Ferguson's models are informed by the biology of diseases—from the molecular-level action of pathogens and host immunity to the environmental, social and behavioral factors that work at the population level. His models of HMD led him to draw some stark conclusions: The existing controls weren't working and the long delays between diagnosis and slaughter of infected animals were fueling the epidemic. A more aggressive eradication program was needed. This would involve the culling of animals on farms suspected of infection within 24 hours, without waiting for confirmation from the lab; all animals on neighboring farms had to be culled within 48 hours.

The other three modeling teams that advised the government came to the same conclusions. The military was called in, and at the peak of the outbreak, some 2000 troops were involved in the eradication nationwide, supported by up to 1000 police officers. Almost 2.5 million animals were slaughtered in 11 weeks, requiring burial pits equivalent to 200 Olympic-size swimming pools.

During a visit to Cumbria, one of the counties hardest hit by the disease (see map), Ferguson participated in one of the army's daily operations meetings. While listening to the soldiers talk about the logistics of the operation, the ramifications of his research became clear. On a separate occasion, he spoke to a group of farmers and came face-to-face with their concerns. "Half the farmers in that room had lost their farms already because of hoof-and-mouth," he says. "I estimated that by the end of the epidemic, up to 70 percent of Cumbrian farms might be affected. That number shocked them, but it is turning out to be about right."

Another member of the Imperial team, Christl Donnelly, found herself confronted by a group of angry, vocal farmers in Devon, another badly hit county in the southwest part of the country. She described feeling "out on a limb." Still, motivated by a desire to offer practical help in a desperate situation, Ferguson and his colleagues defend their recommendations vigorously. "The HMD epidemic has significant economic consequences for the country, which I hope my work might help to minimize," he says.

As soon as the government released its data on infected cases to the Imperial team in March, Ferguson and his colleagues spent an intensive 10 days feeding critical numbers into their models. By calculating the likelihood that one farm would infect another in a particular length of time, given the separation between farms and the dynamics of the nationwide agricultural network, they were able to predict the future course of the epidemic and to rank the potential effects of different interventions.

Reducing R₀
Specifically, they measured the impact of those interventions on R₀, the case-reproduction ratio, or average number of secondary cases generated by one primary case of infection. By definition, when this figure drops below 1.0 the epidemic can no longer sustain itself. "What control measures do, whether it's culling or vaccination or a whole range of other things, is to reduce that quantity, make the disease less infectious, give it less chance to spread to other farms," says Ferguson. He and his team were simply looking for the quickest and most effective way to bring R₀ down.

Publishing their results in the May 11, 2001, issue of Science, they argued that vaccination would be less effective than culling as an emergency response. Although vaccination reduces the population of animals susceptible to the disease, it does not reduce the infectiousness of those animals already contaminated, and so, does not reduce transmission. Similarly, culling of infected farms alone, leaving their seemingly healthy neighbors untouched, would be insufficient. By failing to stem newly acquired though not-yet-apparent infections, such restricted culling would leave R₀ at or around the critical threshold, resulting in a larger epidemic that tailed off more slowly. "Extensive culling is sadly the only option for controlling the current British epidemic," the Imperial team concluded.

The policy has been largely successful. By the end of May, the number of infected farms was dropping by half each fortnight and R₀ was under 1.0. Ferguson's advice was nevertheless to maintain the aggressive cull until the disease was completely eradicated. The second serious outbreak in previously unaffected Yorkshire in mid-May could, he suggests, have been an indirect result of the government's decision to relax the policy, allowing veterinarians more discretion as to which animals were slaughtered on farms that neighbored infected ones. Even taking that outbreak into account, the epidemic has declined in line with the team's predictions.

Oversimplifying Reality?
Still, some leading HMD experts have described the policy as excessive. Alex Donaldson, head of the Institute for Animal Health laboratory in Pirbright, Surrey, and a government adviser himself, has criticized the mathematical models for oversimplifying reality. In a study published in the Veterinary Record on May 19, 2001, he argued that the models were based on an average animal, and did not allow for differences in infectivity, immunity and transmission routes among cows, pigs and sheep.

Fred Brown of the U.S. Department of Agriculture's Plum Island Animal Disease Center in Greenport, New York, who was asked by Science to review the Imperial group's May 11 paper, says that there are so many unknowns in the spread of the disease—including illegal movements of livestock and the behavior of deer and other wildlife populations—that such models could not possibly be accurate.

Ferguson says such criticisms show a lack of understanding of modeling. Infection grows geometrically in a population, giving an epidemic the characteristic shape of an exponential curve. Based on this principle, a mathematical model divides a population into three classes: those who are susceptible, those who are infected and those who have recovered. It consists of a set of equations that represents the epidemic process as a feedback loop where the rate at which susceptible individuals become infected is proportional to the number of those already infected.

At that level, a model is indeed a simplification of reality. The complexities come in with the diverse behaviors of different diseases—the immunity generated by the pathogen, the infectivity and routes of transmission will vary. When any of these parameters are unknown, says Ferguson, or when the relevant data are not available, the skill is to factor them into the equations, with alternative scenarios weighted according to their probabilities. Each alternative generates a different outcome, providing policymakers with a worst-case scenario, a best-case scenario and a spectrum in-between from which to make their decisions. As more hard data come in, the equations can be adjusted to give a better fit and more precise projections.

Inevitably, says Roy Anderson, head of the Imperial group and one of the pioneers of infectious-disease modeling, when you are dealing with an ongoing epidemic the most current data are less precise than you would like. Given the urgency, however, "there is a scientific compromise to be made between detail and something that is sufficiently robust to give a qualitative guide to policy," he says. One of the reasons for studying animal diseases such as HMD, adds Ferguson, is to "offer the opportunity for much more complete data to be collected easily and for experimental and epidemiological studies to be carried out that are not possible in humans."

Merging Models
When it comes to understanding how a disease behaves, modelers are constantly deluged with new information from well-established fields such as immunology—the discovery of new immunological markers of disease, for instance—as well as relatively young fields such as molecular biology. At the moment, different aspects of disease tend to be modeled separately, and the modeling community divides roughly into those who model the spread of a disease within an individual and those who model it within a population. The problem with these partial models, however, is that they tend to overlook interactions among the many different factors that shape an epidemic.

New-variant Creutzfeldt-Jakob Disease (vCJD), the human form of so-called mad cow disease (see Clearing Toxic Clumps from the Brain), is a good example of the difficulty of modeling interactions. In the August 2000 issue of Nature, the Imperial team estimated that the predicted British epidemic would likely affect no more than 136,000 people. That was a preliminary projection, based on data relating to the 70 or so cases that had been recorded at that time. But all of those early cases came from the 40 percent of the population that shares a certain identifiable genetic makeup, or genotype. There was a hope at the time that the other 60 percent might be immune to vCJD.

Since then, molecular biological evidence collected by John Collinge of Imperial College and University College London has indicated that, rather than being immune, people with different genotypes might just incubate vCJD for longer periods. The evidence comes from Collinge's studies of kuru, a similar disease that is still affecting elderly members of the Fore tribe of Papua New Guinea more than 40 years after their practice of eating the brains of dead relatives was banned.

His findings raised fears that the Imperial team had vastly under-estimated the size of the epidemic. But Ferguson points out that they never excluded the possibility that the rest of the population might be susceptible. The only assumption they made, and one he thinks is justified, was that the distribution of incubation periods is likely to be similar to that of other prion diseases. In the case of kuru, for instance, the incubation period varies from 2 years to 35 years, rising to a single peak at around 15 years. By definition, the vCJD cases that have already come to light are those with the shortest incubation periods. It may be 20 years before we see the cases with longer incubations, he says.

The molecular biological evidence is therefore not at odds with the models, Ferguson maintains, or the worst-case estimate his team generated. Still, one thing he may have to correct for as the epidemic unfolds is how those genetic susceptibilities interact with behavior. In vCJD, the incubation period is not determined solely by the underlying genetics but also by the amount of infected material ingested. The more infected meat a person eats, the sooner he or she succumbs. "We don't have any idea of the distribution of doses, but it was probably quite wide, and that will mask a lot of the additional variation you might get from the genetics," says Ferguson.

Martin Nowak of the Institute for Advanced Study in Princeton, New Jersey, who models the spread of viruses within an individual, believes the main challenge facing the field now is to establish a single theoretical framework that will accommodate all the available information and its possible interactions. "In the end, one will build models where a virus spreads within an infected individual and within a population," he says.

Ferguson hopes to start developing the methodology to build these comprehensive models over the next five years to answer basic science questions about evolution as well as public health (see sidebar: Modeling Common Diseases). He believes this next generation of models will make possible more precise predictions, and that modeling will play an increasingly important role in disease control. Even now, he says, more and more scientists are realizing that the value of modeling lies in quantifying uncertainty, in making it manageable for policymakers.

Ask him if there is any disease that can't be modeled, and he smiles: "There is nothing that can't be modeled, but there are lots of things that can't be modeled easily."

Photo: Odd Andersen/AFP

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Reprinted from the HHMI Bulletin,
September 2001, pages 14-17.
©2001 Howard Hughes Medical Institute