Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic

The present paper deals with the numerical solution for a general form of a system of nonlinear Volterra delay integro-differential equations (VDIDEs). The main purpose of this work is to provide a current numerical method based on the use of continuous collocation Taylor polynomials for the numerical solution of nonlinear VDIDEs systems. It is shown that this method is convergent. Numerical results will be presented to prove the validity and effectiveness of this convergent algorithm. We apply models to the COVID-19 epidemic in China, Spain, and Italy and one for the Predator–Prey model in mathematical ecology.


Introduction
In this paper, a numerical method is presented to obtain an approximate solution for the following system of nonlinear delay integro-differential equations y (t) = f (t, y(t), y(t − τ )) + t t−τ k(t, s, y(t), y(s)) ds, ( 1 ) for t ∈ [0, T] and y(t) = (t) for t ∈ [−τ , 0] with : The existence and the uniqueness of the solution of (1) can be found, for example, in [10,11].
The system (1) contains, as particular cases, many important Volterra integro-differential equations, functional equations, delay differential equations, delay integro-differential equations, and functional equations in the literature. Moreover, this method can be used to obtain numerical solutions of Volterra integro-differential equations, delay differential equations and delay integrodifferential equations.
The delay integro-differential equations and their systems have become important in the mathematical modelling of many sciences and engineering fields (see, e.g. [9,13,20,23,43,47,48]). As a particular case in traditional population biology, the predator-prey dynamics system was first modelled by Volterra [45]. Moreover, Liu et al. [25] present a COVID-19 epidemic model, which can be described by a particular form of the system of nonlinear delay integro-differential (1). We will present some applications of this system in Section 4.
The Taylor polynomial method for approximating the solution of integral equations and integrodifferential equations has been proposed. Bellour and Bousselsal [6,7] used the Taylor collocation method for solving delay integral equations and integro-differential equations, Taylor collocation method for the Volterra-Fredholm integral equations is used in [46], Gülsu and Sezer [17] applied a Taylor collocation method for the solution of systems of high-order Fredholm-Volterra integrodifferential equations.
The aim of this paper is to construct an approximate solution for a general form of the system of nonlinear Volterra delay integro-differential equations (1) by using a collocation method based on the use of Taylor polynomials. We use a more straightforward generalization of techniques employed in [7] and [6]. Our method presents some advantages: • It provides a global approximation of the solution • The approximate solutions are given by explicit formulas without needed to solve any algebraic system • High order of convergence • Provides an explicit numerical solution and easy to be implemented.
The paper is organized as follows: In Section 2, we divide the interval [0, T] into subintervals, and we approximate the solution of (1) in each interval by a Taylor polynomial. The convergence analysis is established in Section 3, and the numerical illustrations are provided in Section 4. Theoretical and empirical time complexity is given in Section 5. Finally, a conclusion is given in Section 6.

Description of the method
We suppose that T = rτ , where r ∈ {1, 2, 3, . . .}. Let N be a uniform partition of the interval Moreover, denote by π m the set of all real polynomials of degree not exceeding m. We define the real polynomial spline space of degree m as follows: This is the space of piecewise polynomials of degree (at most) m. Its dimension is rNm + 1, i.e. the same as the total number of the coefficients of the polynomials u i n , n = 0, . . . , N − 1, i = 0, 1, . . . , r − 1. To find these coefficients, we use Taylor polynomial on each subinterval. First, we approximate the exact solution y in the interval σ 0 0 by the polynomial To find y (j) (0), we differentiate Equation (1) j-times, we obtain Second, we approximate y by the polynomial u 0 n (n ∈ {1, 2, . . . , N − 1}) on the interval σ 0 n such that whereû n,0 is the exact solution of the integro-differential equation for t ∈ σ 0 n such thatû n,0 (t 0 n,0 (t 0 n ) are given by the following formulâ whereû 0,p is the exact solution of the integro-differential equation for t ∈ σ p 0 such thatû 0,p (t The coefficientsû (j) 0,p (t p 0 ) are given by the following formulâ whereû n,p is the exact solution of the integro-differential equation for t ∈ σ p n such thatû n,p (t for j ∈ {0, 1, . . . , m} andû n,p (t p n ) = u p n−1 (t p n ).

Analysis of convergence
For ease of exposition, we will consider a feasible linear form of (1), namely More precisely, Equations (4), (5), (7), (8), (10) and (11) may be written in the following linear forms, respectively,û The following three lemmas will be used in this section.

Lemma 3.1 (Discrete Gronwall-type inequality [10]):
Let {k j } n j=0 be a given non-negative sequence and the sequence {ε n } satisfies ε 0 ≤ p 0 and with p 0 ≥ 0.Then ε n can be bounded by

Lemma 3.3 ([19]): Assume that the sequence {ε n } n≥0 of non-negative numbers satisfies
where A, B and K are non-negative constants, then In the following, for a given fonction ψ ∈ C(I, R d ), we define the norm ψ by This lemma will be crucial for establishing the convergence of the approximate solution.
Proof: We use a more straightforward generalization of techniques employed in the case of delay integral and integro-differential equations by using the Taylor collocation method (see, for example, [6,7,24] ).

Numerical examples
This section presents several examples to show the performance of the described method in Section 2 for solving the system (1  [43]. Moreover, we apply two mathematical models, namely the SEIR model (27) in [20] and the SEIRU model (28) in [25,26] that simulate the evolution of the COVID-19 epidemic in China [12], Spain [34], and Italy [22].
for t ∈ [0, 1] and y(t) = (t) for t ∈ [− 1 2 , 0] The exact solution of this system is in the following form y 1 The absolute errors of Taylor Collocation Method (TCM) for m = 10, N = 10 are compared with the absolute error of the Variational Iteration Method (VIM), Adomian Decomposition Method (ADM) and Pseudospectral Legendre Method (PLM) given in [43] in Tables 5 and 6.

Example 4.8 ([20]):
In this example, we found a numerical solution for the SEIR model based on the four nonlinear ordinary differential equations describing the COVID-19 epidemic in China. It has four elements which are S (susceptible), E (exposed), I (infectious) and R (recovered) and can be Table 11. The maximum errors of system (26). represented as follows:  6) and compared the approximate solution obtained with the approximate solution of the variational iteration method (VIM) and the differential transformation method (DTM) given in [20] in Figure 1 and Table 12. For more information about the model refer to [9].

Example 4.9 ([25]):
In this example, we found a numerical solution for the SEIRU model (28) based on a DDEs (delay differential equations), describing the COVID-19 epidemics that use early reported case data from China [12] in Table 13, Spain [34] in Table 14, and Italy [22] in Table 15 to predict the future number of cases.
This system has five elements which are: S is the number of individuals susceptible to infection, E is the number of asymptomatic noninfectious individuals, I is the number of asymptomatic but infectious individuals, R is the number  VIM  TCM   0  1  1  1  0  0  0  0  2  3  3  3  2  0  0  0  4  8  8  6  4  1  1  1  6  2 0  1 9  1 3  6  3  3  3  8  4 2  4 0  2 8  8  7  7  7  10  79  74  59  10  12  12  16  12  133  124  122  12  20  19  34  14  209  193  237  14  30  28  69  16  309  283  425  16  43  40  129  18  438  396  675  18  60  56  223  20  599  537  923  20  82  74  343 Compartment I Compartment R of reported symptomatic infectious individuals, and U is the number of unreported symptomatic infectious individuals, can be represented in the folowing system: For t ≥ t 0 . This system is supplemented by initial fonctions for t ∈ [−τ ,  (28) for the cumulative number of reported symptomatic infectious cases (t → CR(t)) using TCM. Where, the description of parameters and initial values of the model (28) is given in Tables 16 and  17, respectively. In Figures 2(a), 3(a), and 4(a), we compared the cumulative daily reported case data for these three countries with approximate solutions of this model for the cumulative number of reported symptomatic infectious cases (t → CR(t)) using Taylor collocation method (TCM) with m = 6, N = 6,    We plot the graphs of E(t), I(t), R(t), and U(t) from the numerical simulation of this model in both Figures 2(b), 3(b), and 4(b) for China, Spain, and Italy, respectively.  Table 18. Our numerical experimentations were implemented using the Maple version 16 environment and MS Windows 7 operating system on a PC with Intel Core i7-2630QM CPU @2.00 GHz and 8,00 Go of RAM.

Conclusion
In this paper, we have proposed a collocation method based on the use of Taylor polynomials to approximate the solution of the general system of nonlinear delay integro-differential equations (1) in the spline space S (0) m ( N ). We have shown that the numerical solution is convergent. This method is easy to implement, and the coefficients of the approximation solution are determined by iterative formulas without the need to solve any system of algebraic equations. The numerical examples which were introduced have shown that the method is convergent with good accuracy. We have applied models (27) and (28) to predict a COVID-19 epidemics evolution in China, Spain, and Italy based on reported case data from that three countries. As an example, we give the Maple code of Example 26 in Appendix A. Further researches on this kind of problems will be conducted by generalizing the work done to a system of weakly singular delay integro-differential equations.