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

1. Introduction

The 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.

The 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). The SEIR model can represent many human infectious diseases [5-9]. 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 (COVID-19). 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).

The VIM was developed by He [10, 11]. In recent years, a great deal of attention has been devoted to the study of this method. The 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. This 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 [12-14]. This 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. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives [15, 16].

The DTM is a numerical method for solving differential equations. The 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. The DTM provides in a fast manner exact values of the nth derivative of an analytical function at a point in terms of known and unknown boundary conditions. This method constructs, for differential equations, an analytical solution in the form of a polynomial.

2. The SEIR Model

The SEIR model in epidemiology for the spread of an infectious disease is described by the following system of differential equations:

[mathematical expression not reproducible]. (1)

Here, [beta], [alpha], and [gamma] 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].

The SEIR model of the novel coronavirus (COVID-19) can be represented as follows:

(1) The rate of change in the number of susceptible people = the susceptible portion of the population x the average number of people infected by an infectious person over the average duration of infection x the number of people infected by infectious people the susceptible portion of the population x the rate of infectious animal source + travelers entering percentage of population traveling out x the number of susceptible people + natural birth rate x the total number of population-the death rate of susceptible people x the number of susceptible people:

[mathematical expression not reproducible]. (2)

(2) The rate of change in the number of exposed people = the susceptible portion of the population x the average number of people infected by an infectious person over the average duration of infection x the number of people infected by infectious people + the susceptible portion of the population x the rate of infectious animal source-the number of exposed people over the average latency period-percentage of population traveling out x the number of exposed people-the death rate of the exposed people x the number of exposed people-testing and therapy rate x the number of exposed people:

[mathematical expression not reproducible]. (3)

(3) The 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 x the number of infected people-the death rate of the infected people x the number of infected people:

[mathematical expression not reproducible]. (4)

(4) The 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 x the number of recovered people + testing and therapy rate x the number of exposed people Figure 2:

dR/dt (t) = [gamma]I(t)-[mu]R(t) + [sigma]E(t). (5)

The transitions between model classes can now be expressed by the following system of first-order differential equations (Table 1):

with the initial conditions

S(0) = [S.sub.0], E(0) = [E.sub.0], I(0) = [I.sub.0], R(0) = [R.sub.0]. (7)

3. The Variational Iteration Method

To illustrate the basic concepts of the VIM, we consider the following general nonlinear differential equation:

LU(t)+ NU(t) = F(t), (8)

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:

[mathematical expression not reproducible], (9)

where [lambda] is the general Lagrange multiplier [19], which can be identified optimally via the variational theory, [U.sub.n] is the nth approximate solution, and Un denotes a restricted variation, which means [delta]N[[??].sub.n] = 0. Successive approximations, [U.sub.n+1], will be obtained by applying the obtained Lagrange multiplier and a properly chosen initial approximation [U.sub.0]. Consequently, the solution is given by [U = [lim.sub.n[right arrow][infinity]] [U.sub.n]. For solving equation (6) by means of the VIM, we construct the correctional functional as follows:

[mathematical expression not reproducible]. (10)

Here, [[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3], and [[lambda].sub.4] are general Lagrange multipliers. Making the above correction functional stationary with respect to [S.sub.n] (t), [E.sub.n](t), [I.sub.n] (t), and [R.sub.n](t), noticing that [mathematical expression not reproducible], yields

[mathematical expression not reproducible]. (11)

Therefore, the Lagrange multiplier can readily be identified:

[mathematical expression not reproducible]. Consequently, the iteration formula can be obtained as follows:

With initial approximations S(0) = 2500, E(0) = 1, I(0) = 1,R(0) = 0, and N = 2502, which in turn gives successive approximations, and considering the following values for parameters (see [20]) P = 0.8, a = 0.75,

[mathematical expression not reproducible]. (12)

Similarly, we get the following system after two terms:

[mathematical expression not reproducible]. (14)

For the solution after three terms, we can write

4. The Differential Transformation Method

The 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:

The inverse differential transformation of S(k) is defined as follows: when [t.sub.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

S(t) = [[infinity].summation over (k=0)] S(k)[t.sup.k]. (17)

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:

[mathematical expression not reproducible]. (18)

With initial approximations, S(0) = 2500, E(0) = 1, I(0) = 1,R(0) = 0, and N = 2502 and parameters [beta] = 0.8, [alpha] = 0.75, [sigma] = 0.1, y = 0.05, [mu] = 0.009/N, [mu] = 0.01, Z = 0.001, [[rho].sub.I] = 0.15, [[rho].sub.E] = 0.15, [[??].sub.I] = 0.01, and [[??].sub.I] = 0.03, and applying the conditions in equations (16) and (18), we obtain the approximate solution after three terms as follows:

[mathematical expression not reproducible]. (19)

5. 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. The 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 present paper and combine them with the results in [6] (Figures 3-6).

https://doi.org/10.1155/2020/3521936

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

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[2] H. Lu, C. W. Stratton, and Y. W. Tang, "Outbreak of pneumonia of unknown etiology in Wuhan China: the mystery and the miracle," Journal of Medical Virology, vol. 92, no. 4, pp. 401-402, 2020.

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[18] M. H. A. Biswas and L. T. Paiva, "A SEIR model for control of infectious diseases with constraints," Mathematical Biosciences and Engineering, vol. 11, no. 4, 2014.

[19] M. Inokuti, H. Sekine, and T. Mura, "General use of the Lagrange multiplier in nonlinear mathematical physics," in Variational Method in the Mechanics of Solids, S. Nemat-Nassed, Ed., Pergemon Press, Oxford, UK, pp. 156-162, 1978.

[20] T. M. Chen, J. Rui, Q. P. Wang, Z. Y. Zhao, J. A. Cui, and L. Yin, "A mathematical model for simulating the phase-based transmissibility of a novel coronavirus," Infectious Diseases of Poverty, vol. 9, p. 24, 2020.

[21] S. V. Kanth and K. Aruna, "Two-dimensional differential transform method for solving linear and non-linear Schrodinger equations," Chaos, Solitons & Fractals, vol. 41, no. 5, pp. 2277-2281, 2009.

[22] K. Shah, M. A. Alqudah, F. Jarad, and T. Abdeljawad, "Semi-analytical study of Pine Wilt Disease model with convex rate under Caputo-Febrizio fractional order derivative," Chaos, Solitons & Fractals, vol. 135, Article ID 109754, 2020.

[23] K. Shah, F. Jarad, and T. Abdeljawad, "On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative," Alexandria Engineering Journal, 2020, In press.

A. Harir [ID], (1) S. Melliani [ID], (2) H. El Harfi, (1) and L. S. Chadli (1)

(1) Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, B.P. 523, Beni Mellal, Morocco

(2) Laboratory of Flows and Transfers Modelling, Sultan Moulay Slimane University, B.P. 523, Beni Mellal, Morocco

Correspondence should be addressed to A. Harir; atimad.harir@gmail.com

Received 23 April 2020; Revised 29 May 2020; Accepted 3 June 2020; Published 2 July 2020

Academic Editor: Sining Zheng

Caption: FIGURE 1: SEIR compartmental model.

Caption: FIGURE 2: The SEIR compartmental model of the novel coronavirus (COVID-19).

Caption: FIGURE 3: Comparison of solutions of the proposed model (6) for compartment S using VIM and DTM.

Caption: FIGURE 4: Comparison of solutions of the proposed model (6) for compartment E using VIM and DTM.

Caption: FIGURE 5: Comparison of solutions of the proposed model (6) for compartment I using VIM and DTM.

Caption: FIGURE 6: Comparison of solutions of the proposed model (6) for compartment R using VIM and DTM.
Table 1: Parameters and initial conditions of the SEIR model.

Parameter                             Definition

[beta]           Average number of people infected by an infectious
                    person over the average duration of infection
[alpha]           Number of exposed people over the average latency
                                        period
[gamma]          Number of infected people over the average duration
                                     of infection
[mu]                           The death rate of people
v                         The birth rate parameter of people
[sigma]                        The cure rate of people
Z                    Force of infection in the baseline scenario
[[rho].sub.I]       Average daily number of international inbound
                                    air passengers
[[rho].sub.E]       Average daily number of domestic inbound air
                                      passengers
[[??].sub.I]     Average daily number of international outbound air
                                      passengers
[[??].sub.E]     Average daily number of domestic outbound travellers
[S.sub.0]             Number of susceptible people at time t = 0
[E.sub.0]        Number of asymptomatic and noninfectious people at
                                      time t = 0
[I.sub.0]          Number of asymptomatic but infectious people at
                                      time t = 0
[R.sub.0]              Number of recovered people at time t = 0
N                 Total number of population, N (t) = S(t) + E(t) +
                                     I(t) + R(t)
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Title Annotation:Research Article
Author:Harir, A.; Melliani, S.; Harfi, H. El; Chadli, L.S.
Publication:International Journal of Differential Equations
Geographic Code:9CHIN
Date:Jul 31, 2020
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