A numerical solution by alternative Legendre polynomials on a model for novel coronavirus (COVID-19)

Coronavirus disease (COVID-19) is an infectious disease caused by a newly discovered coronavirus. This paper provides a numerical solution for the mathematical model of the novel coronavirus by the application of alternative Legendre polynomials to find the transmissibility of COVID-19. The mathematical model of the present problem is a system of differential equations. The goal is to convert this system to an algebraic system by use of the useful property of alternative Legendre polynomials and collocation method that can be solved easily. We compare the results of this method with those of the Runge–Kutta method to show the efficiency of the proposed method.


Introduction
An outbreak of the 2019 novel coronavirus disease  in Wuhan, China has spread quickly nationwide. The COVID-19 epidemic has spread very quickly from China to all the world [1,2]. Countries continue to battle the novel coronavirus as it has infected more than 28 million around the world [3].
In [4] the COVID-19 mathematical model has been derived as follows, where S p (t) is susceptible people, E p (t) is exposed people, I p (t) is symptomatic infected people, A p (t) is asymptomatic infected people, R p (t) is recovered and dead people, and W (t) is COVID-19 in reservoir in time t. The parameters needed are defined in Table 1 and p = n p × N p , where N p refers to the total number of people: (1) © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The lifetime of the virus in W c The relative shedding rate of A p compared to I p This paper aims to find the transmissibility of the COVID-19 by finding the unknowns S p , E p , I p , A p , R p , and W . In medical sciences, the computation of these variables is vital to measure the progression of disease and to get a better cure. In this paper, for finding these variables, we use alternative Legendre polynomials and their operational matrix of derivative. The proposed method results are compared to those of Runge-Kutta method, which shows the reliability of the proposed method.
There exist some related papers on this topic that have solved the coronavirus model or some differential equation system that appears in the disease model, so we refer the readers to them to see some similar methods on this topic [5][6][7][8].
The remainder of the article is organized as follows. In Sect. 2, we review the properties of alternative Legendre polynomials and approximation of a function with them. Then we present the operational matrix of derivatives of these polynomials. In Sect. 3, we implement the alternative Legendre polynomials method on the coronavirus model. Section 4 shows the applicability of the proposed method through a test problem, also the results are compared with Runge-Kutta method results that confirm the reliability of the proposed method. Then Sect. 5 concludes the paper.

Properties of ALPs
The set P n = {P nk : k = 0, 1, . . . , n} of alternative Legendre polynomials of degree n is defined by an explicit formula on the interval [0, 1] (see [9]) as follows: They are orthogonal on the interval [0, 1] with the weight function w(t) = 1. The ALPs satisfy the orthogonality relationships We can reproduce Eq. (2) with Rodrigues's type as follows: Here, we note that each element of the set P n = {P nk } n k=0 is the polynomial of other n. For example, in the following we introduce the alternative Legendre polynomials In Fig. 1, we display the 4 set of ALPs with n = 3 over the interval [0, 1].

Function approximation
Consider P n = {P nk } n k=0 ⊂ H = L 2 [0, 1] to be a set of ALPs and suppose that Y = Span{P nk (t) : k = 0, 1, . . . , n}. So, Y is a finite dimensional subspace of H. Suppose, f to be an arbitrary function in H. Therefore, based on the Weierstrass theorem, every continuous function f (t) on the interval [a, b] can be uniformly approximated by a polynomial function [9]. So, f has a unique best approximation in Y that we call f * (t). We have Then this implies that where ·, · denotes an inner product. Therefore, any arbitrary function f ∈ H = L 2 [0, 1] may be approximated in terms of ALPs. So, there exists a set of unique coefficients {c k : coefficient c k can be obtained in the following form: and Also, Eq. (8) can be written in a matrix form as follows: where and Let a (n) kj = (-1) j n-k j n+k+j+1 n-k , then Eq. (2) can be written as By using Eq. (14), for k = 0, 1, . . . , n now, we can write Therefore Eq. (13) can be written in the following form: where and is the upper triangular matrix defined by [10] = [q kj ], k, j = 0, 1, . . . , n, Definition The tensor product of two vectors fm = [f i ] and gm = [g i ] is defined as Similarly, for two matrices A = [a i,j ] and B = [b i,j ] ofm ×m, The lemma below will be needed in Sect. 3.

Operational matrix of derivative
In this section, we derive the operational matrix of derivative of the ALPs that plays an important role in simplifying a system of differential equations and implementation of the proposed method.
To compute this operational matrix, we need to introduce the following properties of ALPs that can easily be deduced from the given definitions. Let P ni (t) = n r=0 p (i) r t r , P nj (t) = n r=0 p (j) r t r , and P nk (t) = n r=0 p (k) r t r be ith, jth, and kth of ALPs, respectively. Therefore, we have n-k k + l + r + 1 , k = 0, 1, . . . , n.
The derivative of the vector (t) can be expressed by Here, D (1) is the (n + 1) × (n + 1) operational matrix of derivative.
So, by applying the differential operator with respect to t, we can write D t = d dt (see [11]). By applying the polynomial P nk (t), we obtain Here, by using Eq. (11), one can approximate t k+j-1 in terms of ALPs as follows: The approximation coefficients b Substituting (27) into (26), we have Then, using Eqs. (26) and (27), we have hence Therefore, for the vector (t) defined by (13), we get where D (1) is the (n + 1) × (n + 1) operational matrix of derivative based on the ALPs as follows: kr , k, r = 0, 1, . . . , n.

Implementation of an alternative Legendre polynomials method on the novel coronavirus (COVID-19) problem
Firstly, note that the variable of system (1) becomes normalized as follows [4]: So, the normalized model is changed as follows: with the initial conditions The main objective of this paper is to implement ALPs approach on the system of differential Eqs. (34) with the above initial conditions to find the numerical solution of this system. From Eq. (11), we can approximate our unknown functions as follows: where coefficient vectors C i : i = 1, . . . , 6 that were defined in Eq. (29) are as follows: By using Eqs. (34) and (32), we have By substituting Eqs. (37) and (35) into the system of differential Eqs. (1), we have Also, by considering the initial conditions for main problem (1) and Eq. (35), we have Equation (39) gives six linear equations. Since the total unknowns for vectors C i : i = 1, . . . , 6 are (6n + 5), we collocate Eq. (38) in the set of (6n -1) nodal points t l of the Guass-Chelyshkov [10] as follows: Q n = t l |P n+1,0 (t l ) = 0, l = 0, 1, . . . , n .
For existence and stability of the proposed method with ALPs, we can refer to paper [9]. In our implementation, the calculations are done in Mathematica 11 software, on a personal computer with Core-i5 processor, 2.67 GHZ frequency, and 4 GB memory.

Numerical example
In this section a test problem of the coronavirus model is solved by our proposed method. The values of the initial conditions and parameters are given as [4]: N p = 1,000,000,000, Also, we get the initial values of unknown parameters as follows: We solve this problem by n = 16 in ALPs. The results of proposed method are compared with the results of Runge-Kutta method. Figure 2 and Tables 2-7 show the comparison between them.

Conclusion
The World Health Organization declared the coronavirus (COVID-19) a pandemic on March 11, 2020. This virus spread quickly in more than 200 countries, and up to now      more than 28 million around the world have been infected. This paper aims to solve the mathematical model of coronavirus that can show the transmissibility of this virus that is vital to measure the progression of the disease and to get a better cure. By use of alternative Legendre polynomials and their operational matrix of derivative, we convert the system of coronavirus model to an algebraic model. We compare the results of the present method with those of the Runge-Kutta method, which confirmed the reliability of the proposed method results.