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Page 3 model: When zombies attack

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Zombies

Run? Kill? Ask nicely? (Source: flickr.com/joelf)

Every issue we feature another great model on page 3 of our magazine. This issue it’s this:

\begin{align*}
S’ &= \Pi – \beta S Z – \delta S,\\
Z’ &= \beta S Z + \zeta R – \alpha S Z ,\\
R’ &= \delta S + \alpha S Z – \zeta R.
\end{align*}

This is a model of a zombie outbreak as proposed in When Zombies Attack! by Philip Munz, Ioan Hudea, Joe Imad and Robert J. Smith? (the question mark is not a typo; it is part of his name).

The model features three populations: S, the susceptible or living humans; Z, the zombies; and R, the removed or dead humans. Humans can move between these three groups. The rates of movement are modelled by the following parameters:

α, the rate at which humans can kill zombies,
β, the rate at which live humans are infected and become zombies,
δ, the rate of non-zombie related death in the live humans,
ζ, the rate at which the dead rise again as zombies,
Π, the human birth rate (assumed to be constant).

The terms in the model can be nicely shown in the following diagram:

See how the number of susceptibles feeds the number of zombies, etc.

See how the number of susceptibles feeds the number of zombies, etc.

Munz, Hudea, Imad and Smith? showed that unless a zombie outbreak could be ended quickly by a swift killing of zombies then the human population will eventually become entirely zombie.

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