It's been said that World War I was the chemists' war and World War II the physicists' war, but that World War III is destined to be the mathematicians' war.

With the gravest threat to U.S. security now posed by rogue terrorists who simultaneously hold a grudge and have access to weapons of mass destruction, some of this country's prime human calculators are on the case.

Combatting terrorism not with mortars and missiles but with mathematical models, they intend to prove this theorem: There literally is safety in numbers.

In their vanguard is an amiable Stanford University professor who, by devoting himself to the application of math principles to doomsday scenarios, is beginning to acquire the nickname "Dr. Doom." The attacks of Sept. 11 inspired mathematician Lawrence Wein to channel his expertise into some of the most compelling questions of our time.

He has ciphered the risks of our "woefully inadequate" inspection of container ships, assessed the effectiveness of border-control fingerprint checks to spot terrorists, and performed what may well be the first math analyses of hypothetical botulism, anthrax and smallpox contaminations.

Recall the agitation last year over warnings that the nation's milk supply was vulnerable, based on the calculation that one terrorist with a few grams of botulinum could contaminate a tanker and potentially poison 100,000 gallons of milk? That's just one of many tidings of comfort and joy brought to you by Wein.

Another such upper is that just about 6 percent of containers shipped into U.S. ports will be categorized as suspicious and subjected to tests for a nuclear device, based on a system that relies largely on reporting by the shipper. The rest, Wein noted, "just waltz right into the country without an inspector laying an eye on them."

At Stanford's Graduate School of Business, he teaches a core course in operations, and he says the parallels are strikingly similar: Just as McDonald's needs a well-designed distribution system to get its hamburgers out quickly, so the government needs a well-designed distribution system to get vaccines and antibiotics to citizens who might sicken or die from a bioterror attack. In math lingo, some of these computations rest on "queueing theory" -- the notion that a lot of needy people must line up behind a limited number of distributors.

And what do we do after crunching millions of numbers to arrive at specific prescriptions? Wein has taken on the role of necessary nag to a torpid federal bureaucracy and recalcitrant industries.

In fact, the feds scrambled to try to suppress publication of his damning milk study as a threat to national security -- an argument the journal Proceedings of the National Academy of Sciences rejected before publishing the work.

These days, Wein regularly testifies at congressional hearings, addresses scientific forums and pens op-ed pieces in national newspapers about precisely what the government should be doing -- and where it is falling short.

"I believe my work demonstrates that numbers really matter, and we need to pay attention to what they tell us," said Wein, a critic of the federal government's homeland security failures in key areas, particularly at the ports. "I really don't bring my own personal political views into this -- I would be just as critical of the federal bureaucracy if a Democrat were in the White House.

"Bureaucracy just isn't designed to respond nimbly."

On the other hand, politicians are eager to do something, anything, that might thwart a terror attack or save lives. But in an information vacuum, that makes them easy prey for entrepreneurs hawking all kinds of safety gadgets.

So, much of Wein's math analysis goes beyond documenting threats to assess the most effective, cost-efficient remedies.

Take the milk scare, for example. Wein urged the government to mandate that milk tanks and trucks be locked, that two people be present when it is transferred along each part of the supply chain, and that milk truck drivers use a 15 minute test to detect any toxins. The estimated cost of all this: 2 cents per gallon.

Some other experts disputed Wein's assumptions, and dairy industry groups insisted they already had taken extra steps such as raising pasteurization temperatures to secure the milk supply. They dismissed the scenarios Wein described as "highly unlikely or impossible."

To avoid being victimized by a radioactive device aboard a tanker -- otherwise known as a "poor man's missile" -- he and colleague Stephen Flynn of the Council on Foreign Relations advocate using techniques such as gamma-ray imaging to screen 100 percent of containers. They calculate the cost at $7 per container.

Some of Wein's calculations have translated into real reform.

In 2004, he presented to the White House his findings that the system of collecting only two fingerprints from incoming foreign visitors ran a high risk of missing terrorists because so many prints turned out to be of poor quality. His solution was simple: Take prints from all 10 digits. The government adopted it.

His 2003 research into the most effective response to an anthrax attack prompted pilot programs that are now testing his finding that the best way to widely distribute antibiotics would be by mail carriers, who already go door to door.

Albert Einstein once observed: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

And sure enough, one of the impediments to fighting terrorists with mathematics is the paucity of hard data.

An anthrax scenario illustrates the point. Estimates of fatalities per kilogram of poisonous agent released vary from a low of zero to a high of 660,000. That's because of variables: the dose required to cause infection, the percentage of people who survive infection, the degree of aerosol dispersion, the density of the population, the environmental stability of the agent and the effectiveness of a public health system response.

As Wein acknowledges, "It's not like we can do massive clinical trials on this."

The trail of what-ifs is so convoluted that in 2002 the National Academy of Sciences cited "an irreducible uncertainly of several orders of magnitude in the number of people who will be infected in an open-air release."

This makes some experts suspicious of anybody's calculations, even from a researcher with a pedigree like Wein's.

"With so much uncertainly surrounding the outcome of a bioweapons attack, it does not make sense to plan extensive biodefense programs when more-certain public threats, particularly those involving nuclear weapons, require attention," argued Allison Macfarlane, research associate in the Science, Technology and Global Security Working Group at MIT.

But Wein is undeterred, and is busy on new assessments. Yet he insists his work has not impaired his ability to sleep soundly.

"People really are much more likely to suffer from cancer than a terrorist attack, but that doesn't merit the same attention," he said. "We should do what makes sense to protect ourselves from both."

In 1979, anthrax was accidentally released in the city of Sverdlovsk (pop. 1,200,000) in the former Soviet Union, infecting about 80 to 100 people and killing at least 70. Russian officials claimed at the time that tainted meat sold on the black market was responsible; American officials argued that a nearby biological weapons facility released the killer spores. In the early 1990s, Harvard researchers visited the city to piece together the epidemiology of the outbreak. Their investigation, published in Science magazine in 1994, concluded that the Soviet cover story was false.

Now, physicist Dean A. Wilkening, director of the science program at Stanford's Center for International Security and Cooperation (CISAC), has revisited this Cold War tragedy and used its real-world data to improve our ability to model the medical effects of inhalational anthrax. This, in turn, allows him to model more accurately hypothetical scenarios such as the release of a kilogram of aerosolized anthrax in Washington, D.C., today.

The models researchers have used in such thought experiments "predict very different outcomes," says Wilkening, whose work to better understand the human effects of inhalational anthrax was supported by grants from the John D. and Catherine T. MacArthur Foundation and the Carnegie Corporation. Using real-world data from the Sverdlovsk outbreak and from limited nonhuman primate experiments, he was able to eliminate two of four theoretical models currently used in "what if?" scenarios that inform bioterrorism policies ranging from how much medicine we should have on hand in the Strategic National Stockpile to how rigorous post-attack decontamination efforts need to be. He reports his findings in the May 1 issue of Proceedings of the National Academy of Sciences.

"To date, researchers haven't paid enough attention to which model they use," Wilkening says. "Different models can give predictions that vary by a factor of 10 or more, so it matters which model one uses for predicting the human effects of inhalational anthrax." Wilkening aims to anchor models on the best available data and provide realistic models that the bioterrorism community can employ in policy studies.

The Sverdlovsk outbreak is "a sort of natural experiment," he says. "It's a tragic incident, but it also is a very valuable source of scientific data that one can use to distinguish between the four models currently in use." The upshot of his analysis is that two of the models currently in use are not accurate for predicting the human response to inhalational anthrax.

Insufficient data is available to resolve which of the remaining two models he examined is most accurate. That answer will have to await further data from costly nonhuman primate experiments, should they ever be performed (none are planned). "We have to use both [models] right now, or use them as bounding cases," he advises.

Wilkening explored four policy issues that illustrate the consequences of choosing different models: 1) calculating how many anthrax-exposed people would become infected and how many would die; 2) assessing if decontamination would be needed; 3) determining how soon exposed people would show symptoms and how soon doctors would recognize those symptoms as anthrax; and 4) calculating how soon exposed people need to receive antibiotics to avoid contracting the disease.

"To figure out what happens in a bioterrorist event, you need to know two basic properties about the pathogen you're dealing with," Wilkening says. One is the dose-response curve, which determines the likelihood of becoming infected at different exposure levels--the higher the dose of anthrax you get, the higher the probability that you will become infected. The dose at which 50 percent of an exposed population becomes infected, called the ID50, is around 10,000 spores. The other basic property is the incubation-period distribution, or the time the pathogen takes to grow in the body before symptoms first appear.

Wilkening's study brought dose-dependence to a debate over how long the incubation period is for inhalational anthrax. Published data from vaccine efficacy tests in which nonhuman primates were challenged with high doses of anthrax--up to a million spores--indicate an incubation period of one to five days. Data from Sverdlovsk, which exposed people to low doses probably on the order of 1 to 10 spores, indicate a longer incubation period, about 10 days. Whereas previous authors have debated whether nonhuman primate experiments or the Sverdlovsk data should be used to determine the incubation period for inhalational anthrax in humans, Wilkening demonstrates that both estimates are correct, with the difference between them being due to the dose dependence of the incubation period and the very different doses received in each case.

"If you are exposed to a higher dose, there is a much higher chance that an anthrax spore will germinate quickly, thus leading to a shorter incubation period," he says. "Sverdlovsk was a low-dose exposure event and, consequently, one would expect anthrax spore germination to take a longer time, thus leading to a longer incubation period."

Truth and consequences

Russian officials confiscated the medical records of the Sverdlovsk victims and have so far refused to release details of what happened on April 2, 1979. "It would be nice to know exactly what happened, because that would allow us to model the event more accurately," Wilkening says.

Nevertheless, based on weather and other data from the day of the event, scientists think that around 2 p.m. spores, or dormant cells that revive under the right conditions, were released from a military facility, and the Bacillus anthracis spores spread up to 5 kilometers downwind. People breathed in the spores, which geminated and incubated in the body for between four to 40 days before people began to feel ill or show signs of illness such as sore throat, coughing, pains, aches and runny nose--the same symptoms as flu--that indicated they had entered what doctors call the prodromal phase. Within four days, people passed the point of no return, called the fulminant phase, in which toxins from the bacteria had built up to such an extent that people went into shock and died.

It's impossible to save those who've entered the fulminant phase and difficult to save those who've entered the prodromal phase. But if people can start treatment after exposure but before symptoms appear, there's a good chance that they will survive--a conclusion Wilkening draws from work by colleagues at Stanford's Center for Health Policy. Treatment primarily consists of antibiotics such as ciprofloxacin, doxycycline or penicillin. While a vaccine to prevent anthrax exists, it is not yet available for the general public but would be made available to people exposed to anthrax, according to the Centers for Disease Control and Prevention website.

In his study, Wilkening ruled out two of the four models because they either did not fit the Sverdlovsk data or the nonhuman primate data, or both. "There are two models that people have used that should no longer be used to predict fatalities, models B and C." (The four models used in his analysis are labeled A-D for convenience.)

Using the two remaining models A and D, he predicted that a hypothetical attack releasing 1 kilogram of anthrax spores in Washington, D.C., would infect between 4,000 and 50,000 people, most of whom would die if not treated quickly with antibiotics. The difference of a factor of 10, Wilkening points out, is "an uncertainty with which we must live for the time being until better data can resolve which of the models A or D is more accurate."

Regarding decontamination efforts, the higher the probability of becoming infected at low exposure levels, the greater the need for effective decontamination, especially for indoor environments. Spores "by nature are hardy," Wilkening says. In the soil, out of the way of sunlight, they can last for a decade. "Residual contamination can be a very serious problem in the wake of an attack," Wilkening says. "Unfortunately, both models A and D predict that residual surface contamination from anthrax spores will be a problem. Consequently, we need to come up with effective indoor decontamination strategies."

Analysts such as Professor Lawrence Wein of the Graduate School of Business are considering the issue. Last year, he assessed decontamination and concluded cleaning buildings to make them safe to reoccupy was a billion-dollar proposition.

In addition, the four models make very different predictions about when symptoms would occur. The day after exposure, they predict between 10 and 1,000 people feeling sick, with more people getting sick in the viable versus discredited models.

"In terms of detecting the outbreak rapidly, this is a good thing because it says that doctors could recognize it [sooner]," Wilkening says.

In terms of treating people before they reach the prodromal phase, however, this is a bad thing because people become sick quicker. Wilkening's analysis may help policymakers reassess how fast antibiotics need to reach people. His best model says administering antibiotics by day three saves 90 percent of exposed people. "Today we cannot meet the three-day requirement," he warns.

In his State of the Union address in January, President Bush stressed the importance of improving math education. He proposed to "train 70,000 high school teachers to lead advanced placement courses in math and science, bring 30,000 math and science professionals to teach in classrooms, and give early help to students who struggle with math."

But where will these teachers come from? And will the training of teachers be sufficient to increase the number of students choosing math and science careers? And why does all this matter?

Because mathematics is the foundation of the natural sciences. It is no coincidence that Isaac Newton, the man who formulated the law of gravitational attraction that revolutionized our understanding of the universe, was also the man who popularized the calculus. And the natural sciences, however pure, are what give us airplanes, cable TV and the Internet.

In the 2003 Program for International Student Assessment, a test that measures math literacy, American 15-year-olds performed worse than their peers in 23 countries, as well as those in Hong Kong. It's not hard to see why. According to the National Council of Teachers of Mathematics, 40 percent of the nation's middle school math teachers do not have the equivalent of an undergraduate minor in math. The average starting salary of a teacher is only $30,000, whereas the average starting salary for a recent college graduate in computer science or engineering is $50,000.

Short of following the British, who have proposed paying experienced math teachers more than $100,000, with a guaranteed minimum of $70,000, where will we find a way to attract the thousands of teachers George Bush wants?

New York State initiated an innovative program to bring teachers from Jamaica for two or four years to teach in New York schools. Jamaica, a developing nation where one U.S. dollar equals 65 Jamaican dollars, is nonetheless a stable, English-speaking nation with an unbroken democratic tradition; it stands poised to beat the United States in establishing the world's first Institute for Mathematical Methods in Counterterrorism. When teachers for the New York program were recruited on the campus of the University of the West Indies, recruiters found more experienced math and science teachers than they ever dreamed they would.

But you can have all the teachers in the world and still not inspire kids to learn math. My friend Autumn e-mailed me about her nephew, Joshua: "He's upset because he's asked several of the math teachers why math is important or what are certain formulas used for -- there has to be a use, correct?"

Autumn told her nephew about my work in counterterrorism and for the television crime drama "Numb3rs." Autumn reported, "He's told his math teachers about you as well, and about the show 'Numb3rs.' He's informing them that through something called lattice theory you are managing to fight terrorists -- all with math."

Mathematics is art, and should be appreciated for its beauty, not simply for its utility. But we cannot expect 11 year-olds to cherish totally order-disconnected topological spaces as much as professional mathematicians do.

As I first proposed in January 2005, television shows like "Numb3rs" (or "Medium") -- where the main characters are mathematicians -- could work with the National Council of Teachers of Mathematics to show kids how math is really used; the council and Texas Instruments are now working together to use "Numb3rs" to promote math literacy in schools.

Another way to inspire kids is to relate mathematics to something they see every day. In order to excite students and draw funding to his school, school superintendent Ronald Ross of Roosevelt, N.Y., has begun looking into the idea of creating a curriculum involving math and counterterrorism. What kinds of topics would students learn?

The opening line of the Oscar-winning movie "A Beautiful Mind" is "Mathematicians won the war." During World War II, the mathematics underlying cryptography played an important role in military planning. Winston Churchill admired Alan Turing, the mathematician who had mastered the German codes, recognizing him as the man who had perhaps made the single greatest individual contribution to defeating Hitler.

At Los Alamos, the lab that built the atomic bomb, Cliff Joslyn uses lattice theory to mine data drawn from thousands of reports of terrorist-related activity to discover patterns and relationships that were previously in shadow.

Lattice theoretical methods developed at MIT tell us the probability that we have disabled a terrorist cell, based on how many men we have taken out and what rank they hold in the organization. Lauren McGough, a Massachusetts high school student, tested the accuracy of this model by getting her classmates to pretend they were terrorists, passing orders down a fictitious chain of command, essentially confirming what the theory predicts.

High school students could learn algebra, trigonometry, calculus and logic while also learning concrete applications involving homeland security. No longer would students yawn and ask, "What is math good for?" Beauty could defeat both terror and boredom.

Whatever you may think of the State of the Union address, when it comes to supporting math education, we should all see pi to pi. President Bush is correct when he says that mathematics education in America must improve if the United States is to stay economically competitive, but the stakes are much higher than that. During the Cold War, the United States would not have tolerated a military gap between itself and its adversaries. Yet today, with 61 percent of all U.S. doctorates in math going to foreigners (15 percent to Chinese), we readily accept a "math gap."

Dollar for dollar, the best defense against our adversaries' weapons of mass destruction may be our allies in the Americas, armed with weapons of math instruction.

Improving math education is not merely a smart idea. It is a matter of national security. Algebra is one revolutionary Islamic concept we cannot afford to neglect or ignore.

George Habash, a militant and former secretary-general of the Popular Front for the Liberation of Palestine, once characterized terrorism as a "thinking man's game." Using mathematics, researchers at Stanford University's Center for International Security and Cooperation (CISAC) have made fighting terrorism a thinking man's game as well.

CISAC affiliate Lawrence M. Wein of the Graduate School of Business and CISAC Science Fellow Jonathan Farley are both applying mathematical models to homeland security problems, such as preventing a nuclear detonation in a major U.S. city and determining whether terrorist cells have likely been disrupted.

Wein, who teaches operations classes about different business processes used to deliver goods and services, has focused his research on bioterrorism and border issues. He has performed, he says, the first mathematical analyses of hypothetical botulism poisoning, anthrax outbreaks and smallpox infections.

"One overriding theme of my work is that all these homeland security problems are operations problems," said Wein, the Paul E. Holden Professor of Management Science. "Just as McDonald's needs to get hamburgers out in a rapid and defect-free manner, so too does the government have to get vaccines and antibiotics out and test the borders for nuclear weapons or terrorists in a rapid and defect-free manner."

In collaboration with Stephen Flynn of the Council on Foreign Relations, a nonpartisan research center, Wein recently has conducted research to improve security at U.S. borders and ports. Port security has received significant attention recently owing to the furor over Dubai Ports World's bid to manage six terminals at major U.S. harbors. The aim of Wein and Flynn's work is to prevent terrorists from bringing into the country a nuclear weaponbe it an atomic bomb or a so-called "dirty bomb," or conventional explosive packed with radioactive waste.

"Of all the problems I've studied, this is the most important because the worst-case terrorist scenario is a nuclear weapon going off in a major U.S. city and also it is the one the government has dropped the ball on the most," Wein said. "They have done a very poor job."

Instead of using the existing approach, where U.S. Customs actively inspects a minority of containers based on information from a specialized tracking system designed to identify suspicious containers, Wein and Flynn have recommended the government use a multi-layer, passive screening system for every container entering the country. Under their system, Customs would photograph a shipping container's exterior, screen for radioactive material and collect gamma-ray images of the container's contents. If terrorists shielded a bomb with a heavy metal such as lead to hide it from radiation detectors, gamma-ray imaging would allow inspectors to see the shielding and flag the container for inspection. Wein and Flynn believe this whole process would cost about $7 per container.

"Right now about maybe 6 percent of the containers are deemed suspicious and they will go through some testing and the other 94 percent of the containers just waltz right into the country without an inspector laying an eye on them," Wein said. "What we're proposing to do is 100 percent passive testing."

Wein's earlier work addressed a different threat: bioterrorism. In 2005, Wein revealed the nation's milk supply was vulnerable--a terrorist could potentially poison 100,000 gallons of milk by sneaking a few grams of botulinum into a milk tanker. Although the government and dairy industry have collaborated to intensify the heat pasteurization formula for milk, Wein is still pushing for additional botulinum testing, which he says would cost less than 1 percent of the cost of milk.

Wein also has used math to study smallpox outbreaks, the U.S. fingerprint identification system and U.S.-Mexico border security issues. Wein's congressional testimony on the fingerprint identification system in 2004 led to a switch from a two-finger system to a 10-finger system. His 2003 research on anthrax attacks resulted in a Washington, D.C., pilot program to use the U.S. Postal Service to distribute antibiotics throughout the capital after an outbreak. Seattle is now testing a similar program.

"In Washington, D.C., now, if there is a large-scale anthrax attack, postal workers will be the first to get their Cipro and, on a voluntary basis, they will go door-to-door distributing antibiotics," Wein said.

He said the common thread throughout his research is queuing theory, or the mathematical study of waiting lines, but he also draws upon mathematical epidemiology for his smallpox studies; air dispersion models for the anthrax model; supply chain management for the milk study; probability theory for the fingerprint identification system; and models for nuclear transport and detection for his work with containers.

From tainted lactose to lattice structures

While Wein is working on improving the government's counterterrorism systems, Jonathan Farley is working to figure out when terrorist organizations have been effectively disrupted. His mathematical model is designed to help law enforcement decide how to act once they have captured or killed a terrorist or a number of terrorists in a cell.

A professor at the University of the West Indies who will chair the Department of Mathematics and Computer Science there next year, Farley is on a one-year science fellowship at CISAC. In 2003, he co-founded Phoenix Mathematical Systems Modeling Inc., a company that develops mathematical solutions to homeland security problems.

He is using lattice theory--a branch of mathematics that deals with ordered sets--to determine the probability a terrorist cell has been disrupted once some of its members have been captured or killed.

"Law enforcement has to make decisions about what resources they should allocate to target different cells," Farley said. "The model should provide them with a more rational basis for allocating their scarce resources. ... It will inform you when you're making decisions about how much time and effort and how much money you're going to spend going after a particular cell."

While at Stanford, Farley hopes to unearth the perfect structure, mathematically speaking, for a terrorist cell--or in other words, a cell structure that is most resistant to the loss of members.

"If it's possible to determine the structure of an ideal terrorist cell, you can focus on a much smaller number of possibilities, because it makes more sense to assume the adversary is going to be smart rather than stupid," Farley said.

Farley has suggested it is possible Al-Qaida and other terrorist organizations already may have figured out the perfect structure for a terror cell by trial and error.

"I don't expect Osama bin Laden to be reading lattice theory in his caves in Afghanistan," said Farley. "But if it follows from the mathematics, perhaps heuristically, the terrorists will have come to the same conclusion--that this is the best way to structure a terrorist cell."

Although Farley acknowledges his model is not a panacea for terrorism, he hopes it will help reduce guesswork that might be involved in pursuing terrorists.

"It's not that I think mathematics can solve all of these problems," Farley said. "Because it can't. But it's better to use rational means to make decisions rather than guesswork."

John B. Stafford is a science-writing intern at Stanford News Service.