The field of bacteriology is a wide-reaching one. Blogging about bacteria means that I get to explore many different fields of science; from the highly molecular world of biochemistry and synthetic biology to the larger and more human-centred land of the pathologists and immunologists.

One area that I don't go into so much is epidemiology; the study of how diseases spread through a population. It's an important area of research and leads to vital discoveries about immunisations and population load. By modelling how diseases spread through a population you can look at potential ways to stop them.

Looking at this sort of data for actual diseases is often depressing, so I decided to model a completely non-existent problem - that of werewolves. To help me in this (as epidemiological modelling is NOT my field!) I'll be using the paper referenced below, which examined the epidemiology of a zombie outbreak.

wolf-man

The proper kind of werewolf. Not the kind that are teenage boys in ripped jeans. From wikimedia commons.

Now every different story has its own rules for werewolves, so we'll start off with a few basic rules. First of all: a werewolf turns a human into a werewolf by biting them, only at full moon. A werewolf is therefore contagious once a month. Werewolves are not immortal, but live for far longer than humans and are less likely to catch diseases or injure themselves.

First we need to organise our population. As per the zombie paper, I'm going to divide it into three.

W = werewolves

S = susceptible (i.e people who can become werewolves)

D = dead (either people, or werewolves, from any cause)

Now we need parameters to describe how the populations change between each other. This is where things are different (and easier!) than the zombie paper. Because once either a wolf or a person becomes dead, they are gone. This as it stands means that the population will tend towards dead, but unlike zombies and indeed vampires both humans are werewolves are capable of breeding, and thus replenishing the population.

So parameters:

a = rate at which people become werewolves

b = rate at which werewolves become dead

c = rate at which people become dead

d = rate at which people breed

e = rate at which werewolves breed

This can be illustrated as below:

Those who do understand epidemiological modelling might notice similarities to the SIR model, with this being the SDW model :p Also notice that the parameters are also dependent on the numbers of populations. Thus the rate of humans changing to werewolves is bSW - as it depends on the number of werewolves and the number of people.

This model can be written out in three equations:

S = d - aSW - cS

W = e + aSW - bW

D = bSW+ cS

At this point in the zombie paper, Maths starts to happen, but I thought I'd just use these equations to make a few points about werewolf epidemics and werewolf portrayals in general media. If anyone wants to go ahead and work through the numbers, be my guest!

So what can we see from the equation above? A couple of things are immediately obvious. First of all, unless the birth-rate is spectacular, or the werewolves are very restrained, the level of humans will fall, faster and faster as the number of werewolves increases. Even if the humans get the pitchforks out and properly rout the werewolf population they will still be killing ex-humans, and unless they kill all the werewolves (which would admittedly make the maths a lot easier) their numbers will still be diminishing. With only the childbearing properties of the populace adding to their numbers, the humans quickly tend towards dead.

The werewolves on the other hand, are limited only by the presence of humans and their once-a-month schedule. Even allowing a delay factor before baby werewolves become dangerous doesn't change much as the main supply for new werewolves is the humans. As werewolves take a long time to die, the main reason for death is going to be unsuccessful encounters with humans. Unless the humans are very good at fighting off wolves, the werewolf numbers are going to rise much faster than they fall, to the detriment of the human population.

How does this stack up with the way werewolves are portrayed by the public? It certainly explains why very few depictions of them show a massive upsurge in werewolf numbers throughout the course of the book/film. Back in the older stories werewolves are barely ever shown having families either (so the main source of werewolves is the aSW factor) whereas in some more modern depictions the family (parameter e) is the main input for new werewolves. Werewolves, vampires and all sorts of monsters live among us nowadays. Not as predators, but as complex irritations that must nevertheless be tolerated.

Spike, from Buffy the Vampire Slayer

Some are tolerated more than others ...

They used to be deadly. They used to be killers; silent, unbeatable and among us, ready to strike out at any time. But that was back in the olden days when people did old fashioned things like die a lot. Nowadays a disease is not a reason to stop living, it's a thing to live with. The werewolves keep to themselves, the vampires loose their fangs, the demons got to rave clubs.

The zombies are still pretty dangerous though.

zombie from plants vs. zombies

Possibly because they don't sparkle ... (zombatar (c) plants vs. zombies)

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Ref 1 = Philip Munz, Ioan Hudea, Joe Imad, Robert J. Smith (2009). When Zombies attack!: Mathematical modelling of an outbreak of zombie infection. Infectious Disease Modelling Research Progress Other: 978-1-60741-347-9