The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

by Jennifer Ouellette

  How do we even begin to track those interconnected chains? Nathan Eagle, an engineer at MIT’s justly renowned Media Lab, studies social networking phenomena using an unusual approach for data collection: cell phones. Mobile phones, with their GPS tracking components and call logs, make fantastic behavioral “sensors,” providing a far more accurate record than asking people to record their own behavior in a diary—the traditional methodology for such studies. Self-reported data is notoriously error-prone, in part because people have faulty memories—or are not being entirely honest in their reportage.

  In one study, Eagle and his colleagues provided cell phones to ninety-four test subjects—all students or faculty at MIT—loaded with special software that kept track of their location and logged all calls made and received among those phones. As a control, the study also included self-reportage, with subjects identifying which of the other subjects were friends, acquaintances, or strangers. Based on the calling patterns, the researchers were able to identify correctly whether two given subjects were friends or strangers more than 95 percent of the time.

  That is just one study. Based on three basic parameters—a user’s activity, location, and proximity to other users—Eagle says it is possible to accurately predict someone’s future behavior based on limited observation of their current behavior. And because transmission of disease is strongly correlated to social networks and proximity to others, his method is extremely useful for epidemiological modeling.

  According to Eagle, the typical epidemiological model rests on an erroneous assumption: that the probability of infection is equal for all, that is, the population is well mixed. But social networks are much more complex than that; there is noticeable clustering of social contacts, and people in those clusters would have a higher probability of becoming infected. It makes a strong case for being a hermit. The Bennett family fears social shunning when their younger daughter, Lydia, scandalously elopes with Wickham; had this transpired, they could have taken comfort in the fact that they would have been far less likely to be bitten by zombies than their more socially active neighbors.

  Eagle believes that the data captured by his cell-phone software application gives a much more realistic picture of the dynamics of human social networks,42 thereby arming epidemiologists with “more information to make predictions about our vulnerability to the next SARS, as well as greater insight into preventing future epidemics.” One can only wonder what further insights might be gleaned if Eagle outfitted zombies with cell phones.

  How human beings react to these kinds of threats is another factor that can be tough to calculate. That’s why other researchers are looking to online virtual worlds for modeling the spread of infectious disease, much like the collapse of a virtual bank in Second Life might shed light on economic models. Most epidemiological models use mathematical rules to approximate human behavior, but the modelers must make certain assumptions about how humans are likely to behave— and those assumptions can be inaccurate. Deliberately introducing a deadly pathogen into a controlled population to study the outcome would be immoral, but what if it were possible to design a “disease” specifically for a virtual online community?

  Game designers were a little ahead of the scientists on that front. Blizzard Entertainment, the makers of World of Warcraft—a hugely popular multiplayer game—deliberately introduced a zombie plague into the game to promote World of Warcraft: Wrath of the Lich King. But a far more interesting development was the virtual Corrupted Blood epidemic that broke out in 2005. Blizzard added a new dungeon called Zul’Gurub, controlled by an “end boss” named Hakkar. Only highly advanced players could find Zul’Gurub, where the objective was to kill the end boss. Among the creature’s weapons was a spell called Corrupted Blood, which inflicted damage on infected players at regular, repeating intervals, slowly draining away their vitality until their avatars “died.” Killing Hakkar was the only cure.

  The spell was designed to infect only nearby players, and to remain confined to the Zul’Gurub game space. But things went horribly wrong. Thanks to a glitch in the programming, the animal companions of players’ avatars—technically “nonplayable”—became infected, and even though they showed no symptoms, they spread the disease to the lower levels of the game. While advanced players could survive the infection, the Corrupted Blood plague would kill a lower ranking player very quickly. Widespread panic ensued wherever the plague struck, with game spaces becoming littered with virtual corpses. At least three servers were affected, and Blizzard had to reboot the entire game to fix the glitch.

  A Rutgers University scientist named Nina Fefferman heard about the Corrupted Blood incident and became fascinated by the in-game parallels to real-world epidemics. Human behavior is not necessarily rational, or courageous, and this became obvious in World of Warcraft. True, some players tried to help with “healing spells,” but other players panicked and fled to other game spaces, carrying the disease with them. A few malicious players deliberately spread the disease—behavior that has also been documented in real-world outbreaks—and one hardy soul decided his role was to stand in the town square and narrate the carnage, a self-appointed Doomsday Prophet. There were even thrill-seekers who ignored the warnings and ventured to infected areas out of curiosity, thereby becoming infected as well—similar, says Fefferman, to journalists who travel to war zones and deliberately put themselves in harm’s way to get a story. She went on to coauthor a paper with Eric Lofgren for Lancet Infectious Diseases on the implications of the Corrupted Blood incident for refining epidemiological models.

  Fefferman’s work has its naysayers, who argue that the virtual death of an avatar is not equivalent, in terms of risk, to physical death in the real world. Fefferman counters that players become quite invested in their characters and feel genuine emotional distress when those avatars are injured or killed. “The players seemed to really feel they were at risk and took the threat of infection seriously,” she told BBC News.

  Blizzard, in turn, maintains that World of Warcraft is just a game and was never intended to mirror reality. But the parallels to real-world outbreaks are striking. An epidemiological model based on the Corrupted Blood outbreak would draw on hard data showing how players actually responded to the threat—not on abstract mathematical assumptions. And why not look to video games for insights into the spread of diseases? We’ll need all the help mathematics can give to ward off the coming zombie apocalypse. Just ask the Bennett sisters.

  7

  Body Heat

  Exercise ferments the humors, casts them into their proper channels, throws off redundancies, and helps nature in those secret distributions, without which the body cannot subsist in its vigor, nor the soul act with cheerfulness.

  —JOSEPH ADDISON, Spectator, July 12, 1711

  Comedian Margaret Cho once riffed on the concept of “Stairmaster time”—namely, the fact that time passes much more slowly when one is on the Stairmaster, mindlessly climbing stairs to nowhere. Any gym member can relate: A mere fifteen minutes can feel like an hour if one lacks sufficient distraction. I am just beginning to break a sweat on an elliptical machine under the watchful eye of Adam Boesel, personal trainer and owner of the Green Microgym in Portland, Oregon, but we are not watching the usual graphic display showing pace, calories burned, or distance traveled. Instead, I am laboring to keep a small 60-watt light bulb alight with my exertions, mounted on the front of the machine, with just a single digital readout tracking my output in wattage.

  Open since 2008, the Green Microgym is located in the Alberta Arts district of Portland, just fifteen minutes from the city airport. It’s a very crunchy-granola type of place, where “sustainable” is practically a way of life. The folksy main street boasts quirky little shops, art galleries, and eateries—all owned and operated by local residents. There’s nary a Starbucks in sight; instead, the local bohemians retreat to a funky little café called the Fuel Stop for their caffeine fix after yoga class, where all the drinks, salads, sandwiches, a
nd sweets are made from scratch, and nobody minds if you stay for a few hours to take advantage of the free wireless. The hot brunch spot is the Tin Shack—a small building with aluminum siding and adjacent courtyard for outdoor dining. At 10:30 A.M. on an overcast Saturday, there is already a line of hungry locals snaking around the corner.

  Boesel’s Green Microgym fits perfectly in this neighborhood: a modest, two-story boxlike structure with bright red exterior and no-frills interior. There are the usual ellipticals, stationary bikes, treadmills, and free weights, but look closer and you’ll notice a twist on those fitness staples. Boesel has retrofitted much of his exercise equipment so that gym members can generate a small amount of usable energy during their workouts.

  He is not the first to ponder the potential of human exertion for generating energy. Inmates in nineteenth-century New York prisons were forced to walk on treadmills as punishment, and that energy was used to grind grain for the inmates’ daily bread. Today, a handful of fitness centers around the world are seeking to exploit the same concept—on a purely voluntary basis. California Fitness in Hong Kong has cardio machines to produce energy for the gym’s lighting, while the Netherlands boasts the Sustainable Dance Club in Rotterdam. The dance floor is made up of small modules that move in response to the people dancing, and this movement is converted into electricity that lights up the floor. A Boston gym has a special stationary bike with a laptop built into the handlebars. The laptop has no battery; it is powered entirely by the person pedaling, so someone can get in a decent workout and still surf the Web or answer a few e-mails. It is a multitasker’s delight.

  Back in 1990, before being energy conscious was cool, actor and environmentalist Ed Begley Jr. connected a bicycle to a 24-volt battery to generate the energy needed for small kitchen appliances. He has been known to make toast this way or to run a coffeemaker. However, Boesel and others think there might be the potential for a commercial market as well. Several companies have cropped up in recent years specializing in retrofitted exercise equipment, such as ReRev.com in St. Petersburg, Florida; Henry Works in El Paso, Texas; and entrepreneur Jim Whelan’s Green Revolution. There is an entire academic research program devoted to human-powered energy at the Delft University of Technology in the Netherlands.

  It’s an ingenious idea. We spend hours each week running, cycling, or climbing in place, like hamsters on one of those little wheels, with no other goal than to burn off last night’s indulgence in a hot fudge sundae—all in pursuit of the slim, athletic figure so prized by modern society. (Admit it: For most people, the health benefits of doing so are largely secondary.) So why not try to harness some of that energy otherwise going to waste and turn gym rats into energy generators?43 The human body is essentially a machine—specifically, a heat engine. Boesel’s Green Microgym is based on solid thermodynamic principles, and this makes it an ideal learning environment for exploring the calculus of energy as it relates to diet, exercise, and the economic feasibility of harvesting energy from exercise machines.

  BLOWING OFF STEAM

  Every March in Los Angeles’ funky Echo Park neighborhood, the Los Angeles Wheelmen—a local bicycling club—gather at the foot of Fargo Street for their annual Fargo Street Hill Climb. Members compete to see who can make it up the road’s steep grade between Allesandro and Alvarado Streets the most times in a single day. It’s a daunting challenge: That one-block stretch of Fargo Street boasts a vertigo-inducing 32 percent grade, tying with nearby Baxter Street for the second-steepest grade in the city. (Eldred Street in the Highland Park neighborhood takes top honors with a 33 percent grade.) The current record, set in 2008, is 101 ascents, which took the stalwart cyclist, Steve Gilmore, nine hours to complete. That was an atypical year: Even the toughest Wheelmen (and -women) usually manage only between twelve and thirty climbs.

  The Los Angeles Wheelmen might expend a great deal of energy biking up Fargo Street and back down again, and feel as though they’ve definitely gotten a good workout. But from a physicist’s perspective, nothing has been done. Energy is useless unless it can be harnessed to perform some task. For example, the battery packs in the Green Microgym must be connected to some kind of load before the energy they produce can be useful—say, to operate the fans or the stereo system. Energy, when harnessed, produces work, which has a very specific meaning in physics, namely, a force applied over a given distance (W = fd ). How much work a moving object is capable of performing is precisely equal to its kinetic energy.

  There are many different kinds of energy that can change into each other. For example, you could also get work by burning fuel. Burning coal in power plants produces electricity by converting thermal energy (heat) into mechanical energy in a turbine. Electrical energy can change into mechanical energy. A battery relies upon a series of chemical reactions to produce an electrical current; once all the chemicals have been used up and converted into energy, the battery goes dead. And an electrical generator converts mechanical energy into electrical energy, which can then be used to power most of modern technology. All these conversions are examples of turning stored potential energy into kinetic energy.

  Heat is wasted energy, for the most part, and it’s the reason no machine, no matter how well designed, can ever attain 100 percent efficiency. We know this because of the work of Sadi Carnot. Born in 1796, Carnot was the son of a French aristocrat named Lazare Carnot, who was one of the most powerful men in France prior to Napoléon’s ignominious defeat; the family fortunes rose and fell dramatically throughout young Sadi’s life in conjunction with that of the monarchy. Named for the Persian poet Sadi of Shiraz, Carnot learned mathematics, science, language, and music under his father’s strict tutelage. At sixteen, he entered the École Polytechnique, studying under the likes of Claude-Louis Navier, Siméon Denis Poisson, and André-Marie Ampère.

  It was not a peaceful period in France’s history. Always opposed to the monarchy, Carnot joined in the fighting when Napoléon briefly returned from exile in 1815. When Napoléon was defeated in October of that year, Carnot’s father was exiled to Germany. He never returned to France. Carnot the younger, dissatisfied with the poor prospects offered by his military career, eventually joined the General Staff Corps in Paris and pursued his academic interests on the side.

  In 1821, he visited his exiled father and brother in Germany. Apparently there was very little to do in exile, so the men took to debating the pros and cons of steam engines. Steam power was already used for draining mines, forging iron, grinding grain, and weaving cloth, but the French-designed engines were not as efficient as those designed by the British. (The efficiency of those early French engines was as low as 3 percent.) Convinced that England’s superior technology in this area had contributed to Napoléon’s downfall and the loss of his family’s prestige and fortune, Carnot threw himself into developing a robust theory for steam engines.

  Carnot’s father died in 1823. That same year, Carnot wrote a paper attempting to find a mathematical expression for the work produced by one kilogram of steam; it was never published. In fact, the manuscript was not discovered until 1966. In 1824, he published Reflections on the Motive Power of Fire, which described a theoretical “heat engine” that produced the maximum amount of work for a given amount of heat energy put into the system. The so-called Carnot cycle draws energy from temperature differences between a hot reservoir and a cold reservoir (and became the basis for modern-day refrigerators).

  Carnot knew from endless experimentation that in practice, his design would always lose a small amount of energy to friction, noise, and vibration, among other factors. He knew that in order to approach the maximum efficiency in a heat engine, it would be necessary to minimize the accompanying heat losses that occurred from the conduction of heat between bodies of different temperatures. He also knew no real-world engine could achieve perfect efficiency. These considerations brought him tantalizingly close to discovering the second law of thermodynamics.

  Reflections on the Motive Power of Fire
did not attract much attention when it first appeared. The principle of energy conservation was fairly new and quite controversial among scientists at the time. The work began to gain notice a few years after Carnot’s untimely death from cholera at the age of thirty-six, just one among the myriad casualties of the epidemic that swept through Paris in 1832. Most of his belongings and writings were buried with him, as a precautionary measure to prevent the further spread of the disease. Carnot was twenty years ahead of his time. His work did not immediately lead to more efficient steam engines, but he did set out the physical boundaries so precisely that Rudolf Clausius and William Thomson, Lord Kelvin, would draw on his work to build the foundations of modern thermodynamics in the 1840s and 1850s.

  In the latter half of the nineteenth century, a British scientist named James Prescott Joule toyed with various energy sources to see which ones were most efficient. The choice of fuel can be critical, for different fuels have different conversion rates and produce different amounts of usable energy—and once again, where there is a rate of change, we’re bound to find a derivative. Joule came from a long line of brewers, so chemistry was in his blood, as was scientific experimentation. He and his brother experimented with electricity by giving each other electric shocks, as well as experimenting on the servants.

  Fascinated by the emerging field of thermodynamics, Joule jerry-rigged his own equipment at home (using salvaged materials) to conduct scientific experiments—specifically to test the feasibility of replacing the brewery’s steam engines with the new-fangled electric motor that had just been invented by measuring their conversion rates and how much useful energy they produced. It was his very own simple optimization problem.