# SIR Model Helps Monitor Childhood Obesity in Schools

How much physical activity is necessary to remain fit and active? What quantity and quality of food should I eat to stay healthy? Many of us are faced with these questions as adults. Nowadays several online tools and mobile apps can guide us to an answer. If the questions are posed for a person with preexisting health concerns, a doctor can help him/her plan a healthy lifestyle. However, as over 30% of children in the U.S. continue to be overweight or obese, with the looming possibility of growing into generally unhealthy adults, we clearly need a better understanding of this process to ensure a healthier future [2].

Exposing children to healthier foods and physical activity allows them to build a positive attitude towards healthy lifestyle choices. So, would an additional healthy food option at the school cafeteria help? Would more frequent physical education classes be beneficial? Is longer time for active play a good idea? We can all agree that the answer to these questions is a big YES! But exactly how many, how often, and by how much is not as clear-cut. Any new strategy requires reallocation of resources and funds; e.g., having a salad bar is more expensive than a pizza station, and longer active play might take away from instruction time. These answers also vary between schools based on the communities the schools serve. To determine the right balance for a given school, a tool to explore these thresholds and help authorities make informed decisions is required.

Christopher Kribs-Zaleta acknowledges the benefit of mathematics in situations like this: “Mathematics, a language developed to describe complex relationships concisely through symbols packed densely with meaning, lends itself to this sort of use, and under the label of models, mathematical objects and systems have been fruitfully used to describe phenomena observed in the world around us” [1].

With this in mind, mathematicians, health-care practitioners, and schools have partnered to build a mathematical model that captures the dynamics of obesity in schools. This model will predict the obesity prevalence under the status quo as well as the expected change upon the implementation of a certain food or physical activity modification.

Following the well-known SIR (susceptible-infected-recovered) model structure for infectious diseases, children in a school are categorized as susceptible, \(S\) (children capable of becoming overweight), overweight \((I_1)\), obese \((I_2)\), extremely obese \((I_3)\), and recovered, \(R\) (a child of normal weight who was previously overweight). Each class has constant recruitment \((\lambda)\) of students entering school and a constant graduation rate \((g)\) to account for children leaving the system. The model assumes that a child of normal weight can become overweight spontaneously based on their individual characteristics, \(\alpha\), or from association with other overweight \((k_1)\) or obese \((k_2)\) children who might deter them from healthy lifestyle choices. Thus, the loss in number of susceptibles over a year \((t)\) is given by the equation

\[\frac{dS}{dt}=-(k_1I_1+k_2I_2)\frac{S}{N}-\alpha S.\]

Once overweight, a child can transition to the obese and extremely obese categories by gaining further weight. A child can also move from extremely obese to obese, obese to overweight, and then recover back to the normal weight category by losing weight. This gives us the change in number of overweight children over a year as

\[\frac{dI_1}{dt}=\lambda_{I1}+(k_1I_1+k_2I_2)\frac{S}{N}+\alpha S-(\alpha_1+\rho_1)I_1+\beta_2I_2-gI_1.\]

Here, \(\alpha_1I_1\) represents the number of overweight children who become obese, \(\rho_1I_1\) is the number of children that lose weight to recover, and \(\beta_2I_2\) is the number of obese children who lose weight to become overweight instead. We also formulate equations for the obese, extremely obese, and recovered classes similarly. See Figure 1 for a flowchart representing the core dynamical system. This dynamical system to study childhood obesity is a modification of the adult obesity model developed by Thomas et al. in [3].

**Figure 1.**Flowchart representing the model's core dynamical system. Figure credit: Swati Debroy. Images taken from clipartpanda.com, sweetclipart.com and pixabay.com.

The rates at which the above transitions, called *parameters*, happen are estimated through both direct and indirect techniques. To estimate a parameter that accounts for the total number of children in each of the classes at the beginning of modeling, we would collect the data from school records; that would be a direct estimation. One can indirectly estimate parameters by making the model solution for a particular class mimic the pattern in the data for that class over the years, called *fitting*. Numerical fitting is a very common technique to estimate parameters that are difficult to measure directly.

The parameters in the \(I\)-classes will reflect the differences in groups of children based on gender, race, and socioeconomic status. This will help capture the difference in obesity prevalence and impact of intervention in schools based on the difference in communities they serve. Simulations from this model could inform which combination of strategies will bring about the desired decrease in obesity in the long term. The resulting simulations can also assess whether different combinations of interventions could amount to the same outcome, making it possible for authorities to choose a strategy based on other determinants, like availability of resources.

This project involves several undergraduate students at all levels. Students are involved in data analysis and modeling as well as data collection from the school during related interventions. Undergraduates from different majors create educational material for promoting healthy lifestyles and will soon start visiting a county middle school to spread awareness among school children (as part of an ongoing intervention). These undergraduates serve as educational role models and inspire the grade-school students in their community to aim higher in life with regards to health and achievement.

*This article is based on a minisymposium talk entitled "Modeling to Predict Childhood Obesity Trends in School" at the SIAM Conference on the Life Sciences, held in Boston this July.*

**References**

[1] Kribs-Zaleta, C.M. (2013). Sociological phenomena as multiple nonlinearities: MTBI's new metaphor for complex human interactions. *Math Biosci Eng, 10*(5-6), 1587-1607.

[2] Ogden, C.L., Carroll, M.D., Kit, B.K., & Flegal, K.M. (2014). Prevalence of childhood and adult obesity in the United States, 2011-2012. *Journal of the American Medical Association, 311*(8), 806-814.

[3] Thomas, D.M., Weedermann, M., Fuemmeler, B.F., Martin, C.K., Dhurandhar, N.V., Bredlau, C….Bouchard, C. (2014). Dynamic model predicting overweight, obesity, and extreme obesity prevalence trends. *Obesity (Silver Spring)*, *22(*2), 590-607.

Swati DebRoy is an assistant professor of mathematics at the University of South Carolina Beaufort. Her research lies in the interface of mathematics and biology, specializing in disease dynamics. Current projects focus on modeling childhood obesity in the Lowcountry, Hepatitis C, and Leishmaniasis. She is dedicated to involving undergraduate students in her research. |