# Mathemathical Modeling a Powerful Way to Describe and Predict Nature

When Sebastian Schreiber talks about his work, he uses quite a few references to rock lyrics. He has a PowerPoint presentation titled "Living in the variable world," a nod to a hit by Madonna. Schreiber, distinguished associate professor of arts and sciences at William and Mary, uses quantitative methods to explain and to predict many complex interactions observed in nature. Seen through the lens of Schreiber's mathematical models, our world is a mosh pit of variability. For instance, consider just a single variable factor, temperature.

"If you want to know how temperature affects a certain organism, you'd ask, 'Well, where is the organism?'- because its temperature varies across space," Schreiber said. "And when were you looking at the organism?-because its temperature varies across time. Of course, there's variation among individuals, as well."

If you study enough individuals or record enough temperature readings, you end up with a reliable data set from which to base a model. The concept of "variability" would seem to imply "unpredictability," but not to a mathematical biologist. Schreiber uses different forms of variation to reveal insights about the way diseases spread through human populations or the interactions of predator and prey species. The rock lyrics help to frame some of his findings.

"When I talk about populations persisting and evolving in a variable environment, I use a Clash song because the basic question that every organism is facing is, Should I Stay or Should I Go?," he said. "After all, one might be better or worse. Moving across space influences how fit the population is.

For example, if there's variability only in space, individuals that move less displace those that move more-in other words, the tortoise beats the hare."

Schreiber says he is still looking for a lyric to illustrate his work on disease outbreak. The journal Nature evidently didn't care; Schreiber was a co-author of a paper published in 2005. It proposed a new way of looking at contagious disease outbreaks based on variations in infection rates among individuals, and especially the role of "superspreaders."

The Nature paper examines the phenomenon of superspreading from data collected during eight disease outbreaks, such as the Asian severe acute respiratory syndrome (SARS) incidents of 2003, which made "superspreader" a household word. The diseases studied included pneumonic plague, measles, smallpox and monkeypox, all passed from human to human with no vector in between. Variability turns out to be a crucial epidemiological concept, Schreiber said, because mathematical modeling shows that the degree of variability can predict how the disease is likely to spread.

"Data sets on SARS and Ebola illustrate that diseases can exhibit different degrees of variability in infectiousness," he said. "For SARS, the average infectiousness is achieved by many individuals infecting no one and a handful of people infecting many others. In contrast, for Ebola, the average infectiousness was achieved by most infected individuals infecting the same number of people. Diseases that exhibit more variability are less likely to give rise to an outbreak, Schreiber explained. "However, when an outbreak does occur in a disease with more variability, it tends to spread like wildfire through the population."

High-variability diseases, therefore, tend to erupt in surges, he said. SARS, for example, probably appeared in many locations but did not always progress to an outbreak, because it is one of the more variable diseases.

"But in the places where it did outbreak, it did so very quickly, very explosively, with a lot of people getting infected very quickly," he said. The principles outlined by Schreiber and his co-authors could apply in the case of an outbreak of avian flu among humans or, indeed, any variety of flu.

"Our article is relevant to influenza," Schreiber said. "It's relevant for any disease that's transmitted easily from individual to individual-what we term diseases of casual contact-including the common cold."

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This versatility is one of the prime benefits of mathematical modeling. Just as one size never fits all, one formula won't cover each and every situation, but, as in the casual-contact disease studies, a model can be tweaked to suit circumstances reasonably similar to those presented in the data set. Another example of the versatility of variability comes from Schrieber's examination of the co-evolution of predator-prey-habitat relationships, which he said are applicable to a range of predators that includes-but is by no means limited to-parasitic wasps, wolves, even herbivores "preying" on plants.

Schreiber addressed the effects of predator "handling time"-the time from which the predator catches prey until it begins looking for another meal-in another paper published in 2005 in the Proceedings of the Royal Society. The paper investigates how handling time causes the aggregation of predator and prey species in patchy habitats.

"The song on that is Running with the Devil by Van Halen," he said. "If you think of the predator as the devil, there are some conditions in which the prey actually will go to places preferred by the predator-so they're running with the devil."

Not all prey choose to run with the devil. Sometimes prey will avoid high-quality patches where there's a lot of food and/or better conditions for breeding, choosing patches with a lower quality of life-but fewer predators, he said.

"And the predator will spend most of its time searching for the prey in the high-quality patches despite the fact that most of the prey are in the low-quality patches," Schreiber explained. "So they're doing things that sound quite counter-intuitive at first glance, but not really, because the prey in the low-quality patches are effectively trying to escape the predator and the predator in the high-quality patches are hunting a few prey items, but they're of higher quality."