As a mathematical biologist at the University of Minnesota, Rochester, Claudia Neuhauser studies broadly how living things coexist. Mathematics can predict whether a virus will devastate a community of bacteria and die out itself. The math can also show how a disease sickens a few people, expands into a full-blown epidemic, and then disappears. Similar principles apply to how tumor cells multiply, how microbes evolve, and how different species of animals grow to occupy an ecological niche.
As a teacher, Neuhauser, who is also a Howard Hughes Medical Institute professor, has trained biology majors who are comfortable with math. Faced with bored biology students in the freshman calculus course she was teaching, she wrote a calculus textbook that uses biology examples. The students started asking for more math courses.
Neuhauser provided the courses, then worked with other faculty to incorporate more math into chemistry and biology classes. Now a vice chancellor, she seeks to integrate math into the entire curriculum. She sees math as a necessity for full-fledged biologists and physicians who need to understand the uncertainties in their fields. The third edition of her textbook, Calculus for Biology and Medicine, came out in January 2010.
At the February 2010 meeting of the American Association for the Advancement of Science in San Diego, Neuhauser spoke about the mathematics of diseases. She sat down with SciCom's Olga Kuchment at a hotel bar to discuss the big picture.
What drew you from math to biology?
My degree is in math and physics. In my final year of graduate school at Cornell University, my advisor told me that I should go to this ecology seminar. I was just sitting in the seminar and thought, "Wow, this is complicated. And it would be really interesting to model."
This was the time of the human genome project, and there was all this excitement about molecular biology. I went to the University of Southern California and got involved with doing DNA sequence alignments. That's a way for a mathematician to get a foot in the door. I thought, "I can deal with that, it's just a sequence of letters."
If you're willing to learn biology, you can contribute to a number of different fields because the tools are transferable.
That's the nice thing about math—you understand how everything is related. In biology, you may be focused on a specific system and not see all these parallels.
"The bulk of students who go into biology are not there because they love math. But we're getting massive amounts of data now. The students need to not be scared of data."
Did you ever have math anxiety?
No, but I can understand it. I tutored a lot of students when I was in high school. Math is a foreign language. If you don't become fluent, it can be very, very painful doing these problems. That drove the way I teach now. I want to show students the connections.
The bulk of students who go into biology are not there because they love math. But we're getting massive amounts of data now. We're sequencing entire genomes of humans to figure out ways to diagnose diseases. The students need to not be scared of data.
During your first winter at the University of Minnesota, you taught calculus to biology students. Could you talk about that experience?
The students didn't like it. It wasn't clear why they were taking calculus; there were no biology examples. My conclusion was either to never teach it again or to change it, so I decided to change it.
I wrote a calculus book. I decided it should be at the level of an engineering book, but the examples should come from biology. I wrote it the year I was at the University of Wisconsin, in 1998.
It only took you a year to write the textbook?
Yes, I was quick. I spent a year in the library essentially, sitting between the shelves, pulling out biology books, flipping through until I saw an equation, and thinking, "Can I make a math problem out of this?"
And what was the result?
I mean, it was successful and the students liked it a lot better. They saw the applications, but math was still not integrated into the other courses.
The idea is to introduce other faculty to teaching math in the context of an actual problem. For instance, there's data out there of tumor growth in a patient. They measured how an untreated tumor grew over time, and it fits exponential growth. You can use this even in a pre-calculus course. You fit an appropriate function to the data and see, when did the tumor start or when will the patient die.
These models of cancer growth essentially say that you're at two-thirds of the lifespan of the cancer by the time you detect it. This whole early detection is not early. The cancer has been in the patient for like 10 years, and then the patient has another five years before it reaches a lethal burden. That opens up other discussions that go outside math and biology, into the social sciences.
You described a mathematical model for how a pathogen travels through a host population, in a Nature paper in 2006. Could you tell me more about the situations you modeled?
Close migration means that many of the offspring end up close to the parents. That means the offspring have to be able to utilize the same resources over and over again. And so in this case, using up all the resources really quickly, depleting the resources is a bad strategy. So the species that doesn't do that has more of an advantage.
In another species that has unrestricted migration, they can go places. Their offspring are not going to be where their parents are, where their brothers or sisters are, so they can just really exploit the resources. If you can disperse farther away, you don't have to live with the mess you produce. [Laughs]
Anytime you have species coexisting with each other, there are trade-offs. There are different ways of making a living, essentially. For example, you have plants that produce big seeds that land close by, and you have other plants that produce small seeds that disperse widely.
You predicted that restricting a pathogen's movement would decrease the density of its offspring. Your colleagues tested the prediction experimentally. How did that play out?
We had a virus and an E. coli—the phage is the pathogen and E. coli is the host. And we used a robot to mimic long-range and short-range dispersal to see what that would do to the host and the pathogen. We had certain expectations. And they weren't met in the real system—there was something else going on.
What our colleagues found was that the virus evolved. When the migration is local, we get phage that don't exploit the resources as much. Essentially, evolution generates different species with different ways of making a living. And as soon as we incorporated that heterogeneity into our models, we were able to explain what we saw.
And that's a good example of why you have to do both experiments and modeling.
It sounds as though you enjoy having your predictions proved wrong.
[Laughs] Yes, if you predict everything correctly it's kind of boring.
Mathematicians have been modeling diseases for decades. How does your work fit in with what came before?
Most of the models in ecology are not spatial, and are deterministic [determined by preceding events or natural laws] systems of differential equations. When you have multiple hosts and multiple symbionts, you end up with a large number of differential equations and the analysis becomes almost unmanageable.
We've been interested in adding spatial components and stochasticity [randomness] to the system.
Why is it important to account for stochasticity when modeling diseases?
When you have a small number of individuals and you have an active infection, the infection can just die out because nobody gets infected for a long period of time. And that's a stochastic effect. Whenever you get to small numbers, stochasticity becomes really important.
So why is it important to understand what happens in small numbers of hosts?
That's how an infection gets started; you don't start with millions of people infected. It's the same when you have an invasion of an exotic species into an environment. Whether or not that invasion takes hold is a stochastic effect. It often takes multiple invasions.
What are the differences between modeling diseases in bacteria, in plants, and in people?
The main difference is what I would call the social network of the host. If you have things that move around, like people, you can have a highly connected network. In plants, you might have a more restricted local network.
The models you work on are pretty general. How do they help us fight diseases?
If you want to manage a system, the general patterns give you an idea of what you should be doing. Are there patterns that hold across different systems? If so, then we would not have to start all over again with another species.
But I'm also involved in a project that has to do with viral therapy in cancer, and that's a very concrete system. You infect a cancer with a virus and hope that the cancer cells have a much, much higher probability of infection than normal cells. The cancer cells are essentially the host cells. But the goal is different. There, you actually want to get a really, really bad epidemic going and kill every host cell, which is sort of the opposite from what we usually want—to control the epidemic and eradicate the disease. In this case we want to eradicate the host.
For the viral therapy, you have to kill every single cancer cell to get rid of the cancer. It's not good enough to get down to a thousand cells.
Could a mathematical model predict from the first small numbers of infections how a disease such as the H1N1 flu might spread?
People are trying to do that, but when you have small numbers, you have big error bars too. But there is a value to using models to see how bad it can get. You can find out how infectious it is, and if you know what the mortality rates are, you can be developing the right level of concern.
But the models aren't perfect.
That's where the randomness comes in. [Laughs] The models aren't perfect, but they can give you a sense of what to expect.
What do you see as the biggest problems in developing mathematical models of disease?
The behavior of humans is hard to predict. Controlling for the movement of humans is always quite difficult. We're not modeling the actual country, we don't know where people are going. And when you have a new disease, many parameters are just not known—infection rates, for instance.
With H1N1, we certainly overestimated the mortality rate, but that's a good problem to have.
Olga Kuchment, a graduate student in the Science Communication Program at UC Santa Cruz, earned her bachelor's degree from the University of Kansas and her Ph.D. from the University of California, Berkeley, both in chemistry. She has worked as a reporting intern at the Santa Cruz Sentinel and the news office of the SLAC National Accelerator Laboratory; this spring, she will report for the San Jose Mercury News.
© 2010 Olga Kuchment
