MATHEMATICAL DETECTIVES DETAIL A DEADLY DISEASE

Five years ago, Mac Hyman began to worry about AIDS. “I was convinced that the problem was very much larger than the people around me were reacting to,” recalls Hyman, a mathematical modeler at Los Alamos (N.Mex.) National Laboratory. “There just wasn’t any other problem that was crying out like this.” Driven by this conviction, Hyman convinced fellow mathematicians, social scientists, computer specialists, and medical experts to help him build a theoretical model of the disease. Their efforts have enlarged the work of a cadre or biostatisticians tracking the course or the disease and evaluating ways to slow the epidemic. Although they take different approaches—statisticians work within the confines of existing data, while mathematicians juggle the more theoretical elements of the epidemic—the numbers may save lives by pinpointing flaws in our understanding of the disease. Models also identify what is not yet known about AIDS. “Trying to model how the virus spreads in a population is very important,” says John P. Phair, professor of medicine at Northwestern University Medical School in Chicago. “If the one million people infected are likely to infect 10 million others, you have to implement greatly different policies than if [they are likely to infect] just one million others.” But coming up with those numbers hasn’t been easy. The Centers for Disease Control keeps count of the number of people diagnosed with AIDS. However, the figure says little about the number of people infected or about the length of time between infection and development of symptoms. Working with data that have so far been serendipitously gathered, Mitchell Gail of the National Cancer Institute and Ron Brookmeyer, associate professor of biostatistics at Johns Hopkins School of Hygiene and Public Health, have devised a method for pinning down these unknowns. Using a system called back calculation, the pair are able to estimate incubation periods from the approximate dates of infection of a small group of people. They began with persons who were infected with AIDS via blood transfusions, as well as gay men in San Francisco whose blood was sampled as part of a hepatitis study and then resampled at a later date. They applied that information to the number of cases for each year of the epidemic and worked backward to estimate the number infected for the years prior to the epidemic. Based on those numbers, Brookmeyer and Gail can make projections of future cases. Although back calculation and other creative statistical approaches have been hailed as ingenious, “these methods don’t really give you an understanding of the mechanisms that are going on,” says Herbert Hethcote, a professor of mathematics at the University of Iowa. Hethcote has used information about the AIDS epidemic in San Francisco to develop a transmission model. His computer model, like that developed by the Los Alamos team, helps scientists rank the factors contributing to the spread of the virus. These computer simulations help evaluate how vaccines, treatments, changes in behavior, or different degrees of mixing of people from various risk groups affect the course of the epidemic. Says Hyman, “we can use this knowledge to help set priorities in research.” Much of the modeling involves three or four equations, but a dozen modeling scientists, including Hyman, are also tackling AIDS with partial differential equations. Rather than simplifying risk into two categories, high or low, Hyman’s model represents risk as a continuum. In addition, the model accounts for social bias—the fact that the disease spreads unevenly because of the nature of the contacts made by those who are infected. To provide accurate predictions, the modelers must incorporate a half-dozen variables, which means doing multidimensional computations that are still beyond the power of todays supercomputers. “They will bring a Cray to its knees in no time at all,” says Hyman. Most modeling efforts are still trying to expand from two dimensions to three. In addition, the simulations lack such critical data as the size and growth of risk groups and details about sexual activities. Nevertheless, the simulations have revealed that the spread of AIDS is determined largely by an infected person’s number of sexual partners, choice of partner, and patterns of needle sharing. The Los Alamos model also revealed an important difference in the rate of spreading between this epidemic and others. In most epidemics, the number of cases grows exponentially overtime, hut the number of cases of AIDS grew cubically. This rate of growth was consistent even when the researchers looked at subgroups by race, sex, or sexual preferences. Their conclusion is that the AIDS epidemic has spread more slowly than expected—not because of education and changes in behavior, as health officials and the media have claimed, but because of the tendency of those infected with the HIV virus to associate with their peer groups. While the mathematicians and statisticians agree that about one million people currently are infected with HIV, there’s no consensus about how the epidemic will look in five years. “It’s analogous to trying to predict weather long-term,” says Phair. “We don’t know enough about the biology.” The disease has too many variables, and all the models depend on too many assumptions. In the meantime, many experts agree that these mathematical sleuths have made tremendous strides. And the sleuths themselves are not discouraged by their rate of progress. “I take the position that you have to do the best you can,” says Brookmeyer. “The role that we need to continue to play is to really quantify our uncertainty and to communicate that to the public.” —E.P.

MATHEMATICAL DETECTIVES DETAIL A DEADLY DISEASE

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