Night of the Living Model

18Aug09

It started in Miami – or at least, that’s where the first reports came from. Haiti, Cuba and the Dominican Republic were overrun so quickly that there was some speculation that those first cases were just imports from abroad. Of course, by that time, with 56 million dead and twice that many undead, the vaugaries of who to blame seemed…inconsequential.

That’s right folks, today we’re going to talk about zombies. And epidemiology. At the same time. Why? Because we can, and because there’s a new paper out on it.

There’s a paper that appears in the book ‘Infectious Disease Modelling Research Progress’, which is either newly published or due to be published shortly. The title?: When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection. The full text of the paper can be found here: http://www.mathstat.uottawa.ca/~rsmith/Zombies.pdf

I’ve talked a little bit about modeling, and it happens to be my research interest. And if you haven’t gathered this by now, I’m also a tremendous geek. Two great tastes that taste…well, a bit like decayed flesh.

The paper itself focuses on the emergence of a “classical” zombie – infectious, prone to eating humans, dumb and slow moving. Old school, in opposition to say, the fast moving super-charged predators of 28 Days Later. And good for them – stick with the classics.

They attempt to do this via a classic deterministic, compartmentalized model (which I will from now on call an SIR model because its much, much faster to type), which is a well-established, pretty solid way to model the dynamics of an infectious disease. Incidentally, for those interested in a decently approachable primer for dynamic transmission models of diseases, this paper isn’t a bad one, especially if you skip the middle (which we’ll get to). How these models work is that a population moves between different compartments, or stages of infection as time does on, and this movement is dependent on the numbers of individuals in each of the compartments. For example, a population can start out entirely Susceptible (S), with one person being Infected (I). Based on various rates the model has in it, this one person will infect others, who will in turn infect others, etc. etc. Some may Recover (R) from the disease, and it gets more complex from there. The way you govern these models is, typically, using a series of ordinary differential equations, which sound way more complex and intimidating than they actually are in practice.

What makes them interesting is that you capture both the dynamics of infection (the probability of being infected goes up the more infectious people there are, infections may die out because they exhaust their pool of susceptible, etc.) and model interventions and diseases you can’t do in observational epidemiology. Like, dare I say it, Zombies.

The model they chose is pretty straightforward. People are Susceptible (S), Zombies (Z) or Dead (R – “recovered” can also be “removed”). The one twist over the usual epidemic model is that the Dead, which in most models are unassailably dead, can be reanimated as Zombies and move back into the Z class. What follows in Section 2 of the paper is a rather dry mathematical explanation for why, in their particular model, “coexistance”, or an endemic level of zombie infection, is impossible. Some mathematicians will undoubtedly disagree with me, but I find the preoccupation with finding the equilibria of a model with analytic approaches to be somewhat tedious and unnecessary, especially as, in this case, the findings aren’t all that interesting.

The authors go through a couple classic next-steps, introducing latent infections (you have some time in-between bite and full on zombie time), quarantine and treatment, and find that with the exception of the last one, humanity is wiped out to the person. Pity that. Apparently treatment lets us poor living types skulk about in bunkers, isolated farms and ramshackle convoys in low numbers. Up until this point, I’m not all that impressed. Straightforward, but largely well, plain, model that with the exception of allowing the existance of zombies, doesn’t really *model* them. After all, while quarantine and treatment are common tactics for infectious diseases, I’ve never really seen them in the zombie-outbreak cultural awareness. Lets be honest, there is only one public health intervention that needs to be tried for zombies:

Aim for the head

Then, however, towards the end we get to the part that I think is the most interesting. We go Zombie huntin’. The authors examine a “Impulsive Eradication” scheme, where, every one in awhile, when resources allow, there’s a mass zombie cull. Think an occasional counter-attack from the surviving humans, pulling open the doors of their fortified houses and holdouts long enough to let armored trucks through, bristling with guns.

This, it appears, might work.

Generally, the gist of the model is to show that mathematical modeling is a flexible tool that can be adapted to a variety of scenarios – even somewhat fantastic ones. Which is an excellent point. But I’ve got some unanswered questions, and really do wish the authors would have gone whole hog, embraced their inner zombie, and modeled some more things:

1. Zombies in this model can, it appears, cycle between being Zombies and being Dead. This is, to any genre fan, clearly cheating. Once a zombie’s down – in the decapitation, blown to bits, etc. sense of the word – it should stay down.

2. The authors have Max Brook’s World War Z several times in their citations – it would have been nice if they had put a touch more realism in their pulsed eradication scenario. After all, in addition to going to kill zombies, you are going *to* the zombies – you should expect more than your baseline number of casualties. As such, the zombie-human mixing rate (β in the model) should go up during the zombie culls.

3. With quarantine, the author’s assume that we’re talking about quarantine the infected zombies. It would also have been interesting to see about quarantining the uninfected humans – the classic we hide in a mall/mansion/stately manor/military base – scenario, to see if you could get a co-existence of man and zombie.

4. Parameter choices. Nowhere do they seem to mention their parameter choices, and how they got them. All this math, elegant and complex as it is, and they don’t discuss the fundamental numbers that make their model run. This drives me *nuts*.

So basically, it was a cool paper (because zombies are cool) and honestly, I wrote a paper on video games, so who am I to judge. It would have been nice however if the models felt a little bit more like a true attempt to model zombie outbreak scenarios – embracing that you’re now talking about mostly fantasy, and a little less like someone scratched out “Influenza” and wrote “Zombie” before submitting it. I think that would also have served the author’s stated purpose of showing the flexibility of this approach much, much better.

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4 Responses to “Night of the Living Model”

  1. I love Infectious Diseases/Virology…hate Math!
    But this was a fun explanation.
    JTD

  2. In general, let me just say that I’m very happy if people are questioning our assumptions. They should, because assumptions in modelling are very important. And many of the ideas you mention would be excellent variations. Indeed, I presented this work at a recent conference and had a number of people wanting to do followup work, which I think it great.

    However, to address your points:

    1. In the basic model you’re right: when zombies are killed, they go to the
    dead (removed) class and can later be reanimated.

    However, in the final intervention scenario (what we call impulsive
    eradication), the zombies are destroyed and do not return to the removed
    class.

    We were thinking that you might kill a zombie that can then come back to
    life (as Shaun does with his car in Shaun of the Dead) but also considered
    the case where you can kill the zombie for good.

    2. For the pulsed eradication, we were thinking more of the situation at the end of Shaun of the Dead, where the army wipes out zombies with very few casualties to themselves. Basically a heavy artillery scenario. So I’m not convinced that this applies.

    3. I like this idea, it would be a great extension of the model.

    4. Since we had analytic results, the parameter values are illustrative (we tried to get a grant to go observe zombies in the wild, but for some reason it didn’t fly…). But parameter values don’t make the model run, the differential equations do. However, we included the Matlab code at the end, so anyone is welcome to copy and paste it into a program and pick different parameter values to see what happens.

    And, just to be clear: it was zombie modelling from the start! No other models were adapted and we thought a lot about the choices we were making. By all means disagree with those chocies, but that’s where we were coming from.

    Hope that’s helpful.

    – Robert Smith?

  3. 3 Epi_Junkie

    Well, it’s certainly cool to hear from an author on the paper. Its always great to hear where people are coming from, and the choices you made in building the model… one often has to divine the authors intent, and from your post, some of my divinations were wrong. Who’d have thought it?

    It appears most of our disagreements stem from different ways one could compose a model – and indeed, probably different approaches to model building. I very rarely focus on analytical results, so I’m often a fan of more complex, subdivided model structures, and admittedly rather obsessed with parameter choices, because when it comes down to it, most of my results are numerical.

    To address some of your comments.

    1. Admittedly, there is the “We got him, and he got back up” – usually the result of not aiming for the head :). I’d just consider this someone “staying” in the Z class, rather than going from Z to R back to Z. But yeah, that’s philosophical.

    2. Yeah, I understand your thinking there. But for every Shaun of the Dead’s smooth cleanup, there’s a “Battle of Yonkers”-scale fiasco. I think it would be an interesting sensitivity analysis to have beta vary over a range of values during the impulsive eradication periods, to see how much human casualties during the eradication effect the results – and what the projected cost of the campaigns will be.

    Thanks very much for your comments, and I enjoyed reading your paper. For all the parts we disagree on, it is on Zombies, which makes it automatically pretty great in my book.


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