The SZR model of the zombie apocalypse

You’ve watched all the movies.  You’ve read all the books.  You’ve even practiced tactial skirmishes with lifesize zombie targets.  But now, all of a sudden, you are thinking, “I didn’t know there would be math!”

Actually, if you’re a regular reader of this blog, I imagine you are thinking, “I didn’t know there would be zombies.”  For those who want a math-free overview of Munz et al.‘s “When Zombies Attack: Mathematical Modelling of an Outbreak of Zombie Infection,” please see this piece by Wired.  For the rest of you, I promise Greek letters and differential equations after the jump.

Acknowledgement: I would like to thank blame this post on Dave Clarke, who brought the Munz et al. paper to my attention in an answer he gave to a question I asked about the computational theory of simulation of diseases.

The SIR Model

Suppose we want to model the effect of a rapid infection on a group of people who have contact with one another.  (For example, everyone waits at the same large train station to get to work.)  We approximate the social behavior by assuming everyone sees everyone else with exactly the same probability — and the social network graph is a complete graph — so we can focus our attention on the behavior of the infection itself.  Some diseases are slow, or potentially have long latency periods, so the population can change due to births and deaths unrelated to infection.  We limit ourselves to diseases that act fast, so the model does not need to consider both or non-infecton-related death parameters.

The SIR Model, a classic mathematical model in epidemiology, divides the population into three “compartments”: Susceptible, Infectious, and Recovered.  We want to predict the behavior of the infection over the course of time, $t$.  We define the following parameters.

$\beta$ – the contact rate.  If a susceptible and an infectious meet, the susceptible becomes infected with probability $\beta$.

$\nu$ – the rate of recovery.  This is expressed as 1/(duration of infection), i.e., in reciprocal units of time.

We then obtain the following differential equations to express the behavior of the infection over time.

$\frac{dS}{dt} = -\beta I S$

$\frac{dI}{dt} = \beta I S - \nu I$

$\frac{dR}{dt} = \nu I$

This model has been intensively studied, and many variations have been considered.  Note, for example, that in the model described, once a subject recovers, that subject will never be susceptible again.  This is a good assumption for an infection like the mumps, but there are of course situations where a subject could recover and then be susceptible again.  Further, one could imagine an infected subject moving to either of two possible compartments: recovered, or deceased.  Those variations have been well studied.  We now turn to the model introduced in 2009 by Munz et al., which includes the possibility that the deceased may come back to life.

The SZR Model

We assume the rise of zombieism is rapid, so births and population death by natural causes do not need to be modeled.  We have three compartments, as before, but the transitions between them are more complex.  The compartments are now: Susceptible, Zombie, and Removed.  We define the following parameters.

$\beta$ – the probability that a susceptible becomes a zombie when the two encounter each other.

$\alpha$ – the rate at which zombies are slain, by removing their heads or destroying their brains.

$\zeta$ – the rate at which humans in the removed class resurrect to become zombies.

This then gives us the following differential equations.

$\frac{dS}{dt} =-\beta S Z$

$\frac{dZ}{dt} = \beta S Z + \zeta R - \alpha S Z$

$\frac{dR}{dt} = \alpha S Z - \zeta R$

There are two equilibria for this set of equations, at $Z=0$ (human victory), and at $S=0$ (apocalypse).  Munz et al. show that only the apocalypse equilibrium is stable.  They also provide MATLAB code that solves the differential equations, so one can graph the exact rate of zombie takeover.

This paper made a splash in the popular media when it first came out, because “scientists were finally taking the threat of zombieism seriously.”  However, I’d like to conclude with this quote from the paper, which points out the model has potential for applications well beyond fantasy fiction.

The key difference between the models presented here and other models of infectious disease is that the dead can come back to life.  Clearly, this is an unlikely scenario if taken literally, but possible real-life applications may include allegiance to political parties, or diseases with a dormant infection.

This is, perhaps unsurprisingly, the first mathematical analysis of an outbreak of zombie infection.  While the scenarios considered are obviously not realistic, it is nevertheless instructive to develop mathematical models for an unusual outbreak.  This demonstrates the flexibility of mathematical modelling and shows how how modelling can respond to a wide variety of challenges in “biology.”

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, 133-150