The math behind measles

You do the math: In 1963, a highly effective measles vaccine became widely available. In 1997, the Infectious Diseases Society of America reported 996,479 measles-related deaths worldwide.

Ninety-nine percent of the deaths were in the developing world, about half in Africa, where vaccination programs reached as little as 70% of target populations. In contrast, the United States, which has almost universal vaccination, had in 1998 a total 99 cases of measles - mere occurrences of the illness, not deaths. Most of these cases were contracted abroad.

Annual worldwide spending of about $1.5-billion for prevention and treatment of measles has pretty much eradicated the infectious disease from many developed countries, but epidemics still sweep through the developing world.

David Earn did the math. Earn, a professor of applied mathematics at McMaster University in Hamilton, Ont., built an elegantly simple mathematical model that simulates measles epidemic patterns. The results from the model, published in a recent edition of the journal Science, suggest that new strategies for deploying those limited vaccination resources might finally lead to the eradication of measles from the whole world.

Infectious diseases such as measles come in waves: Epidemics rise and recede in what had until now been thought to be unpredictable patterns. Earn's model uses readily available data to decipher these patterns, enabling epidemiologists to predict when the next epidemic will occur.

"Measles was thought to be an example of chaos in epidemiological or ecological systems," he says. "If that were true, then it would not be possible to predict the times and places where these transitions between different patterns occur. But the analysis that we did in this paper showed that we could, in fact, predict all these changes."

The patterns Earn refers to involve the frequency of measles epidemics. Some cities have epidemics every year, some every other year. Others seem to follow no discernible schedule. Earn's mathematical model explains how these patterns form and change.

The idea behind a mathematical model is to create an equation, or a set of equations, that will mimic statistical changes in the real world. If the model corresponds to the historical data, then odds are it will also be a good predictor of future events. In this case, Earn and a team of researchers from McMaster, Cambridge University and the University of Florida, were trying to design a model that fit historical patterns of measles epidemics.

They started with 30 years of data on measles outbreaks in four representative cities: London and Liverpool in Britain, and New York and Baltimore, Md., in the U.S. This data, especially in the years after vaccination programs had commenced, appeared to describe an incoherent, random series of epidemics with no consistency over time, or from location to location.

Their breakthrough came when they began to use information about birth rates along with vaccination rates as parameters for their equations. With the inclusion of this data, their model became stunningly lifelike.

"The insight is that if you make a change in the birth rate, even very slowly, when it goes beyond a certain level it will cause a big shift in the dynamics. That's not intuitively obvious. That requires some mathematical justification," Earn says.

"We dug out the birth rates in the places for which we had measles data and on the basis of the birth rates and the vaccination levels, we were able to predict precisely when changes in the pattern would occur."

Earlier attempts by other mathematicians had failed to create such a model that fit real-life transitions from, say, annual to biannual epidemics. Using this new model, epidemiologists should be able to use changes in birth rates and vaccination rates for a given area to predict when the next outbreak will hit.

Sir Robert May, a mathematician and professor of zoology at Oxford University who is also the chief scientific advisor to the British government, says Earn's model uses textbook-standard mathematical techniques but that it transforms epidemiological knowledge about infectious diseases.

"It's useful in that it illuminates the trajectory of the course to eradication," he says. "Intuition might suggest that if you've got some roughly steady level of infection, that as you reduce it you will just move smoothly to eradication. But if you look more carefully at the dynamics, it can often be that in fact you get quite violent fluctuations."

Earn's model demonstrates how the introduction of a vaccination program can change a simple pattern of epidemics into a far more complicated one. He said that being able to predict these changes will lead to changes in thinking about how the vaccine should be distributed.

"Maybe [by] not continuously vaccinating kids at six months of age, maybe [by] waiting and vaccinating people of all ages at a particular time of year - that's one idea that we're playing with - it might allow us mathematically to show that with the same investment in vaccine and the infrastructure for distributing the vaccine, but organized differently, we might be able to eradicate measles