Mathematical modeling of adulthood obesity epidemic in Spain using deterministic, frequentist and Bayesian approaches

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Highlights

  • Modeling of adulthood obesity in Spain using surveys from 1987 to 2017.

  • Compartmental model considering social contacts based on differential equations.

  • Nonlinear regression is conducted when white noise error is included.

  • Bayesian inference is performed using the Metropolis algorithm.

  • Sensitivity analyses highlight the importance of prevention strategies.

Abstract

The excess weight population is growing riskily in Spain. This is confirmed by the Spanish National Health Survey 2017, which gathers the percentage of overweight and obese adults in Spain from 1987 to 2017. In this paper we propose a mathematical model based on differential equations to calibrate the incidence of excess weight in the Spanish adulthood population. The main principle is that fatness is an epidemic, where there are three stages, namely normal weight, overweight and obese, and healthy individuals may become “infected” by social contact. We start with a well-posed deterministic formulation of the model, where the parameters are fitted by minimizing the mean square error objective function. The long-term behavior of the system shows that 37% and 24% of Spanish adults will be overweight and obese in the long run, respectively. Due to the incomplete knowledge of the underlying phenomenon and error measurements, randomness must be incorporated into the model formulation. A white noise error term is thus added to conduct nonlinear regression. The response becomes a stochastic process, whose prediction interval captures the variability of the overweight and obese populations correctly. We go a step beyond by treating the model parameters as random variables. The Bayesian inference method allows for quantifying the propagation of uncertainty in this setting, by running the Metropolis algorithm (both brute-force and Adaptive). The numerical simulations show that it provides similar results to the frequentist analysis. The sensitivity analyses from the different methods agree and suggest that prevention strategies are more important than treatment strategies to control adulthood obesity. Specially, the treatment of the transition from the obese to the overweight stages is the least recommendable for reducing the obesity epidemic.

Introduction

Obesity can be considered as a worldwide epidemic [1]. In Spain, the last survey from the Spanish National Health Survey (ENSE) shows that obesity has been increasing at worrying rate in the last decades for both infants and adults [2]. Obesity is not only a serious health concern, but also a public economic problem for the country [3]. Hence effective policies and control measures are necessary in order to mitigate the augment of excess weight in Spain.

Mathematical models are a tool to analyze the evolution of the epidemic, its future dynamics, manage and control its increase and prioritize policies that permit preventing and reducing or at least alleviating the spread. Most of the approaches in mathematical modeling are based on using compartmental models [4], [5], [6], [7]. The population is divided into compartments according to the status and the individuals flow between the compartments as time passes. The mathematical model is usually based on systems of ordinary differential equations in which homogeneous mixing is assumed.

These types of models have been applied to explain the evolution of excess weight in Spain. In the case of the region of Valencia, Spain, we find two models: one for infants [8] and another one for adults [9]. They are based upon systems of ordinary differential equations with input parameters, where the Valencian population is divided into compartments according to weight. Persons move between the compartments as time goes by. Of course, many factors may play a role in body weight: genes, metabolism, environment, culture, etc. In [8], [9], the authors focused on the fact that fatness is “transmitted” by social pressure and contacts that entail unhealthy habits (this includes social environment, media, culture, access to food, economical status, etc.). Overweight and obese individuals are viewed as infected, which transmit the disease to healthy persons (susceptible) by contact. This idea was developed in [10], [11]. In the seminal paper [10], the authors evaluated a densely interconnected social network to analyze the extent of the person-to-person spread of obesity as a possible factor contributing to the obesity epidemic; they explicitly concluded that obesity appears to spread through social ties. Paper [11] and its abundant references from the introduction defend the idea that gains in weight are spread through the population when social surrounding influences healthy weight individuals, in a way reminiscent of a contagious disease of social transmission. In their words, it may be easier to be fat in a society that is fat. Hence the transition from healthy weight to overweight in a certain moment of one’s life may be justified by his/her contacts in the society with overweight and obese persons (the infectives).

This paper proposes a mathematical model for the number of overweight and obese adults in Spain as a disease of social transmission, by taking into account data gathered by the Spanish National Health Survey from 1987 to 2017. In general, finding abundant reliable data is difficult because of the hard preparation and storage and the economic cost. We use a nonautonomous nonlinear system of ordinary differential equations, whose parameters are estimated by minimizing the mean square error. Accounting for modeling and data measurement errors, randomness is incorporated into the model, and frequentist nonlinear regression and Bayesian inference are carried out [12]. A sensitivity analysis is conducted for designing the optimal strategy for reducing the excess weight prevalence in Spain. In epidemiology, the quantification of the propagation of input uncertainties has been investigated in [13], for some prototypical continuous compartmental models, in [14], for discrete compartmental models, and in [15], for the obesity model proposed in [9], by utilizing generalized polynomial chaos expansions [16], [17], but no inverse parameter estimation method. In our recent work [18], we inferred consistent probability distributions for the input parameters of the obesity model proposed in [9] according to their deterministic estimates, by employing the maximum entropy principle. The use of the maximum entropy principle may be justified in the following situations: there are replicated data (which is not the case for an epidemic scenario); or the amount of data at hand is very limited (like four instants of time or less) and the model parameters are therefore fixed empirically or from earlier technical reports. On the other hand, Bayesian inference has been carried out for the analysis of epidemics such as respiratory syncytial virus and chickenpox infections in the region of Valencia [19], [20].

The structure of this paper is as follows. In Section 2, we explain the study from the Spanish National Health Survey and present the data that will be used. In Section 3, we formulate the deterministic mathematical model for adulthood excess weight in Spain, we fit the parameters, study the dynamics and conduct a sensibility analysis. Section 4 adds a random white noise error to the governing model accounting for model errors, with constant parameters, and frequentist nonlinear regression is carried out to estimate the error variance and to construct prediction intervals for the incidence of obesity. In Section 5, we work with random input parameters to apply the Bayesian methodology, accounting for model and data errors at the same time. Different versions of the Metropolis algorithm (namely standard and Adaptive) are implemented to simulate from the posterior distribution and the posterior predictive distribution. Finally, Section 6 draws the main conclusions and points towards potential avenues of future research.

The code implementations and the numerical computations have been developed in the software Mathematica, version 11.2 [21].

Section snippets

Survey and data

The Spanish National Health Survey (ENSE) is a periodic study conducted by the Spanish Ministry of Health, Consumption and Social Welfare, with the collaboration of the National Statistics Institute of Spain (INE), since 1987. It is a massive scale survey based on home personal interview (CAPI) that collects transversal data regarding the resident population in Spain, on aspects relating to health information, namely medical conditions, use of health services, and health determinants. It makes

Deterministic model

In this section we propose an epidemic-based mathematical model for the data from Fig. 1. The construction is based on a system of ordinary differential equations, where the time-dependent variables refer to the overweight and obese individuals in Spain who are at least 18 years old.

Frequentist-based model

In mathematical modeling, there are three sources of error: model error, numerical errors, and data errors. The first one is due to the incomplete knowledge of the underlying physics. The second one comes from the resolution of differential equations, optimization problems, etc. Finally, the third one accounts for error measurements. Numerical errors can be reduced as much as desired, at least theoretically, by employing finer discretizations, better algorithms, etc. However, model error and

Bayesian model

As explained at the beginning of Section 4, randomness should be included into the model accounting for modeling and data errors. In Section 4, randomness was incorporated through an error term of white noise form, keeping the input parameters as constants, accounting for model errors. The validation of the model output consisted of a pointwise estimation, which coincided with the deterministic estimation from Section 3, and a prediction interval that captured all the actual data. In this

Discussion and perspectives

In this paper, we have modeled successfully the number of excess weight adults in Spain for the last thirty years. We have started with a well-posed deterministic model of ordinary differential equations, in which excess weight is treated as an epidemic where the individuals flow between different compartments according to their fatness. The parameters of the model have been estimated by minimizing the objective function defined through the mean square error. Accounting for model and data

CRediT authorship contribution statement

Julia Calatayud: Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Supervision. Marc Jornet: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare that there is no conflict of interests regarding the publication of this article.

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