## 2009-08-20

### Braaaaaaains! Braaaaaains!

WHEN ZOMBIES ATTACK!: MATHEMATICAL
MODELLING OF AN OUTBREAK OF ZOMBIE
INFECTION

Abstract
Zombies are a popular figure in pop culture/entertainment and they are usually portrayed as being brought about through an outbreak or epidemic. Consequently, we model a zombie attack, using biological assumptions based on popular zombie movies. We introduce a basic model for zombie infection, determine equilibria and their stability, and illustrate the outcome with numerical solutions. We then refine the model to introduce a latent period of zombification, whereby humans are infected, but not infectious, before becoming undead. We then modify the model to include the effects of possible quarantine or a cure. Finally, we examine the impact of regular, impulsive reductions in the number of zombies and derive conditions under which eradication can occur. We show that only quick, aggressive attacks can stave off the doomsday scenario: the collapse of society as zombies overtake us all.
We assume that a susceptible can avoid zombification through an altercation with a zombie by defeating the zombie during their contact, and each susceptible is capable of resisting infection (becoming a zombie) at a rate . So, using the same idea as above with the probability Z=N of random contact of a susceptible with a zombie (not the probability of a zombie attacking a susceptible), the number of zombies destroyed through this process per unit time per susceptible is:
α( N)(Z=N)S = αSZ :
The ODEs satisfy
S0 + Z0 + R0 = Π
and hence
S + Z + R → ∞
as t → ∞, if Π ≠ 0. Clearly S NOT → ∞, so this results in a ‘doomsday’ scenario: an outbreak of zombies will lead to the collapse of civilisation, as large numbers of people are either zombified or dead. If we assume that the outbreak happens over a short timescale, then we can ignore birth and background death rates. Thus, we set Π = δ = 0.

Setting the differential equations equal to 0 gives
-β SZ = 0
βSZ + ζR - α SZ = 0
αSZ - ζR = 0 :
From the first equation, we have either S = 0 or Z = 0. Thus, it follows from S = 0 that we get the ‘doomsday’ equilibrium
( S , Z , R ) = (0; Z; 0) :
When Z = 0, we have the disease-free equilibrium
( S , Z , R) = (N; 0; 0) :

Best of all, at the bottom you can get the code to run the Zombie formulas at home with any parameters you feel like entering.

If you're really want, you can use the Runge-Kutta method, rather than Euler, to solve the ODEs in the paper.