Physics gravity model applicable to disease spread

Epidemics are like forest fires, says Ottar Bjornstad. “They tend to burn through a host population, deplete it, and move on.

“We have fairly good theories to explain the local dynamics,” adds Bjornstad, an assistant professor of entomology and biology at Penn State who also holds an appointment in statistics. “The size of an outbreak depends on three things: The number of susceptible hosts, the rate of transmission, and the length of the infective period. You can actually calculate, given these three values, the number of individuals you would expect to be infected in the course of an outbreak.”

map of England with red, yellow, and green contours
Ottar Bjornstad

Contour lines on a map of England show the spatial spread of childhood viral diseases, overlaying color depicting population density. The composite image shows that the highest risk of disease outbreak is associated with big cities.

With measles, for example, one of the most contagious diseases known to humankind, “You can expect about 20 secondary infections from one initial infection,” Bjornstad says. “That means an outbreak will hit 95 percent of the population before it dies out. So it was said about measles before mass vaccinations started in the 1960s that everyone got it.

“Smallpox, by comparison, is about one-third as infectious as measles, which means that in a completely susceptible population—think of native Americans before the Europeans arrived—you'd expect 80 percent of the population to get hit before the outbreak winds down. Influenza is about 70 percent.” In practical terms, this percentage of the population must be vaccinated in order to bring about what epidemiologists call “herd immunity.”

We know a good deal, Bjornstad reiterates, about how an infection sweeps through the local “herd.” What we don't understand is how it gets from the herd in Boston to the one in Seattle, or Tokyo. “In the old days, pre-automobile, infectious disease would have to travel locally,” he explains. “To cross a long distance it had to make very many intermediate steps.” Even in more recent times, in the fight to eradicate smallpox from developing countries, “spatial spread was typically local.” What that meant, Bjornstad says, was that once the infectious agent was close to being wiped out, health authorities could set up a cordon sanitaire —a vaccination ring around the infected area —;and thereby extinguish it.

“Compare that to the situation in this country today, where air travel is a part of daily life,” Bjornstad says.

One result of increased mobility is much more complex patterns of human contact, and, therefore, of disease transmission. These transmission “networks,” Bjornstad says, remain largely invisible to us—a special concern in an age faced with the threat of bioterrorism. “Until we know what the networks look like, we can't design appropriate control strategies.

“Ideally, we'd want to understand the different commuter patterns so we could create a kind of risk map,” he says, “so that if an infectious agent hits a given area we know where it's most likely to hit next. We would want to have a situation where, say, we knew that more people living in Baltimore than in Philadelphia work in D.C.”

Risk-mapping is complicated, however, by other variables. “With a sexually transmitted disease, for example, travel patterns are only a good indicator of a network if everyone who travels has identical sexual behavior.” Similarly, for a disease that affects only children, adult movements are meaningless.

To get a better grip on the basic dynamic, Bjornstad and his colleagues at the University of Cambridge turned to the historical record—in particular, the record of measles outbreaks in Great Britain during the middle of the 20th century. Measles make a great case study, Bjornstad says, and not only because they are so contagious. “From 1944, when the register general made measles a notifiable disease, every M.D. in every town had to report the number of cases encountered each week,” he explains. This information was compiled into a national disease database of rare completeness and resolution.

“Before mass vaccination, there were large outbreaks every two years in the large cities,” Bjornstad says. “You could have 50,000 cases nationwide in a single week, as opposed to a maximum of 1,000 a year now.” By plotting weekly statistics from a thousand cities and towns and applying a new statistical technique called wavelet analysis, Bjornstad and his colleagues were able to tease out other, more subtle patterns. A video clip on Bjornstad's computer demonstrates what he calls this “spatial mosaic.” On a map of the British Isles, with red circles designating measles outbreaks, London, with its population of three million people, is instantly recognizable as a giant cluster of red in the south. At the other end of the scale, towns with populations below 1,000 are represented by black and red pinpoints. As the clip runs and the weeks roll by, the red areas mushroom and blink as outbreaks pass through different locations.

“What we find,” Bjornstad narrates, “is that in cities above half a million the infectious agent is always present. When an epidemic breaks out, it hits the big cities first, then intermediate, then small, moving out in waves at a rate of about 5 kilometers a week.” The delay in reaching very small, isolated communities is sometimes over a year, knocking these far-flung communities altogether out of the larger biennial pattern.

As an epidemic dies out, Bjornstad notes, “the infectious agent retreats back down to the big cities.” In general, two infection centers—London and the Northwest, including Manchester and Liverpool—govern the timing of outbreaks all over the British Isles.”

Bjornstad and colleagues Bryan T. Brenfell and Jens Kappey reported these results in the journal Nature in December of 2001. “This is very much a work in progress,” Bjornstad stresses. Plotting all that space-time data, he says, “is a great challenge to our set of tools.” In addition to advanced numerical techniques, “you need monster computers.” For now, Bjornstad relies on Penn State's LION-XE, a cluster computer housed in the basement of the Center for Academic Computing that holds the equivalent of 250 desktop processors, but he is looking to upgrade soon to a system four times as powerful.

“The hope is that once we've developed all the technology to do this kind of analysis and achieved a good understanding of measles, we can go back and look at other infectious diseases. If we can understand their spatial networks, we will be able to come up with better containment strategies. “Ultimately, we want to be able to design a new type of cordon sanitaire.”

Ottar Bjornstad, Ph.D., is assistant professor of entomology, College of Agricultural Sciences, 515 Agricultural Sciences and Industries Bldg., University Park, PA 16802; 814-863-2983; onb1@psu.edu. His work was supported by the Wellcome Trust and the National Science Foundation.

Last Updated January 01, 2003