On the Analytical Solution of Fractional SIR Epidemic Model

This article presents the solution of the fractional SIR epidemic model using the Laplace residual power series method. We introduce the fractional SIR model in the sense of Caputo's derivative, it is presented by three fractional differential equations, in which the third one depends on the first coupled equations. The Laplace residual power series method is implemented in this research to solve the proposed model, in which we present the solution in a form of convergent series expansion that converges rapidly to the exact one. We analyze the results and compare the obtained approximate solutions to those obtained from other methods. Figures and tables are illustrated to show the efficiency of the LRPSM in handling the proposed SIR model.


Introduction
Epidemiology is a discipline of biology that examines the prevalence and causes health-related problems in particular groups or communities. It may be used to manage community health concerns. Frequency, distribution, and the factors that cause diseases are the elements of epidemiology. By establishing priorities among these services, the Department of Epidemiology seeks to ofer the data required for the planning, execution, and assessment of services aimed at disease prevention, control, and treatment. Epidemiology is used to investigate the historical development and decline of illnesses in the population, community diagnosis, planning and evaluation, assessment of an individual's risks and chances, identifcation of syndromes, and completion of the disease's natural history [1][2][3][4][5].
Te study of fractional calculus depends on computing integrals and derivatives of noninteger orders. Te importance of these studies has appeared in the various applications in physics and engineering, in which these derivatives can describe them more realistically in the felds of science. A more accurate explanation of the challenges in the actual world can be provided by the fractional application models [6][7][8][9][10][11].
Trough the process of mathematical modeling, one may examine, anticipate, and ofer insight into issues that arise in the actual world. It is advantageous since conventional theoretical approaches are inadequate for the study of technological, ecological, economic, and other systems examined by modern research. Fractional diferential equations and systems have been used to simulate a variety of real-world issues [12][13][14][15][16][17][18].
Te SIR model is one of the simplest fractional models, and many models are derived depending on its form. Te model consists of three components: Te study of the SIR model was frst introduced in 2009 by Ahmet and Cherruault [19], then in 2011 they present new research on analytical solutions of some related models. After that, many authors have investigated the susceptible-infected recovered models of integer fractional orders [20][21][22][23].
Te integer-order SIR epidemic model is given by under the initial conditions: where φ(τ) denotes the susceptible individuals, ψ(τ) denotes the infected individuals, ϕ(τ) denotes the removed individuals, and τ denotes the time. N is the total number of the studied population such that φ(τ) Te rate of changes between the previous and the new values are q 1 , q 2 and q 3 , for more details see [24,25]. Tere are some constraints on the model concerning the population number N, which should be large enough, the parameters of the system should be ft, and fnally, the healing does not provide immunity. Tis research presents the LRPSM, which is a new analytical method that combines the Laplace transform with the residual power series method; it was frst introduced in [26], and it is implemented by researchers to solve several models of fractional ordinary and partial diferential equations and systems. Tis method shows its efciency and applicability in solving similar problems [27][28][29][30][31][32][33][34][35].
Te main aim of this article is to present an analytical series solution of the fractional SIR model. We use the LRPSM to get the solution in the form of a rapidly convergent series. We introduce the method and state the convergence analysis of the method, then we apply it to solve the proposed model. Numerical simulations of the results are discussed, and comparisons are made with the results obtained from other numerical methods. Te strength of the LRPSM arises in the ability of solving similar models and presenting many terms of the series solutions with fewer calculations and without the need of linearization, discretization, or diferentiation as with other numerical methods.
Te layout of this article is as follows: in Section 2, we present some defnitions and theorems related to fractional power series and the analysis of LRPSM. In Section 3, we construct the series solution of the fractional SIR model in the sense of Caputo's derivative by LRPSM. In Section 4, we introduce some numerical simulations of our results and comparisons to other numerical methods. Finally, the conclusion section is presented in Section 5.

Fractional Power Series
In this section of the article, we introduce some basic definitions and characteristics of the Caputo fractional derivative and the Laplace transform. Also, we present theorems about the fractional Taylor's series of expansions. Defnition 1. Caputo fractional derivative of the function φ(τ) of order c is given by the following equation: where r ∈ N and J c τ is the Riemann-Liouville integral of the fractional order c to the function φ(τ), provided the integral exists.
Tere are many properties of Caputo's derivative, that might be found in [23,24], and we mention some of them as follows: is of exponential order, then φ(τ) has the Laplace transform which is defned as follows: Te inverse Laplace transform is given by for some, c provided the integral exists. In the following arguments, we list some of the most popular properties of the Laplace transform.
Te next theorem illustrates a new form of Teorem 1 in the Laplace space considering δ � 0.

Theorem 2.
Te fractional power series in (6) has three possibilities for convergence: (i) If τ � δ, then the series is convergent and the radius of convergent is zero (ii) If τ ≥ δ, the series is convergent and the radius of convergent is ∞ (iii) If δ ≤ τ < δ + ζ, for some possible real number ζ, then the series converges, and if τ > δ + ζ, then the series (1) diverges Theorem (see [23]). If the fractional power series expansion of the function Φ(s) � L[φ(τ)] is expressed as follows: then the coefcient α n can be obtained from the following formula: Moreover, the inverse Laplace transform of the series expansion (8) in Teorem 3 has the form: Theorem 4 (see [23]) (Convergence analysis). Let Φ(s) � L[φ(τ)] be a function that has the fractional power series in equation (2).
where 0 < c ≤ 1 and V > 0, then the remainder R n (s) of the series representation (2) has the following bound:

Fractional SIR Epidemic Model.
Tere are several phenomena in the real world in engineering and physics, which can be reformulated by fractional initial value problems. Not all of these problems can be solved exactly, so, they are challenging researchers around the whole world. Our aim in this section is to introduce the main idea of the LRPSM in solving systems of nonlinear fractional diferential equations that might be difcult to solve by usual techniques. Now, we present the fractional SIR model: where c i ∈ (0, 1], ∀i � 1, 2, 3, D c i denotes the Caputo derivative, q 1 , q 2 , and q 3 are real positive numbers, q 1 denotes the infection rate, q 2 denotes the removal rate, and q 3 is the recovery rate. Te given initial conditions for system (12) are as follows: Also, we have: and the relation: Te relation (15) gives an extra condition that enable us to solve only two equations of three variables.

LRPSM for Solving Fractional SIR Model.
Te basic idea of LRPSM is to apply the Laplace transform on the target equations, then defne the so-called Laplace residual functions. After that, multiply each equation by s kα+1 by the truncated Laplace residual functions and take the limit at infnity to get the required values of the series coefcients recursively.
To get the series solution of the system (12) using the proposed method, we frst operate the Laplace transform on both sides of each equation in system (5), to get: Running Laplace transform on system (16), we get:

Applied Computational Intelligence and Soft Computing
Simplifying the equations in system (17) and substituting the initial conditions (13), we get: Suppose that the solution of system (18) can be presented in the following series forms: Using the property that lim s⟶∞ s Φ(s) � φ(0), it is obvious that Te series expansion in (19) can be written as follows: Now, defne the Laplace residual functions of system (18) as follows:

Applied Computational Intelligence and Soft Computing
Te kth truncated series of (21) has the form: and the kth Laplace residual functions as follows: Now, the frst truncated series of (23) are as follows: Substituting the above values in the frst Laplace residual functions to get the following: Using the following facts of the LRPSM, which can be found in [23], Now, multiplying each equation in (26) by s c 1 +1 , s c 2 +1 , and s c 3 +1 , respectively, and then taking the limit as s ⟶ ∞, we get the frst coefcients of the series expansion (23) as follows: Repeating the previous steps, we can get other coefcients.
In addition, if we take c 1 � c 2 � c 3 � ϑ, one can get the coefcients of the series solution as follows: Te k th coefcients of the series solutions (21) have the form: Tus, the kth solution of systems (12) and (13) in Laplace space can be expressed as follows: Operating the inverse Laplace transform on each equation in (32), we obtain the kth solution of system (12) as follows: As k ⟶ ∞, the kth truncated solutions converge to the exact solutions in the integer orders: Hence, we get the required solution.

Numerical Simulation
Tis section presents the solution of a numerical example of a SIR epidemic system using the LRPSM, and it introduces numerical simulations and fgures to show the efectiveness of the suggested approach. Te outcomes demonstrate the dependability and strength of LRPSM in solving such problems.
We consider the epidemic SIR model as follows: subject to the initial conditions: Te total number of populations is N � 700.
To solve system (33) by the proposed method, we apply the Laplace transform on each equation in the system, and using the initial conditions (36), we get: Now, the kth truncated expansion of the solution of system (35) is as follows: We defne the kth Laplace residual functions as follows: Multiplying each equation in (39) by s kϑ+1 , k � 1, 2, · · · recursively, and taking the limit as s ⟶ ∞, we get the coefcients of the series solutions as follows: Applied Computational Intelligence and Soft Computing 13 Te 20th terms solution of systems (33) and (34) at ϑ � 1 is given by the following equation:
Te following tables, Tables 1-3 present the LRPSM solution of systems (33) and (34) at various values of ϑ(ϑ � 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3). In Table 1, we propose diferent values of φ 20 (τ) with diferent values of ϑ, in Table 2, we propose diferent values of ψ 20 (τ) with diferent values of ϑ, and in Table 3, we propose diferent values of ϕ 20 (τ) with diferent values of ϑ. We notice the efciency and strength of the method from the agreement of the values, and they all coincide and converge to the exact solution in the integer order when ϑ � 1.
In the following, we introduce Figure 3, which illustrates the absolute error between the Runge-Kutta method solution and the LRPSM solution at ϑ � 1.
Te following tables (Tables 4-6), present comparisons between the obtained results φ 20 (τ), ψ 20 (τ), and ϕ 20 (τ) from the LRPSM and the fourth-order Runge-Kutta method (RK4). Tese comparisons prove the strength and convergence of the presented method. We notice from Tables 4-6 the efciency of the proposed method, since the diference between the methods is too small.

Conclusions
In this study, we introduce the fractional SIR epidemic model in the sense of Caputo's fractional derivative. Te LRPSM is used to solve the proposed system. We present a series of solutions of the model and compare our results with those obtained by the fourth-order Runge-Kutta method. In addition, we sketch the graphs of the solutions with diferent values of ϑ and analyze the results. Finally, we conclude that the LRPSM is efcient in solving the SIR epidemic system and similar models. Moreover, as the method is applicable and easy in treating similar problems, it could provide many terms of the series solution with less calculations and efort compared to other numerical methods. In the future, we intend to solve new models by LRPSM and make comparisons to other analytical methods. We conclude the following points from our study: (i) LRPSM is a powerful technique for solving fractional models (ii) LRPSM is simple and could provide many terms of the series solution without requiring discretization, linearization, or special assumptions on the conditions (iii) Te method could give exact solutions when the exact one is a polynomial

Data Availability
No underlying data were collected or produced in this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.