Werewolf Mathematics, Spooky Scary

A werewolf has been spotted in a nearby village, the paladin Sir Model has been dispatched to try and prevent an outbreak.

Hold on though, we never hear about werewolf epidemics, do we? Which is odd when you think about it, lycanthropy (the pseudo-medical term for werewolf-disease) is spread by being bitten by a werewolf, something that presumably the werewolf could do a lot in a short amount of time, even if they only get one chance a month to really let their hair down. But usually we only see one or a few werewolves at a given time. What’s going on then? Sir Model will help us illustrate, and will simultaneously serve as the most egregious pun I’ve ever perpetrated.

Upon arrival at the village Sir Model begins to think of the folk in two ways. People who could become werewolves, and people who already are werewolves. Once he gets started, he’ll make a third group, people who used to be werewolves. Traditionally that involves quite a bit of silver, generally very sharp. But it doesn’t take a professional to deal with a werewolf problem, with enough torches and pitchforks your average mob of villagers can take a stab at it. However, according to the Tao of Pratchett, the intelligence of a mob is that of it’s stupidest member divided by the total number of mobsters, so the mob ends up picking someone basically at random. So while they do have a chance of getting a werewolf, the odds are pretty low. In fact, the odds of the mob actually getting one are equal to the number of werewolves per potential werewolves (villagers).

Sir Model, while not perfect himself, is a bit more effective. In the Van Helsing scale of monster killing he’s about 0.1VH, not great but not too bad. One Van Helsing can dispatch 10 monsters in a given time, so 0.1VH means Sir Model can effectively remove 1 werewolf a month, the same timescale the werewolves operate on, for the sake of simplicity.

The next big question then is how many people can the werewolf turn? Well, traditionally werewolves are fairly savage, so they end up killing more villagers then they turn. Remember that to become a werewolf you have to be bitten, not eaten. So let’s say that every month on the night of the full moon, the werewolf catches one villager and eats them nine times out of ten. That last time is the kicker though, because while that ‘lucky’ villager survives the encounter, by the time the next full moon comes around we’ll have two werewolves galavanting about, each with the chance to make a new werewolf.

Now that I’ve introduced some numbers and driven off at least half the readers I can reveal my dastardly plot. Sir Model is in actuality an initialism and pun, my unimaginative stand in for the SIR model. The S, I, and R each stand for different groups. Epidemiologists call this a compartment model, because of the grouping of people into compartments, and this one in particular is called the SIR model (which is an initialism like FBI, so you read the letters and don’t say ‘sir’) and thus the worst pun I’ve committed is revealed. We can think of ours as as the VWD model, Villagers, Werewolves, and Dead. In the real world we have Susceptible, Infected, and Removed. Susceptible and Villagers are the group who can succumb to the disease (lycanthropy or some real-world equivalent), Infected and Werewolf are the group that actually have the disease, and Removed or Dead are the group that got better or died. Surprise, you’ve just learned some basic epidemiology, the study of how diseases spread.

And because we can count the number of people in each group, and estimate how many people the werewolf can turn and how many werewolves Sir Model can eliminate, we can learn some things. Chiefly, we can learn whether or not we will have an outbreak of werewolves. An outbreak, or epidemic, occurs when the disease spreads rapidly through a population. Basically if peopple get sick faster than they get better or die then you get an epidemic. Which seems like it would be good to know; whether the problem will go away or whether everyone will be turned into monsters or die. And horror of horrors, we’ll use math to learn something useful. What few readers remain, don’t worry, I won’t ask you to do any yourselves but I do hope to demonstrate that your middle school algebra teacher was right all along. Even if they didn’t think to use math to keep you safe from an epidemic of werewolves.

First, some pictures:

So here we have the groups Sir Model has binned everyone into. Everyone in the village starts out as a villager, then we get a werewolf. Then the werewolf gets to work and some people die, and some become werewolves. Mob ‘justice’ or Sir Model will try and move the werewolves into the dead group, but they might be wrong. The werewolves will move villagers into their own group or will kill them trying. So no we’ll add some arrows to show these movements:

An we can estimate or just plain make up some numbers to put on those arrows, because they describe the rate of flow of people (or werewolves) between each group. In this model we’re assuming no zombies and no cure for lycanthropy, but if we wanted to we could add those groups and arrows, but we’ll keep things simple for now. The rates of movement between groups can be described with some pretty simple calculus, so there’s a prime example of how math can be used in everyday life. For fighting werewolves. So our numbers are:

Werewolves attack 10 people and 9/10 times they kill the villager, but 1/10 times they turn them into a werewolf. So the Villager -> werewolf arrow is worth 1/10, and the villager -> dead arrow is worth 9/10.

Mob justice is about as good as random, which means that the chance of actually killing a werewolf is proportional to the number of werewolves in the total population, or werewolves/everyone, and the chance of killing a villager is villagers/everyone. So the villagers -> dead arrow is worth villagers/everyone, and the werewolf -> dead arrow is worth werewolves/everyone.

Let’s make some assumptions to simplify things:

1. We’ll start out with 999 villagers and 1 werewolf. We’ll call the number of villagers V and the number of werewolves W.
2. Nobody will leave town, or show up unannounced, or be born, so we’ll stay with 1000 total until people start dying, and even then they’ll just move to the dead box. We’ll call this total N, by convention.
3. Nobody dies from anything but murder, from either werewolves, mobs, or Sir Model.
4. Anyone not already a werewolf can become one by being bitten. There is no cure or immunity for lycanthropy.
5. We’ll call the villagers -> werewolves arrow L for lycanthropy, the werewolves -> dead arrow we’ll call M_J for Mob Justice and the villagers -> dead by the mob arrow M_I for Mob Injustice. We can call the rate of werewolf killings A for attacks. So now we’ve got labels, and I’ll add a second arrow between villagers and dead to differentiate the two paths to get there.

So we see the same picture with a few added complications. And we can add in the numbers ourselves:

V = 999

W = 1

N = 1000

L = 0.1

A = 0.9

M_J = W/N = 1/1000 = 0.001

M_I = V/N = 999/1000 = 0.999

Now you might ask, what can we learn from this? Well after some algebra and calculus that we don’t need to get into here but which is super interesting and that I entreat you to look into if you’re interested in epidemiology, we find that one number can tell us whether or not an epidemic will occur. That number is called R₀, which tells us how many times a werewolf spreads lycanthropy before dying. If the number is greater than 1 then we get an epidemic, which intuitively makes sense - if more werewolves are made than are killed, then the total number of werewolves gets bigger. That ends up being just L/M_J, which is 0.1/0.001, or 100. Which means that a single werewolf will make another 100 werewolves before mob justice successfully kills it. That’s a super bad outbreak, for reference here is the same number for a few real world diseases:

Influenza (the seasonal flu): 2-3

Ebola: 1.5-2.5

HIV/AIDS: 2-5

Measles: 12-18

The range is because conditions differ, we could produce a similar range by varying the effectiveness of the werewolf, from 1 in 10 to maybe 1 in 100, or 2 in 3, or that of Sir Model or the mob.

So without a professional werewolf hunter like Sir Model this village is doomed. But once you bring in a professional you can dispense with the mob, the people have been pacified and will trust the paladin to get the job done. So we eliminate M_J and M_I. We replace M_J with S_M (for Sir Model, the joke I just will not let die) and assign it a value, in this case lets go with 1, Sir Model can kill 1 monster per unit time. In our case the unit of time is the month, because once a month we get a werewolf attack. So we crunch the numbers again and R₀ = L/S_M = 0.1/1 = 0.1, which is less than 1 and so we don’t have an outbreak! Sir Model has been enough of an intervention to save the day.

That checks out for the early stages of the outbreak, but what happens if Sir Model doesn’t show up until there are already 25 werewolves? What happens if we change the effectiveness of the mob, werewolves, or Sir Model? What happens if the village is instead a city, or a country, or a single house? What if there’s a cure for lycanthropy, or a vaccine? What if some villagers are inherently immune? What if villager behavior changes, and they stop leaving their houses around the night of the full moon? These questions and many more are answerable through a bit more epidemiology, which for now is just a bit more than we’ll explore. If these questions interest you I encourage you to look further into epidemiology, the SIR model and compartment models in general, and disease ecology.

I’m not the first or only person to write about werewolves from an epidemiological point of view, for a bit more math check this out. Other spooky topics used to explore real world biology include zombies and vampires, and probably just about any other example you can think of!