Thoughts about mathematical models might not keep you up at night. But, in fact, they may ultimately let you rest easier about your health. Without these tools, scientists might have taken a lot longer to confirm that the HIV virus is killed and replenished at a high rate during what was thought to be a latent stage of infection. Or that childhood vaccination for congenital rubella can delay the onset of the disease, leading to more serious cases in children of mothers who contract it during pregnancy. In the past decade, a growing number of epidemiologists in the United States have exploited the power of mathematical models, combining statistical analysis with epidemiologic techniques to boost our understanding of infectious diseases.
This approach took root in the early 1900s, when scientists mathematically represented the spread of malaria, in part to determine what intervention against the disease might work best. One common term in their models, the mosquito biting rate, was squared. So it became clear that to interrupt biting by protecting individuals with bed nets or sprays would have a greater effect on curtailing infection than other non-barrier intervention plans. Still more effective would be to systematically reduce the mosquitoes' lifespan, so that large numbers of infected insects died off before they could, in turn, cause infections in humans. The idea behind the process was to generate some general hypotheses about a complex disease that could then be tested in the field. "You can get valuable information by looking at models, even when they're not perfect or some of their assumptions are wrong," says Megan Murray, assistant professor in the Department of Epidemiology at the Harvard School of Public Health. "We don't use models to estimate exactly how good or bad interventions are but to get an overall understanding of what might be the case. Down the line, we can get more specific answers from follow-up observational studies or clinical trials."
From a model she recently developed, Murray has derived important insights into what we can and cannot infer from molecular epidemiologic studies of tuberculosis (TB). Since 1990 epidemiologists have conducted these studies to find out more about the disease and its transmission, but they have frequently read more into the data than was warranted, cautions Murray. For example, an epidemiologist may find a large cluster of TB cases that share a particular DNA fingerprint and conclude that the strain infecting these people is more transmissible than one found in a smaller cluster. But cluster size is often determined by other factors, such as whether or not an individual or his neighbor has HIV or how people come in contact with one another in a community. "You can't make these kinds of inferences from these kinds of data," Murray insists. "If you want to measure the transmissibility of an organism, you have to design a study pretty differently."
To illustrate this point, Murray created a mathematical model that types individuals according to a variety of exposure-related conditions, such as proximity to an area with a high incidence of TB infection. The model simulates a process by which TB infects individuals in a community, goes into latency, and later reactivates. Input parameters that determine chance interactions among members of the community over time were varied automatically in hundreds of computer runs, resulting in a cluster distribution of TB cases. Among other things, the model showed that multiple host-related factors--HIV prevalence, chemotherapy and vaccination against TB, and latent TB infection--contribute to a wide variation in how clusters of the disease are distributed.
Murray, who is also studying ways to mathematically represent the control of drug-resistant strains of TB, stresses that modeling of infectious diseases differs from that of other illnesses tracked by epidemiologists because individual outcomes are not independent of one another. "Your chance of getting heart disease is not much affected by whether or not your neighbor gets heart disease," she explains. "But your chance of getting a respiratory infection may be very much affected by your neighbor, especially if you get the infection from him. So there's this non-linear element of infectious disease dynamics that depends on transmission, and that's what we're modeling."
Murray adds that her models are not intended to forecast a TB epidemic or decline. "There are a lot of TB modelers out there saying we can reduce TB incidence by 80 percent or eradicate TB through various interventions," she says. "These estimates are based on parameters that are not well defined. But even if we don't come up with an exact answer, mathematical models give us a much better sense of what could happen or what kinds of things drive epidemics."
Assistant Professor of Epidemiology Marc Lipsitch, a colleague of Murray's who models antibiotic resistance in different pathogens, agrees. "A major role of models in the antibiotic setting is extrapolation," he notes. "Clinical intervention trials are expensive and logistically difficult, so you can't ask whether a particular antibiotic use or infection control policy works in every possible setting. What models can do is to say what are the likely characteristics where this worked, evaluate why it worked, and how we can extrapolate those findings to other places where we don't have the time and resources to do multiple complicated trials."
Based on a model he recently developed in collaboration with former Emory University colleagues Bruce Levin and Carl Bergstrom, Lipsitch has found that effective interventions to control anti- biotic resistance in hospitals should show results in a matter of weeks or months, rather than years or decades in the general community. Interventions can succeed much more quickly in hospitals than in the community, he explains, because hospitals rapidly and continuously import and export new bacteria and people. This fast-paced exchange, coupled with the fact that incoming patients are much more likely to carry drug-sensitive (drug-susceptible) bacteria than existing patients who have been exposed to antibiotics already, enables drug-sensitive bacteria to quickly outcompete drug-resistant bacteria for hosts. Eventually, the drug-resistant carriers become vastly outnumbered.
"The model shows that such a dilution effect can happen," says Lipsitch, "and is consistent with what people have seen: if you intervene to reduce resistance in a hospital, you often get a very rapid reduction in resistance." Using what he calls a compartmental model, Lipsitch tracks individuals who carry a drug-resistant strain of bacteria, a drug-sensitive strain, or no strain. People switch compartments if they enter or leave the hospital, or if they become colonized or cleared of a strain. The rate at which people move from one compartment to another depends on how many people are in each compartment at a given time. For example, the rate at which people move from the no-strain compartment to the resistant-strain compartment depends on how many people currently carry the resistant strain. Runs of the model show that the high admission rate of people with drug-sensitive strains leads to a quick reduction of those with resistant strains.
Lipsitch, who has also applied this technique to simulate the transmission of bacteria that cause pneumonia, envisions multiple appli- cations for the kinds of models he and Murray are developing. "One way mathematical models can be used is to help you extrapolate from one disease, population, or organism to another disease, population, or organism," he says. "They can identify the factors that are critical in determining the outcome, and you can ask whether those are the same in the two cases you're comparing"--which may mean a few more sleepless nights for hard-working epidemiologists but many healthier days to come for us all.
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