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

Authors: A. Harir; S. Melliani; H. El Harfi; L. S. Chadli
Journal: International Journal of Differential Equations
 (2020)

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## Abstract

The 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. The 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).


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## Body

## 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:(1)dSdtt=−βStIt,dEdtt=βStIt−αEt,dIdtt=αEt−γIt,dRdtt=γIt.Here, β,α, and γ 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 Figure1 using the SEIR model.Figure 1
SEIR compartmental 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× 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)dSdtt=−βStNIt−ZNSt+ρI+ρE−ϱIN+ϱENSt+νNt−μSt.(2)
The 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)dEdtt=βStNIt+ZNSt−αEt−ϱIN+ϱENEt−μEt−σEt.(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 × the number of infected people − the death rate of the infected people × the number of infected people:(4)dIdtt=αEt−γIt−ϱIN+ϱENIt−μIt.(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 × the number of recovered people + testing and therapy rate × the number of exposed people Figure 2:(5)dRdtt=γIt−μRt+σEt.The transitions between model classes can now be expressed by the following system of first-order differential equations (Table1):(6)dSdtt=−βStNIt−ZNSt+ρI+ρE−ϱIN+ϱENSt+νNt−μSt,dEdtt=βStNIt+ZNSt−αEt−ϱIN+ϱENEt−μEt−σEt,dIdtt=αEt−γIt−ϱIN+ϱENIt−μIt,dRdtt=γIt−μRt+σEt,with the initial conditions(7)S0=S0,E0=E0,I0=I0,R0=R0.Figure 2
The SEIR compartmental model of the novel coronavirus (COVID-19).Table 1
Parameters and initial conditions of the SEIR model.
ParameterDefinitionβAverage number of people infected by an infectious person over the average duration of infectionαNumber of exposed people over the average latency periodγNumber of infected people over the average duration of infectionμThe death rate of peopleνThe birth rate parameter of peopleσThe cure rate of peopleZForce of infection in the baseline scenarioρIAverage daily number of international inbound air passengersρEAverage daily number of domestic inbound air passengersϱIAverage daily number of international outbound air passengersϱEAverage daily number of domestic outbound travellersS0Number of susceptible people at timet=0E0Number of asymptomatic and noninfectious people at timet=0I0Number of asymptomatic but infectious people at timet=0R0Number of recovered people at timet=0NTotal number of population,Nt=St+Et+It+Rt


## 3. The Variational Iteration Method

To illustrate the basic concepts of the VIM, we consider the following general nonlinear differential equation:(8)LUt+NUt=Ft,where L is a linear operator, N is a nonlinear operator, and Ft is a known analytical function. We can construct a correction functional according to the variational method as follows:(9)Un+1t=Unt+∫0tλLUns+NU˜ns−Fsds,where λ is the general Lagrange multiplier [19], which can be identified optimally via the variational theory, Un is the nth approximate solution, and U˜n denotes a restricted variation, which means δNU˜n=0. Successive approximations, Un+1, will be obtained by applying the obtained Lagrange multiplier and a properly chosen initial approximation U0. Consequently, the solution is given by U=limn⟶∞Un. For solving equation (6) by means of the VIM, we construct the correctional functional as follows:(10)Sn+1t=Snt+∫0tλ1τdSnτdτ+βS˜nτNI˜nτ+ZNS˜nτ−ρI+ρE+ϱIN+ϱENS˜nτ−νN˜nτ+μS˜nτdτ,En+1t=Ent+∫0tλ2τdEnτdτ−βS˜nτNI˜nτ−ZNS˜nτ+α+ϱIN+ϱEN+μ+σE˜nτdτ,In+1t=Int+∫0tλ3τdInτdτ−αE˜nτ+γ+ϱIN+ϱEN+μI˜nτdτ,Rn+1t=Rnt+∫0tλ4τdRnτdτ−γI˜nτ+μR˜nτ−σE˜nτdτ.Here, λ1,λ2,λ3,andλ4 are general Lagrange multipliers. Making the above correction functional stationary with respect to Snt,Ent,Int,andRnt, noticing that δS˜nτ=δE˜nτ=δI˜nτ=δR˜nτ=0, yields(11)δSn+1t=δSnt+δ∫0tλ1τdSnτdτ+βS˜nτNI˜nτ+ZNS˜nτ−ρI+ρE+ϱIN+ϱENS˜nτ−νN˜nτ+μS˜nτdτ=0,δEn+1t=δEnt+δ∫0tλ2τdEnτdτ−βS˜nτNI˜nτ−ZNS˜nτ+α+ϱIN+ϱEN+μ+σE˜nτdτ=0,δIn+1t=δInt+δ∫0tλ3τdInτdτ−αE˜nτ+γ+ϱIN+ϱEN+μI˜nτdτ=0,δRn+1t=δRnt+δ∫0tλ4τdRnτdτ−γI˜nτ+μR˜nτ−σE˜nτdτ=0.Therefore, the Lagrange multiplier can readily be identified:λ1τ=−1,λ2τ=−1,λ3τ=−1,andλ4τ=−1. Consequently, the iteration formula can be obtained as follows:(12)Sn+1t=Snt−∫0tdSnτdτ+βS˜nτNI˜nτ+ZNS˜nτ−ρI+ρE+ϱIN+ϱENS˜nτ−νN˜nτ+μS˜nτdτ,En+1t=Ent−∫0tdEnτdτ−βS˜nτNI˜nτ−ZNS˜nτ+α+ϱIN+ϱEN+μ+σE˜nτdτ,In+1t=Int−∫0tdInτdτ−αE˜nτ+γ+ϱIN+ϱEN+μI˜nτdτ,Rn+1t=Rnt−∫0tdRnτdτ−γI˜nτ+μR˜nτ−σE˜nτdτ.With initial approximationsS0=2500,E0=1,I0=1,R0=0, and N=2502, which in turn gives successive approximations, and considering the following values for parameters (see [20]) β=0.8,α=0.75, σ=0.1,γ=0.05,ν=0.009/N,μ=0.01, Z=0.001,ρI=0.15,ρE=0.15,ϱI=0.01,andϱI=0.03, we obtain(13)S1t=2500−25.53132774t,E1t=1−0.059656274t,I1t=1+0.689984012t,R1t=0.15t.Similarly, we get the following system after two terms:(14)S2t=2500−25.53132774t−0.143869926t2+1.87756013310−3t3,E2t=1−0.059656274t+0.29733881t2−1.87756013310−3t3,I2t=1+0.689984012t−0.043076138t2,R2t=0.15t+0.013516786t2.For the solution after three terms, we can write(15)S3t=2500−25.53132774t−0.143869926t2+0.013851166t3−8.48550280810−5t4−4.79158897410−7t5+4.31004739710−9t6,E3t=1−0.059656274t+0.29733881t2−0.09860941t3+4.8381106310−4t4−4.79158897410−7t5+4.31004739710−9t6,I3t=1+0.689984012t−0.043076138t2+0.074410709t3−3.52042524910−4t4,R3t=0.15t+0.013516786t2+9.1483020810−3t3−4.69390033310−5t4.


## 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:(16)Sk+1=1k+1−βN∑m=0kSmIk−m−ZNSk+ρI+ρEδk+νNk−ϱIN+ϱEN+μSk,Ek+1=1k+1βN∑m=0kSmIk−m+ZNSk−α+ϱIN+ϱEN+μ+σEk,Ik+1=1k+1αEk−γ+ϱIN+ϱEN+μIk,Rk+1=1k+1γIk−μRk+σEk.The inverse differential transformation ofSk is defined as follows: when t0 is taken as zero, the given function St is declared by a finite series, and the above equation can be written in the form(17)St=∑k=0∞Sktk.By solving the above equations forSk+1,Ek+1,Ik+1, and Rk+1 up to order 3, we obtain the functions of Sk,Ek,Ik, and Rk, respectively:(18)St=∑k=03Sktk,Et=∑k=03Ektk,It=∑k=03Iktk,Rt=∑k=03Rktk.With initial approximations,S0=2500,E0=1,I0=1,R0=0, and N=2502 and parameters β=0.8,α=0.75, σ=0.1,γ=0.05,ν=0.009/N,μ=0.01, Z=0.001,ρI=0.15,ρE=0.15,ϱI=0.01,andϱI=0.03, and applying the conditions in equations (16) and (18), we obtain the approximate solution after three terms as follows:(19)St=2500−25.53132774t−0.143869688t2+0.013851335t3,Et=1−0.059656274t+0.297292652t2−0.099439782t3,It=1+0.689984012t−0.043076138t2+0.075184915t3,Rt=0.15t+0.013516786t2+9.1467634810−3t3.


## 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).Figure 3
Comparison of solutions of the proposed model (6) for compartment S using VIM and DTM.Figure 4
Comparison of solutions of the proposed model (6) for compartment E using VIM and DTM.Figure 5
Comparison of solutions of the proposed model (6) for compartment I using VIM and DTM.Figure 6
Comparison of solutions of the proposed model (6) for compartment R using VIM and DTM.



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