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

by Jennifer Ouellette

  Grahame-Smith felt Austen’s original text was a natural fit for zombie horror. “You have this fiercely independent heroine, you have this dashing heroic gentleman, you have a militia camped out for seemingly no reason whatsoever nearby, and people are always walking here and there and taking carriage rides here and there,” he told the Daily Beast. “It was just ripe for gore and senseless violence.” And ninjas—don’t forget the ninjas. It makes even more sense when one considers that Regency England was no stranger to deadly outbreaks of disease and that the modern zombie genre pioneered by George Romero’s Night of the Living Dead routinely treats the spread of rampant zombification as an epidemic. As such, zombies provide an excellent case study in epidemiology.

  Epidemiologists study the rate at which disease outbreaks spread and how various intervention strategies—vaccination or quarantines, for example—can help slow the transmission rate. They study this in the context of general population dynamics: the number of infected individuals and the rate at which a population grows or declines are connected. If there’s a varying rate of change between two connected factors, there must be a derivative to be taken somewhere. So calculus is very useful in epidemiology and therefore in the analysis of zombie outbreaks. It just so happens that nature has its own microcosmic version of a zombie epidemic, which lends itself very nicely to illustrating a fundamental epidemiological model.

  A FUNGUS AMONG US

  Deep in the forests of West Central Africa lurks a species of parasitic fungus that targets a particular kind of ant. The fungus belongs to the Cordyceps family, scattering spores into the air, which then attach to the ant’s body to germinate. The spores work their way inside the poor insect’s body, sprouting long tendrils called mycelia that eventually reach into the ant’s brain and release chemicals that make the ant the fungus’s zombie slave.

  The chemicals change how the ant perceives critical pheromones, altering its behavior. In this case, the ant feels less inclined to devour delicious brains and is instead compelled to climb to the top of the nearest plant and clamp its tiny jaws around a leafy stem. It is the fungus that plays the role of zombie now, devouring what little remains of the insect’s brain, then sprouting through the ant’s head as one final indignity. Those sprouts burst and release even more spores into the air, which go forth to infect even more unsuspecting ants. The entire horrific process can take four to fourteen days. Fear the fungus, my friends.

  There are over four hundred different species of Cordyceps fungi, each targeting a particular species of insect, whether it be ants, dragonflies, cockroaches, aphids, or beetles. Consider Cordyceps an example of Nature’s own population control mechanism to ensure that ecobalance is maintained. The fungus proliferates when there is a large supply of hosts—that is, when the ant population flourishes and becomes so large that it threatens to overwhelm the resources available to the colony. As more ants fall victim to zombifying spores, their numbers dwindle until (a) there are once again sufficient resources to support what remains of the colony, and (b) there are far fewer ants available to serve as hosts, making it more difficult for the fungi to reproduce, so their numbers dwindle as well. And the whole population growth-and-decline cycle begins all over again. That is the essence of population dynamics in a nutshell.

  An English clergyman named Thomas Robert Malthus was one of the earliest pioneers in modeling population dynamics. Malthus was born with a harelip and cleft palate—defects that ran in the family—and was intensely self-conscious about his appearance as a result. He had an unremarkable childhood in the Surrey countryside, earning a mathematics degree from Cambridge University before being ordained as an Anglican curate.

  Malthus bemoaned the decline of living conditions in late eighteenth-century England and observed that in nature, plants and animals were capable of reproducing at far greater rates than the surrounding resources could support. This led him to develop his classic theory on population: If human population were allowed to grow unchecked, it would do so exponentially, and we would all too quickly outstrip our limited resources for subsistence. He believed this fundamental truth had been obscured by catastrophic events like disease, famines, or wars, which serve periodically to cull the herd, so to speak. “Epidemics, pestilence and plague advance in terrific array, and sweep off their thousands and ten thousands,” he wrote with considerable dramatic flourish. “Should success be still incomplete, gigantic famine stalks in the rear, and with one mighty blow, levels the population with the food of the world.”

  In 1798, Malthus published The Principle of Population, in which he outlined his model for population growth. It’s based on the notion that the population for a given generation is dependent on the size of the previous generation, and that this number will be a multiple. We can denote population size (p) as a function of time (t), where t can represent any unit of time we choose: days, months, years, and so forth. The key parameter is known as the Malthusian factor (r), denoting the multiple that determines the growth rate. We can plot different values for p when r = 1.19, 1.20, and 1.21 to see how a slight change in the value for r (denoted by the variable a in the figure) results in significant differences in the overall population size. The resulting graph produces three different exponential growth curves. Even a difference as small as 0.02 causes population to double after 40 units of time (whether it be 40 days or 40 years).

  Maybe it had something to do with being one of eight children, but Malthus’s proposed solution to overpopulation included restricting the family size of the lower classes to ensure that parents did not produce more children than they could support. It sounds more elitist than he perhaps intended: Malthus thought having too many children doomed the lower classes to poverty, making it impossible for them to rise above those conditions and improve their lot in life. Then again, he also flirted with the notion of eugenics—a term not coined until 1883—by pondering whether the techniques of animal husbandry might be applied to breed out undesirable qualities in people, although he didn’t feel this was a realistic goal: “As the human race, however, could not be improved in this way without condemning all the bad specimens to celibacy, it is not probable that an attention to breed should ever become general.” (Yes, even in the 1800s, people realized that abstinence alone is not a viable solution for family planning.)

  This sort of pessimistic thinking did not win Malthus any popularity contests at a time when fervent social reformers preached the gospel of erasing all the ills of man if only one could implement the proper social structures. The model is not without merit, but the Malthusian growth equation is only applicable under specific conditions, such as scientists growing bacteria in a lab in a perfectly controlled environment. Even then, while growth occurs exponentially for a time, it does not continue forever.

  It fell to his contemporary, the Brussels-born Pierre Verhulst, to improve upon the basic idea by devising a more sophisticated model that more accurately reflected real-world population dynamics. Verhulst said there are forces at work to prevent exponential growth in the population, and these forces increase in direct proportion to the ratio of the excess population to the total population. In other words, population growth depends not just on the size of the population, but also on how far that size is from its upper limit.

  The crux is something called carrying capacity (which we can denote by K): the maximum population size that any given habitat can support. If the population of ants starts to double every year—grows exponentially—there will be two thousand ants the first year, and the next year there will twice as many. But there is a limited supply of food and other resources, so if that exponential growth rate continues unchecked, the population of ants will rapidly consume all the available resources. Exponential growth simply cannot be sustained indefinitely. Once the food runs out, the ants will begin to die out too. Verhulst’s model34 shows that if the population is less than the maximum, the population will increase rather steeply because people have lots of food. But then when it gets clos
er to the maximum sustainable population, the rate at which the population increases slows down.

  Look closely and you will recognize the telltale sign of a derivative. In the Verhulst model, the derivative of the population with respect to time—that is, the rate of change in population ( p) or the number of additional people over time—would be proportional to the number of people at time 0 (now) multiplied by the maximum sustainable population minus the current population. If the population actually followed that equation, it would start out low and show exponential growth at first. But then the rate of growth would begin to slow as it approached the maximal population, eventually leveling off to become stable as it reaches the carrying capacity for that particular habitat. Plot this out on a graph and you end up with a smooth S-shaped, or sigmoid, curve.

  Say you’re part of a colony of a particular species of ant, going about your business in the forest: gathering food, doing your little communication dance, and of course, reproducing. We can use the derivative to analyze the rate at which your little colony is adding to its population. We’ll keep things simple by assuming an initial population of 100, which increases to 120 ants after one year. How long will it take your little ant colony to grow from 100 to the critical threshold, or carrying capacity (K ), of 300 ants? Just plug in the relevant numbers to the Verhulst equation: 100 for the initial population, and (in this case) a growth rate per year (r) of 1.2. The answer: six years.

  The Verhulst model is useful for limited applications, but the realities of population dynamics are far more complex, with innumerable variables. Even the carrying capacity (K) is not a constant (fixed over time); it fluctuates depending on conditions. Furthermore, instead of being continuous, as in the Verhulst model, population change often occurs in discrete shifts. Instead of the population changing continuously in tiny increments each day, there may be a major event that will cause the population to either explode or rapidly decline. An earthquake that wipes out an entire village would result in a sudden rapid decrease, while an influx of immigrants or refugees would give rise to a sudden spike in population. Then we are no longer dealing with a straightforward calculus problem, but something akin to a chaotic system, like the stock market’s wild fluctuations, making predictions extremely difficult.

  When it comes to our zombifying fungi, the situation resembles a predator-prey model: as the fungi (predators) proliferate, the ant population (prey) diminishes; when the ant population flourishes, so does the predator population, so you have equations for both populations. Nature always finds a way to maintain balance. These fungi are so effective at controlling certain pests that they have been used to control the numbers of wheat grain beetles. In fact, researchers are investigating the use of one particular species of fungus (Metarhizium anisopliae) against African mosquitoes to control the spread of malaria, because the disease is often spread through mosquito bites. That’s another useful application of calculus: assessing the rate of the spread of a disease, and determining how effective various intervention strategies might be.

  MATH IN THE TIME OF CHOLERA

  Cholera is a nasty way to die. It starts with horrible bouts of vomiting and diarrhea and a slowed pulse, plus cramps. Those cramps become more severe as the disease progresses, the victim’s entire body convulsing in pain. Eventually the lips, face, hands, and feet turn blue, purple, or even blackish in hue. The skin becomes cold and damp. Respiration slows, but instead of a telltale death rattle in the throat, victims often die quietly, with a whimper. At least the disease progression is rapid, so one’s misery is short-lived. That’s about all that can be said for it.

  In the nineteenth century, England’s physicians, scientists, and political leaders watched with trepidation as cholera morbus moved from India through Eastern Europe to Germany and the shores of England, officially “arriving” in London in 1831. Cholera killed over 10,000 people in one year alone. In 1854, London’s Soho District was hit by an especially virulent outbreak of the disease, killing 127 people in the first three days. By the time it was over, 616 people had died.

  The means by which a disease spreads throughout a population is known as a vector; the most common vector is person-to-person transmission, such as with the flu or measles—or a zombie bite. With cholera in the nineteenth century, the vector was less clear. Medical opinion was divided, because the evidence was contradictory, sometimes indicating transmission through contact, sometimes indicating transmission through squalid unsanitary conditions. The streets of Soho in the 1850s were filled with animal droppings, runoff from slaughterhouses, and primitive sewers. Had anyone checked under the floor-boards of their cellars, they would have found fetid cesspits.

  The man who solved the mystery was Dr. John Snow, a pioneer of modern epidemiology. He lived locally, on Fifth Street, and monitored the epidemic’s progress on-site. He was convinced that cholera was spread by a poison passed from victim to victim through tainted water; he’d already traced an earlier outbreak of contaminated water supplied by the Vauxhall Water Company. But authorities didn’t believe him, and the water company refused to admit culpability. He figured this was his chance to prove his theory was right.

  Snow patrolled the district, interviewing the families of those who had died, and found that nearly all the deaths had occurred near a water pump on the corner of Broad Street and Cambridge Street—the epicenter of the outbreak. Houses closer to an alternate pump had only experienced ten deaths, and five of those were schoolchildren who occasionally drank from the Broad Street pump. Ever the scientist, Snow took a sample of the pump’s water, examined it under a microscope, and noted that it contained “white flocculent particles,” which he deemed the cause of the infection.

  The Board of Guardians in St. James Parish reluctantly followed his advice and removed the pump handle as an experiment. The spread of the disease stopped dramatically. There were still a few unexplained deaths from cholera that appeared unrelated to the Broad Street pump. The most damning was a widow who lived in Hampstead, and her niece, neither of whom lived anywhere near Broad Street. Snow proved quite the detective: He found that the widow had once lived in Broad Street and liked the taste of that well water sufficiently that she had a servant bring her back a large bottle from it every day. The last bottle had been fetched on the day the Soho outbreak began.

  Yet authorities were still doubtful of Snow’s findings. A local vicar, Reverend Henry Whitehead, thought the outbreak was the result of divine intervention—a very vicarlike approach to human calamity—and set about “proving” his case. In the end, Whitehead actually helped confirm a single probable cause of the outbreak: A young child living on Broad Street had been ill with cholera symptoms, and the child’s soiled diapers had been soaked in a tub of water that was then emptied into a cesspool three feet from the Broad Street pump. Underground leakage did the rest.

  How do we model an outbreak of a disease? Let’s assume that a nasty flu virus strikes a university dormitory. The rate of infection will vary, depending on the nature of the disease and how it is transmitted. The flu is spread when an infected person, during the contagious period, coughs or sneezes near another person or touches another person. We can chart how the number of infected people (I ) changes over time (t)—in other words, I is a function of t, and for our purposes t will be measured in days. For epidemiology, there are two other parameters: how many people an infected person can infect per day, or rate of infection (r), and the rate at which the outbreak fizzles, as infected people recover—or die (a). So there will be a single equation for I(t), in which r and a will appear as parameters.

  The end result is almost always the same: As more people recover or succumb and as precautionary measures kick in—quarantine, hand-washing, or just removing the handle of the offending pump—there are fewer new cases of infection. When r is less than 1, each infected person is, on average, transmitting the virus to fewer than one other person. This will not be sufficient to sustain the outbreak, and it will end. As for the flu, so fo
r cholera.

  Fourteen years after Snow’s discovery, a cholera epidemic hit Buenos Aires, Argentina. An account of the outbreak can be found in Charles Darbyshire’s My Life in the Argentine Republic 1852-1894. He moved his household to the countryside because he worried about the unsanitary conditions of town life, having seen the impact of cholera in London before he came to Argentina. He described the conditions in alarming detail:I felt positive that sooner or later there must be an epidemic. There was no drainage. The soil on which the houses were built was becoming infected. The defecations, the waste water from kitchens, etc. went into wells 30 feet deep in the back patios. When one of these wells became full of filth and could hold no more, what was called a sangria (a bleeding) was made. A well was sunk to the same depth . . . and the sangria took place by pushing an iron bar through the full well . . . as the old well began to drain into the new. This went on for years, and some of the patios in the old houses were honeycombed by wells.

  Darbyshire’s fears proved well founded when an epidemic broke out in the summer of 1868, brought about (he believed) by Brazilian ships tossing the bodies of those who had died from cholera into the River Paraná, contaminating the water supply. People fled to the countryside, bringing the disease with them, and Darbyshire advised his neighbors not to drink the water unless it was boiled, to bury all refuse, and to keep floors and patios clean. His own household did not contract the disease, which lent credence to his advice. Despite all the deaths, there was one positive outcome: The Argentine government overhauled the city’s drainage system and installed a proper water supply.