The Calculus Diaries - Jennifer Ouellette [58]
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