Variational Iteration Method and Differential Transformation Method for Solving the SEIR Epidemic Model

&e aim of the present study is to analyze and find a solution for the model of nonlinear ordinary differential equations (ODEs) describing the so-called coronavirus (COVID-19), a deadly and most parlous virus. &e mathematical model based on four nonlinear ODEs is presented, and the corresponding numerical results are studied by applying the variational iteration method (VIM) and differential transformation method (DTM).


Introduction
e whole world is experiencing hardship due to coronavirus (COVID- 19), which was first identified in Wuhan, China, in the month of December 2019. It has been considered that COVID-19 originated from wild animals (bats [1]) and transmitted to humans as numerous infected patients claimed that they had been to a local wet market in Wuhan during the end of November [2]. Later, some investigators confirmed that the virus transmission occurs from person to person [3].
Mathematical models can simulate the effects of a disease at many levels, ranging from how the disease influences the interaction between cells in a single patient (within-host models) to how it spreads across several geographically separated populations (metapopulation models). Models simulating the disease spread within and among populations, such as those used to forecast the COVID-19 outbreak [4], are typically based on the SEIR model. e SEIR model is based on the division of the population under study into four compartments: an individual can either be susceptible (S), exposed to the disease but not yet infectious (E), infectious (I), or recovered (R). e SEIR model can represent many human infectious diseases [5][6][7][8][9]. In this paper, we focus, analyze, and find a solution for the model of nonlinear ordinary differential equations (ODEs) describing the deadly and most parlous coronavirus . A mathematical model based on the four nonlinear ODEs is presented, and the corresponding numerical results are studied by applying the variational iteration method (VIM) and differential transformation method (DTM). e VIM was developed by He [10,11]. In recent years, a great deal of attention has been devoted to the study of this method. e reliability of the method and the reduction in the size of the computational domain make this method applicable to a wide range of model predictions. is method is based on the use of restricted variations and a correction functional, and it was found to have wide applications in finding a solution for the nonlinear ordinary and partial differential equations [12][13][14]. is method does not depend on small parameters in the differential equation and provides a solution (or an approximation to it) as a sequence of iterations. e method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives [15,16]. e DTM is a numerical method for solving differential equations. e concept of the differential transformation was first proposed by Zhou [17], and its main application therein is solving both linear and nonlinear initial value problems in electric circuit analysis. e DTM provides in a fast manner exact values of the n th derivative of an analytical function at a point in terms of known and unknown boundary conditions. is method constructs, for differential equations, an analytical solution in the form of a polynomial.

The SEIR Model
e SEIR model in epidemiology for the spread of an infectious disease is described by the following system of differential equations: Here, β, α, and c are positive parameters and S, E, I, and R denote the fractions of the population that are susceptible, exposed, infectious, and recovered, respectively. A schematic diagram of the disease transmission among the individuals is shown in Figure 1 using the SEIR model.
For more information about the model refer to [18]. e SEIR model of the novel coronavirus (COVID-19) can be represented as follows: (1) e rate of change in the number of susceptible people � the susceptible portion of the population × the average number of people infected by an infectious person over the average duration of infection × the number of people infected by infectious people − the susceptible portion of the population × the rate of infectious animal source + travelers entering − percentage of population traveling out × the number of susceptible people + natural birth rate × the total number of population − the death rate of susceptible people × the number of susceptible people: (2) (2) e rate of change in the number of exposed people � the susceptible portion of the population × the average number of people infected by an infectious person over the average duration of infection × the number of people infected by infectious people + the susceptible portion of the population × the rate of infectious animal source − the number of exposed people over the average latency period − percentage of population traveling out × the number of exposed people − the death rate of the exposed people × the number of exposed people − testing and therapy rate × the number of exposed people: (3) e rate of change in the number of infected people � the number of exposed people over the average latency period − the number of infected people over the average duration of infection − percentage of population traveling out × the number of infected people − the death rate of the infected people × the number of infected people: (4) e rate of change in the number of recovered people � the number of infected people over the average duration of infection − the death rate of the recovered people × the number of recovered people + testing and therapy rate × the number of exposed people Figure 2: e transitions between model classes can now be expressed by the following system of first-order differential equations (Table 1): with the initial conditions

The Variational Iteration Method
To illustrate the basic concepts of the VIM, we consider the following general nonlinear differential equation: where L is a linear operator, N is a nonlinear operator, and F(t) is a known analytical function. We can construct a correction functional according to the variational method as follows: where λ is the general Lagrange multiplier [19], which can be identified optimally via the variational theory, U n is the n th approximate solution, and U n denotes a restricted variation, which means δNU n � 0. Successive approximations, U n+1 , will be obtained by applying the obtained Lagrange multiplier and a properly chosen initial approximation U 0 . Consequently, the solution is given by U � lim n⟶∞ U n . For International Journal of Differential Equations solving equation (6) by means of the VIM, we construct the correctional functional as follows: Here, λ 1 , λ 2 , λ 3 , and λ 4 are general Lagrange multipliers.

The Differential Transformation Method
e basic definition and the fundamental theorems of the DTM and its applicability to various kinds of differential equations are given in [17,21]. According to the operations of differential transformation given in Table 1 in [21], we have the following recurrence relation: e inverse differential transformation of S(k) is defined as follows: when t 0 is taken as zero, the given function S(t) is declared by a finite series, and the above equation can be written in the form By solving the above equations for S(k + 1), E(k + 1), I(k + 1), and R(k + 1) up to order 3, we obtain the functions of S(k), E(k), I(k), and R(k), respectively: International Journal of Differential Equations 5 With initial approximations, S(0) � 2500, E(0) � 1, I(0) � 1, R(0) � 0, and N � 2502 and parameters β � 0.8, α � 0.75, σ � 0.1, c � 0.05, ] � 0.009/N, μ � 0.01, Z � 0.001, ρ I � 0.15, ρ E � 0.15, ϱ I � 0.01, and ϱ I � 0.03, and applying the conditions in equations (16) and (18), we obtain the approximate solution after three terms as follows:

Conclusions
In this paper, we have developed the SEIR model of the COVID-19 epidemic in China that incorporates key features of this pandemic. For solving this model, we used the variational iteration method (VIM) and differential transformation method (DTM). It is found that these methods are effective in providing analytic form solutions for such problems. e comparison of the results obtained by these two methods is in excellent agreement.
For further research, we propose the study of the fractional-order model using the Caputo-Fabrizio derivative [22,23]. In addition, we propose to extend the results of the   International Journal of Differential Equations present paper and combine them with the results in [6] ( Figures 3-6).

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.