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https://doi.org/10.5831/HMJ.2018.40.4.795

DYNAMIC MODEL PREDICTING OVERWEIGHT AND OBESITY IN KOREAN ADOLESCENTS

Chunyoung Oh

Abstract. Adolescent obesity has a high risk of developing into adult obesity and may cause many psychological problems. This paper aims at giving a mathematical model for the obesity of Ko- rean adolescents and predicting how much the prevalence of obesity among adolescents will increase using real data. We estimate that the obesity rate of boys will increase until about 28 ∼ 29% in 2025.

1. Introduction

Adolescents have been steadily increasing in weight due to stress from overworking and a decrease in physical activity [22]. Adolescent obesity has a high risk of developing into adult obesity and may cause many physical and psychological problems. In addition, traditional eating habits have changed. A lots of food and food advertisements make people are attracted to encouraging people to intake more than they need. Psychological atrophy, depression, and anxiety due to obesity are observed in adolescents, and these psychological problems are known to affect overall health in negative way [24]. The negative effects of peer pressure with regard to weight is greater among females and adolescents with a high body mass index [5]. It is especially difficult to lose weight, as it is a complex condition created by diverse genetic, environmental, cultural, and socioeconomic influences.

Obesity goes beyond personal problems. The global impact of obesity is increasing. Obesity is a major global economic problem caused by a multitude of factors [9]. The root causes of rising obesity are highly

Received September 9, 2018. Revised October 31, 2018. Accepted November 26, 2018.

2010 Mathematics Subject Classification. 92B05.

Key words and phrases. mathematical modeling, obesity, overweight.

This study was financially supported by Chonnam National University, 2016.

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complex, spanning evolutionary, biological, psychological, sociological, economic, and institutional factors.

Primary obesity refers to general obesity resulting from an imbalance in energy intake and consumption, which accounts for more than 90% of all obesity. Secondary obesity refers to obesity caused by a specific dis- ease, drug, or genetic disorder [24]. Additionally, individuals exposed to the effects of obesity at birth are considered born in an obesogenic envi- ronment and are thereby considered more susceptible to becoming over- weight and, later, obese. In middle and high schools in Korea, although boys have a higher rate of obesity than girls, girls higher rate of over- weight than boys. However, female students are frequent dieters have a lower quality of health than male students health. In 2000 the Western Pacific Regional Office of WHO(WPRO) proposed an alternative defini- tion of overweight (BMI 23.0 ∼ 24.9) and obesity (BMI 25.0) for Asian populations [3]. Obese student standards are combined with the BMI method and the relative weight calculation for the standard weight by height [7]. Korea Centers for Disease Control & Prevention(KCDC) de- fined 85 ∼ 95 percentiles or 23 < BM I ≤ 24.9kg/m2 for overweight, and 95 percentiles or 25kg/m2 or more for obesity by the 2007 Korean Na- tional Growth Charts [17]. In this article, adolescents mean the middle and high school students. The infected population comprised two differ- ent classes, namely, an overweight population (23 < BM I ≤ 24.9kg/m2) and an obese population (25 ≤ BM Ikg/m2).

Based on the WHO classification, adults with a BMI from 25 to 30 are defined as overweight, and those with a BMI of 30 or over as obese, in Korea, obesity is defined as a body mass index (BMI) of 25 or more [18], [23]. Based on WHO 2015, 33.5% of adults are overweight, and 5.8% are obese [16].

Weight loss is known to improve or resolve these cormorbid disorders, and it is generally known that in the overweight and obese, weight loss will improve health. Hence, most countries are monitoring the preva- lence of obesity. We are trying to predict the trend of the prevalence of obesity in Korean adolescents.

This paper seeks to provide a mathematical model for the obesity of Korean adolescents and, using the data, to predict how much the prevalence of obesity among adolescents will increase [17].

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2. Mathematical Model

For modeling, the adolescent population is divided in four sub-populations:

individuals with normal weight S(t), people who are overweight W (t), obese individuals O(t), overweight and obese individuals on a diet D(t).

We developed four differential equations that describe interactions and transitions between sub-populations with different body mass in- dex(BMI) classifications. We note that only the flow rates between compartments are key factors in this model.

The well-known SIR approach divides a population into compart- ments of infected and non-infected individuals and model terms are constructed to describe the flow to and from each compartment.

Following the SIR model structure for infectious diseases, adolescents are categorized as susceptible(becoming overweight), overweight, and obese. A student who has recovered(of normal weight who was pre- viously overweight) by losing weight belongs to the susceptible class.

Students in the diet compartment belong to the overweight or obese class(Figure 1).

This model will predict the obesity prevalence under the status quo.

A socially influenced transition to being overweight and obese was mod- eled by a mass action term βS(W + O). Normal students can become overweights βS(W + O) or obese δW by being deterred from following healthy lifestyle choices. Thus, the loss in number of those suscepti- ble during a six-year period(middle school and high school) is given by equation (1).

Once overweight, a student can transition to the obese compartment by gaining additional weight. A student can also lose weight, moving from obese to overweight, and then to the normal weight class.

The susceptible compartment has a constant recruitment rate µ of students entering school and a constant graduation rate µ to account for students leaving this system. This model assumes that some of the normal weight students are overweight pS or obese qS spontaneously based on their individual characteristics. This gives us a change in the number of overweight students over the years as represented by the following equations.

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S0 = µN − βS(W + O) + αD − (µ + p + q)S W0 = βS(W + O) − (δ + γ + µ)W + pS + εD O0 = δW − (σ + µ)O + ρD + qS

D0 = σO + γW − (α + ε + ρ + µ)D N = S + W + O + D = 1.

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Here, δW represents the number of overweight children who become obese, and αD is the number of obese and overweight children who lose weight to become normal weight. We formulate equations for the over- weight, obese, and recovered classes in a similar way. Figure 1 represents the flowchart of a dynamical system. Parameters of the system are as follows.

• µ average time staying in the system,

• α recovered rate at which an overweight and obese individual on a diet becomes one of normal weight,

• β transmission rate due to susceptibility to being overweight and obese, factor contributing to the development of being overweight and obese,

• δ rate at which an overweight individual becomes an obese indi- vidual,

• γ rate at which an overweight individual is on a diet,

• ε rate at which an overweight individual on a diet fails, or an obese individual becomes an overweight individual due to a diet,

• ρ rate at which an obese individual on a diet fails,

• σ rate at which an obese individual being a diet,

• p spontaneous rate at which an overweight individual enters an obesogenic environment,

• q spontaneous rate which an obese individual remains in an obe- sogenic environment.

In order to be able to run simulations, parameters must be estimated.

We estimate the values for model parameters in order to determine model predictions. As some of the parameters can only be estimated very roughly, our principal objective shall be to see how closely model behavior corresponds to observations.

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Figure 1. The dynamics of transitions among sub- populations is depicted graphically.

3. Parameter Estimation

Some parameters, such as the rate of effective interaction between overweight and obese individuals, are impossible to know; however, the range is fixed by the knowledge of the total prevalence in each category and thus dynamics can be examined by fluctuating these parameters within their ranges. According to the data [22], [Appendix], the ratio of an overweight person is lower than that of obese person. The rate of increase in overweight persons is low for male students.

The following parameters are computed for time t in months as:

• β is socially influenced transition to being overweight and obese modeled by a mass action term. Parameter β will be estimated by fitting the model with the data.

• δ means the transit rate weight increased of the overweight per- son. It is estimated using the growth of the average weight of the overweight person 0.09kg/mon. Male adolescents increased aver- age weight growth was about 33.1667 × 10−2 month−1, and female adolescents increased average weight growth about 17.7222 × 10−2 per month [22]. The parameter δ is sensitive, so we adopt an appropriate range.

• µ is the average time spent by adolescents in the system, which is six years.

µ = 1/72 month−1 = 1.3889 × 10−2 month−1.

• γ and σ are the rate at which overweight and obese individuals who are put on diets. The rate of female and male students who are trying to reduce is about 43% and 23%, respectively [22]. These

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proportions are higher than the proportion of students combined with obesity and being overweight. The percentage of obese stu- dents trying to reduce was assumed to be different from that of overweight students.

The case of boys is

γ = 1

0.33 × 12 × 0.60 = 0.1515 month−1,

σ = 1

0.33 × 12 × 0.80 = 0.2020 month−1. The case of girl is

γ = 1

0.33 × 12 × 0.70 = 0.1768 month−1,

σ = 1

0.33 × 12 × 0.90 = 0.2273 month−1.

We assume students use vacation time or after school time to lose weight. A moderate intensity effort to lose weight has the effect of creating a loss of 2 ∼ 4kg in 6 ∼ 12 months [21].

• ε and ρ are the percentages of failure to lose weight of overweight and obese individuals on diets, respectively. The success rate of the diet is 10% [19], [25], and the diet success rate of obese people is 5% [20]. An individual trying to decrease his or her weight takes weeks to transit from obese to overweight to normal weight. The period of effort to lose weight is 5.4 weeks [6].

obese failure : ρ = (1 − 0.05) × 1

1.35= 7.0370 × 10−1month−1. overweight failure : ε = (1 − 0.05) × 1

1.35= 7.0370 × 10−1month−1.

This parameter is estimated using the average time for a person on a diet to see the effects of obesity management.

• α, the rate at which an obese or overweight person moves to the normal weight population. The success rate of the diet is 10% [19], [25]. That is,

boy : α = 3.3167 × 10−1× 0.10 = 0.3317 × 10−1 month−1. girl : α = 1.7222 × 10−1× 0.10 = 0.1722 × 10−1 month−1.

• p and q have a family history(obesogenic environment) which tran- sition to being overweight and obese among students and the per- centages are p = 7.75% and q = 13.05%, respectively [4]. In case

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of male, the percentages are p = 0.385 × 0.0775 × 1

12 = 0.2486 × 10−2 month−1, and q = 0.385 × 0.1305 × 1

12 = 0.4187 × 10−2month−1 and in case of female, the percentages are

p = 0.385 × 0.071 × 1

12 = 0.3167 × 10−2 month−1, and q = 0.385 × 0.1 × 1

12 = 0.2278 × 10−2month−1.

In case of individuals with obese mothers, the percentage that are overweight and obese 7.1% and 15.2%, respectively. In case of students with obese fathers, the percentage who are overweight and obese are 8.4% and 10.9%, respectively [4]. In [4], the obesity of the mother was influential on the child’s obesity. Maternal BMI among those in the obese group was significantly higher than among those of the normal- weight group [15]. Parental obesity contributes to obesity in the children [13]. Among the parental obesity groups, the odds ratio of offspring obesity was the greatest when both parents were obese. There was no significant difference in the strength of the odds ratio between sons and daughters [14].

Table 1. The estimated values of parameters for the model

boy girl

Parameter Value Value Refs

µ 1.3889 × 10−2 1.3889 × 10−2 6 years α 0.3317 × 10−1 0.1722 × 10−1 cf.[22]

σ 0.2020 0.2273 cf.[12], [22]

γ 0.1515 0.1768 cf.[12], [22]

ε 0.7185 0.7185 cf.[19], [22], [25],

ρ 0.7037 0.7037 cf.[19], [22], [25]

p 0.2486 × 10−2 0.3167 × 10−2 cf.[4]

q 0.4187 × 10−2 0.2278 × 10−2 cf.[4]

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4. Numerical Simulation for Dynamic

Each compartment, S0, W0, O0, are the proportions of the normal- weight, overweight and obese populations, respectively, in the first- year group. We obtain the initial condition (year 2011, first year, i.e.

t = 0) and the final condition (year 2016, last year, i.e. t = 6) of the differential equations system. The initial conditions of males were S(0) = 0.8026, W (0) = 0.0238, O(0) = 0.1735, and the initial conditions of females were S(0) = 0.8142, W (0) = 0.0791, O(0) = 0.1066.

With the obtained parameters from surveys and data and the initial and final conditions, we performed several simulations for different values of β and δ in an appropriate range 0.06 ≤ β ≤ 0.08(0.03 ≤ β ≤ 0.05 : girl), 0.07 ≤ δ ≤ 0.09(0.07 ≤ δ ≤ 0.10 : girl) in order to find the value of β and δ that minimize the error of the difference between the solutions of the model for the proportions of sub-populations of normal weight, overweight and obese and the real data in the first year 2011(t = 0) to 2016(t = 5) final condition.

We performed several simulations in which we varied the parameters of the model in order to find out what influence the changes had on the final solution. We observed the most sensitive parameters in β, δ.

For the case of obesity, when the parameters are β = 0.07, δ = 0.6 and β = 0.05, δ = 0.8, as Figure 2 shows, the obese male students in this model will increase by about 28 ∼ 29% but the rate of overweight of male students will under 10% in 2025.

We observed that the parameters were β = 0.03, δ = 0.10 and β = 0.04, δ = 0.10 in the figure of female students. Among female students, the increase in obesity and being overweight increase in a linear fashion, and the increase rate is insufficient. Figure 3 shows that prevalence predictions in this model increase by about 12 ∼ 13% and the rate of overweight of male students will around 10% in 2025. Real data were not exactly linear, but the figures of simulation show an almost linear increase in the obesity and very little increase in the condition of being overweight.

We consider the sensitivity of R0 to changes in parameter values. We adapt the next generation method [1] to determine the basic reproduc- tion number R0. We got a basic reproduction number

R0= β(µ + δ + σ) (µ + δ + γ)(µ + σ),

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0 5 10 15 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time(year)

Proportions

Boys Population Dynamic

normal overweight obese

0 5 10 15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time(year)

Percentages

Boys Population Dynamic

normal overweight obese

Figure 2. The graphs on the left show male students for parameters β = 0.07, and δ = 0.6. The graphs on the right represent male students for parameters β = 0.05, and δ = 0.8.Time

We examined the sensitivity of two parameters β and δ that affect obe- sity rates. we have

Sβ = ∂R0

∂β β

R0 = µ + γ + δ µ + γ + σ, and Sδ= ∂R0

∂δ δ

R0 = δ γ − σ

(µ + δ + γ)(µ + δ + σ),

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0 5 10 15 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time(year)

Proportions

Girls Population Dynamic

normal overweight obese

0 5 10 15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time(year)

Proportions

Girls Population Dynamic

normal overweight obese

Figure 3. The graphs on the left show female students for parameters β = 0.03, and δ = 0.10. The graphs on the right represent female students for parameters β = 0.04, and δ = 0.10.

which provide an approximate measure of the relative change ratios be- tween R0 and the parameters considered. The sensitivity of δ is Sδ and the value of Sδ is affected by the values of σ and γ. When γ > σ, Sδ

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becomes sensitive to δ, and when σ > γ, Sδ becomes negative. The sen- sitivity of β is Sβ and the value of Sβ always positive on the parameter values σ and δ of the model.

5. Conclusion and Discussion

From the beginning of 2011 to the final value of 2016, we used real data to predict the future spread of obesity. Because there are many differences between female and male students in terms of rates of obesity and being overweight, we observed female students and male students separately.

This model estimates that the obesity rate of male students will in- crease until about 28 ∼ 29% and the prevalence predictions of the obese female students will about 10% in 2025. The rate of overweight of male students will under 10%, but the rate of overweight of female students will around 12 ∼ 13% in 2025.

Female students had low rates of obesity and low growth rates. But the proportion of overweight to normal-weight females was higher than that of boys. The rate of increase being overweight and obese were very similar.

Male students had a high rate of obesity and a high rate of growth.

However, the proportion of overweight to normal-weight males was low, and the rate of increase to being overweight and obese was low. Among males, normal students have developed into obese rather than overweight students.

Obesity is not a serious problem among female students, but among males, it should be considered serious. So schools and society should be looking at ways to prevent the spread obesity

It is very difficult to lose weight after becoming obese [2]. Despite the efforts of a large portion of the population, the prevalence of obesity has remained high. Therefore, it is important to prevent obesity. Finding ways to increase physical activity across all age groups is important in the quest for good health, but options for increasing energy expenditure through physical activity may be limited in physical education programs in schools.

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References

[1] P. van den Driessche, and James Watmough:Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180 (2002) 29 − 48.

[2] Franz MJ, VanWormer JJ, Crain AL, Boucher JL, Histon T, Caplan W, et al. Weight-loss outcomes: a systematic review and meta-analysis of weight-loss clinical trials with a minimum 1-year follow-up. J Am Diet Assoc 2007; 107 : 1755 − 67.

[3] genre.com, Business School, Blog: genreperspective.com, Edition 2, 2014.

[4] HJ. Im, HR. Park: A Study on the Obesity Policy of Children and Adolescents, National Youth Policy Institute(NYPI), Report 09 − R10, (2009).

[5] Justin G. Trogdon, James Nonnemaker, Joanne Pais: Peer effects in adoles- cent overweight, Journal of Health Economics, Journal of Health Economics, 27 (2008) 1388 − 1399.

[6] L.Jodar, F.J. Santonja, G.Gonzalez-Parra: Modeling dynamics of infant obesity in the region of Valencia, Spain, Computers and Mathematics with Applications 56, (2008), 679 − 689.

[7] KCDC: Development of a standardized model for revising the Korean growth standards, Policy Research Service Report,(2016), 11 − 1352159 − 000350 − 01.

[8] Manal I. Al-Kloub, Erika S. Froelicher: Factors contributing to adolescent obesity, Saudi Med. J., Vol.30(6) (2009), 737 − −749.

[9] McKinsey Global Institute: Overcoming Obesity: An Initial Economic Analysis, Discussion paper, Nov.(2014).

[10] Kim HK, Lee HJ.: Effects of obesity management program for obese elementary school children, J. Korean Acad. Child Health Nurs., 12 (2006), 451 − 61.

[11] Jeongeun Kim et. al: S line secrect know-how for women, mookhouse, (2011).

[12] Young Ho Lee, Si Young Heo: The Benefits and Adverse Effects of Weight Loss, J. Korean Acad. Fam. Med., 25 (2004), 721 − 739.

[13] WonKyun Lee: The relationship between Parental Obesity and obesity, Physical Fitness of Children, SNU Graduate School Master’s Thesis, (2001).

[14] Bokyung Park, Hee-Cheol Kang, Dae Ryong Kang, Sun Ha Jee: Associations be- tween Parental and Adult-offspring Obesity in South Korean Population, Journal of The Korea Society of Health Informatics and Statistics, 37(2) (2012), 1 − 11.

[15] Jeong Ah Shin, Sang Pil Bae, Hyo Soon Kim, Hye Soon Park: Psychosocial Factors and Familial Environments in Adolescent Obesity. J. Korean Acad. Fam.

med., Vol.23(8) (2002), 1024 − 1032.

[16] The 2016 Global Nutrition Report(from promise to impact ending malnutrition by 2030), www.globalnutritionreport.org,(2016), 126-128.

[17] 2007 Korean National Growth Charts: review of developmental process and outlook, Korea Center for Disease Control and Prevention, The Korean Pe- diatric Society, 2007 Korean Children and Adolescents Growth Standard (commentary for the development of 2007 growth chart). Available from:

URL:http://www.cdc.go.kr/

[18] https://data.oecd.org/healthrisk/overweight-or-obese-population.

[19] https://ppss.kr/archives/98664

[20] https://stayfit.co.kr:45505/articles diet/2078, (assessed 2017.08.28) [21] https://342436.com/cyda/contents, (assessed 2017.08.28)

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[22] https://yhs.cdc.go.kr/(Youth Health Behavior Online Survey Statistics, 7th ∼ 12th (2011 ∼ 2016year)), 2016(assessed 2017.05.01)

[23] http://www.index.go.kr/unify, (assessed 2018.07.28) [24] http://www.kosso.or.kr/general/general/sub03.html

[25] http://www.seehint.com/r.asp.no=11188, (assessed 2017.07.28)

6. Appendix

These data are downloaded from the site(https://yhs.cdc.go.kr). The following Tables are the average weight, overweight and obesity of ado- lescents.

Table 2. The average weight of adolescents year 2011 2012 2013 2014 2015 2016 boy(m) 61.8 61.8 62.2 62.2 62.8 63.7 boy(h) 51.99 57.6 62.1 65.7 67.8 70 girl(m) 53.5 53.55 54.1 54.1 54.3 54.5

girl(h) 48.57 51.7 54.2 55.6 56.4 57.2

Table 3. The average overweight rate of adolescents year 2011 2012 2013 2014 2015 2016 boy(m) 2.38 2.55 2.55 2.5 2.55 2.65

boy(h) 7.93 4.7 2.4 1.1 0.2 0

girl(m) 7.91 8.4 9.4 8.7 8.65 8.65 girl(h) 12.94 9.4 9.9 7.2 6.2 5.7

Table 4. The average obesity rate of adolescents year 2011 2012 2013 2014 2015 2016 boy(m) 17.35 18.05 18.75 19.05 20.15 18.05 boy(h) 12.92 16.5 18.5 20.8 22.8 21.8 girl(m) 10.66 11.15 12.1 12.45 12.9 11.15

girl(h) 7.47 10.2 12 13 15.5 12.9

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Table 5. The average attempt rate of Weight Loss year 2011 2012 2013 2014 2015 2016

boy 23.9 22.4 22.7 23.1 22.7 23 girl 45.5 43.5 44.7 45.1 42.2 43.4

We write the middle school as m and the high school as h. Table 5.

is the average attempt rate for students to lose weight.

Chunyoung Oh

Department of Mathematics Education, Chonnam National University, Gwangju 61186, Republic of Korea.

E-mail: cyoh@jnu.ac.kr

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