschitz condition is fulfilled. In addition, if0<α1l1≤1, then k1 is a contraction.
Similarlyk2,k3,k4,k5 satisfies Lipschitz condition as stated(8)k2t,Et−k2t,E1t≤λ2Ex−E1x,k3t,Ct−k3t,C1t≤λ3Cx−C1x,k4t,Dt−k4t,D1t≤λ4Dx−D1x,k5t,Pt−k5t,P1t≤λ5Px−P1x.
Using the following equation into equation (10)(9)St=S0+21−η2−ηMηk1t,S+2η2−ηMη∫0tk1x,SxdxEt=E0+21−η2−ηMη+k2t,E2η2−ηMη∫0tk2x,ExdxCt=C0+21−η2−ηMηk3t,C+2η2−ηMη∫0tk3x,CxdxDt=D0+21−η2−ηMηk4t,D+2η2−ηMη∫0tk4x,DxdxPt=P0+21−η2−ηMηk5t,P+2η2−ηMη∫0tk5x,Pxdx.
Thus, consider the following recursive formula:(10)Snt=21−η2−ηMηk1t,Sn−1+2η2−ηMη∫0tk1x,Sn−1dxEn−1t=21−η2−ηMη+k2t,En−1+2η2−ηMη∫0tk2x,En−1dxCn−1t=21−η2−ηMη+k3t,Cn−1+2η2−ηMη∫0tk3x,Cn−1dxDn−1t=21−η2−ηMη+k4t,Dn−1+2η2−ηMη∫0tk4x,Dn−1dxPn−1t=21−η2−ηMη+k5t,Pn−1+2η2−ηMη∫0tk5x,Pn−1dx.
Now consider the differences between successive terms as follows:(11)An=Sn−Sn−1=21−η2−ηMηk1t,Sn−1−k1t,Sn−2+2η2−ηMη∫0tk1x,Sn−1−k1x,Sn−2dxBn=En−En−1=21−η2−ηMη+k2t,En−1−k2t,En−2+2η2−ηMη∫0tk2x,En−1−k2x,En−2dxFn=Cn−Cn−1=21−η2−ηMη+k3t,Cn−1−k3t,Cn−2+2η2−ηMη∫0tk3x,Cn−1−k3x,Cn−2dxHn=Dn−Dn−1=21−η2−ηMη+k4t,Dn−1−k4t,Dn−2+2η2−ηMη∫0tk4x,Dn−1−k4x,Dn−2dxGn=Pn−Pn−1=21−η2−ηMη+k5t,Pn−1−k5t,Pn−2+2η2−ηMη∫0tk5x,Pn−1−k5x,Pn−2dx.
Equivalently we can write as(12)Sn=∑j=1nAj,En=∑j=1nBj,Cn=∑j=1nFj,Dn=∑j=1nHj,Pn=∑j=1nGj.
Taking norm to the differences between successive recursive terms(13)An=Sn−Sn−1=21−η2−ηMηk1t,Sn−1−k1t,Sn−2+2η2−ηMη∫0tk1x,Sn−1−k1x,Sn−2dx≤21−η2−ηMηk1t,Sn−1−k1t,Sn−2+2η2−ηMη∫0tk1x,Sn−1−k1x,Sn−2dx.
Sincek1 satisfies Lipschitz condition,(14)Sn−Sn−1≤21−ηλ12−ηMηSn−1−Sn−2+2ηλ12−ηMη∫0tSn−1−Sn−2dx,An≤21−ηλ12−ηMηAn−1+2ηλ12−ηMη∫0tAn−1dx.
Similar results would be obtained as follows:(15)Bn≤21−ηλ22−ηMηBn−1+2ηλ22−ηMη∫0tBn−1dxFn≤21−ηλ32−ηMηFn−1+2ηλ32−ηMη∫0tFn−1dxHn≤21−ηλ42−ηMηHn−1+2ηλ42−ηMη∫0tHn−1dxGn≤21−ηλ52−ηMηGn−1+2ηλ52−ηMη∫0tGn−1dx.Theorem 2.
The Caputo–Fabrizio fractional derivative model in (16) has system of solutions if there exists t such that 21−ηλi/2−ηMη+2η/2−ηMηλit<1, for i=1,2,3,4,5.Proof.
By using recursive method and result from the following equation and (17)(16)An≤S021−η2−ηMηλ1+2η2−ηMηλ1tnBn≤E021−η2−ηMηλ2+2η2−ηMηλ2tnFn≤C021−η2−ηMηλ3+2η2−ηMηλ3tnHn≤D021−η2−ηMηλ4+2η2−ηMηλ4tnGn≤P021−η2−ηMηλ5+2η2−ηMηλ5tn.
This result proved the existence and smoothness of solution in (17).
To show thatS(t), E(t), C(t), D(t), and P(t) are solutions of (18), consider the following equations:(17)St−S0=Snt−R1nEt−E0=Ent−R2nCt−C0=Cnt−R3nDt−D0=Dnt−R4nPt−P0=Pnt−R5n,where Rjn, j = 1, 2, 3, 4, 5 defines the remainder term after nth iteration.(18)R1n=St−Sn=21−η2−ηMηk1t,S−k1t,Sn−1+2η2−ηMη∫0tk1x,S−k1x,Sn−1dx≤21−η2−ηMηk1t,S−k1t,Sn−1+2η2−ηMη∫0tk1x,S−k1x,Sn−1dx≤21−η2−ηMηλ1S−Sn−1+2ηλ12−ηMη∫0tS−Sn−1dxR1n≤21−η2−ηMηλ1+2η2−ηMηS−Sn−1λ1t.
Applying the above process recursively,(19)R1n≤21−η2−ηMηλ1+2η2−ηMηtn+1λ1n+1q,where q is positive constant.
Whenn⟶∞,R1n⟶0.
Similarly, whenn⟶∞,Rjn⟶0 for 21−ηλi/2−ηMη+2η/2−ηMηλit<1. Hence the proof is completed.Theorem 3.
The Caputo–Fabrizio fractional derivative model in the following equation has a system of unique solutions provided that(20)1−21−ηλi2−ηMη−2η2−ηMηλit≥0,i=1,2,3,4,5.Proof.
Suppose the following equation has another solutions say:S1t, E1t,C1t,D1t,P1t(21)St−S1t=21−η2−ηMηk1t,S−k1t,S1+2η2−ηMη∫0tk1x,S−k1x,S1dxSt−S1t=21−η2−ηMηk1t,S−k1t,S1+2η2−ηMη∫0tk1x,S−k1x,S1dx≤21−η2−ηMηk1t,S−k1t,S1+2η2−ηMη∫0tk1x,S−k1x,S1dx≤21−η2−ηMηλ1St−S1t+2η2−ηMηλ1tSt−S1tSt−S1t≤21−η2−ηMηλ1+2η2−ηMηλ1tSt−S1tSt−S1t1−21−η2−ηMηλ1−2η2−ηMηλ1t≤0.
From (22) and (23)(22)St−S1t1−21−η2−ηMηλ1−2η2−ηMηλ1t=0.
This implies that;St−S1t=0
EquivalentlySt=S1t
In the same manner,(23)Et−E1t=0⇒Et=E1t,Ct−C1t=0⇒Ct=C1t,Dt−D1t=0⇒Dt=D1t,Pt−P1t=0⇒Pt=P1t.
Hence, the proof completed.


### 3.2. Iterative Numerical Scheme

For the desired numerical solution of the proposed model in (24), we apply the technique of fractional Adams-Bash-forth for Caputo–Fabrizio fractional derivative. To develop the desired iterative numerical scheme, consider the first equation (25):(24)CFDtηSt=−α1SE+α2CCFDtηSt=K1t,St.Applying Caputo–Fabrizio fractional integral to both sides gives the following equation:(25)St−S0=CFItηk1t,StSt=S0+1−ηMηk1t,S+ηMη∫0tk1x,Sxdx.Att=tn+1(26)Stn+1=S0+1−ηMηk1tn,Sn+ηMη∫0tn+1k1t,Stdt.Att=tn(27)Stn=S0+1−ηMηk1tn−1,Sn−1+ηMη∫0tnk1t,Stdt.Subtracting the following equation from (29) leads to the following equation (30).(28)Stn+1=Stn+1−ηMηk1tn,Sn−k1tn−1,Sn−1+ηMη∫tntn+1k1t,Stdt.Takingh=tn+1−tn, approximating the integral ∫tntn+1K1t,Stdt with the help of Lagrange interpolation polynomial of degree two passing through three points tn−2,k1tn−2,Sn−2, tn−1,k1tn−1,Sn−1,tn,k1tn,Sn.(29)∫tntn+1k1t,Stdt=∫01s−2s−31−21−3k1tn,Sn+s−1s−32−12−3k1tn−1,Sn−1+s−2s−13−23−1k1tn−2,Sn−2ds∫tntn+1k1t,Stdt=h2312k1tn,Sn−43k1tn−1,Sn−1+512k1tn−2,Sn−2.Substituting this approximated value in (31),(30)Stn+1=Stn+1−ηMη+23ηh12Mηk1tn,Sn−1−ηMη+4ηh3Mηk1tn−1,Sn−1+5ηh12Mηk1tn−2,Sn−2.Similarly the others can be manipulated as;(31)Etn+1=Etn+1−ηMη+23ηh12Mηk2tn,En−1−ηMη+4ηh3Mηk2tn−1,En−1+5ηh12Mηk2tn−2,En−2Ctn+1=Ctn+1−ηMη+23ηh12Mηk3tn,Cn−1−ηMη+4ηh3Mηk3tn−1,Cn−1+5ηh12Mηk3tn−2,Cn−2Dtn+1=Dtn+1−ηMη+23ηh12Mηk4tn,Dn−1−ηMη+4ηh3Mηk4tn−1,Dn−1+5ηh12Mηk4tn−2,Dn−2Ptn+1=Ptn+1−ηMη+23ηh12Mηk5tn,Pn−1−ηMη+4ηh3Mηk5tn−1,Pn−1+5ηh12Mηk5tn−2,Pn−2.


## 3.1. Existence and Uniqueness of Solutions

In this section the main result of the paper was investigated applying Caputo–Fabrizio fractional integral to both sides of equation (2) gives the following equations:(3)St−S0=CFItη−α1SE+α2C,Et−I0=CFItη−α1SE+α2−α3C+α4−α5D+α6PE,Et−I0=CFItη−α1SE+α2−α3C+α4−α5D+α6PE,Ct−I0=CFItη−α1SE−α2+α3C+α4D,Dt−I0=CFItηα3C−α4+α5D+α6PE,Pt−I0=CFItηα5D−α6PE.Using the definition of Caputo–Fabrizio fractional integral, we obtain(4)St−S0=21−η2−ηMη−α1SE+α2C+2η2−ηMη∫0t−α1SxEx+α2Cxdx,Et−E0=21−η2−ηMη−α1SE+α2−α3C+α4−α5D+α6PE+2η2−ηMη∫0t−α1SxEx+α2−α3Cx+α4−α5Dx+α6PxEx−α1SxEx+α2−α3Cx+α4−α5Dx+α6PxExdx,Ct−I0=21−η2−ηMη−α1SE−α2+α3C+α4D+2η2−ηMη∫0t−α1SxEx−α2+α3Cx+α4Dxdx,Dt−I0=21−η2−ηMηα3C−α4+α5D+α6PE+2η2−ηMη∫0tα3Cx−α4+α5Dx+α6PxExdx,Pt−S0=21−η2−ηMηα5D−α6PE+2η2−ηMη∫0tα5Dx−α6PxExdx.For the sake of convenience, we consider the following equations:(5)k1t,S−α1SE+α2C,k2t,E=−α1SE+α2−α3C+α4−α5D+α6PE,k3t,C=−α1SE−α2+α3C+α4D,k4t,D=α3C−α4+α5D+α6PE,k5t,P=α5D−α6PE.Theorem 1.
The kernelk1t,S satisfies the Lipschitz condition and a contraction if the following inequality holds: 0<α1l1≤1.Proof.
Consider functions andSt and S1(t);(6)k1t,St−k1t,S1t=−α1SxEx+α2Cx−−α1S1xEx+α2Cx=−α1ExSx−S1x=−α1ExSx−S1x=α1ExSx−S1x≤α1l1Sx−S1x=λ1Sx−S1x,where λ1=α1l1 and l1=Ex and this is abounded function, then we have(7)k1t,St−k1t,S1t≤λ1Sx−S1x.
Thus, the Lipschitz condition is fulfilled. In addition, if0<α1l1≤1, then k1 is a contraction.
Similarlyk2,k3,k4,k5 satisfies Lipschitz condition as stated(8)k2t,Et−k2t,E1t≤λ2Ex−E1x,k3t,Ct−k3t,C1t≤λ3Cx−C1x,k4t,Dt−k4t,D1t≤λ4Dx−D1x,k5t,Pt−k5t,P1t≤λ5Px−P1x.
Using the following equation into equation (10)(9)St=S0+21−η2−ηMηk1t,S+2η2−ηMη∫0tk1x,SxdxEt=E0+21−η2−ηMη+k2t,E2η2−ηMη∫0tk2x,ExdxCt=C0+21−η2−ηMηk3t,C+2η2−ηMη∫0tk3x,CxdxDt=D0+21−η2−ηMηk4t,D+2η2−ηMη∫0tk4x,DxdxPt=P0+21−η2−ηMηk5t,P+2η2−ηMη∫0tk5x,Pxdx.
Thus, consider the following recursive formula:(10)Snt=21−η2−ηMηk1t,Sn−1+2η2−ηMη∫0tk1x,Sn−1dxEn−1t=21−η2−ηMη+k2t,En−1+2η2−ηMη∫0tk2x,En−1dxCn−1t=21−η2−ηMη+k3t,Cn−1+2η2−ηMη∫0tk3x,Cn−1dxDn−1t=21−η2−ηMη+k4t,Dn−1+2η2−ηMη∫0tk4x,Dn−1dxPn−1t=21−η2−ηMη+k5t,Pn−1+2η2−ηMη∫0tk5x,Pn−1dx.
Now consider the differences between successive terms as follows:(11)An=Sn−Sn−1=21−η2−ηMηk1t,Sn−1−k1t,Sn−2+2η2−ηMη∫0tk1x,Sn−1−k1x,Sn−2dxBn=En−En−1=21−η2−ηMη+k2t,En−1−k2t,En−2+2η2−ηMη∫0tk2x,En−1−k2x,En−2dxFn=Cn−Cn−1=21−η2−ηMη+k3t,Cn−1−k3t,Cn−2+2η2−ηMη∫0tk3x,Cn−1−k3x,Cn−2dxHn=Dn−Dn−1=21−η2−ηMη+k4t,Dn−1−k4t,Dn−2+2η2−ηMη∫0tk4x,Dn−1−k4x,Dn−2dxGn=Pn−Pn−1=21−η2−ηMη+k5t,Pn−1−k5t,Pn−2+2η2−ηMη∫0tk5x,Pn−1−k5x,Pn−2dx.
Equivalently we can write as(12)Sn=∑j=1nAj,En=∑j=1nBj,Cn=∑j=1nFj,Dn=∑j=1nHj,Pn=∑j=1nGj.
Taking norm to the differences between successive recursive terms(13)An=Sn−Sn−1=21−η2−ηMηk1t,Sn−1−k1t,Sn−2+2η2−ηMη∫0tk1x,Sn−1−k1x,Sn−2dx≤21−η2−ηMηk1t,Sn−1−k1t,Sn−2+2η2−ηMη∫0tk1x,Sn−1−k1x,Sn−2dx.
Sincek1 satisfies Lipschitz condition,(14)Sn−Sn−1≤21−ηλ12−ηMηSn−1−Sn−2+2ηλ12−ηMη∫0tSn−1−Sn−2dx,An≤21−ηλ12−ηMηAn−1+2ηλ12−ηMη∫0tAn−1dx.
Similar results would be obtained as follows:(15)Bn≤21−ηλ22−ηMηBn−1+2ηλ22−ηMη∫0tBn−1dxFn≤21−ηλ32−ηMηFn−1+2ηλ32−ηMη∫0tFn−1dxHn≤21−ηλ42−ηMηHn−1+2ηλ42−ηMη∫0tHn−1dxGn≤21−ηλ52−ηMηGn−1+2ηλ52−ηMη∫0tGn−1dx.Theorem 2.
The Caputo–Fabrizio fractional derivative model in (16) has system of solutions if there exists t such that 21−ηλi/2−ηMη+2η/2−ηMηλit<1, for i=1,2,3,4,5.Proof.
By using recursive method and result from the following equation and (17)(16)An≤S021−η2−ηMηλ1+2η2−ηMηλ1tnBn≤E021−η2−ηMηλ2+2η2−ηMηλ2tnFn≤C021−η2−ηMηλ3+2η2−ηMηλ3tnHn≤D021−η2−ηMηλ4+2η2−ηMηλ4tnGn≤P021−η2−ηMηλ5+2η2−ηMηλ5tn.
This result proved the existence and smoothness of solution in (17).
To show thatS(t), E(t), C(t), D(t), and P(t) are solutions of (18), consider the following equations:(17)St−S0=Snt−R1nEt−E0=Ent−R2nCt−C0=Cnt−R3nDt−D0=Dnt−R4nPt−P0=Pnt−R5n,where Rjn, j = 1, 2, 3, 4, 5 defines the remainder term after nth iteration.(18)R1n=St−Sn=21−η2−ηMηk1t,S−k1t,Sn−1+2η2−ηMη∫0tk1x,S−k1x,Sn−1dx≤21−η2−ηMηk1t,S−k1t,Sn−1+2η2−ηMη∫0tk1x,S−k1x,Sn−1dx≤21−η2−ηMηλ1S−Sn−1+2ηλ12−ηMη∫0tS−Sn−1dxR1n≤21−η2−ηMηλ1+2η2−ηMηS−Sn−1λ1t.
Applying the above process recursively,(19)R1n≤21−η2−ηMηλ1+2η2−ηMηtn+1λ1n+1q,where q is positive constant.
Whenn⟶∞,R1n⟶0.
Similarly, whenn⟶∞,Rjn⟶0 for 21−ηλi/2−ηMη+2η/2−ηMηλit<1. Hence the proof is completed.Theorem 3.
The Caputo–Fabrizio fractional derivative model in the following equation has a system of unique solutions provided that(20)1−21−ηλi2−ηMη−2η2−ηMηλit≥0,i=1,2,3,4,5.Proof.
Suppose the following equation has another solutions say:S1t, E1t,C1t,D1t,P1t(21)St−S1t=21−η2−ηMηk1t,S−k1t,S1+2η2−ηMη∫0tk1x,S−k1x,S1dxSt−S1t=21−η2−ηMηk1t,S−k1t,S1+2η2−ηMη∫0tk1x,S−k1x,S1dx≤21−η2−ηMηk1t,S−k1t,S1+2η2−ηMη∫0tk1x,S−k1x,S1dx≤21−η2−ηMηλ1St−S1t+2η2−ηMηλ1tSt−S1tSt−S1t≤21−η2−ηMηλ1+2η2−ηMηλ1tSt−S1tSt−S1t1−21−η2−ηMηλ1−2η2−ηMηλ1t≤0.
From (22) and (23)(22)St−S1t1−21−η2−ηMηλ1−2η2−ηMηλ1t=0.
This implies that;St−S1t=0
EquivalentlySt=S1t
In the same manner,(23)Et−E1t=0⇒Et=E1t,Ct−C1t=0⇒Ct=C1t,Dt−D1t=0⇒Dt=D1t,Pt−P1t=0⇒Pt=P1t.
Hence, the proof completed.


## 3.2. Iterative Numerical Scheme

For the desired numerical solution of the proposed model in (24), we apply the technique of fractional Adams-Bash-forth for Caputo–Fabrizio fractional derivative. To develop the desired iterative numerical scheme, consider the first equation (25):(24)CFDtηSt=−α1SE+α2CCFDtηSt=K1t,St.Applying Caputo–Fabrizio fractional integral to both sides gives the following equation:(25)St−S0=CFItηk1t,StSt=S0+1−ηMηk1t,S+ηMη∫0tk1x,Sxdx.Att=tn+1(26)Stn+1=S0+1−ηMηk1tn,Sn+ηMη∫0tn+1k1t,Stdt.Att=tn(27)Stn=S0+1−ηMηk1tn−1,Sn−1+ηMη∫0tnk1t,Stdt.Subtracting the following equation from (29) leads to the following equation (30).(28)Stn+1=Stn+1−ηMηk1tn,Sn−k1tn−1,Sn−1+ηMη∫tntn+1k1t,Stdt.Takingh=tn+1−tn, approximating the integral ∫tntn+1K1t,Stdt with the help of Lagrange interpolation polynomial of degree two passing through three points tn−2,k1tn−2,Sn−2, tn−1,k1tn−1,Sn−1,tn,k1tn,Sn.(29)∫tntn+1k1t,Stdt=∫01s−2s−31−21−3k1tn,Sn+s−1s−32−12−3k1tn−1,Sn−1+s−2s−13−23−1k1tn−2,Sn−2ds∫tntn+1k1t,Stdt=h2312k1tn,Sn−43k1tn−1,Sn−1+512k1tn−2,Sn−2.Substituting this approximated value in (31),(30)Stn+1=Stn+1−ηMη+23ηh12Mηk1tn,Sn−1−ηMη+4ηh3Mηk1tn−1,Sn−1+5ηh12Mηk1tn−2,Sn−2.Similarly the others can be manipulated as;(31)Etn+1=Etn+1−ηMη+23ηh12Mηk2tn,En−1−ηMη+4ηh3Mηk2tn−1,En−1+5ηh12Mηk2tn−2,En−2Ctn+1=Ctn+1−ηMη+23ηh12Mηk3tn,Cn−1−ηMη+4ηh3Mηk3tn−1,Cn−1+5ηh12Mηk3tn−2,Cn−2Dtn+1=Dtn+1−ηMη+23ηh12Mηk4tn,Dn−1−ηMη+4ηh3Mηk4tn−1,Dn−1+5ηh12Mηk4tn−2,Dn−2Ptn+1=Ptn+1−ηMη+23ηh12Mηk5tn,Pn−1−ηMη+4ηh3Mηk5tn−1,Pn−1+5ηh12Mηk5tn−2,Pn−2.


## 4. Numerical Simulation

Simulation was implemented by using the following parameters value subjected to initial conditions. Some of the parameters values were taken from different literature and others are assumed. In the following Table2 we assumed and provide the values of parameters.Table 2
Parameters and their values.
Parametersα1α2α3α4α5α6Values0.0650.0160.010.030.0550.00056Initial conditions;S0=0.8,E0=0.3,C0=0.3,D0=0.2,P0=0.1.


## 5. Result and Discussion

Figures2 and 3revealed that substrate concentration increases and decreases as the rate at which enzyme substrates associate to form enzyme substrate complexes increases and the rate at which enzyme substrate complexes dissociate to form enzyme substrate increases, respectively. Figures 4 and 5indicate that enzyme concentration decreased and increased as the rate at which enzyme substrates associate to form enzyme substrate complexes increased and the rate at which enzyme substrate complexes dissociated to enzyme substrates increased, respectively. Figures 6 and 7 indicate that product concentration increased and decreased as the rate at which enzyme substrates associate to form enzyme product complexes increased and the rate at which enzyme substrate complex dissociate to enzyme product decreases, respectively.Figure 2
Graph of substrate verses time for different values ofα1 by keeping others parameters constant.Figure 3
Graph of substrate verses time for different values ofα2 constant by keeping others parameters.Figure 4
Graph of enzyme concentration verses time for different values ofα1 others parameters constant.Figure 5
Graph of enzyme concentration verses time for different values ofα2 others parameters constant.Figure 6
Graph of product verses time for different values ofα5 constant.Figure 7
Graph of product verses time for different values ofα6 constant.


## 6. Conclusion

In this paper, a mathematical model of enzyme kinetics in the sense of Caputo–Fabrizio fractional derivative was developed and investigated. The existence and uniqueness of the solution to the model was lucidly shown. Finally, simulation was implemented to verify the applicability of the results.



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