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.
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 , suppose the population consists of(i): the number of susceptible, who do not have the disease but could get it.(ii): the number of infectives, who have the disease and can transmit it to others.(iii): 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 time , the susceptible becomes infective:where is a positive constant. If is the rate at which current infectives become isolated, thenand the number of new isolated is given by
If we let , then the following nonlinear system of ODEs determines the progress of the disease:with initial conditions
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 efficiently
The 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 is defined asfor all , and the fractional derivative at 0 is defined as
Let and be -differentiable at a point , and if is differentiable, then
The fractional integral of order is defined bywhere 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 size is given by the following conformable fractional differential system:with initial conditions
3.1. The Conformable Fractional Differential Transformation Method
In [9, 11], assume is infinitely -differentiable function, for some . can be expanded in fractional power series expansion about a point as
Here, denotes the application of the fractional derivative for times. Conformable fractional differential transform of is defined as
Then, the inverse conformable fractional differential transform of is defined as
The fundamental mathematical operations performed by conformable fractional differential transform method are listed.
Proposition 1. (i)If , then .(ii)If , then .(iii)If , then .(iv)If , then .(v)If , then , where
3.1.1. Application of Conformable Fractional Differential Transform Method
Equation (10) can be rewritten as follows:
Hence, recurrence relation is obtained as
With initial conditions and parameters , , apply the condition in (17); then, the closed form of the solution where can be written as
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:where is a linear operator, is a nonlinear operator, and is any real function which is called the inhomogeneous term. Then, the corresponding correction functional for equation (19) is given bywhere is the general Lagrange multiplier [26], which can be identified optimally via the variational theory [27] and is considered as restricted variation, i.e., . 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 bywhere are the th approximation, is the conformable fractional derivative of order , and is the fractional integral of order .
Proof. Equation (10) may be rewritten equivalently asMultiplying equation (22) by a general Lagrange multiplier , and yieldsNow, upon applying to the both sides of equation (23) will giveThen, the correction functional of equation (10) will be read as follows:In this case, the value of , cannot be evaluated easily from equation (25), which will be a functional with fractional integral. Equation (25) can be expressed as follows:where and are the restricted variations with and . From equation (26), we obtainUsing integration by parts, equation (27) becomes (for equations (28) and (29) are similar to the proof of equation (27))The Lagrange multipliers , and can be obtained by for all with boundary condition: , for all . Solving the last initial value problem for for all , the general Lagrange multiplier is found to beHence, substituting the value of into the corresponding correction functional (25) will give the following iteration formula:
3.2.1. Application of the Variational Iteration Method
Applying the variational iteration method using Theorem 1,
With initial conditions and parameter , , we obtain
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.
4. Conclusion
In this study, we have found out approximate solutions with two numerical methods for the epidemic model. These methods are based on conformable derivative which is extremely popular in the last years. Firstly, by using the -derivative, we have redefined the conformable differential transformation method () and variational iteration method (). Then, we have demonstrated the efficiencies and accuracies of the proposed methods by applying them to the epidemic model. It is found that the approximate solution generated the by our method which is in complete agreement with the corresponding approximate solution . Besides, in view of their usability, our methods are applicable to many epidemic models / and of fractional order.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.