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Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 1084769, 12 pages
https://doi.org/10.1155/2017/1084769
Research Article

Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion

1Maestría en Ciencias de la Complejidad, Universidad Autónoma de la Ciudad de México, 03100 Ciudad de México, Mexico
2Maestría en Ciencias de la Salud, Escuela Superior de Medicina, Instituto Politécnico Nacional, Plan de San Luis S/N, Miguel Hidalgo, Casco de Santo Tomas, 11350 Ciudad de México, Mexico
3Maestría en Matemáticas Aplicadas, Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas CU, 39087 Chilpancingo, GRO, Mexico

Correspondence should be addressed to Cruz Vargas-De-León

Received 1 October 2016; Accepted 13 December 2016; Published 24 January 2017

Academic Editor: Tetsuji Tokihiro

Copyright © 2017 Enrique Lozano-Ochoa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study an epidemiological mathematical model formulated in terms of an ODE system taking into account both social and nonsocial contagion risks of obesity. Analyzing first the case in which the model presents only the effect due to social contagion and using qualitative methods of the stability analysis, we prove that such system has at the most three equilibrium points, one disease-free equilibrium and two endemic equilibria, and also that it has no periodic orbits. Particularly, we found that when considering (the basic reproductive number) as a parameter, the system exhibits a backward bifurcation: the disease-free equilibrium is stable when and unstable when , whereas the two endemic equilibria appear from (a specific positive value reached by and less than unity), one being asymptotically stable and the other unstable, but for values, only the former remains inside the feasible region. On the other hand, considering social and nonsocial contagion and following the same methodology, we found that the dynamic of the model is simpler than that described above: it has a unique endemic equilibrium point that is globally asymptotically stable.

1. Introduction

Obesity has gone from being an isolated health problem, related to some people, to a global problem. Considered as “The Pandemia of the 21st Century” [1], it is present in both developed and underdeveloped countries, eclipsing in the latter the problem of malnutrition to become today one of its main priorities [2]. Around the world countries spend a huge amount of their year budgets as well as qualified human resources to fight this disease, which is frequently associated with serious health pathologies such as diabetes mellitus, high blood pressure, and lung and heart diseases and, moreover, it is the cause of several kinds of cancer [3, 4]. Also it affects the psychological condition of the individuals, because it can damage their self-esteem and social relationships [1, 2, 5, 6].

Clearly, obesity is a huge and difficult problem, with the aggravating factor that it is present in all sectors of society regardless of the income, ethnicity, age, gender, or another socioeconomic status of their individuals. Besides, it is often associated with the wrong diet, sedentary lifestyle, or genetic predisposition of individuals, and if it was not enough, it has also been found that obesity can be produced by a large variety of causes that are linked to cultural, social, and economic conditions of the environment in which people develop. Today, the way in which the latter influence the origin of obesity is far from being understood. In summary, the problem of pandemic obesity is complex and multifactorial and it has increased in recent years over all the world. Thus, obesity has become a relevant current research topic in which different fields of human knowledge converge for the purpose of understanding its causes, knowing its consequences, and, as far as possible, keeping it under control or eradicating it.

In this way, mathematical modelling is a means to provide a general insight for the dynamics of obesity and, as such, could hopefully become a useful device to develop control strategies. With regard to its causes of social origin, the dynamics of obesity can be well modelled by epidemic-type models as a process of social contagion, as was evidenced by Christakis and Fowler who studied the spread of obesity in a large social network over 32 years and established that obesity can spread through social ties [7]. This approach has resulted in a wide range of papers of mathematical modelling in which obesity is studied as a social epidemic [815]. Social obesity epidemic models typically divide the population into two or several classes or subpopulations. In [12] the classical model is extended, where infection occurs by nonsocial mechanisms as well as through social transmission. There are models in which it has been considered a bilinear incidence rate [14] (for subpopulations of normal weight, overweight, and obese individuals), obtaining as a result a unique stable equilibrium point; in [911, 13] this effect was considered, but for six subpopulations: normal weight, latent, overweight, obese, becoming overweight, on diet, and obese on diet individuals. Other models have incorporated the effects of the time delay [15] and have also formulated nonautonomous obesity epidemic models [9], in which periodic positive solutions were found under some sufficient conditions using a continuation theorem based on coincidence degree theory.

In this paper we analyze the model proposed by Ejima et al. [8]: a variant of the SIRI model in which the individuals who recover temporarily may get recurrence to infectious state and is formulated on the premise that obesity is caused by both social and nonsocial contagion routes [16]. The objective of this work is to analyze this system using the methods of the qualitative theory of ordinary differential equations.

The rest of this paper is organized as follows: in Section 2, we present the Ejima et al. model [8] and we reduce it in a two-dimensional system. Section 3 focuses on the case in which only the risk of social contagion of obesity is considered: we perform a local analysis in order to establish the equilibrium points and their corresponding local stabilities as well as bifurcations; we also obtain a global analysis by means of an appropriate Lyapunov function to establish the global stability of the endemic equilibrium point and, by using Dulac’s criterion, the nonexistence of periodic orbits. In Section 4 we will study the case with risk of both social and nonsocial contagion of obesity: we prove the existence of a unique endemic equilibrium point that is globally asymptotically stable, and also through a suitable Dulac function, we determine the nonexistence of periodic orbits. Section 5 investigates several important aspects of our model from a numerical point of view. Finally, in Section 6 we collect some observations and conclusions.

2. Obesity Mathematical Model

The model proposed by Ejima et al. [8] for the dynamics of obesity is given by the following set of three differential equations:wherein , , and , respectively, denote the susceptible (never-obese), infectious (obese), and recovered (ex-obese) individuals in a population. In (1) the natural death and birth rates are assumed to be equal and denoted by ; thus, we have for all time (the population size is constant). Also, the parameter is the transmission rate due to social contagion risk of obesity, describes the rate at which the infectious individuals become recovered individuals, is the hazard of obesity due to nonsocial contagion reasons, and is the relative risk of weight regain among ex-obese individuals which typically takes a value greater than unity () due to high risk of coming back to the obese state. All the involved parameters are positive.

In Figure 1 the flow diagram is shown in which the model assumptions are based and system (1) is deduced. There may be seen how it is a variant of a SIRI model, since the term , given in the second equation (1), represents the effect of relapse; that is, the recovered people (ex-obese), after some time, become infected (obese) again. It should be noted that, in total absence of the relapse term, (1) is reduced to the corresponding well-known SIR model.
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