halloween-candy

Halloween: zombies walk versus random walk

I don’t know if it’s because of Halloween (or the new season of walking dead), but I had weird thoughts recently. For instance, I wondered if zombies walk was a random walk… Note that I am not the only one: there are papers on that topic on Wired or even Nature (at least on one of their blogs). Before modeling that walk, it might be better to describe the process, and to generate zombies walk through simulations.

I don’t know if it’s because of Halloween (or the new season of walking dead), but I had weird thoughts recently. For instance, I wondered if zombies walk was a random walk… Note that I am not the only one: there are papers on that topic on Wired or even Nature (at least on one of their blogs). Before modeling that walk, it might be better to describe the process, and to generate zombies walk through simulations. Consider a human being, walking. At time http://freakonometrics.blog.free.fr/public/perso4/z02.png, his location is http://freakonometrics.blog.free.fr/public/perso4/z08.png. Assume that his walk is a random walk, as illustrated below,

http://freakonometrics.free.fr/zombies-1.gif
Note that the random walk is limited to the square city, with a reflection on borders. And actually, there might be a few more humans, e.g. 3 here,
http://freakonometrics.free.fr/zombies-2.gif
A zombie is a very simple non-human being. Assume that at time http://freakonometrics.blog.free.fr/public/perso4/z02.png, his location is denoted http://freakonometrics.blog.free.fr/public/perso4/z03.png. From time http://freakonometrics.blog.free.fr/public/perso4/z02.png and http://freakonometrics.blog.free.fr/public/perso4/z01b.png. he seeks for the closest human being, i.e.

http://freakonometrics.blog.free.fr/public/perso4/z06.png

and then, he moves in his direction (always at the same speed),

http://freakonometrics.blog.free.fr/public/perso4/z07.png

In that case, the trajectory is the following (the zombie is the white cross + that enters from the upper corner, and starts following the purple human, and then the red one, etc)
http://freakonometrics.free.fr/zombies-3.gif
Of course, it is possible to be more specific about what happens when the zombie reaches a human being. So far, he keeps following him, and we have the following path
http://freakonometrics.free.fr/zombies-4.gif
But we might assume that the human-being become himself (or itself) a zombie when he has been catch, or at least that he is no longer a target. So that the zombie will be seeking another fresh human,
http://freakonometrics.free.fr/zombies-5.gif
Here is the new trajectory….
http://freakonometrics.free.fr/zombies-6.gif
Note that in my model, humans are more stupid than zombies: zombies seek humans, but humans do not try to avoid zombies… Anyway, it is also possible to consider another model for human trajectories: instead of a random walk, assume that they keep moving straight forward (with reflection on border, I do not allow them to escape from the city…). Here are trajectories of our three humans.
http://freakonometrics.free.fr/zombies-s-2.gif
If I add a zombie (that kills humans once they reach them), this will be his trajectory (note that humans go faster than zombies here),
http://freakonometrics.free.fr/zombies-s-5.gif
If we focus on zombies, this is what we get,
http://freakonometrics.free.fr/zombies-s-6.gif
I have to admit that I am not very familiar with particle models, and I do not know if this kind of model is known (perhaps changing a little bit the walks of human beings, since we might assume that they have a goal, and that they do not only walk randomly, or straight, and perhaps that they might try to avoid zombies). Anyway, my model looks awesome… and I can’t wait to try my zombie walk on Monday !


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