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 (
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 [
Obesity has reached the epidemic proportions since recent decades [
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 noncommunicable disease and developing global strategy on diet, physical activity, and health through broad consultation processes [
Some mathematical models have been developed to understand this disease like in [
Unlike the authors in [
Mathematical model for obesity in a closed population will be constructed in this section. We assume that the human population is divided into three different compartments, that is, healthy compartment
According to [
Transitions from healthy to overweight compartment depends on daily social interaction between healthy people with overweight and/or obese people with interaction coefficient given by
We assume that recovery rate from overweight compartment to healthy compartment is
From the assumptions above which is illustrated by the transmission diagram given in Figure
Parameters description and value.
Par.  Description  Value 


Natural recruitment rate (per day) 


Natural mortality rate (per day) 


Interaction coefficient (per day) 


Infection rate because bad life habit (per day) 


Natural recovery rate from overweight to healthy comp. (per day) 


Total of human population (individuals) 


Portion of health recruitment rate from overweight comp. 


Portion of health recruitment rate from obese comp. 


Health life campaign rate (per day) 


Treatment rate for obese people (per day) 

Transmission diagram for obese disease.
The parameters of
In order to obtain the equilibrium points of system (
The diseasefree equilibrium point is given by
Using the next generation matrix operator approach in [
Figure
Parameter sensitivity of
Figure
In the next section, characterization of optimal control problem to reduce the number of overweight and obese compartments with intervention of health life campaign and treatment intervention will be given.
Our purpose in this model is to minimize number of overweight and obese compartments and the corresponding control functions. Together with the mathematical model of obese disease in (
We would like to find the value of control variable
To find the optimal control for
The adjoint equation variables
To obtain the optimality condition in order to minimize the cost function (
Now, we point out that our optimal control system consists of the state system (
The dynamical model in this paper considered the obese spread via daily social interaction and the data for simulation is given in Table
This condition describes the situation where the cost for application in the field is different between campaign and rehabilitation intervention. We balance the human populations and control functions in cost functional while choosing the weight cost such that
Simulations in this subsection are given to accommodate the different initial condition for state equation, let us call it a prevention scenario where the number of obese people is relatively low
The numerical results for this subsection are obtained for different values of
Numerical simulations of overweight (a) and obese (b) compartments and control variable (c) with different conditions of
This phenomenon appears because some people from obese compartment move to overweight compartment as a result of rehabilitation programme. From Figure
From Figure
Numerical simulations of overweight (a) and obese (b) compartments and control variable (c) for prevention and endemic reduction scenario.
We perform the last simulations in this subsection for the autonomous system (
Numerical simulations of healthy (a), overweight (b), and obese (c) compartments when
Other alternative way to reduce
Mathematical model for obesity in a closed population has been constructed in this paper. Vertical transmission from infectious parents has been accommodated into the model as well as the intervention program like healthy life campaign and treatment/rehabilitation program. Diseasefree equilibrium and endemic equilibrium have been shown analytically. Basic reproductive ratio as the endemic indicator has been obtained for local stability criteria of diseasefree equilibrium.
With the optimal rate, it is shown that the intervention program with healthy life campaign and rehabilitation has significantly reduced number of obese people. The numerical results show that it is much better to start the program in the early situation where number of obese people is relatively low rather than to wait until the number of obese people is relatively high.
Further work should be done to include different social interaction rates for each compartment and different age classes as well as to accommodate the crossmarriage among the parents.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the reviewer for all the constructive comments. This research is funded by the research grant of the Indonesian Directorate General for Higher Education.