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# The Math Behind Decision-Making

Mathematics is a language interwoven in the everyday movements of the world around us. It is universal, often surprising in its applications, and can reveal previously hidden patterns and information about the workings of the world. Tim Reluga is an applied mathematician using the language of mathematics across disciplines in his work with the Center for Infectious Disease Dynamics (CIDD) at Penn State.

Reluga, associate professor of mathematics and biology, took an interest in biology at a very young age. As a child, he spent summers wading through tidal pools at the beach with his father, who was a high school biology teacher.

“My father would teach a marine biology class, and so when we would go to the beach, he would point out, ‘Look at this type of seaweed or that type of animal or shellfish.’ So I spent a lot of my time wandering through tidal pools collecting things and catching things. And I was always sort of fascinated by the huge variety of different things we would find there. And some years there would be new animals there that had never been there before. I was always fascinated by the life that we would find around us,” Reluga said. There on the beach, Reluga knew biology was in his blood.

But when he showed an aptitude for calculations and computers in school, he began to follow a different path.  He found that, in high school, the science math paired with was physics, not biology: “You would go into class and learn about Newton’s laws, then you would go and figure out the trajectory of a cannon ball and get the answer. Then, you’d go into the lab and figure out that ‘Yeah, that answer is about the same’ and the math helped you predict something about it. But, in biology, there never was really any math,” said Reluga.

In college, he skipped over biology his first semester because he felt he never quite fit with the traditional biology major following a pre-med track. So, he leaned more toward physics.  Reluga said, “I was sort of debating ‘Do I major in physics or math’ and I was sort of leaning toward physics because it seemed like an exciting area.” Then, an entertaining biology class during his second semester at Tufts University made him decide to double major in his two interests.  He recounted, “I took a biology class my second semester.  There was a guy who was giving a lecture on Mendelian Genetics the first day of class and it was just fun.  It was so much more entertaining than the physics stuff I had been doing.  And so I said, ‘Okay, well I’m going to do biology and I’m going to do math.  Maybe I can find a way to do both.’”
Merging his roots in biology and his aptitude for math led him to graduate work at the University of Washington, where he was surrounded by faculty with the same interests he had. The research being done at the University of Washington applied mathematics to biology, and the faculty there were discovering how living things work in new and exciting ways. Reluga recalled, “My adviser, Mark Kot, studied ecology. There was a statistician named Elizabeth Thompson who was involved with saving the condors.  There was another guy who studied insect flight.  There were all these sort of mathematically inclined people who were applying math to biology.” Reluga had found his track, and his calling.

After postdoctoral work at the Yale School of Medicine researching disease dynamics, Reluga studied how disease works in the body at Los Alamos National Laboratory, performing research to help make pharmaceutical drugs more effective.

One day, a dream opportunity arose. The Center for Infectious Disease Dynamics (CIDD) at Penn State was forming and they were looking for a mathematician to join the team. Reluga answered the call.

With expertise that crosses the disciplines of math and biology, Reluga adds to the world-renowned expertise of the scholars in CIDD. Some of his research involves applying the language of mathematics, through game theory, to create models that better explain human behavior and how it affects the spread of disease. And his award-winning research in this area is helping to create more effective public policy around the world.

What is Game Theory?

But he sees what he does in a much more simple way: “What I’m really interested in is how living things work. That goes from people down to diseases, and everything in between. I use math as a language to try to describe these things and better understand how they work,” he said.

#### Using Math to Describe Human Behavior

Recently, Reluga’s research has focused on using game theory to create a framework for describing how human behavior can impact public health. Reluga explained, “Science gets you so far, but human nature kicks in at a certain point and makes things much more difficult.  If you want to actually eradicate diseases, understanding how people react…is going to be essential to future public health efforts.”

To illustrate this point, Reluga cited an infamous policy failure.  The federal government approached the problem of the public health crisis nicotine addiction was creating by mandating a lower concentration of nicotine in cigarettes.  The thinking was that less nicotine per cigarette meant less exposure to the drug and would result in better national health.  But, the government did not account for “policy resistance” created by individuals looking out for their own best interests.  Instead of less exposure to cigarettes, smokers were actually smoking more  to get the same dose of nicotine and the policy resulted in poorer overall health.

“Good policy takes into account how people will react.  It tries to understand the mechanisms of disease transmission and behavior,” says Reluga.  His research gives policy makers the tools to create good policy and avoid situations of policy resistance.  It is a systematic way of comparing policy options.  “The mathematics allows us to set up a framework to compare data instead of a system where the loudest person wins the argument,” Reluga explained.

#### Winning Research

Along with Dr. Alison Galvani of Yale University, Reluga was recently awarded the prestigious Bellman Prize, for their research paper, “A General Approach for Population Games with Application to Vaccination,” published in Mathematical Biosciences during 2010-2011.   The Bellman Prize is judged by a panel of mathematical biologists and is awarded every two years to the best paper published in the journal.

The winning paper explores the interaction of public health initiatives that focus on the best interests of the community and the behaviors of individuals who will act in their own best interests with regard to vaccination.  However, Reluga and Galvani created their model to enable more extensive use in the future.

In the past, theoretical epidemiologists would look at ideal vaccination coverage by looking at the entire community and assuming that everyone would act in the best interests of the group. They failed to account for an individual’s choice not to adhere to medical advice and make decisions based on his own personal best interests.

Generally, when there is prevalent disease, the risk is obvious and individuals are more willing and likely to get vaccinated.  It makes sense for the group and also for the individual.  However, when we begin to eradicate disease, or at least effectively treat it so that it is no longer as visible, the perceived risk diminishes and the choice to vaccinate becomes less obvious.  People begin to deviate from medical best practices and act in their own perceived best interest.  Traditional theories that ignore individual behavior could not explain mounting resistance to health initiatives and diminishing coverage, nor could they advise how to counteract it.

This is becoming increasingly obvious with diseases like measles and whooping cough which are both experiencing resurgence even though they were nearly wiped out and are both extremely preventable.  When everyone knew someone who had been infected, was sick, or who had died, individuals were clamoring for the vaccines. Now there are fewer infected people and the diseases are no longer widely visible to the public, so the perceived risk of infection is dramatically lowered. Individuals are now weighing the personal choice of vaccination.  Some have found reasons of personal interest not to vaccinate, are deviating from medical advice, and opening the door for these diseases to make a comeback.
When Reluga and Galvani began their research, they realized that both the population-scale models and the individual-scale models were needed to describe disease management problems. Disease prevalence depends on the behaviors and interventions of the policy-makers and individual behaviors can depend on disease prevalence.  Their approach allows the two models to be expressed in parallel to “create population games with explicit ecology dynamics” according to Reluga and Galvani.  Reluga explained, “The act of accounting for all the different incentives and choices and how our decisions influence what happens next should give us better insight into what consequences our decisions have, how to hold ourselves accountable for our decisions, and how to make better decisions so that we can deal with these problems that we can foresee but for some reason can’t quite avoid.”
The framework Reluga and Galvani developed in their 2011 research paper has since been expanded and has been used to investigate the management of several infectious diseases, including HPV and chickenpox, in a variety of contexts.  And new possibilities are being explored outside of vaccination.

In 2013, Reluga published a research paper in Bulletin of Mathematical Biology entitled, “Equilibria of an Epidemic Game with Piecewise Linear Social Distancing Cost.” The paper builds upon the framework Galvani and he created to show how individuals change their behaviors (known as “social distancing”) to balance the cost of prevention against the risk of infection when faced with an epidemic.

Reluga talked about applying his framework in new areas: “As an applied mathematician, I’m not an expert in all fields, but I bring it all together and create a framework that we can analyze, study and make sense of using the mathematics.  Pulling it all together is where the math comes in.” Realizing the massive impact human behavior has on public health, it will be exciting to see just how far-reaching the applications of this framework may be.