Solutions of Conformable Fractional-Order SIR Epidemic Model

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). (ese models are nonlinear system of conformable fractional differential equation (CFDE) that has no analytic solution. (eVIM is based on conformable fractional derivative and proved.(e result revealed that bothmethods are in agreement and are accurate and efficient for solving systems of OFDE.


Introduction
e 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. e 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) s(t): the number of susceptible, who do not have the disease but could get it. (ii) i(t): the number of infectives, who have the disease and can transmit it to others. (iii) r(t): 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. e 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. en, in time δtandδs, the susceptible becomes infective: where β is a positive constant. If c > 0 is the rate at which current infectives become isolated, then and the number of new isolated δr is given by If we let δt ⟶ 0, then the following nonlinear system of ODEs determines the progress of the disease: with initial conditions s(0) � N s , e following simple SIR model [2][3][4][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][8][9][10][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. e main advantages of the CFD can be summarized as follows [16][17][18][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 efficiently e reason for considering a fractional-order system instead of its integer order counterpart is that fractionalorder 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.
e 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.

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][21][22][23][24][25]. e conformable fractional derivative of order α is defined as for all t > 0, α ∈ (0, 1), and the fractional derivative at 0 is defined as Let α ∈ 0, 1 and f be α-differentiable at a point t > 0, and if f is differentiable, then e fractional integral of order α is defined by where the integral is the usual Riemann improper integral.

Mathematical Modeling of the Conformable Fractional SIR Model
e conformable fractional model of actual evolution of this epidemic in a population of large size N is given by the following conformable fractional differential system:

e Conformable Fractional Differential Transformation
Method. In [9,11], assume f is infinitely α-differentiable function, for some 0 < α ≤ 1. f(t) can be expanded in fractional power series expansion about a point t � 0 as Here, [(T α f) (k) ] t�0 denotes the application of the fractional derivative for k times. Conformable fractional differential transform of f(t) is defined as en, the inverse conformable fractional differential transform of F(k) is defined as e fundamental mathematical operations performed by conformable fractional differential transform method are listed.

Application of Conformable Fractional Differential
Transform Method. Equation (10) can be rewritten as follows: Hence, recurrence relation is obtained as With initial conditions N s � 2000, N i � 300, and N r � 200 and parameters N � 2500, β � 0.00012, and c � 0.1, apply the condition in (17); then, the closed form of the solution where k � 4 can be written as

Variational Iteration Method.
To illustrate the basic idea of the variational iteration method, we consider the following nonlinear differential equation in the operator form: where L is a linear operator, N is a nonlinear operator, and g is any real function which is called the inhomogeneous term. en, the corresponding correction functional for equation (19) is given by u n+1 (t) � u n (t) + t 0 λ Lu n (s) + Nu n (s) − g(s) ds, (20) 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). en, the variational iteration formula is given by where s n , i n , andr n are the nth approximation, T α is the conformable fractional derivative of order α, and I α is the fractional integral of order α ∈ 0, 1.

International Journal of Differential Equations
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.

Conclusion
In this study, we have found out approximate solutions with two numerical methods for the SIR epidemic model. ese 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). en, 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.