# In AP Calculus, Students Explore Epidemics

Ebola. Yellow Fever. Nipah Virus. Pneumonic Plague. Most people would rather not think about these frightening scourges. Yet, understanding an infectious disease’s path is an essential step toward prevention, treatment and cure. Routinely, scientists rely on complex math to help advance their research. They could undoubtedly learn a thing or two from Chapin’s AP Calculus BC class.

Through a recent investigative project, Lauren Riva’s Class 12 students have become experts in a far-reaching system known as mathematical modeling, which helps professionals in a range of disciplines analyze various circumstances and predict outcomes through a series of variables. In this particular undertaking, the students applied their knowledge of modeling to the fascinating field of epidemiology, the study of health-related states or events, including diseases.

“I wanted the students to see how calculus is being used in a variety of fields,” explained Dr. Riva, who is the Head of Upper School Mathematics and also teaches algebra. “Over the past weeks, they learned about compartmental models to study the spread of infectious disease.”

To begin, the students were asked to pick a disease they found interesting and wanted to explore in depth. Two websites – the Centers for Disease Control and Prevention (www.cdc.gov) and the World Health Organization (www.who.int) – provided them with valuable information. Next, they searched for a research article (published within the last decade) that described a model of the spread of the disease they had chosen to examine.

Dr. Riva then introduced her class to the ODEToolkit, an open-source modeling software from Harvey Mudd College that provides a method for calculating, visualizing and investigating solutions to the models. Using this sophisticated software, the girls re-created the mathematical model — a framework in which to understand the diseases — from their articles.

Simply stated, a compartmental model assigns the population to segments based on certain assumptions. For example, a model in epidemiology divides individuals into two groups: those susceptible to infection from the disease (denoted by “S”) and those infected by the disease (“I”). To expand the boundaries of the probe, additional compartments can be added, such as those recovered or immune to the infection (“R”); identified infectious individuals for whom treatment was effective (“Q”) and identified infectious individuals for whom treatment was not effective (“D”).

After they finished the research portion of their projects, Dr. Riva’s students tackled an imagined scenario: What, aside from mass panic, would happen if there was an outbreak of your disease in Manhattan? The young mathematicians discovered how each disease would behave, based on the determinations of their model.

For the final step in the process, they presented their findings to their class and to several visitors, most of whom (including this writer) were not familiar with calculus or mathematical modeling. For 10 engrossing minutes, the students – working mostly in pairs – spoke with impressive authority and confidence. They began by giving an overview of their disease, including its cause (such as a bite from an infected mosquito); the symptoms (ranging from fever to unexplained hemorrhaging); how it spreads (through bodily fluids, for example); the geographic location of outbreaks; as well as the incubation period, treatment options and long-term impact.

In preparing to present, the students considered such fundamental questions as: Would the disease die out? If so, when? How many people would be affected? To support their research, they created a flow chart demonstrating a disease’s path and a model of its course over 50 days.

“Modeling epidemics was especially interesting to me because it reached across so many disciplines,” commented one student who analyzed the Nipah Virus. “We studied the scientific context of a disease, the social implications of said disease and the mathematical representation of its spread.”

Whether an outbreak of Yellow Fever in the Republic of Congo or an isolated case of Ebola in New York City, the AP Calculus students are exceptionally well prepared to analyze these deadly diseases, one mathematical model at a time.

Browse photos from the presentations below: