## Archive for March 20, 2020

### Coronavirus and Mathematical Modeling

In Oregon, Governor Kate Brown banned gatherings of more than 250 people. Similar restrictions have been imposed in other states, too, and the Center for Disease Control and Prevention (CDC) recommends that organizers cancel or postpone any event that consists of 50 or more people. Moreover, the CDC recommends that you “put distance between yourself and other people,” because *social distancing* is believed to inhibit the spread of coronavirus. The virus is thought to spread between “people who are in close contact with one another (within about 6 feet).”

All of this information leads, of course, to **an incredible opportunity for students to engage in mathematical modeling**.

What size space would be appropriate for a large gathering to ensure that all attendees could maintain an adequate distance from one another?

This is a variation of a classic packing problem, a mathematical optimization problem that involves packing objects (in this case, people) into containers (concert halls, restaurants, or some other social gathering spaces).

To create a reasonable model, some assumptions must be made. For instance, one assumption might be that each person is treated as the center of a circle with radius 3 feet, and circles are not allowed to overlap when packed into the container. Consequently, no two people will ever be within 6 feet of one another.

Statistician George Box noted, “All models are wrong, but some of them are useful.” It’s reasonable to assume that each person is surrounded by a protective cylinder, but how could these cylinders fit together? What about this model could be improved? What aspects of this model are appropriate for analysis but don’t quite work in the real world?

One configuration that could work is arranging the 250 people into 10 rows of 25 people each. With 3 feet above and below, to the right and to the left, of every person, that arrangement could fit into a rectangle that measures 30 feet × 150 feet, which has an area of 4,500 square feet.

Is a better arrangement possible?

A corollary problem, of course, is considering the maximum number of attendees that a particular space could handle to maintain social distancing. For instance, our local synagogue suggested that congregants not attend Friday services, but they will accommodate those who feel strongly about attending. Their website states, “All services will be held in the main sanctuary, and we will encourage any participants to sit at a distance from others.” How many congregants could be seated in the sanctuary and still maintain safe distance?