Nine Worlds: The Mathematics of Zombie Epidemics

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The flesh-hungry shambling horrors of George Romero and beyond are clearly no simple virus. That said, there are many ways in which a zombie outbreak behaves much like a disease outbreak, and epidemiologists and statisticians have spent more than a century modelling such incidents to better understand how infections spread.

In this talk, I shall outline in layperson’s language some of the foundational concepts of these mathematical models for the spread of infection, and explore how they operate when the specific properties of a zombie-virus are assumed. How long can humanity survive? Can we actually fight back and defeat the horde? And how exactly can we mathematically account for Rick Grimes?
Speaker: Dr. Ric Crossman

I attended this talk in the excellent company of Andrew Knighton, and we two historians hid at the back whilst the mathematically inclined got their undead equation-groove on. Honestly, whilst I found the talk absolutely fascinating I got lost in the algebra about halfway through. Dr. Crossman very kindly sent me his slides though, so hopefully where my notes stop making sense, his original content will see you through.

The SIR Model

There are, unsurprisingly, existing models for measuring the spread of transmissible diseases. They were first developed in 1915-17 by Sir Ronald Ross (who discovered that malaria was carried by mosquitos) and Hilda Hudson (who used maths to revolutionise aircraft design during WW1). The theory was named Kermack-McKendrick after some chaps who came along in the 1920s and formalised it, because history’s like that. It’s also called the SIR model, after the three values involved in the equation:

  • S – number of Susceptible people who could become infected but aren’t yet
  • I – number of Infective people who could pass on the disease (carriers, but not necessarily infected)
  • R – number of Removed people who aren’t infected and cannot pass on the disease (immune, recovered and developed resistance, dead of disease, dead of other factors)

At any give point in time, S + I + R = N where N is the total number of population. In short-term models (which is usually the assumption for zombieism) the value of N doesn’t change because the spread of disease is too fast to allow for reproduction.

b38cce77-31c7-4113-851f-146716fb5b3f

Move slow, spread fast

Rates of Infection

There’s a number of speed factors to take into account. The first is rate of contact, which measures the proportion of total population encountered by one person in a set unit of time. In other words, how many people in your village do you bump into per week? To spread the infection, you need both a Susceptible and an Infective person at the meeting. You also need to know what percentage of the population is Susceptible. If you give the rate of contact the value of ß, the equation looks like this:

ß x N x I x S/N = ß x S x I

The second speed factor is the rate of removal – the proportion of infected people who stop being Infective per unit of time. Let’s call that ∝. The higher the value of ∝, the faster the disease runs its course.

The Bubonic Plague

In 1666, the village of Eyam in Derbyshire totally quarantined itself during the Black Death, hoping that would spare its population of 350. Since the plague was carried by fleas on rats, that didn’t work out so well for them but it did provide statisticians with a useful self-contained example of epidemic spread. The records of disease progression look like this:

Date (1666) Susceptibles Infectives Removed
Mid-May 254 7 89
July 4th 235 14 101
July 19th 201 22 127
August 4th 153 29 168
August 19th 121 22 207
September 4th 108 8 234
October 20th 83 0 267

Data therefore puts the infective period at 11 days. So ∝ = 1/0.3667 = 2.73, and we can work out from the table that the rate of contact was ß = 0.0178. (Honestly, this is where I got lost and I’m taking those calculations entirely on faith since I don’t understand how they were reached. If you do, feel free to explain in the comments!)

The Rick Grimes Effect

Let’s now apply the equation to zombies. For that, we need to swap I(nfectives) for Z(ombies). They don’t recover or die – they have to be destroyed. That means the rate of Zombies becoming Removed is heavily dependent on the number of S(urvivors).

You also have a spike in death rates from non-zombie factors, due to apocalypse chaos. There’s a lack of access to medical facilities and supplies, food shortages, and an increase in human-on-human violence over contested resources. So you can go straight from S ⇒ R without passing Z. And, of course, the Removed can come back. So the movement of population looks like this:

SIR

Reproduced with kind permission from Ric Crossman

We also need to add a factor for the proportion of Susceptibles who die of natural causes (∂) and a factor for the proportion of Removed who rise from the grave (µ). We also need to change the rate of removal (∝) to just account for zombie elimination rather than recovery. To calculate the rate of removal (aka zombie slaying, which as previously mentioned is determined by the number of survivors), the equation is as follows:

ß x N x S x Z/N = ß x Z x S

 

This assumes that ß always results in the elimination of either the Zombie or the Susceptible.

Achieving Equilibrium

Equilibrium is when all forces are in balance and the rates continue at a constant pace. There’s two kinds of equilibrium environment:

  1. Stable: when the system is moved (i.e. a factor is changed temporarily or the environment changes), equilibrium restores itself at roughly the same place
  2. Unstable: when the system is moved, equilibrium completely collapses

The only stable equilibrium achievable in a zombie apocalypse is the removal of all Susceptibles. In other words, humans die and zombies inherit the earth. Quarantine is just delaying the inevitable because the moment it fails – and many many media have proven that it will – the environment becomes unstable.

What about regular zombie culls that become increasingly effective with experience? Well, there’s maths for that too. The trouble is that the rate of infection gets the zombie population back up to the same or a higher level between each cull, so again you’re just delaying the inevitable.

culls

Thank you, Dr. Crossman, for this graph of hopelessness

As Crossman said, if Rick Grimes can’t take out all the zombies in one cull he’s just wasting everyone’s time.

So there you have it, folks – when the zombie apocalypse happens, there’s only one possible way for the human race to survive, and that’s to have babies faster than zombies.

Next week: the principles of designing spaceships

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