We used a novel method developed in system dynamics (17) to connect the micro-level dynamics of BMI of individuals to the macro-level dynamics of the distribution of BMI in the population. Since, based on the Belgian Health Interview Surveys (4), overweight and obesity trends were different for male and female adults (aged 20 to 74 years old) and across the major regions, we divided the Belgium population, first, into two subpopulations based on their gender (male and female) and then into six subpopulations based on their major region (Flanders, Wallonia, Brussels) and gender. We estimated the trends of EIG and MEG for each subpopulation.
For each subpopulation, we first divided the range of possible values of BMI into 14 different partitions or classes (e.g., (15,18], (18,20], (20,23], …., [65,68]), each of which represented a distinct stock. Each stock contained the part of the population whose BMI fell within the BMI interval associated with that stock.
We assigned a hypothetical representative individual to each stock, where the BMI of the representative individual was the average of the BMI interval associated with that stock.
We modeled the dynamics of body weight gain and loss of each representative individual using the Hall et al. model (12) of adult metabolism and body weight dynamics. Weight gain/loss of representative individuals was modeled as the result of imbalance between their energy intake and energy expenditure, represented by their EIG.
As the representative individual associated with each BMI class (stock) gained/lost weight, he/she pushed the population of that stock into the neighboring BMI classes, thus, causing a shift in the distribution of BMI over time. The rate of change of the BMI of representative individuals provided the speed by which population BMI distribution shifted to the right or left.
For each subpopulation j, the EIG of the representative individual associated with each BMI class k (k=1, …, 14) at any time t (t=1997, …, 2018) was modeled as a function of their equilibrium energy expenditure (EEjk) and energy gap multiplier (\({\mu }_{jkt}\)), as shown in Equation (1). The equilibrium energy expenditure of each representative individual associated with BMI class k shows the energy required for normal activity and maintenance of the body weight.
\({EIG}_{jkt}={EI}_{jkt}-{EE}_{jk}={{\mu }_{jkt}*EE}_{jk}\) | (1) |
We then calculated the energy intake of the representative individual of BMI class k (\({EI}_{jkt}\)) by adding the energy imbalance gap \({EIG}_{jkt}\) to the equilibrium energy expenditure \({EE}_{jk}\). We modeled the energy imbalance gap \({\mu }_{jkt}\) as a function of time, BMI class, and interaction of them as shown in Equation (2). We specified general models to allow for flexible and non-linear relationships between time and BMI in the model.
\({\mu }_{jkt}={BMI Effect}_{jk}+{Time Effect}_{j}+{Interaction of Time and BMI Effect}_{jk}\) \({BMI Effect}_{jk}={\beta }_{1j+}{\beta }_{2j}{BMI}_{jk}+{\beta }_{3j}({BMI}_{jk}{)}^{{\beta }_{4j}}\)
\({Time Effect}_{j}={\beta }_{5j}Time+{\beta }_{6j}(Time{)}^{2}+{\beta }_{7j}(Time{)}^{3}\)
\({Interaction Effect}_{jk}={\beta }_{8j}{BMI}_{jk}Time\)
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(2)
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To make sure the system dynamics model was demographically representative for the Belgium adult population, we also modeled the death rate and the rate of transition from childhood (19-year-old individuals) into adulthood for each BMI class. To take into account the differential mortality attributable to very low or high BMI, we used the mortality adjustment curves developed by Gray (18).
Calibration
We used the cross-sectional data from the six Belgian Health Interview Surveys from 1997, 2001, 2004, 2008, 2013 and 2018 to obtain population level distribution of BMI for age 20 to 74 years for different subpopulations in this study. The first edition of the HIS was conducted in 1997 for the general population. Since then it has been repeated periodically over time up to its 6th edition in 2018. One of the main objectives of the HIS is to measure the health status of the population in Belgium, accounting also for the three regional sub-populations (in Flanders, Wallonia and Brussels-Capital). Based on sample size calculations, the total number of successful participants for the basic sample is generally set to 10,000 (3500 for Flanders, 3500 for Wallonia, 300 for East Belgium and 3000 for Brussels). To select the sample, a stratified clustered multi-stage design was used. More details on the design and sampling of the survey can be found elsewhere (19). Data collection in the HIS takes place using two standardized questionnaires: 1) a questionnaire administered in a face-to-face interview setting, and 2) a paper questionnaire handed out to participants for self-completion. Height and weight of participants is self-reported and collected through the face-to-face questionnaire. Data of transition rates from adolescence (19 years old) to adulthood (age 20 to 74) were also obtained from the HIS. The data for all-cause mortality was obtained from the Belgian Mortality Monitoring (Be-Momo) project (20).
For each subpopulation, we initialized the model using the BMI distribution data obtained from the first HIS (i.e., 1997). The model was then simulated through 2018. We used the maximum likelihood method to estimate the beta parameters of EIG (and consequently estimate the EIG) so that the distribution of BMI generated by the model was as close as possible to the distribution of BMI obtained from the HIS in all years for which data was available (i.e., 1997, 2001, 2004, 2008, 2013, 2018). The overall log-likelihood function summed up the logarithm of likelihood values across survey years.
We used a non-linear optimization method to find the beta parameters of EIG so that the overall log-likelihood function was maximized. The estimated beta parameters are reported in Table 1. We also calculated the MEG for each subpopulation to represent the increase in energy intake needed to maintain the higher average body weight comparted to the one in 1997.
To validate our results, we used the one-sample Kolmogorov–Smirnov Goodness of Fit test to examine whether the BMI distribution simulated by the system dynamics model was different from the BMI distribution observed in the surveys. Table 2 shows the Kolmogorov–Smirnov test results. We conducted the data processing by Stata version 14 (StataCorp, College Station, TX, USA) and all simulations and optimizations by Vensim™ (Ventana Systems, Inc., Harvard, MA, USA).