RELIABLE ITERATIVE METHODS FOR MATHEMATICAL MODEL OF COVID-19 BASED ON DATA IN ANHUI, CHINA

RELIABLE ITERATIVE METHODS FOR MATHEMATICAL MODEL OF COVID-19 BASED ON DATA IN ANHUI, CHINA SAWSAN MOHSIN ABED, M.A. AL-JAWARY Department of Mathematics, College of Education for Pure Science (Ibn AL-Haytham), University of Baghdad, Baghdad, Iraq Copyright © 2020 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In this paper, five reliable iterative methods: Daftardar-Jafari method (DJM), Tamimi-Ansari method (TAM), Banach contraction method (BCM), Adomian decomposition method (ADM) and Variational iteration method (VIM) to obtain approximate solutions for a mathematical model that represented the coronavirus pandemic (COVID -19 pandemic). The accuracy of the obtained results is numerically verified by evaluating the maximum error remainder. In addition, the approximate results are compared with the fourth-order Runge-Kutta method (RK4) and good agreement have achieved. The convergence of the proposed methods is successfully demonstrated and mathematically verified. All calculations were successfully performed with MATHEMATICA10.


INTRODUCTION
One of the disaster events in 2020 was the spread of COVID -19 around the world. The term COVID -19 can be divided into three sections, the first CO one representing the first letters of the 3
Also, other iterative method called the Banach contraction method (BCM) based on the Banach contraction principle [25]. The BCM, was used to solve different types of differential and integral equations [26]. Moreover, the Adomian decomposition method (ADM), is analytic method that was introduced and developed by George Adomian 1976 [27][28]. ADM is a reliable method to solve many various kinds of problems in applied science. This method has been used by many researchers and has extensive applications of linear and nonlinear ordinary differential equations, partial differential equations and integral equations [29][30]. In addition, the variational iteration method (VIM), is an iterative analytic method that was established by He in (1999) [31,32] and used to solve a wide variety of linear and nonlinear, homogeneous and inhomogeneous equations.
In this paper, the five iterative methods: DJM, TAM, BCM, ADM and VIM will be used to solve the COVID-19 models to obtain approximate solutions. These solutions will be numerically compared with the fourth order Runge-Kutta method. The convergence and some error indicator will be introduced and discussed.
We have organized this paper as follows: The mathematical biological model will be introduced in section 2. In section 3 the basic ideas of the three iterative methods are given for the COVID-19 models. The convergence of the methods will be presented in section 4. Section 5, the solution of the COVID-19 models by the proposed methods will be given. Numerical results and convergence will be presented in section 6. Finally, the conclusion is presented in section 7.

MATHEMATICAL MODELS OF COVID-19
In this section, the epidemic in Anhui, China can be divided into three phases, from January 10 to February 11, 2020 as given in [1].
Phase I (prior to January 23, 2020): The population can be divided into four categories: susceptible ( ), exposed and pre-symptomatic population ( ), symptomatic population ( ), and recovered population ( ), which can be presented as follows: (1) The initial conditions are where, is the transmission rate of disease, controls the infectiousness of exposed presymptomatic individuals relative to symptomatic individuals, is the rate at which exposed individual showing symptom, is the cure rate at which symptomatic individuals moving to recovered class and the total population .

Phase II (between January 23 and February 6, 2020):
On the basis of Eq. (1), we assume all symptomatic cases were quarantined in quarantined class ( ) and change the exposure rate to 1 , cure rate at which quarantined cases moving to recovered class.
which can be presented as follows, see [1].

3.THE BASIC CONCEPTS OF THE PROPOSED ITERATIVE METHOD
In this section the five iterative methods will be introduced to solve a system of ordinary differential equations contains linear and nonlinear terms. Then, in section five the methods will be used to solve the models given in equations (1), (3) and (5) with the given initial conditions.
with the initial conditions where represents the independent variable, the unknown function, are the linear operators, are the nonlinear operators. Let us start by introducing the basic ideas of the following five iterative methods.

The basic concept of the VIM
The VIM can be presented for Eq. (7) in the following form [36]. Where , is a general Lagrange multiplier that can all be optimal way identified via the Variation Theory, and , , = 1,2,3,4 as a restricted variation. The Lagrange multiplier , may be a value or a function, and given the general formula [37].
For the first order = −1.

CONVERGENCE OF THE PROPOSED METHODS
In this section, we introduce the convergence principles of the proposed methods by using some theorems [38]. For the DJM and ADM the convergence can be directly proved. However, to demonstrate the convergence for the TAM, VIM and BCM, we must follow the following procedure: Let, where, is the operator which can be defined by where, is the solution for the problem in the following form, For the TAM: For the BCM or VIM: By using the same conditions which will be used for the approximate iterative method. We get, Therefore, by using (37) and (38), one can get the solution by In the recursive algorithm of DJM, TAM, BCM, ADM and VIM, the following theorems [38] will provide the sufficient condition for achieving the convergence of our proposed methods.
Theorem 4.1 [38]. ''Let introduce in Eq. (38), be an operator from Hilbert space This theorem is a special case of a fixed point theorem which is a sufficient condition for the study of convergence. In another meaning, for each rank , if the parameters are defined ∃0 < < 1 such that ., if the parameters are used for each iteration converges to the exact solution ( ), when 0 ≤ < 1, ∀ = 0, 1, 2 ,…., = 1, 2,3,4. We evaluate the .

SOLVING MATHEMATICAL MODELS OF COVID-19 BY THE PROPOSED METHODS
In this section, the five iterative methods introduced in section three will be used to solve the three phases mathematical models of the COVID-19.
By using the steps given in subsection 3.1, and using the Eqs. (12), (13) and (14), we get : The obtained approximated solution for six iterations by the DJM will be:

Solving phase I by using the TAM:
In order to apply the TAM to solve the model of phase I, we follow similar steps given in subsection 3.2, the we get the following approximate solution for six iterations: (48)

Solving phase I by using the BCM:
To solve the phase I by using the BCM, we follow similar steps given in subsection 3.3, and the approximate solution will be the same to the TAM given in Eq. (48).

Solving phase I by using the ADM:
Let us consider the Eq. (1) with the given initial conditions the Eq. (2). Integrating both sides of Eq. (1) from 0 to and using the given initial conditions, we have where , are the Adomian polynomials, which evaluated from the nonlinear terms and , as follows:

Solving phase I by using VIM:
To solve the phase I by using the VIM, we follow similar steps given in subsection 3.4, and the approximate solution will be the same to the TAM given in Eq. (48).
In this phase, the results for all the DJM, TAM, BCM and VIM are the same. Therefore, we selected the TAM to solve this phase.

Solving phase II by using TAM:
In order to apply the TAM to solve the model of phase II, we follow similar steps given in : and so on. By continuing in this way, we get the following approximate solution for six iterations: where , is the Adomian polynomial, which evaluated from the nonlinear terms , as follows: Then, we get approximate solutions So, the obtained approximated solution for six iterations by the ADM will be:
Once again it was observed that the obtained series approximate solutions via the four methods proposed are the same. So, let solve this phase by using the BCM.

Solving phase III by using BCM:
To solve the phase III by using the BCM, we follow similar steps given in subsection 3.3, by implementing the inverse operator given in (10) and using the given initial conditions, we get: : and so on. We obtain the following approximate solution for six iterations:

Solving phase III by using ADM:
Let us consider the Eq. (54) with the initial conditions given in the Eq. (55). Integrating both sides of Eq. (54) from 0 to and using the given initial conditions, we have Then, we get approximate solutions

Numerical results
In this section, the accuracy of the approximate solutions obtained is investigated for the proposed methods for the three phases. the maximum remaining error will be calculated [18].

For Phase I:
The error remainder of the Phase I can be defined as

RELIABLE ITERATIVE METHODS
The values are less than one, for ≥ 0, 0 < ≤1. Hence, the BCM and ADM are satisfied the convergence condition.

CONCLUSION
This paper implemented five iterative methods: the DJM, TAM, BCM, VIM and ADM to solve the mathematical models that represented the coronavirus pandemic COVID-19 in three phases.
The obtained approximate solutions were presented in a series terms. Moreover, the maximum errors remainder were calculated to verify the convergence of the obtained solutions and it appeared the errors for the DJM, TAM, BCM and VIM are the same and less than the ADM.
Furthermore, the convergence of the proposed methods was demonstrated based on used the Banach fixed point theorem. In addition, the obtained numerical results were compared with the fourth order Runge-Kutta (RK4) method and good matches were achieved.