Monday, 11 July 2011

Diffusing zombies.

As mentioned in a previous post, I  have written a mathematical article on zombiism as an infectious disease. Since Robert Smith? et al. had already done this already you maybe wondering what we added to the zombie theory.

Ask yourself the following question. What are zombies known for? You may come up with a long list of properties, but personally I can think of two defining characteristics:
• they prey on humans;
• the move slowly and unsteadily.
Although Robert et al. covered the first one very well they had ignored zombie movement. Now, although the speed of zombie movement is arguable, as recently they've started to run (and even ride motorbikes), we decided to stick with the slow moving variety as they are easier to understand.

So, how do we model zombie movement? Well, we based their movement on the idea of a "drunkard's walk"; the agent lurches to and fro in a random fashion with no bias in direction. Of course, you may argue that zombies head directly towards humans, which would be true. And, although we could mathematically model this, it is simpler to assume random movement. Thus, we can think of our equations portraying the earlier stages of a zombie infestation, i.e. when the zombies first arise, they will be very confused and will be moving around randomly.

From this assumption we can model the zombie population as a diffusive substance. Now diffusion has two primary properties. Firstly, the agents move without directional bias and, secondly, the zombies move from places of high density to places of low density. This is shown in the figure above and the movie below which illustrates how a population of zombies all starting at the left hand side would move across the domain.

So what can this formulation tell us? Firstly, it implies that running away is better than trying to slow a zombie down. The reason behind this is that by doubling your distance between yourself and a zombie, you multiply the first meeting time by four. However, if you slow the zombie down by half, then you only multiply the first meeting time by two. Thus, here is my advice for today. If you see a zombie...
RUN!

1. Interesting article! I'm going to try doing the same they did but in 2D for a class project

1. Hi Hernan,
Many thanks for the comment. Are you going have the zombies invading from all sides with a human population in the middle? Or are you going to start with the separated on either side of a square? Whilst you're constructing your model try and add new kinetics to allow us survive. So far every model I've seen causes our extinction!

2. I'm going to model their movement in a straight line, something simple really. I would like to calculate how long it takes for them to arrive from one city to mine.

Following a paper by the Japanese in epidemiology, I want to add then the SIR model to my diffusion equation. Then modify this model to encounter our best chance of survival, such as adding a quarantine class, infected class, intervention by the army, etc.

We'll see how it goes!

1. A couple of your ideas have already been applied by Robert Smith? (the question mark is part of his name). Have a look at his original paper to see how he did it and to give you more ideas:
http://mysite.science.uottawa.ca/rsmith43/Zombies.pdf