A Mathematical Model for Vaccinations

Erin Ellefsen ’17, Major: Math

John Doorenbos ’16, Majors: Math, Computer Science

Kyle Fey, Assistant Professor of Mathematics

About the Project

"‘Herd immunity’ to an infectious disease occurs when a large enough proportion of individuals in the population have acquired immunity so that those individuals who are not immune nevertheless are protected against the disease. It’s attained since many of the possible transmission routes are interrupted by immune individuals,” Fey says.

The project’s goal was to better model disease spread with different vaccination levels in a population and determine how much of a population requires vaccination to achieve herd immunity. “Much of what we’re doing includes modeling a population and running a disease through it, and comparing our results to a current and very widely used model,” Ellefsen says.

“With the 2015 measles outbreak, this research is particularly pertinent and important,” says Doorenbos.

Unique Roles Translate to Varied Challenges

Throughout the research, Fey felt it was difficult to construct model populations which had contact networks that contained the essential features of contact networks in real populations.

Ellefsen felt it was most crucial to be patient during the research process. “When it comes to running programs and waiting for our results, I didn't make cool discoveries as quickly as I'd like,” she says. “I also thought learning how to program was a challenge since it isn’t something I’m as experienced in.”

Doorenbos found communication with his research partners most challenging since it was always necessary to make sure everyone was on the same page during the project. “Coming to compromises was often necessary but usually led to better results in the end,” he says.

Interesting Discoveries When Using a Model

“It was interesting to see how a very popular epidemiological model had results that were inconsistent with our results,” Doorenbos says, “Our findings indicated that this popular model vastly underestimated the number of individuals that needed to be vaccinated to achieve herd immunity.”

Ellefsen was fascinated by the fact that as their model took out some key assumptions that a current and popular model does, the current model does not represent the spread of disease nearly as well. “When the current model would tell us we had achieved herd immunity with our vaccinations, our programs could still have 30-40 percent of the population infected,” she says.

Fey found the research interesting, too. “Our model predicts that populations with contact networks that are highly clustered tend to have a higher likelihood of outbreaks than populations with less clustered contact networks, but that the size of the outbreak tends to be smaller in highly clustered populations.”