An Efﬁcient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by Using Bernstein Wavelets

: In this paper, the operational matrix based on Bernstein wavelets is presented for solving fractional SIR model with unknown parameters. The SIR model is a system of differential equations that arises in medical science to study epidemiology and medical care for the injured. Operational matrices merged with the collocation method are used to convert fractional-order problems into algebraic equations. The Adams–Bashforth–Moulton predictor correcter scheme is also discussed for solving the same. We have compared the solutions with the Adams–Bashforth predictor correcter scheme for the accuracy and applicability of the Bernstein wavelet method. The convergence analysis of the Bernstein wavelet has been also discussed for the validity of the method.


Introduction
The construction of mathematical models for real-world phenomena and development of efficacious techniques to define them is one of the most critical issues in applied mathematics biology, engineering, physics and other fields of science. In the 19th century, SIR epidemiological was first introduced by Kermack et al. [1]. In an SIR epidemic model, the population has three components, those susceptible (S), those infected (I) and those recovered (R) from the disease [1]. Presently childhood diseases are most significant infectious diseases. Rubella, measles, poliomyelitis, and hepatitis B have serious concerns among them [2][3][4][5][6][7]. Recently, coronavirus diseases have been discussed in [8]. Mathematical models play a key role in analyzing the mechanism of transmission of disease and provide different approaches to control the propagation of disease. In the current article, we examine the subsequent mathematical model of arbitrary order: The associated initial conditions (ICs) are given as S(0) = S 0 , I(0) = I 0 and R(0) = R 0 . Bernstein polynomials (BP) were suggested by Sergei Natanovich Bernstein in 1912 which is a polynomial in the Bernstein form that is a linear combination of Bernstein basis polynomials. There are several research articles on Bernstein polynomial for solving fractional differential equations [9][10][11].
Wavelet theory is a relatively incipient and dominating area in research and innovation. Wavelets have been used in numerous fields such as image compression, signal processing, time frequency analysis, data compression and fast algorithm for easy implementation [12,13]. Wavelet sanctions the precise depiction of a variety of functions and operators [14,15]. Recently, many research papers are published on different types of wavelets, the aim of these research papers to provide the numerical solutions to differential equations of integer order as well as fractional order with the aid of wavelets [16][17][18][19][20][21]. Recently, Boonrod and Razzaghi discussed a numerical approach based on Legendre wavelets for examining fractional differential equations (FDEs) by the exact formula for Riemann-Liouville (RL) [22]. Rahimkhani and Ordokhani discussed a numerical scheme for solving FDEs by Bernoulli wavelets [23]. Recently several other analytical and numerical schemes have been used to examine fractional order models [24][25][26].
The key aim of present investigation is to a discuss an efficient computational approach for solving Equation (1). The suggested computational approach is based upon Bernstein wavelets approximation. An exact formula for the RL fractional integral operator for the Bernstein wavelets is computed. Moreover, the derived formula is then employed to convert the linear or non-linear fractional differential equations into the system of algebraic equations. Next, Adams-Bashforth predictor correcter scheme is also discussed for solving the same [27,28]. We have compared the solutions with Adams-Bashforth predictor correcter scheme for the accuracy and applicability of the Bernstein wavelets technique.
The rest of paper is presented as follows: Some results, basic definitions and fractional calculus (FC) are provided in Section 2 which are used in the proposed work. The basic idea of normalized Bernstein wavelets and its properties are presented in Section 3 which is base of proposed work. In Section 4, a Bernstein wavelet operational matrix using Riemann−Liouville integral operator is discussed and presented. In Section 5, we presented convergence and error analysis theorem on Bernstein wavelets. In Section 6, we have implemented Bernstein wavelets and Adam's-Bashforth-Moulton methods. Numerical results and discussions for the fractional SIR epidemic model are completely discussed in Section 7 which is main part of the proposed work.

Definition 2.
The following two Eqs. presents the RL and Caputo fractional derivatives of order a, respectively, where τ > 0 and m is an integer. It has the following two basic properties for m − 1 < α ≤ m and Υ ∈ L 1 [a, b],

The Normalized Bernstein Wavelets and Its Properties
In the present section, we construct Bernstein wavelet using the orthonormal Bernstein polynomial. Some important properties of Bernstein wavelets are considered in this section.
A family of wavelet functions build up from dilation and translation of a single function ψ are constituted wavelets. if the dilation parameter ρ and the translation parameter δ are continuous, the family of continuous wavelets is presented as If it is restricted that the parameters a and b take discrete values as ρ = ρ −k 0 , δ = nb 0 ρ −k 0 , ρ 0 > 1, δ 0 > 0, then we get the subsequent family of discrete wavelets, where the sequence ψ kn form a wavelet basis for L 2 (R), and if ρ 0 = 2 and δ 0 = 1 then we have an orthonormal basis. The aforesaid Bernstein wavelet ψ nm (τ) = ψ(k, n, m, τ) have four parameters, defined over [0, 1] by, where n = 0, 1, . . . , 2 k − 1, m = 0, 1, . . . , M, τ is the normalized time and m is the degree of the orthonormal Bernstein polynomial. Furthermore, Any function Υ(τ) defined over [0, 1) may be expressed in terms of Bernstein wavelet as where Λ nm =< Υ, ψ nm >= 1 0 ψ nm (τ)Υ(τ)dτ, with < ., . > as the inner product defined on L 2 [0, 1]. If the infinite series is truncated, then the above Eq. is presented as where T denotes the transposition and, C and Ψ(τ) are them = 2 k (M + 1) column vectors.
Here, we define Bernstein wavelet matrix Φm ×m as The above matrix Φm ×m is Bernstein matrix at given collocation points 2i−1 2m , where k = 2 and M = 2.

Bernstein Wavelet Operational Matrix Using Riemann-Liouville Integral Operator
The principal target of this part is to derive the operational matrix for Bernstein wavelet without using block pulse functions. For this, we operate I α τ operator directly into Ψ(τ) as follows where where µ a (τ) is the unit step function given as Here, operating the Laplace transform to calculate I σ τ ψ nm (τ) using Equation (11) for m = 0, 1, . . . , M, where Furthermore, operating the inverse Laplace transform into Equation (16), we get If, we consider the fixed value as k = 2, M = 2, σ = 0.5 and the collocation points 2i−1 2m . However, we get the operational matrix given below The above square matrix Q 0.5 is operational matrix based on Bernstein wavelet at σ = 0.5. However, we can also find Bernstein operational wavelet matrix for arbitrary 0 < σ ≤ 1.

Convergence and Error Analysis
Theorem 1. The solution obtained by Bernstein wavelets method is converges. (4) forms an orthonormal basis.

Proof. Since Bernstein wavelets in Equation
Let Consequently, When n → ∞, ∑ ∞ i=0 |Λ ni | 2 is convergent by Bessel's inequality. Hence, P n is Cauchy's sequence converges to P (say). Therefore, Thus, series solution of the Bernstein wavelets is convergent.
is the approximate solution using the Bernstein wavelets then error bound would be given as Proof. In view of the concept of norm in inner product space, we have Now, dividing into the 2 k sub-intervals I n = n 2 k , n+1 2 k , n = 0, 1, 2, . . . , 2 k − 1.
where p M+1 (τ) is the interpolating polynomial of M + 1 degree which approximate Υ(τ) on the interval I n . With the aid of the maximum error estimate for the polynomial on I n , we obtain

Bernstein Wavelets for the Numerical Solution of SIR Epidemic Model
Consider the SIR epidemic model (1), we assume higher fractional derivatives in terms of the Bernstein wavelets as where A T r = [Λ r 00 , Λ r 01 , . . . , Λ r 0,M , Λ r 1,0 , . . . , Λ r 1M , Λ r (2 k −1)0 , . . . , Λ r (2 k −1)M ] are the unknowns and r = 1, 2, 3. Now, we operate fractional integral operator into Equation (18) in the sense of Riemann-Liouville, we obtain also, Then where only A T r are unknowns. Putting these values of S, I and R into the main Equation (1) and using the collocation points 2i−1 2m , where i = 1, 2, . . . , 2 k (M + 1), we can get the system of non-linear algebraic equations with 3m number unknowns. By solving these equations with the avail of the Newton iteration method by MATLAB software, we can ascertain the unknown Bernstein coefficients. By superseding unknown coefficients into Equation (21), we may obtain the desired solutions.

Numerical Results and Discussion
Here, we obtain the numerical results for fractional SIR epidemic model 1 to verify the applicability and efficiency of the Bernstein wavelets.
with initial conditions S(0) = 620, I(0) = 10 and R(0) = 70. Furthermore, we use aforesaid operational matrix to convert system of non-linear FDEs into the system of algebraic equations. To show the validity of the Bernstein wavelets, we compared the obtained solutions with the solutions obtained by Adam's-Bashforth-Moulton methods (ABM). When M = 2 and k = 5, it is obvious from Figures 1-3 that the obtained solutions by the Bernstein wavelet scheme has an excellent agreement with the ABM.
Moreover, we have obtained relative error and absolute error between solutions obtained by both methods, where relative error and absolute error is defined as and Relative error and absolute error are depicted through Figures 4-9. Consequently, we have shown the effect of fractional derivative on the SIR model. From Figures 10-12, it is clear that the fractional derivative gives more freedom compare to integer order derivative, by the proposed method we are able to find the solutions for arbitrary order of derivative. From Figures 13-15 we have shown the 3D plot of the susceptible, infected and recovered people. In Tables 1-3, we have compared the obtained solutions with RK4 and RPS numerical methods, we found that our solutions are convergent. If we increase the values of M and k we can obtain more appropriate results compare to another numerical methods.

Conclusions
In this study, to get the numerical solutions of the non-linear fractional SIR epidemic model, a pattern of Bernstein wavelet method was discussed. In addition, an error estimation and convergence analysis of the function approximation based on aforesaid wavelets was discussed. Furthermore, an exact approximation for arbitrary order Riemann-Liouville integral operator was also discussed. The operational matrix together with collocation points was used to diminish the non-linear FDEs into several algebraic equations. Another numerical method known as Adams-Bashforth-Moulton was additionally discussed to show the precision and applicability of the suggested method. Convergence and error analysis assert the validity of proposed method.
Funding: This research was funded by SERB, Government of India, grant number EEQ/2017/000385.

Conflicts of Interest:
The authors declare no conflict of interest.