Optimal Control Problem of Treatment for Obesity in a Closed Population

Variety of intervention programs for controlling the obesity epidemic has been done worldwide. However, it is still not yet available a scientific tool to measure the effectiveness of those programs. This is due to the difficulty in parameterizing the human interaction and transition process of obesity. A dynamical model for simulating the interaction between healthy people, overweight people, and obese people in a randomly mixed population is discussed in here. Two scenarios of intervention programs were implemented in the model, dietary program for overweight people with healthy life campaign and treatment program for obese people. Assuming all control rates are constant, disease free equilibrium point, endemic equilibrium point, and basic reproductive ratio ( ) as the epidemic indicator were shown analytically. We find that the disease free equilibrium point is locally asymptotical stable if and only if . From sensitivity analysis of , we obtain that larger rate of dietary program and treatment program will reduce significantly. With control rates are continuous in time, an optimal control approach was applied into the model to find the best way to minimize the number of overweight and obese people. Some numerical analysis and simulations for optimal control of the intervention were shown to support the analytical results. 1. Introduction Obesity is an overweight situation in human body as a result of excessive accumulation of fat situation. Every person needs some calories to save them energy, as well as to keep their body warm, and for many other purpose. The high consumption of high calorc food, over nutrition, and fast food combining with less physical activity to burn the calories become the main factors that cause obesity. The normal comparison between body fat with peoples weight is 18–23 percent for men and 25–30 percents for women [1]. Obesity has reached the epidemic proportions since recent decades [2]. It became a worldwide problem as stated in the WHO report [3]; there are more than 1 billion overweight adults and at least 300 million of them obese. Obesity could lead to chronic diseases, including diabetes type 2, cardiovascular disease, hypertension, stroke, and certain forms of cancer [3]. Several factors have been identified in determining the susceptibility for obesity such as human genes and also energy balance. Calory intake and also physical activity are the main factors in energy balance. Many programs have been initialized by WHO to solve the obesity pandemic such as recognizing the heavy criteria and growing burden of