Category Archives: Math

Friends, Who Needs Them?

I heard a lot of great talks at the TCM conference last weekend, but a few in particular stood out to me.  One of my favorites, presented by Dan Teague of the NCSSM, was about using network/graph theory to revolutionize existing models of the spread of infectious diseases.

The existing models makes an assumption that all people interact randomly as if they were walking about the earth in a random fashion, bumping into and interacting with people without any pattern.  However, as we’re all aware, we’re more likely to interact with people that we’re friends with, or at least have some connection to.  If you were able to visualize a group of peoples connections, you’d have a graph similar to the one below:

Each numbered node represents an individual, and the line between them represents a connection (relationship).  The more lines connecting you to other people mean increases your popularity, and thus you are more likely to interact with other people.  When talking about the case of the spread of disease, higher rates of interaction lead to (on average) a higher chance of becoming infected.

One interesting fact discovered when you run the numbers here: your friends are more likely to (on average) have more friends than you do.  This is called the “Friendship Paradox” [wikipedia link].  This is actually useful to catch a potential epidemic early.  Here’s an example:

In 2010 Harvard University conducted a study of their student body to see if this paradox could help predict the spread of the influenza virus on campus.  319 undergraduates were selected and asked to name students that they were friends with.  A total of 425 friends were named.  The university then followed these two groups, the selected undergraduate group and the “friend” group.  If the friend paradox were to hold true, the “friend” group would have more interaction with other individuals on campus, and thus be more likely to catch the flu, and most likely catch it sooner than the other group.  Using self reported data, it was found that the “friend” group contracted the flu about 2 weeks prior than the selected group, and almost 45 days before the flu peaked for the season.  (Full story and more at:

Using this method, and continuing to advance the ideas held within the graph, researchers should be able to mathematically determine where and when epidemics might emerge and take the necessary steps to prevent the spread of disease before it reaches a tipping point.  Even better, through theory and simulation, researchers should be able to objectively determine which people provide the biggest gain in overall health to a community if they were to be vaccinated (either very “popular” people who interact with a lot of others, or people that link two large groups together).

Stay safe out there, and just remember, your friends and family will get you sick!


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