# Solutions of Conformable Fractional-Order SIR Epidemic Model

Authors: Atimad Harir; Said Malliani; Lalla Saadia Chandli
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

In this paper, the conformable fractional-order SIR epidemic model are solved by means of an analytic technique for nonlinear problems, namely, the conformable fractional differential transformation method (CFDTM) and variational iteration method (VIM). These models are nonlinear system of conformable fractional differential equation (CFDE) that has no analytic solution. The VIM is based on conformable fractional derivative and proved. The result revealed that both methods are in agreement and are accurate and efficient for solving systems of OFDE.


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## Body

## 1. Introduction

The problem of spreading of a (non) fatal disease in a population which is assumed to have a constant size over the period of the epidemic is considered in COVID-19. The goal of the epidemic model is to understand and if possible control the spread of disease [1]. At time t, suppose the population consists of(i)
st: the number of susceptible, who do not have the disease but could get it.(ii)
it: the number of infectives, who have the disease and can transmit it to others.(iii)
rt: the number of removed, who cannot get the disease or transmit it; either they have a natural immunity or they have recovered from the disease and are immune from getting it again or they have been placed in isolation or they have died. The mathematical model does not distinguish between these possibilities.Assume there is a steady constant rate between susceptible and infectives and that a constant proportion of this constant results in transmission. Then, in timeδtandδs, the susceptible becomes infective:(1)δs=−βsiδt,where β is a positive constant. If γ>0 is the rate at which current infectives become isolated, then(2)δi=βsiδt−γiδt,and the number of new isolated δr is given by(3)δr=γiδt.If we letδt⟶0, then the following nonlinear system of ODEs determines the progress of the disease:(4)ddtst=−βstit,ddtit=βstit−γit=0,ddtrt=γit,with initial conditions(5)s0=Ns,i0=Ni,r0=Nr.The following simple SIR model [2–5] is transformed to conformable fractional differential equation and is tested to show the efficiency of the variational iteration method [6] and differential transformation method [7–11] to solve such models.Fractional differentiation and integration operators have different kinds of definitions which we can mention, the Riemann–Liouville definition [12, 13], the Caputo definition [14], and so on. Lately, Khalil et al. [15] introduced a new simple definition of the fractional derivative named the conformable fractional derivative (CFD), which can redress shortcomings of the other definitions.The main advantages of the CFD can be summarized as follows [16–19]:(1)
It satisfies all concepts and rules of an ordinary derivative such as: quotient, product, and chain rules, while the other fractional definitions fail to meet these rules(2)
It can be extended to solve exact and numerical fractional differential equations and systems easily and efficientlyThe reason for considering a fractional-order system instead of its integer order counterpart is that fractional-order differential equations are generalizations of integer order differential equations. Also, using fractional-order differential equations can help us to reduce the errors arising from the neglected parameters in modelling real-life phenomena.We like to argue that fractional-order equations are more suitable than integer order ones in modelling biological, economical, and social systems (generally complex adaptive systems) where memory effects are important.The main objective of our work is to introduce the conformable fractional-order approach for the study of a particular SIR model in a constant population. In this case, the conformable fractional-order system of the SIR model will be transformed to one conformable fractional equation and are solved using the variational iteration method and the conformable differential transformation method for numerical comparison.


## 2. Conformable Fractional Derivative and Some Properties

In this part, we review some definitions and some results of conformable fractional derivative. For more details, the reader can refer to [15, 18, 20–25]. The conformable fractional derivative of order α is defined as(6)Tαft=limε⟶0ft+εt1−α−ftε,for all t>0,α∈0,1, and the fractional derivative at 0 is defined as(7)Tαf0=limt⟶0+Tαft.Letα∈0,1 and f be α-differentiable at a point t>0, and if f is differentiable, then(8)Tαft=t1−αddtft.The fractional integral of orderα is defined by(9)Iαft=Itα−1f=∫0tsα−1fsds,for allα∈0,1,where the integral is the usual Riemann improper integral.


## 3. Mathematical Modeling of the Conformable Fractional SIR Model

The conformable fractional model of actual evolution of this epidemic in a population of large sizeN is given by the following conformable fractional differential system:(10)Tαst=−βstit,Tαit=βstit−γit,Tαrt=γit,with initial conditions(11)s0=Ns,i0=Ni,r0=Nr.


### 3.1. The Conformable Fractional Differential Transformation Method

In [9, 11], assume f is infinitely α-differentiable function, for some 0<α≤1. ft can be expanded in fractional power series expansion about a point t=0 as(12)ft=∑k=0∞tαkαkk!Tαfkt=0,0<t<R1/α,R>0.Here,Tαfkt=0 denotes the application of the fractional derivative for k times. Conformable fractional differential transform of ft is defined as(13)Fαk=1αkk!Tαfkt=0.Then, the inverse conformable fractional differential transform ofFk is defined as(14)ft=∑k=0∞tαkFαk=∑k=0∞tαkαkk!Tαfkt=0.The fundamental mathematical operations performed by conformable fractional differential transform method are listed.Proposition 1.
(i)
Ifft=ut±vt, then Fαk=Uαk±Vαk.(ii)
Ifft=cut,c∈ℝ, then Fαk=cUαk.(iii)
Ifft=utvt, then Fαk=∑l=0kUαlVαk−l.(iv)
Ifft=Tαut, then Fαk=αk+1Uαk+1.(v)
Ifft=t−t0p, then Fαk=δk−p/α, where(15)δk=1,ifk=0,0,ifk≠0.


#### 3.1.1. Application of Conformable Fractional Differential Transform Method

Equation (10) can be rewritten as follows:(16)αk+1Sαt=−β∑l=0kSαlIαk−l,αk+1Iαt=β∑l=0kSαlIαk−l−γIαk,αk+1Rαt=γIαk.Hence, recurrence relation is obtained as(17)Sαt=1αk+1−β∑l=0kSαlIαk−l,Iαt=1αk+1β∑l=0kSαlIαk−l−γIαk,Rαt=1αk+1γIαk.With initial conditionsNs=2000,Ni=300,andNr=200 and parameters N=2500, β=0.00012,andγ=0.1, apply the condition in (17); then, the closed form of the solution where k=4 can be written as(18)st=∑l=0kSktkα=2000−72tαα−3.744t2αα2+0.034368t3αα3+0.013309t4αα4,it=∑l=0kIktkα=300+42tαα+1.644t2αα2−0.089168t3αα3−0.011080t4αα4,rt=∑l=0kRktkα=200+30tαα+2.1t2αα2+0.0548t3αα3−0.002229t4αα4.


### 3.2. Variational Iteration Method

To illustrate the basic idea of the variational iteration method, we consider the following nonlinear differential equation in the operator form:(19)Lut+Nut=gt,where L is a linear operator, N is a nonlinear operator, and g is any real function which is called the inhomogeneous term. Then, the corresponding correction functional for equation (19) is given by(20)un+1t=unt+∫0tλLuns+Nu˜ns−gsds,where λ is the general Lagrange multiplier [26], which can be identified optimally via the variational theory [27] and Nu˜n is considered as restricted variation, i.e., δNu˜n=0. Consider the stationary condition of the above correction functional.Theorem 1.
Consider the conformable fractional differential equation (10). Then, the variational iteration formula is given by(21)sn+1t=snt−IαTαsnt+βsntint,in+1t=int−IαTαint−βsntint+γint,rn+1t=rnt−IαTαrnt−γint,where sn,in,andrn are the nth approximation, Tα is the conformable fractional derivative of order α, and Iα is the fractional integral of order α∈0,1.Proof.
Equation (10) may be rewritten equivalently as(22)Tαsnt+βsntint=0,Tαint−βsntint+γint=0,Tαrnt−γint=0.
Multiplying equation (22) by a general Lagrange multiplier λ1t,λ2t, and λ3t yields(23)λ1tTαsnt+βsntint=0,λ2tTαint−βsntint+γint=0,λ3tTαrnt−γint=0.
Now, upon applyingIα to the both sides of equation (23) will give(24)Iαλ1tTαsnt+βsntint=0,Iαλ2tTαint−βsntint+γint=0,Iαλ3tTαrnt−γint=0.
Then, the correction functional of equation (10) will be read as follows:(25)sn+1t=snt+Iαλ1tTαsnt+βs˜nti˜nt,in+1t=int+Iαλ2tTαint−βs˜nti˜nt+γi˜nt,rn+1t=rnt+Iαλ3tTαrnt−γi˜nt.
In this case, the value ofλjt,forj=1,2,3, cannot be evaluated easily from equation (25), which will be a functional with fractional integral. Equation (25) can be expressed as follows:(26)sn+1t=snt+∫0tτα−1λ1ττ1−αddτsnτ+βs˜nτi˜nτdτ,in+1t=int+∫0tτα−1λ2ττ1−αddτinτ−βs˜nτi˜nτ+γi˜nτdτ,rn+1t=rnt+∫0tτα−1λ3ττ1−αddτrnτ−γi˜nτdτ,where s˜n and i˜n are the restricted variations with δs˜n=0 and δi˜n=0. From equation (26), we obtain(27)δsn+1t=δsnt+δ∫0tλ1τddτsnτ+βτα−1s˜nτi˜nτdτ,(28)δin+1t=δint+δ∫0tλ2τddτinτ−βτα−1s˜nτi˜nτ+γτα−1i˜nτdτ,(29)δrn+1t=δrnt+δ∫0tλ3τddτrnτ−γτα−1i˜nτdτ.
Using integration by parts, equation (27) becomes (for equations (28) and (29) are similar to the proof of equation (27))(30)δsn+1t=δsnt+δλtsnt−∫0tλ1′τsnτdτ=1+λ1tδsnt−δ∫0tλ1′τsnτdτ,δin+1t=1+λ2tδint−δ∫0tλ2′τinτdτ,δrn+1t=1+λ3tδrnt−δ∫0tλ3′τrnτdτ.
The Lagrange multipliersλ1t,λ2t, and λ3t can be obtained by λj′τ=0 for all j=1,2,3 with boundary condition: 1+λjt=0, for all j=1,2,3. Solving the last initial value problem for λj for all j=1,2,3, the general Lagrange multiplier λj is found to be(31)λj=−1,for allj=1,2,3.
Hence, substituting the value ofλj into the corresponding correction functional (25) will give the following iteration formula:(32)sn+1t=snt−IαTαsnt+βs˜nti˜nt,in+1t=int−IαTαint−βs˜nti˜nt+γi˜nt,rn+1t=rnt−IαTαrnt−γi˜nt.


#### 3.2.1. Application of the Variational Iteration Method

Applying the variational iteration method using Theorem1,(33)sn+1t=snt−IαTαsnt+βs˜nti˜nt,in+1t=int−IαTαint−βs˜nti˜nt+γi˜nt,rn+1t=rnt−IαTαrnt−γi˜nt.With initial conditionsNs=2000,Ni=300,andNr=200 and parameter N=2500, β=0.00012,andγ=0.1, we obtain(34)s1t=2000−72tαα,i1t=300+42tαα,r1t=200+30tαα,s2t=2000−72tαα−3.744t2αα2+0.12096t3αα3,i2t=300+42tαα+1.644t2αα2−0.12096t3αα3,r2t=200+30tαα+2.1t2αα2,s3t=2000−72tαα−3.744t2αα2+0.034368t3αα3+0.014437t4αα4−0.000183t5αα5−0.000013t6αα6+2.51∗10−7t7αα7,i3t=300+42tαα+1.644t2αα2−0.089168t3αα3−0.014413t4αα4+0.000183t5αα5+0.000013t6αα6+2.51∗10−7t7αα7,r3t=200+30tαα+2.1t2αα2+0.0548t3αα3−0.003024t4αα4.It can be observed that the result of the epidemic system of equation (10) is in complete agreement with the result obtained by the conformable fractional differential transformation method. Figures 1, 2 and 3.Figure 1
Comparison of VIM solutions with the CDTM solutions ofSt for system (10) at different α.Figure 2
Comparison of VIM solutions with the CDTM solutions ofIt for system (10) at different α.Figure 3
Comparison of VIM solutions with the CDTM solutions ofRt for system (10) at different α.


## 3.1. The Conformable Fractional Differential Transformation Method

In [9, 11], assume f is infinitely α-differentiable function, for some 0<α≤1. ft can be expanded in fractional power series expansion about a point t=0 as(12)ft=∑k=0∞tαkαkk!Tαfkt=0,0<t<R1/α,R>0.Here,Tαfkt=0 denotes the application of the fractional derivative for k times. Conformable fractional differential transform of ft is defined as(13)Fαk=1αkk!Tαfkt=0.Then, the inverse conformable fractional differential transform ofFk is defined as(14)ft=∑k=0∞tαkFαk=∑k=0∞tαkαkk!Tαfkt=0.The fundamental mathematical operations performed by conformable fractional differential transform method are listed.Proposition 1.
(i)
Ifft=ut±vt, then Fαk=Uαk±Vαk.(ii)
Ifft=cut,c∈ℝ, then Fαk=cUαk.(iii)
Ifft=utvt, then Fαk=∑l=0kUαlVαk−l.(iv)
Ifft=Tαut, then Fαk=αk+1Uαk+1.(v)
Ifft=t−t0p, then Fαk=δk−p/α, where(15)δk=1,ifk=0,0,ifk≠0.


### 3.1.1. Application of Conformable Fractional Differential Transform Method

Equation (10) can be rewritten as follows:(16)αk+1Sαt=−β∑l=0kSαlIαk−l,αk+1Iαt=β∑l=0kSαlIαk−l−γIαk,αk+1Rαt=γIαk.Hence, recurrence relation is obtained as(17)Sαt=1αk+1−β∑l=0kSαlIαk−l,Iαt=1αk+1β∑l=0kSαlIαk−l−γIαk,Rαt=1αk+1γIαk.With initial conditionsNs=2000,Ni=300,andNr=200 and parameters N=2500, β=0.00012,andγ=0.1, apply the condition in (17); then, the closed form of the solution where k=4 can be written as(18)st=∑l=0kSktkα=2000−72tαα−3.744t2αα2+0.034368t3αα3+0.013309t4αα4,it=∑l=0kIktkα=300+42tαα+1.644t2αα2−0.089168t3αα3−0.011080t4αα4,rt=∑l=0kRktkα=200+30tαα+2.1t2αα2+0.0548t3αα3−0.002229t4αα4.


## 3.1.1. Application of Conformable Fractional Differential Transform Method

Equation (10) can be rewritten as follows:(16)αk+1Sαt=−β∑l=0kSαlIαk−l,αk+1Iαt=β∑l=0kSαlIαk−l−γIαk,αk+1Rαt=γIαk.Hence, recurrence relation is obtained as(17)Sαt=1αk+1−β∑l=0kSαlIαk−l,Iαt=1αk+1β∑l=0kSαlIαk−l−γIαk,Rαt=1αk+1γIαk.With initial conditionsNs=2000,Ni=300,andNr=200 and parameters N=2500, β=0.00012,andγ=0.1, apply the condition in (17); then, the closed form of the solution where k=4 can be written as(18)st=∑l=0kSktkα=2000−72tαα−3.744t2αα2+0.034368t3αα3+0.013309t4αα4,it=∑l=0kIktkα=300+42tαα+1.644t2αα2−0.089168t3αα3−0.011080t4αα4,rt=∑l=0kRktkα=200+30tαα+2.1t2αα2+0.0548t3αα3−0.002229t4αα4.


## 3.2. Variational Iteration Method

To illustrate the basic idea of the variational iteration method, we consider the following nonlinear differential equation in the operator form:(19)Lut+Nut=gt,where L is a linear operator, N is a nonlinear operator, and g is any real function which is called the inhomogeneous term. Then, the corresponding correction functional for equation (19) is given by(20)un+1t=unt+∫0tλLuns+Nu˜ns−gsds,where λ is the general Lagrange multiplier [26], which can be identified optimally via the variational theory [27] and Nu˜n is considered as restricted variation, i.e., δNu˜n=0. Consider the stationary condition of the above correction functional.Theorem 1.
Consider the conformable fractional differential equation (10). Then, the variational iteration formula is given by(21)sn+1t=snt−IαTαsnt+βsntint,in+1t=int−IαTαint−βsntint+γint,rn+1t=rnt−IαTαrnt−γint,where sn,in,andrn are the nth approximation, Tα is the conformable fractional derivative of order α, and Iα is the fractional integral of order α∈0,1.Proof.
Equation (10) may be rewritten equivalently as(22)Tαsnt+βsntint=0,Tαint−βsntint+γint=0,Tαrnt−γint=0.
Multiplying equation (22) by a general Lagrange multiplier λ1t,λ2t, and λ3t yields(23)λ1tTαsnt+βsntint=0,λ2tTαint−βsntint+γint=0,λ3tTαrnt−γint=0.
Now, upon applyingIα to the both sides of equation (23) will give(24)Iαλ1tTαsnt+βsntint=0,Iαλ2tTαint−βsntint+γint=0,Iαλ3tTαrnt−γint=0.
Then, the correction functional of equation (10) will be read as follows:(25)sn+1t=snt+Iαλ1tTαsnt+βs˜nti˜nt,in+1t=int+Iαλ2tTαint−βs˜nti˜nt+γi˜nt,rn+1t=rnt+Iαλ3tTαrnt−γi˜nt.
In this case, the value ofλjt,forj=1,2,3, cannot be evaluated easily from equation (25), which will be a functional with fractional integral. Equation (25) can be expressed as follows:(26)sn+1t=snt+∫0tτα−1λ1ττ1−αddτsnτ+βs˜nτi˜nτdτ,in+1t=int+∫0tτα−1λ2ττ1−αddτinτ−βs˜nτi˜nτ+γi˜nτdτ,rn+1t=rnt+∫0tτα−1λ3ττ1−αddτrnτ−γi˜nτdτ,where s˜n and i˜n are the restricted variations with δs˜n=0 and δi˜n=0. From equation (26), we obtain(27)δsn+1t=δsnt+δ∫0tλ1τddτsnτ+βτα−1s˜nτi˜nτdτ,(28)δin+1t=δint+δ∫0tλ2τddτinτ−βτα−1s˜nτi˜nτ+γτα−1i˜nτdτ,(29)δrn+1t=δrnt+δ∫0tλ3τddτrnτ−γτα−1i˜nτdτ.
Using integration by parts, equation (27) becomes (for equations (28) and (29) are similar to the proof of equation (27))(30)δsn+1t=δsnt+δλtsnt−∫0tλ1′τsnτdτ=1+λ1tδsnt−δ∫0tλ1′τsnτdτ,δin+1t=1+λ2tδint−δ∫0tλ2′τinτdτ,δrn+1t=1+λ3tδrnt−δ∫0tλ3′τrnτdτ.
The Lagrange multipliersλ1t,λ2t, and λ3t can be obtained by λj′τ=0 for all j=1,2,3 with boundary condition: 1+λjt=0, for all j=1,2,3. Solving the last initial value problem for λj for all j=1,2,3, the general Lagrange multiplier λj is found to be(31)λj=−1,for allj=1,2,3.
Hence, substituting the value ofλj into the corresponding correction functional (25) will give the following iteration formula:(32)sn+1t=snt−IαTαsnt+βs˜nti˜nt,in+1t=int−IαTαint−βs˜nti˜nt+γi˜nt,rn+1t=rnt−IαTαrnt−γi˜nt.


### 3.2.1. Application of the Variational Iteration Method

Applying the variational iteration method using Theorem1,(33)sn+1t=snt−IαTαsnt+βs˜nti˜nt,in+1t=int−IαTαint−βs˜nti˜nt+γi˜nt,rn+1t=rnt−IαTαrnt−γi˜nt.With initial conditionsNs=2000,Ni=300,andNr=200 and parameter N=2500, β=0.00012,andγ=0.1, we obtain(34)s1t=2000−72tαα,i1t=300+42tαα,r1t=200+30tαα,s2t=2000−72tαα−3.744t2αα2+0.12096t3αα3,i2t=300+42tαα+1.644t2αα2−0.12096t3αα3,r2t=200+30tαα+2.1t2αα2,s3t=2000−72tαα−3.744t2αα2+0.034368t3αα3+0.014437t4αα4−0.000183t5αα5−0.000013t6αα6+2.51∗10−7t7αα7,i3t=300+42tαα+1.644t2αα2−0.089168t3αα3−0.014413t4αα4+0.000183t5αα5+0.000013t6αα6+2.51∗10−7t7αα7,r3t=200+30tαα+2.1t2αα2+0.0548t3αα3−0.003024t4αα4.It can be observed that the result of the epidemic system of equation (10) is in complete agreement with the result obtained by the conformable fractional differential transformation method. Figures 1, 2 and 3.Figure 1
Comparison of VIM solutions with the CDTM solutions ofSt for system (10) at different α.Figure 2
Comparison of VIM solutions with the CDTM solutions ofIt for system (10) at different α.Figure 3
Comparison of VIM solutions with the CDTM solutions ofRt for system (10) at different α.


## 3.2.1. Application of the Variational Iteration Method

Applying the variational iteration method using Theorem1,(33)sn+1t=snt−IαTαsnt+βs˜nti˜nt,in+1t=int−IαTαint−βs˜nti˜nt+γi˜nt,rn+1t=rnt−IαTαrnt−γi˜nt.With initial conditionsNs=2000,Ni=300,andNr=200 and parameter N=2500, β=0.00012,andγ=0.1, we obtain(34)s1t=2000−72tαα,i1t=300+42tαα,r1t=200+30tαα,s2t=2000−72tαα−3.744t2αα2+0.12096t3αα3,i2t=300+42tαα+1.644t2αα2−0.12096t3αα3,r2t=200+30tαα+2.1t2αα2,s3t=2000−72tαα−3.744t2αα2+0.034368t3αα3+0.014437t4αα4−0.000183t5αα5−0.000013t6αα6+2.51∗10−7t7αα7,i3t=300+42tαα+1.644t2αα2−0.089168t3αα3−0.014413t4αα4+0.000183t5αα5+0.000013t6αα6+2.51∗10−7t7αα7,r3t=200+30tαα+2.1t2αα2+0.0548t3αα3−0.003024t4αα4.It can be observed that the result of the epidemic system of equation (10) is in complete agreement with the result obtained by the conformable fractional differential transformation method. Figures 1, 2 and 3.Figure 1
Comparison of VIM solutions with the CDTM solutions ofSt for system (10) at different α.Figure 2
Comparison of VIM solutions with the CDTM solutions ofIt for system (10) at different α.Figure 3
Comparison of VIM solutions with the CDTM solutions ofRt for system (10) at different α.


## 4. Conclusion

In this study, we have found out approximate solutions with two numerical methods for theSIR epidemic model. These methods are based on conformable derivative which is extremely popular in the last years. Firstly, by using the α-derivative, we have redefined the conformable differential transformation method (CDTM) and variational iteration method (VIM). Then, we have demonstrated the efficiencies and accuracies of the proposed methods by applying them to the SIR epidemic model. It is found that the approximate solution generated the VIM by our method which is in complete agreement with the corresponding approximate solution CDTM. Besides, in view of their usability, our methods are applicable to many epidemic models SEIR/SEIRS and SIS of fractional order.



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