Journal of Applied Mathematics

Volume 2015 (2015), Article ID 217808, 9 pages

http://dx.doi.org/10.1155/2015/217808

## Optimal Intervention Strategies for the Spread of Obesity

^{1}Department of Mathematics Education, Chonnam National University, Gwangju 500-757, Republic of Korea^{2}Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

Received 16 March 2015; Accepted 3 June 2015

Academic Editor: Zhen Jin

Copyright © 2015 Chunyoung Oh and Masud M A. 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

The present study considers a deterministic compartmental model for obesity dynamics. The model exhibits forward bifurcation at basic reproduction number, , that is; for , obesity is not sustained. However for the model approaches a locally asymptotically stable endemic equilibrium. To control this epidemic and reduce the obesity at the endemic equilibrium, we considered intervention strategies for the spread of overweight and obesity, where Pontryagin’s Maximum Principle is applied. The numerical technique was used to show that there are effective control strategies that include minimizing the social contact rate with the overweight and obese population and campaigning. Numerical results indicated the effects of the two controls (prevention and education/campaigning) to be different. In societies with lower obesity, the social contact rate with the overweight and obese population plays a more prominent role in spreading obesity than lack of educational programs/campaigns. However, for societies with very high obesity burden, education/campaigning proved to be highly effective strategies. Reducing the social contact rate can result in other results such as a depression and an invasion of their individual rights. The appropriate approach to obesity is needed to lower obese societies.

#### 1. Introduction

Excessive or abnormal fat gain in the body which produces risk factors for life is categorized as a spectrum of illness termed as obesity or overweight. To define obesity problem, Body Mass Index (BMI) is widely used. BMI more than 25 is considered to be hazardous to health and is labelled as overweight. Further, a person with BMI greater than 30 is considered as obese [1]. Obesity is not a cosmetic problem, but it rather has been proved to enhance the risk of life-threatening diseases like coronary heart disease, high blood pressure, stroke, type 2 diabetes, metabolic syndrome, cancer, osteoarthritis, and many more [2]. Besides health issues, obesity has notable impact on economy, as it was identified that it is more than $215 billion in the United States [3], which was estimated in early years to be $147 billion [4]. As risk of severe diseases and growing economic drain are rooted into obesity, it deserves to be analyzed thoroughly.

Nowadays in many countries obesity has become a prevalent problem which is mostly overlooked at the beginning, as it happens slowly and does not cause any immediate health hazard. When someone is already in, s/he urges to recover, which is not possible overnight. Obesity has been identified as a contagious problem which is spreading over social networks [5] and was studied theoretically using epidemic models [6–9]. Epidemiology along with optimal control theory provides us with tools to assess the evolution of the problem through social network, identify major facts to control the epidemic, and finally and most importantly establish optimal control strategy. The authors in [6] studied infant obesity and through numerical simulation they pointed to food consumption behavior as a propeller for childhood obesity. Considering adult obesity, the authors in [7] claimed that prevention strategies are more effective by analyzing a variant of SIS epidemic model numerically. The authors in [8] experimented a system of SIS difference equation model and drew a similar conclusion. But in these studies constant controls have been considered which might not be optimal and practically unfeasible. In [9], time-dependent controls for obesity including dietary program for healthy life campaign and treatment have been considered in a deterministic model. It was suggested that intervention program should be implemented as early as possible to attack comparatively small epidemic, which was also proved for another social contagious problem [10].

However, nowadays in some countries obesity captured more than 70% of the total population [11, 12] and in some countries it is not so high; that is, obesity is affecting different societies in different scale. We have shown in this paper that intervention strategies for societies with different obesity burden are different. With this aim in Section 2 we presented our model. In Section 3, the dynamics of system was discussed. We pointed out the factors to be considered as intervention measures, proved the existence of optimal control strategy, and presented numerical results in Sections 4 and 5. Conclusion has been drawn in the last section.

#### 2. Mathematical Model

The model is modified from the model of [13] as follows: In the model of [13], the social contacts with the overweight and obese population are and , respectively. However, in model (1), we used ; that is, we considered that both parameters are the same, as they are close enough to each other.

In this model, the adult population is divided into three subpopulations: the normal population , overweight population , and obese population . The adult population sizes at time are normalized to unity; that is, , . All parameters used in the model are assumed to be strictly positive constants.

The transitions between the subpopulations , and are governed by terms proportional to the sizes of these subpopulations. The transitions from the normal compartment to the overweight compartment occurred at a rate of . The rate at which overweight adults with an unhealthy lifestyle and inactivity become obese individuals is . The obese individuals become overweight adults at a rate of and the overweight individuals become normal at a rate of due to healthy lifestyles such as activity and less food consumption. The basic reproduction number [14] is given by

#### 3. Equilibrium and Stability

To study the dynamics of the model, taking into account , the following reduced version is considered:

Equating the right-hand sides of (3) to zero, we get obesity-free equilibrium (OFE), , and endemic equilibrium (EE), , where

Lemma 1. *System (3) admits OFE, , which is locally asymptotically stable for and unstable if .*

*Proof. *To check the stability, we compute the Jacobian of system (3) as At OFE, , we have where is the Jacobian at OFE. If , then and and consequently OFE is a stable node. On the other hand, when , which makes OFE unstable.

Lemma 2. *If , system (3) admits unique endemic equilibrium (EE), , which is locally asymptotically stable for .*

*Proof. *If , then which is impossible. Therefore, EE does not exist for .

At EE, , we have where is the Jacobian at EE. If , then and and as a result EE is locally asymptotically stable.

Combining Lemmas 1 and 2, we observe that there exists only one OFE for , which is locally asymptotically stable. For , there exists an OFE along with a unique EE between which the former is unstable while the latter is asymptotically stable. So, we can conclude that the system has a forward bifurcation at . This fact is summarized in the following theorem.

Theorem 3. *System (1) exhibits a forward bifurcation at *

To illustrate the bifurcation phenomena, simulations have been carried out with the parameters listed in Table 1. The bifurcation diagram is presented in Figure 1, where the solid and the dashed lines correspond to the stable and unstable solutions, respectively. To numerically verify the existence of EE, the phase portrait for the endemic equilibrium (EE) at is shown in Figure 2, where the nearby points () of the EE approach the EE ().