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

by Jennifer Ouellette

  Darbyshire correctly identified a contaminated water source as the source of the outbreak. Initially the disease spread at a very rapid rate, and people panicked. With no quarantine in effect, those already infected brought the contamination to the countryside. Had that first city been quarantined, the outbreak would have been contained as the rate of removal (a) increased. Just as with an economic market, at some point, a critical threshold is reached, and the exponential growth rate of removal would level off as fewer and fewer people remained to be infected. Darbyshire had the foresight to put protective measures in place that limited the spread of infection. That caused the rate of removal to increase even faster, and thus the outbreak leveled off and died out much more quickly.

  In the case of cholera, there was a single vector: the Broad Street pump, or, more specifically, the white flocculent particles contained in the water that came out of that pump. The disease spread to whomever drank from that particular source. A disease like the Black Death is much more complicated to model because there is more than one vector.

  MASQUE OF THE BLACK DEATH

  One week before Christmas in 1664, a comet streaked across the sky over England. Astrologers claimed it was an omen of impending apocalypse. One William Lilly predicted that this, combined with a lunar eclipse in January 1665, would bring “the sword, famine, pestilence, and mortality or plague.” Lilly was really hedging his bets—why not throw in a prediction of a zombie invasion or an asteroid strike while he was at it?—but his dire prediction came partially true. Pestilence was common in the rat-infested urban centers of England, and this led to a deadly outbreak of bubonic plague in London in the summer of 1665.

  By October, one in ten Londoners had succumbed to the disease—over sixty thousand people. The government banned public meetings, but the epidemic spread to Cambridge, where the young Isaac Newton was in his second year of studies. The university closed, and Newton was forced to return to his country home in Grantham for over a year until the plague had run its course and the university opened its doors again in April 1667. And in that short time, he invented calculus, with no idea that it would one day be applied to study the spread of disease.

  This was not the plague’s first appearance. Back in the Middle Ages, the plague decimated Western Europe, wiping out roughly one third of the population, some 25 million people. It could sweep through a region and wipe out entire villages in a matter of weeks. During the 1630s, various outbreaks of plague killed half the populations of affected cities. Similar numbers perished in an outbreak in Holland in the 1660s: A thousand people were dying each week in Amsterdam at the height of the outbreak. And the plague significantly culled the population of France during an outbreak between 1647 and 1649.

  The plague spread rapidly and was so virulently infectious that even doctors feared treating victims. At the time, they believed the disease spread via “bad air,” or miasmas. Those who did treat patients took what precautions they could, donning large beaked hats made of bronze and stuffing the “beak” with strong herbs and spices to purify the air the doctor breathed. (As an added bonus, the aroma from the herbs helped mask the stench of rot that invariably accompanied the plague.) Plague doctors dressed in pants and a long gown and wore leather gloves, as well as crystal eyepieces for added protection— anything to ward off contamination, even though the source of the plague was not identified until the nineteenth century. All clothing, even undergarments, were doused in camphor oil or treated with wax to further seal the doctor from bad air.

  These precautions might have been partially effective. We now know that plague is caused by a bacillus called Yersinia pestis and is spread by rodents and their fleas to humans.35 Protecting the eyes, nose, and mouth made it harder for Y. pestis to get into the body via mucous membranes, and coating one’s clothing with wax made it more difficult for fleas to penetrate to the skin and transmit the disease with their bites. And the herbs stuffed into the beak of the mask at least partially blocked breathing holes, so the doctor would be less likely to inhale the bacillus. Their Achilles’ heel was actually the ankles, which remained exposed and therefore vulnerable to flea bites.

  The credit for this momentous discovery goes to a French scientist named Alexandre Yersin, a former student of Louis Pasteur.36 Yersin went to Hong Kong in 1894 to investigate an outbreak of the plague there. He extracted some of the pus from a dead soldier’s bubo (swollen lymph node) and injected it into guinea pigs; all the guinea pigs died. He examined the pus from both the dead soldier and the doomed rodents and noticed both samples contained the same type of bacteria. Yersin also noted the large number of dead rats around the city, examined those bodies, and once again observed the same bacteria. Conclusion: Y. pestis was the culprit for the spread of plague.

  Yersin did not determine the means of transmission, however. That honor fell to his fellow scientist, Paul-Louis Simond, who experimented with infected rats and fleas. He noticed that even if he placed an infected rat into a jar with healthy rats, the healthy ones only became sick if fleas were present. Just how virulent is this plague-causing Y. pestis? In lab experiments, mice died after being infected with just 3 bacilli; your average flea can transmit 24,000 in a single bite.

  Plague has many different vectors: It can spread person to person or via the rats and fleas; which type you get depends on how the bacillus invades your body. The Black Death came in three forms: bubonic, pneumonic, and septicemic. After the bite of an infected flea, the first site of infection is generally the lymph nodes. In this form, bubonic plague, your lymph nodes swell to form enormous buboes. Lancing the buboes releases oozing, foul-smelling pus. The bubonic plague was the most common form, with a mortality rate of 30 to 75 percent. In addition to enlarged and inflamed lymph nodes around the armpits, neck, and groin, victims were subject to headaches, nausea, aching joints, high fever, and vomiting, and symptoms took from one to seven days to appear.

  The pneumonic plague, infecting the lungs, is particularly virulent, capable of killing an infected person within twenty-four hours. You would catch this merely by breathing Y. pestis into your lungs. The mortality rate for the pneumonic plague was 90 to 95 percent (if treated today, that would be reduced to 5 to 10 percent). Symptoms included slimy sputum (a saliva-and-mucus concoction) tinted with blood. As the disease progressed over one to seven days, the sputum turned bright red.

  If Y. pestis entered your bloodstream directly through the bite of a flea or via a cut or sore in contact with diseased tissue, you would get septicemic plague and would be almost certain to die. The septicemic plague was the rarest form of all, but the mortality rate was close to 100 percent; even today there is no treatment. Victims ran a high fever, and the skin turned deep shades of purple, almost black, hence the name Black Death. Victims usually died the same day symptoms appeared; in some cities, as many as eight hundred people died every day.

  Septicemic plague was rarer than the other two forms of plague because people died so quickly that they had little opportunity to transmit the disease to others. That’s why any good epidemiological model must take into account the latency period between infection and death. Pneumonic plague was easily transmitted from person to person, but death usually occurred within a day or two, so it, too, did not propagate as rapidly. Bubonic plague gets it just right, from the perspective of Y. pestis, whose sole purpose is to infect as many hosts as possible. It is not as virulent as pneumonic plague. Once infected, the victim could appear healthy for as long as a week, merrily passing the disease on to others, and death occurred much more slowly.

  “Because of its infectious nature, the disease may be spread by apparently healthy people who harbour the disease but have not yet exhibited the symptoms,” Daniel Defoe wrote in A Journal of the Plague Year, which appeared in 1722. “Such a person was in fact a poisoner, a walking destroyer perhaps for a week or a fortnight before his death, who might have ruined those that he would have hazarded his life to save.” Defoe may have been writing about the real-lif
e plague that decimated London in the 1600s, but he could just as easily have been describing Grahame-Smith’s alternate version of the village of Meryton, where residents who were bitten would seem normal but were in fact gradually turning into zombies.

  ASSUME A SPHERICAL ZOMBIE

  Pride and Prejudice and Zombies is rife with graphic battle scenes, as Elizabeth Bennett travels the countryside with her aunt and uncle, leaving a path of zombie casualties in her wake. She teams up with Darcy to defeat a horde of zombies at his Pemberley estate, and after accepting his proposal of marriage, the newly engaged couple dispatches one final group of zombies to plight their troth. But is all this bloody violence toward zombies really necessary? Can’t humans and zombies learn to get along and coexist in harmony?

  According to a 2009 paper by a group of Canadian epidemiologists: no way, nohow. The lead researcher is Robert Smith?37 of the University of Ottawa, who specializes in modeling the spread of infectious disease. He and three students adapted their models to the spread of a fictitious zombie infection, starting out with a simple model and gradually adding elements to make it more complex.

  “The key difference between the models presented here and other models of infectious disease is that the dead can come back to life,” the authors write, tongues firmly in cheeks. According to Smith? and his students, people fall into three basic categories: susceptibles (S), those who are not infected; zombies (Z); and removed (R), susceptibles who have died of other causes. The key factor is not the actual numbers in each category, but how those numbers change with time as new zombies are made and existing zombies are killed. Anytime we have a rate of change, we have a derivative situation on our hands. The rate of change in zombies is the net increase or decrease in their numbers during a given period of time.

  There are well-established rules governing the zombification process.38 Zombies can be killed by cutting off their heads and destroying their brains. Susceptibles can become zombies if they are bitten by one, but zombies can also be created by resurrecting the removed—those who are already dead. If we have six humans turned into zombies every hour and four dead people resurrected into zombies every hour, the result is ten new zombies every hour. Now let’s say we manage to kill three zombies every hour. The net result is an increase in the zombie population of seven zombies per hour. And at that rate, there is no chance of maintaining what’s known as an endemic state—one of peaceful coexistence, or at least a comfortable equilibrium.

  Smith?’s model doesn’t end there; that’s just the process for calculating the rate of change in the zombie population. We also have to run equations for how the number of dead and the number of uninfected humans change, which means factoring in the birth and death rates of humans as well. This is called a coupled system of ordinary differential equations, which is really just a fancy way of saying that the system must be described by not one, but three connected equations: one for how the number of humans changes, one for how the number of zombies changes, and one for how the number of dead changes. Furthermore, Elizabeth Bennett’s good friend Charlotte has been bitten but is not yet a zombie, although doomed to become one in a matter of weeks—as good an explanation as any for her marriage to the odious Mr. Collins. Smith? and company call these people Latents, giving us a total of four coupled equations. The coupling occurs because the same variables appear in all four equations—or, practically speaking, because the different populations interact with one another.

  Assuming the zombie infection occurs quickly, the birth and death rates of humans will be insignificant during the time over which the infection occurs, so we still have the same scenario: Everyone will be turned into zombies very quickly, at which point the population will become unsustainable. In the worst-case scenario, Smith? estimates it would take a mere four days to wipe out the humans. The outcome remains the same: The zombies get us all in the end.

  Quarantining the few healthy humans could help—the standard “hole up in a basement somewhere and hope the zombie hordes don’t find you” approach employed in classic zombie horror films. We’ve seen how (in)effective that strategy can be onscreen, and Smith?’s numbers back up those observations. However, another study by an Italian scientist named Davide Cassi implies that hiding out at the mall (à la Dawn of the Dead) could vastly improve one’s chances of survival. Cassi wasn’t analyzing zombies specifically, but his version of a predator/prey model applies to any kind of “predatory random walker”: organisms (like zombies!) that stumble around without any obvious purpose or direction, destroying any human that comes into their path. The larger and more complex the structure—such as a large mall with many twists and turns—the lower the chances that the predator will stumble upon the prey.

  Alternatively, we can quarantine the zombies by herding them into some sort of holding pen, but if we don’t isolate enough of them fast enough, once again, the zombies will win. Both options are rather passive strategies, and most likely will only postpone the inevitable annihilation of the human race.

  Smith? and his students suggest that our only hope is an “impulsive eradication” scheme. A series of fierce, concentrated attacks could sufficiently cull the number of zombies over time so that the outbreak would finally die out. “The most effective way to contain the rise of the undead is to hit hard and hit often,” the paper concludes. “As seen in the movies, it is imperative that zombies are dealt with quickly, or else we are all in a great deal of trouble.”39 Enter the Bennett sisters and respective paramours, with their wild, weapon-wielding ways, to make quick work of any rampaging zombie hordes.

  Applying epidemiological modeling to a zombie invasion might seem silly, but it is not very different from modeling the spread of swine flu or the HIV virus. In November 2009, Smith? published another paper in the open-access journal BMC Public Health, arguing against spending $60 million in funding to combat the spread of HIV over fifteen to twenty years. Smith? recommended a far more aggressive five-year program—a variation on his “impulsive eradication” scheme for combating zombies—insisting that a gradual approach is doomed to fail because HIV/AIDS spreads so rapidly through travel and migration.

  Smith?’s group also studies the kinds of slow-moving, chronic diseases in less developed countries that tend to be neglected by newspapers and funding agencies alike: things like leishmaniasis and dracunculiasis—both parasitic diseases that give rise to festering skin sores, among other symptoms—which can have long-term socioeconomic impacts on large populations. Dracunculiasis, or guinea worm disease, is particularly nasty. Drinking contaminated water will introduce the larva into your body, where it will hatch and grow for about a year until it forms a blister on your skin, which then ruptures so the worm’s wriggling form sticks out.40 With these types of diseases, as with zombies, the infected don’t die: They live on, and thus have far more opportunity to transmit the disease to others.

  SIX DEGREES OF ZOMBIFICATION

  Most epidemiological models follow the basic format of separating the host population into those who are susceptible, infected, or immune to a particular pathogen. The assumption is that the rate at which new infections occur is proportional to the number of encounters between susceptible and infected individuals. That reproductive ratio doesn’t merely depend on latent and infectious periods, but also on how much contact there is between those who are infected and those in the healthy-but-susceptible population.

  This means that social networks play a big role in how quickly (or slowly) an outbreak propagates. The good citizens of Meryton go to balls, congregate in drawing rooms, and visit friends and relatives in other townships for a fortnight or more, providing ample opportunity for zombie infection to spread. So the more we know about the social networks involved in an outbreak, the better we can refine our epidemiological models.

  Social networking is related to the small-world phenomenon, better known to most of us as “six degrees of separation” and epitomized by the popular game Six Degrees of Kevin Bacon, in which player
s try to make a series of connections to the actor based on those who have been associated in some way with his movies. In the original 1967 study by psychologist Stanley Milgram, information packets were sent to randomly selected people in Omaha, Nebraska, and Wichita, Kansas, containing a letter describing the purpose of the experiment, providing basic information about the target contact in Boston, Massachusetts, and asking them to forward an enclosed letter. If the recipient knew the target, he or she would forward the letter directly. If not, the recipient would forward it to a friend or relative more likely to know the target. While the number of connections it took for the letters to reach the target varied, the average was around 5.5—hence, six degrees of separation.

  Milgram’s study fell into some disrepute when it was revealed that his famous experiment and conclusions were based on a minuscule data sample. In one experiment, out of sixty letters, fifty people responded to his challenge to forward the letter via their social networks, but only three letters eventually reached their destination. A far greater number of people didn’t bother to participate in the experiment at all. That said, the study does offer intriguing evidence that smaller communities, such as those of actors and mathematicians41 are densely connected by chains of personal or professional associations.