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Mahdi A. Sabaa                                                          Sinan Hatif Abd Almjeed 

                                       
   

 

 

 

 
 

Abstract 

     In this paper, our aim is to solve analytically a nonlinear social epidemic model as an 

initial value problem (IVP) of ordinary differential equations. The mathematical social 

epidemic model under study is applied to alcohol consumption model in Spain. The economic 

cost of alcohol consumption in Spain is affected by the amount of alcohol consumed. This 

paper refers to the study of alcohol consumption using some analytical methods. Adomian 

decomposition and variation iteration methods for solving alcohol consumption model have 

used. Finally, a compression between the analytic solutions of the two used methods and the 

previous actual values from 1997 to 2007 years is obtained using the absolute and relative 

errors. The analysis results obtained have been discussed tabularly and graphically. 
 

Keywords: Ordinary differential equations, Adomian decomposition method, Variation    

iteration method, social epidemiology.                                                                                                     

1. Introduction 

     The type of epidemiological models had been used in the study of many social diseases. In 

recent years, several researchers were interested to study and analyze the social epidemics, 

ecstasy or heroin addiction [1, 2]. Smoking habit evaluation in Spain [3-5]. a cocaine abuse in 

Spain [6, 7]. Campaigns on reducing excess weight in Valencia [8-10]. 

     Alcohol consumption habit is considered a disease that spread out rapidly by social 

pressure or social contact which influences the spread of social disease. Recently, the rate of 

alcohol consumption has increased more with the developing countries, so alcohol 

consumption represents a big problem that efforts not only on the human health, but also on 

the community economy. The public health care becomes more expensive when the number 

of people who are suffering from such diseases increased, so the community economy will be 

damaged. With regard to the health effects of chloride that impact on the healthy body, the 

chloride can damage some parts of the body, such as heart and liver, influence also on the 

other functions. An addition that, the cost of alcohol affects the economy [11, 12].  

     Adomian decomposition method (ADM) is a reliable method to solve many various kinds 

of problems so it is a trusty method which emerging in applied science, linear and nonlinear 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi:10.30526/32.3.2292 

  Maha A. Mohammed    

   

sinan.h.a@ihcoedu.uobaghdad.edu.iq 

Approximate Solutions for Alcohol Consumption Model in Spain 

 

Article history: Received 28 March 2019, Accepted 12 May 2019,Publish September 

2019 

 

Department of Mathematics, College of Education for Pure Science, Ibn Al-Haitham 

University of Baghdad, Baghdad, Iraq. 

 
mahdiabd281@gmail.com       mahasssa@yahoo.com                            

                             

mailto:sinan.h.a@ihcoedu.uobaghdad.edu.iq
mailto:sinan.h.a@ihcoedu.uobaghdad.edu.iq
http://en.uobaghdad.edu.iq/?page_id=15060
mailto:mahdiabd281@gmail.com
mailto:mahdiabd281@gmail.com
mailto:mahasssa@yahoo.com
mailto:mahasssa@yahoo.com


  

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as well. The Adomian decomposition method was introduced and developed by George 

Adomian. This method has been used by many researchers as well as it has extensive 

applications of linear and nonlinear ordinary differential equations, partial differential 

equations and integral equations [13, 14]. The modified ADM and its applications on some 

equations have also been given; see examples in [15-18]. ADM was introduced and developed 

by George Adomian, 1976. This method took a large part of the researcher's work, where it 

was applied widely on the linear and nonlinear differential equations, as well as on the 

integral equations [19, 14]. Some researchers used the ADM to solve a system of ordinary 

differential equations [20]. and apply at epidemic model [21, 22]. As well as, ADM solved a 

system of integral–differential equations [23]. Recently, the accuracy of nonlinear singular 

initial value problems was discussed using a semi-analytic [24]. ADM is applied to solve 

fuzzy fractional order differential algebraic equations [25]. Modified Adomian 

Decomposition Method was applied on Integro-Differential Inequality [26]. 

     The Variational Iteration method (VIM) was established by Ji-Huan in 1997 where it is 

considered as one of the reliable iterative methods that give approximate solutions for the of 

differential equations [27, 28]. This method is used widely in scientific and engineering 

applications, where it is used to solve linear, nonlinear and homogeneous and inhomogeneous 

equations. VIM is used for the analytical purpose because it is an effective and reliable way in 

the approaching quickly to the exact solution. The VIM method differs from the ADM, where 

VIM does not require specific treatments for nonlinear problems [14]. Many works to solve 

nonlinear problems using VIM [27]. With autonomous ordinary differential systems [29]. 

And to solve differential equations that have fractional order [30]. 

      The application of real social epidemic alcohol consumption model is considered in this 

paper, in order to study the behavior of this social epidemic during sometimes under study. 

The feature of such model under study that the model has multiple parameters and 

multivariate. As well as, it is a nonlinear system of first-order ordinary differential equations 

for autonomous initial value problem that don't have mostly exact solution. Our interested is 

to solve analytically such these models. This study is organized as follows: in Section 2, the 

mathematical model of alcohol consumption is described; Section 3 derived the analytic 

methods ADM and VIM to solve the nonlinear system of alcohol consumption model in 

Spain. Following, in Section 4, the results of the presented methods is discussed. Finely, 

Section 5 is devoted to the conclusion of the study. 

2.  Mathematical Model 

     Alcohol Ireland and the United Kingdom, this problem was discussed and the data were 

reported from the first year of 2002 until the end of 2014 [31]. In Spain, the effects of the 
different usage of alcohol between the female and male in  Spanish university alumni was 

studied [32].The mathematical model of alcohol consumption was explained and described in 

the current study by Santonja et al., (2010). This model consists of three subpopulations of 

Spanish population of the age of 15-64 years old from 1997 to 2007 years that represented as 

the nonlinear system of three ordinary differential equations of the first order. This system is 

referred to analyze the changing in the social epidemic stages (non-drink alcohol people, non-

risk-drink alcohol people and risk-drink alcohol people), see Table 1. The parameters of the 

model are described in Table 2. Alcohol consumption model for the Spanish public appears 



  

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as the following nonlinear system of ordinary differential equations, [12]. consumption is a 

dangerous social habit that spread in many countries such as  

𝑎ˊ(𝑡) = 𝛼 + 𝛽 ∗ 𝑟 − 𝜇 ∗ 𝑎 − 𝜉 ∗ 𝑎 ∗ (𝑚 + 𝑟) − 𝑎 ∗ (𝛼 − 𝜇 ∗ 𝑎 − 𝜎 ∗ 𝑚 − 𝜎 ∗ 𝑟)                (1) 

𝑚ˊ(𝑡) = 𝜉 ∗ 𝑎 ∗ (𝑚 + 𝑟) − 𝜂 ∗ 𝑚 + 𝜇 ∗ 𝑎 ∗ 𝑚 − 𝜎 ∗ 𝑎 ∗ 𝑚 − 𝛼 ∗ 𝑚                                    (2) 

𝑟ˊ(𝑡) = 𝜂 ∗ 𝑚 − 𝛽 ∗ 𝑟 + 𝜇 ∗ 𝑎 ∗ 𝑟 − 𝜎 ∗ 𝑎 ∗ 𝑟 − 𝛼 ∗ 𝑟                                                                (3) 

The initial conditions of equations (1-3) in 1997 are 𝑎(𝑡 = 0) = 0.362, 𝑚(𝑡 = 0) = 0.581, 

𝑟(𝑡 = 0) = 0.057,  with the predicted parameters are given as: 𝛼 = 0.01, 𝛽 = 0.0014, 

𝜇 = 0.008, 𝜉 = 0.0284534, 𝜎 = 0.009, and  𝜂 = 0.000110247,[12]. 

 

Table 1. Variables of alcohol consumption model, [12]. 

 

𝒂(𝒕) Non-drink alcohol people are subpopulations who never drink alcohol in their life. 

𝒎(𝒕) Non-risk-drink alcohol people are subpopulations who drink a little liquid of alcohol alcohol that 

means the men who drink less than 50 cc and women who drink less than 30 cc of alcohol every day.  

𝒓(𝒕) Risk-drink alcohol people are subpopulations who drink a lot of alcohol that means the men who drink 

more than 50 cc and the women who drink more than 30 cc of alcohol every day.  

 
Table 2. Parameters of alcohol consumption model, [12]. 

 

Birth rate in Spain 𝜶 
The rate at which a risk-drink alcohol people becomes a non-drink alcohol people 𝜷 
Death rate in Spain 𝝁 
Transmission rate because social pressure that leads to increasing the alcohol drinking 𝝃 
Growing death rate due to alcohol drinking 𝝈 
A rate that transmits a non-risk-drink alcohol people moves to the risk-drink alcohol 

people 
𝜼 

 

3. Analytic Methods 

    In this section some analytic methods are introduced such  

3.1 Adomian Decomposition Method 

     We can solve the nonlinear system of equations (1-3) by using the Adomian decomposition 

method with the above initial condition and the given parameters. Let 𝑙 be an operator that is 

given by =
𝑑

𝑑𝑡
 , the inverse of this operation is 𝑙−1 = ∫ (. )𝑑𝑡

𝑡

0
, then by applying 𝑙−1 for both 

sides of equations (1-3), we obtain: 

 𝑎(𝑡) − 𝑎(0) = 𝑙−1 (𝛼 + 𝛽 ∗ 𝑟(𝑡) − 𝜇 ∗ 𝑎(𝑡) − 𝜉 ∗ 𝑎(𝑡) ∗ (𝑚(𝑡) + 𝑟(𝑡)) − 𝑎(𝑡) ∗

                             (𝛼 − 𝜇 ∗ 𝑎(𝑡) − 𝜎 ∗ 𝑚(𝑡) − 𝜎 ∗ 𝑟(𝑡))), where 𝑎0(𝑡) = 0.362. 



  

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 Similarity,                                                                           

𝑚(𝑡) − 𝑚(0) = 𝑙−1 (𝜉 ∗ 𝑎(𝑡) ∗ (𝑚(𝑡) + 𝑟(𝑡)) − 𝜂 ∗ 𝑚(𝑡) + 𝜇 ∗ 𝑎(𝑡) ∗ 𝑚(𝑡) − 𝜎 ∗ 𝑎(𝑡) ∗

                               𝑚(𝑡) − 𝛼 ∗ 𝑚(𝑡)), where 𝑚0(𝑡) = 0.581, and 

𝑟(𝑡) − 𝑟(0) = 𝑙−1 (𝜂 ∗ 𝑚(𝑡) − 𝛽 ∗ 𝑟(𝑡) + 𝜇 ∗ 𝑎(𝑡) ∗ 𝑟(𝑡) − 𝜎 ∗ 𝑎(𝑡) ∗ 𝑟(𝑡) − 𝛼 ∗ 𝑟(𝑡)), 

where 𝑟0(𝑡) = 0.057.  

The above equations equivalent the following equations (4-6): 

𝑎𝑘+1(𝑡) = 𝑙
−1(𝛼 + 𝛽 ∗ 𝑟𝑘 (𝑡) − 𝜇 ∗ 𝑎𝑘 (𝑡) − 𝜉 ∗ 𝐴𝑘 (𝑡) − 𝜉 ∗ 𝐵𝑘 (𝑡) − 𝛼 ∗ 𝑎𝑘 (𝑡) + 𝜇 ∗

                    𝐶𝑘 (𝑡) + 𝜎 ∗ 𝐴𝑘 (𝑡) + 𝜎 ∗ 𝐵𝑘 (𝑡)), forall𝑘 ≥ 0.                                                             (4)                                                                

𝑚𝑘+1(𝑡) = 𝑙
−1(𝜉 ∗ (𝐴𝑘 (𝑡) + 𝐵𝑘 (𝑡)) − 𝜂 ∗ 𝑚𝑘 (𝑡) + 𝜇 ∗ 𝐴𝑘 (𝑡) − 𝜎 ∗ 𝐴𝑘 (𝑡) − 𝛼 ∗ 𝑚𝑘 (𝑡)), for 

all  𝑘 ≥ 0.                                                                           (5)                                                                                                                                                                                                                                                  

 𝑟𝑘+1(𝑡) = 𝑙
−1(𝜂 ∗ 𝑚𝑘 (𝑡) − 𝛽 ∗ 𝑟𝑘 (𝑡) + 𝜇 ∗ 𝐵𝑘 (𝑡) − 𝜎 ∗ 𝐵𝑘 (𝑡) − 𝛼 ∗ 𝑟𝑘 (𝑡)),  

for all  𝑘 ≥ 0.                                                                                                                            (6) 

 

The general form of the non-linear borders 𝐴𝑘 (𝑡), 𝐵𝑘 (𝑡) and 𝐶𝑘 (𝑡) have to be: 

 

𝐴𝑘 (𝑡) = (∑ 𝑎𝑛(𝑡)
2
𝑛=0 ) ∗ (∑ 𝑚𝑛(𝑡)

2
𝑛=0 ),    for  all 𝑘 = 0,1,2                                                  (7)  

 

𝐵𝑘 (𝑡) = (∑ 𝑎𝑛(𝑡)
2
𝑛=0 ) ∗ (∑ 𝑟𝑛 (𝑡)

2
𝑛=0 ),      for  all 𝑘 = 0,1,2                                                  (8) 

 

𝐶𝑘 (𝑡) = (∑ 𝑎𝑛
2
𝑛=0 (𝑡)),

2                             for  all 𝑘 = 0,1,2                                                 (9) 

 

The non-linear borders of 𝐴0(𝑡), 𝐵0(𝑡) and 𝐶0(𝑡) as the following: 

𝐴0(𝑡) = 𝑎0(𝑡) ∗ 𝑚0(𝑡), 𝐵0(𝑡) = 𝑎0(𝑡) ∗ 𝑟0(𝑡),  𝐶0(𝑡) = (𝑎0(𝑡)).
2    

Substituting for all 𝑎0(𝑡), 𝑟0(𝑡), 𝐴0(𝑡), 𝐵0(𝑡) and 𝐶0(𝑡)  by Eq.(4), to obtain 

𝑎1(𝑡) = −0.00009881𝑡, 

Substituting for all 𝑚0(𝑡), 𝐴0(𝑡) and 𝐵0(𝑡)  by Eq.(5), to find 

𝑚1(𝑡) = 0.00048711𝑡, 

Substituting for all 𝑚0(𝑡), 𝑟0(𝑡) and 𝐵0(𝑡) by Eq.(6), to get 

𝑟1(𝑡) = −0.00060638𝑡, 

Now we will find 𝑎2(𝑡), 𝑚2(𝑡) and 𝑟2(𝑡).  

The non-linear borders of 𝐴1(𝑡), 𝐵1(𝑡) and 𝐶1(𝑡) are given in the following formula: 

𝐴1(𝑡) = 𝑎0(𝑡) ∗ 𝑚1(𝑡) + 𝑚0(𝑡) + 𝑎1(𝑡), 

𝐵1(𝑡) = 𝑎0(𝑡) ∗ 𝑟1(𝑡) + 𝑟0(𝑡) ∗ 𝑎1(𝑡), 



  

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𝐶1(𝑡) = 2 ∗ 𝑎0(𝑡) ∗ 𝑎1(𝑡), 

Substituting for all  𝑎1(𝑡), 𝑟1(𝑡).  𝐴1(𝑡), 𝐵1(𝑡) and 𝐶1(𝑡) by Eq.(4), to obtain 

𝑎2(𝑡) = 1.2117815 ∗ 10
−8𝑡2 

Substituting also for all 𝑚1(𝑡), 𝐴1(𝑡) and 𝐵1(𝑡) by Eq.(5), to get 

𝑚2(𝑡) = −0.00000403𝑡
2, 

Substituting for all  𝑚1(𝑡), 𝑟1(𝑡)𝑎𝑛𝑑 𝐵1(𝑡) by Eq.(6), to have 

𝑟2(𝑡) = 0.00000359𝑡
2, 

At the same previous steps, the non-linear borders of 𝐴2(𝑡), 𝐵2(𝑡) and 𝐶2(𝑡) are given as: 

𝐴2(𝑡) = 𝑎0(𝑡) ∗ 𝑚2(𝑡) + 𝑎1(𝑡) ∗ 𝑚1(𝑡) + 𝑚0(𝑡) ∗ 𝑎2(𝑡), 

𝐵2(𝑡) = 𝑎0(𝑡) ∗ 𝑟2(𝑡) + 𝑎1(𝑡) ∗ 𝑟1(𝑡) + 𝑟0(𝑡) ∗ 𝑎2(𝑡), 

𝐶2(𝑡) = 2 ∗ 𝑎0(𝑡) ∗ 𝑎2(𝑡) + (𝑎1(𝑡))
2

, 

Substituting for all  𝑎2(𝑡), 𝑟2(𝑡). 𝐴2(𝑡), 𝐵2(𝑡) and 𝐶2(𝑡) by Eq.(4), to be 

𝑎3(𝑡) = 2.55439073 ∗ 10
−11𝑡3, 

Substituting for all 𝑚2(𝑡), 𝐴2(𝑡) and 𝐵2(𝑡) by Eq.(5), to find 

𝑚3(𝑡) = 1.27758808 ∗ 10
−8𝑡3, 

Substituting for all  𝑚2(𝑡), 𝑟2(𝑡) and 𝐵2(𝑡) by Eq.(6), to get 

𝑟3(𝑡) = −1.42663059 ∗ 10
−8𝑡3. 

The Adomian decomposition method assumes that the unknown functions 𝑎(𝑡), 𝑚(𝑡) and 

𝑟(𝑡) that can be written by series as follows: 

 

𝑎(𝑡) = ∑ 𝑎𝑘 (𝑡),
∞
𝑘=0  𝑚(𝑡) = ∑ 𝑚𝑘 (𝑡)

∞
𝑘=0 , 𝑟(𝑡) = ∑ 𝑟𝑘 (𝑡),

∞
𝑘=0           

                                                    

∑ 𝑎𝑘 (𝑡)

∞

𝑘=0

= 𝑎1(𝑡) + 𝑎2(𝑡) + 𝑎3(𝑡) + ⋯ 

𝑎(𝑡) = 0.362  − 0.00009881𝑡 + 1.21178155 ∗ 10−8𝑡2 + 2.55439073 ∗ 10−11𝑡3 +       (10)  

∑ 𝑚𝑘 (𝑡)

∞

𝑘=1

= 𝑚1(𝑡) + 𝑚2(𝑡) + 𝑚3(𝑡) + ⋯ 

𝑚(𝑡) = 0.581  + 0.00048711𝑡 − 0.00000403𝑡2 + 1.27758808 ∗ 10−8𝑡3 +                    (11)                   



  

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∑ 𝑟𝑘 (𝑡)

∞

𝑘=1

= 𝑟1(𝑡) + 𝑟2(𝑡) + 𝑟3(𝑡) + ⋯ 

𝑟(𝑡) = 0.057  − 0.00060638𝑡 + 0.00000359𝑡2 − 1.42663059 ∗ 10−8𝑡3 +                   (12)                     

 𝑎(𝑡), 𝑚(𝑡) and 𝑟(𝑡) of ADM results are unsettled terms.                   

3.2 Variation Iteration Method 

      The VIM gives a better approximate solution by constructing a correctional functional that 

uses an initial function. Where Lagrange multiplier considers the key of the correction 

functional which can be specified via variation theory [14]. The nonlinear system of alcohol 

model under study, [12]. can be solved by using the VIM with given initial condition and 

parameters. The correction functional of the system (1-3) becomes: 

𝑎𝑘+1(𝑡) = 𝑎𝑘 (𝑡) + ∫ 𝜆 (𝑎ˊ𝑘 (𝑡) − (𝛼 + 𝛽 ∗ 𝑟𝑘 (𝑡) − 𝜇 ∗ 𝑎𝑘 (𝑡) − 𝜉 ∗ 𝑎𝑘 (𝑡) ∗ (𝑚𝑘 (𝑡) +
𝑡

0

𝑟𝑘 (𝑡)) − 𝑎𝑘 (𝑡) ∗ (𝛼 − 𝜇 ∗ 𝑎𝑘 (𝑡) − 𝜎 ∗ 𝑚𝑘 (𝑡) − 𝜎 ∗ 𝑟𝑘 (𝑡)))) 𝑑𝑡, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘 ≥ 0.               (13) 

𝑚𝑘+1(𝑡) = 𝑚𝑘 (𝑡) + ∫ 𝜆 (𝑚ˊ𝑘 (𝑡) − (𝜉 ∗ 𝑎𝑘 (𝑡) ∗ (𝑚𝑘 (𝑡) + 𝑟𝑘 (𝑡)) − 𝜂 ∗ 𝑚𝑘 (𝑡) + 𝜇 ∗ 𝑎𝑘 (𝑡) ∗
𝑡

0

𝑚𝑘 (𝑡) − 𝜎 ∗ 𝑎𝑘 (𝑡) ∗ 𝑚𝑘 (𝑡) − 𝛼 ∗ 𝑚𝑘 (𝑡))) 𝑑𝑡, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘 ≥ 0.                                            (14)                                           

𝑟𝑘+1(𝑡) = 𝑟𝑘 (𝑡) + ∫ 𝜆 (𝑟ˊ𝑘 (𝑡) − (𝜂 ∗ 𝑚𝑘 (𝑡) − 𝛽 ∗ 𝑟𝑘 (𝑡) + 𝜇 ∗ 𝑎𝑘 (𝑡) ∗ 𝑟𝑘 (𝑡) − 𝜎 ∗ 𝑎𝑘 (𝑡) ∗
𝑡

0

𝑟𝑘 (𝑡) − 𝛼 ∗ 𝑟𝑘 (𝑡))) 𝑑𝑡, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘 ≥ 0                                                                                   (15) 

    The Lagrange multiplier is 𝜆 = −1 in equations (10-12). By substituting this value in 

equations (10), (11) and (12). The zero borders will be:  

𝑎0(𝑡) = 0.362, 𝑚0(𝑡) = 0.581, 𝑟0(𝑡) = 0.057. 

In equations (13-15), if we substitute (k=0), we obtain the following 𝑎1(𝑡), 𝑚1(𝑡) and 𝑟1(𝑡). 

𝑎1(𝑡) = 0.362  + 0.00011927𝑡. 

𝑚1(𝑡) = 0.581  + 0.00048711𝑡. 

𝑟1(𝑡) = 0.057  − 0.00060638𝑡. 

By the same way, if we have (k=1), in equations (13-15) we get the following 

𝑎2(𝑡), 𝑚2(𝑡) and 𝑟2(𝑡). 

𝑎2(𝑡) = 0.362  + 0.00011927𝑡 − 0.00000147𝑡
2 + 1.30183484 ∗ 10−10𝑡3,   

𝑚2(𝑡) = 0.581  + 0.00048711𝑡 − 0.00000212𝑡
2 − 1.54291666 ∗ 10−10𝑡3,    

𝑟2(𝑡) = 0.057  − 0.00060638𝑡 + 0.00000359𝑡
2 + 2.41081825 ∗ 10−11𝑡3,        

Continuing in the same manner when (k=2), we can a achieved the following 

𝑎3(𝑡), 𝑚3(𝑡) and 𝑟3(𝑡). 

𝑎3(𝑡) = 0.362  + 0.00011928𝑡 − 0.00000147𝑡
2 + 1.04339442 ∗ 10−8𝑡3 − 2.97475940 ∗

                10−12𝑡4 + 1.20789692 ∗ 10−14𝑡5 − 1.75446808 ∗ 10−18𝑡6 + 6.64675814 ∗

                10−23𝑡7                                                                                                                    (16)                                         

𝑚3(𝑡) =

 0.581  + 0.00048710𝑡 − 0.00000212𝑡2 + 3.66528836 ∗ 10−9𝑡3 +

                  3.38204598 ∗ 10−12𝑡4 − 1.31514568 × 10−14𝑡5 + 1.82643463 ∗ 10−18𝑡6 −

                  6.60192260 ∗ 10−23𝑡7                                                                                         (17)                                                                                                        



  

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𝑟3(𝑡) = 0.057  − 0.0006064𝑡 + 0.00000359𝑡
2 − 1.40992325 ∗ 10−8𝑡3 − 4.07286579 ∗

              10−13𝑡4 + 1.07248763 ∗ 10−15𝑡5 − 7.19665484 ∗ 10−20𝑡6 − 4.48355311 ∗

              10−25𝑡7                                                                                                                     (18)                                                                                                         

And so on, continue in order to get better approximations:  

𝑎(𝑡) = lim𝑘→∞ 𝑎𝑘 (𝑡), 𝑚(𝑡) = lim𝑘→∞ 𝑚𝑘 (𝑡) and 𝑟(𝑡) = lim𝑘→∞ 𝑟𝑘 (𝑡). 

4. Results and Discussion 

     The analytical solutions for nonlinear alcohol consumption model in Spain are analyzed 

and discussed in this section then listed in Table 3. The predicted values of variables 𝑎(𝑡), 

𝑚(𝑡) and 𝑟(𝑡) for alcohol consumption model, [12]. had been given. Since the exact solution 

is not available for the current model, the predicted values are used to compare between the 

current analytic solutions of ADM and VIM with the predicted values [12]. in the interval 

(0,10) from 1997 to 2007. For comparison purpose, the corresponding absolute error of 𝑎(𝑡), 

𝑚(𝑡) and 𝑟(𝑡) for ADM and VIM methods are shown numerically in Table 4. where the 

absolute error in this study is the absolute value of the difference between the analytic 

solutions and the predicted values. Moreover, the relative error of 𝑎(𝑡), 𝑚(𝑡) and 𝑟(𝑡) for 

ADM and VIM methods is also listed in Table 5. which is defined as the absolute value of 

dividing the absolute error on the predicted values in this study. In Table 4,5. the absolute and 

relative errors of ADM for 𝑎(𝑡) have smaller values from 2003 to 2005 and 2007 than the 

VIM, while the absolute and relative errors of ADM for 𝑚(𝑡) are smaller than VIM from 

1999 till 2001 and 2007. For 𝑟(𝑡), the absolute and relative errors of ADM in the interval 

(0,10) from 1997 to 2007 are oscillatory with VIM errors. 

     The figures describe the behavior of alcohol drinking habit from 1997 to 2007. Figure 1a. 

of 𝑎(𝑡) shows the ADM and VIM obtained results near to some predicted values in 2001 until 

2005.While Figure 1b. Of 𝑚(𝑡) shows the predicted values around both ADM and VIM 

obtained results. Regarding to Figure C. of 𝑟(𝑡), both ADM and VIM curves obtained results 

converge to the predicted values in 1999 until 2005. For Figure 1a. That related to non-drink 

alcohol people 𝑎(𝑡), the ADM curve keeps on its level from 1997 to 2007. More other, there 

exists a variation between them such that the VIM curve is higher level than ADM curve. On 

the other hand, both ADM and VIM curves of non-risk-drink alcohol people 𝑚(𝑡) have 

higher that appears during the ten years from 1997 till 2007 in Figure 1b. Figure 1c. 

illustrates the decrease in the risk-drink alcohol people 𝑟(𝑡) through the ten years under study 

for both ADM and VIM curves. The results are calculated by Mathematica software, the 

figures are drawn by the Magic Plot program. Finally, the percentage of non-drink alcohol 

people 𝑎(𝑡) and the risk-drink alcohol people 𝑟(𝑡) are almost decrease, but increase with the 

non-risk-drink alcohol people 𝑚(𝑡). 

Table 3.  Analytic solutions and predicted values [12] .of alcohol consumption model (* means 

present result).  

2007 2005 2003 2001 1999 1997 Method Subpop

. 

0.400 0.354 0.359 0.363 0.383 0.362 Santonj
a at. el. 

𝒂(𝒕) 



  

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Table 4. Absolute error for ADM and VIM solutions as relative the predicted values [12].  

 

 

Table 5. Relative error for ADM and VIM solutions as relative the predicted values [12]. 

 

     

in 2010 

0.36101316 0.36121033 0.36140759 0.36160497 0.36180243 0.362 ADM* 
 

0.36305586 0.36286526 0.36266487 0.36245419 0.36223274 0.362 VIM* 
 

0.566 0.591 0.588 0.581 0.578 0.581 Santonj
a at 

al.in 

2010 

𝒎(𝒕) 

 0.58548056 0.58464529 0.58378022 0.58288472 0.58195819 0.581 ADM* 
 

0.58566309 0.58476327 0.58384724 0.58291479 0.58196578 0.581 VIM* 
 

0.034 0.055 0.053 0.056 0.039 0.057 Santonj
a at el. 

in 2010 

𝒓(𝒕) 

0.051281517 0.05237178 0.05348808 0.05463109 0.05580150 0.057 ADM* 
 

0.05128105 0.05237147 0.05348789 0.05463101 0.05580148 0.057 VIM* 
 

2007 2005 2003 2001 1999 Absolut 

error 

Subpop. 

0.03898684 0.00721033 0.00240759 0.00139503 0.02119757 ADM 𝒂(𝒕) 
0.03694414 0.00886526 0.00366487 0.00054581 0.02076726 VIM 

0.01948056 0.00635471 0.00421978 0.00188472 0.00395819 ADM 𝒎(𝒕) 

0.01966309 0.00623673 0.00415276 0.00191479 0.00396578 VIM 

0.017281517 0.00262822 0.00048808 0.00136891 0.01680150 ADM 𝒓(𝒕) 

0.01728105 0.00262853 0.00048789 0.00136899 0.01680148 VIM 

2007 2005 2003 2001 1999 Relative 

error 

Subpop. 

0.0974671 0.02036816 0.00670638 0.00384306 0.05534613 ADM 𝒂(𝒕) 
0.09236035 0.02504311 0.01020855 0.00150361 0.05422261  VIM 

0.03441795 0.01075247 0.00717649 0.00324392 0.00684808 ADM 𝒎(𝒕) 

0.03474044 0.01055284 0.00706252 0.00329567 0.00686121 VIM 

0.50827991 0.04778582 0.00920906 0.02444482 0.43080769 ADM 𝒓(𝒕) 

0.50826617 0.04779145 0.00920547 0.02444625 0.43080718 VIM 



  

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Figure 1. Variation of analytic solutions for ADM and VIM around predicted values [12].1 of (a) 𝑎(𝑡), 

(b) 𝑚(𝑡) and (c) 𝑟(𝑡) from 1997 to 2007 years. 

5. Conclusion  

     The current study, the convergence of the results for the analytic methods which are 

Adomian decomposition and variation iteration methods are examined in the nonlinear case. 

These methods have been known as a powerful device for solving a system of ordinary, partial 

differential equations or Integral equations and so on. In our work, they are used for solving a 

system of non-linear ordinary differential equations. The behavior of unhealthy social habit 

which is alcohol consumption in Spain are analyzed, based on the epidemiological model 

through ten years under study. The ADM and VIM methods help to analyze the effects of the 

unhealthy social habit of alcohol consumption. The obtained results are shown that there is 

increasing in alcohol consumption with non- risk-drink consumers and declining the risk-drink 

consumers during the ten years under study. For the number of the non-drink consumers has a 

small increase with the variation iteration method and maintains its level with respect to the 

Adomian decomposition method. The most predicted values [12]. around the ADM and VIM 

curves. Other analytical methods can solve such system under study like homotopy 

perturbation method, homotopy analysis method and Semi Analytical Iterative Method 

Temimi and Ansari. 

 

 



  

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https://doi.org/10.3389/fpsyg.2017.00756