Zombie Mathematics

Earlier this year a group of mathematician from Canada published a paper in the journal “Infectious Disease Modelling Research Progress” examining the ability to model a zombie apocalypse using current infectious disease models (with a few modifications). What they determined was that if zombies were to exist (without a cure or massive military action, which they also looked at) then the only outcome was everyone becoming a zombie. To use more technical terminology, the only stable equilibrium was all zombie. In their analysis they found this to be true with a virus (or whatever) that acted quickly, slowly and if they included quarantine of the infected/zombies as well. The only situations where it might be possible (according to their study) to survive a zombie apocalypse was if 1) there was a cure developed (although the world would be mostly zombies) or 2) if you had a group of survivors and were able to use the military to attempt to kill off every zombie (which I would have liked the authors to bulk up a little more).

Now, while I appreciate that someone took the time to actually study this (yes, this does appear to be a real, published article) I believe that there are a few things that were missing in their analysis that severely limit the validity of their results. In a word: decay. The authors seem to believe that a zombie, once “made” is an eternal being of evil, amazingly transformed from either a corpse or a person into a shambling automaton of doom. While this might be a useful model for older “magical” zombies, it conflicts with the assumptions made by modern zombie stories that a zombie is still an organic being. Specifically a dead organic being. And being an organic being zombie should be expected to rot.

Not to seem overly pedantic, but I believe that this is an important point that needs to be consider and would shift the equilibrium point in our (the humans) favor. To be clear, I’m not considering “magic zombie,” as those 1) don’t usually seem to “infect” the living as much as they do the dead and 2) are in the same class of monsters as living skeletons, and thus are not likely to be affected by decay. I’m more interested in the modern interpretation, where zombies are caused by a toxin, virus or extra-terrestrial equivalent that converts the once living in the walking dead. In most of these scenarios the “cause” (virus, etc.) takes over the CNS of the victim and causes them to seek out and attack the living.

(An aside – the ATP issue. One of the big issues with zombies as being dead and yet still moving is the fact that the lack of circulation would mean that muscles would run out of chemical energy (ATP) fairly quickly, immobilizing the zombie. In the cases of fungi which actually do “zombify” insects and the like, the victim is still alive. For the sake of simplicity I will ignore this issue, assuming that for the sake of writing this that the zombie are physically dead, but able to move. The fungus/living victim will be a discussion for another post.)

After a cursory glance at wikipedia I found that bones will start to be exposed on a body (showing high levels of decay) at between 10-20 days, of course depending on location. As wetness and warmness help decay one would expect that in the tropical parts of the world one would expect decay to occur faster than elsewhere. What this would indicate for me is that one would have to expect that a zombie has a maximum “deathtime” (think lifetime) before they would be unable to move and thus no longer a threat. As such I believe that the Canadian model needs an additional population (along with the S, R and Z) of D, the decayed. D’ (the decay population growth rate) would take the form:

D’ = θR + ψZ

Where θ is the decay rate of the removed group (the dead, R) and ψ is the decay rate of the zombie group (Z). Each of those population’s rates would also need to have these terms added to show the population loss to decay.

So, what does this mean? Well, without getting too technical (and without spending any time to go and plot these graphs) these added rates should offers two additional solutions 1) after a certain amount of time the world would become safe again as all zombies or members of the removed (dead) have decayed (allowing any survivors to camp out until it dies out) or 2) it is possible that for some combination of rates and populations (βSZ + ζR -αSZ – ψZ) would be less than zero while having a standing survivor population (S), indicating a shrinking zombie population and the possibility of survival. While these would need a full mathematical analysis to be sure, I think that this more accurate model is not as grim as the Canadian one.

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One Comment

  1. Posted April 6, 2010 at 17:10 | Permalink

    Dude, that’s just awesome! (Though all of the extra math stuff makes my head hurt …)