# Figure 3. Basic model outbreak scenario. Susceptibles are quickly eradicated and zombies take over, infecting everyone.August 13, 2009 12:03 PM   Subscribe

When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection [pdf] (via)
posted by brundlefly (65 comments total) 27 users marked this as a favorite

My poor math skills do not allow me to catch the actual "maths" involved here; but I smell and amazing plot device in a low-budget Zombie flick right here.
posted by NiteMayr at 12:07 PM on August 13, 2009

"A zombie is a reanimated human corpse that feeds on living human ﬂesh [1]."

Comedy gold.
posted by escabeche at 12:14 PM on August 13, 2009 [1 favorite]

The key difference between the models presented here and other models of infectious
disease is that the dead can come back to life. Clearly, this is an unlikely scenario if taken
literally, but possible real-life applications may include allegiance to political parties...

I've never noticed the similarity between political party members and zombies before, but yeah... that explains a lot.
posted by twoleftfeet at 12:20 PM on August 13, 2009

I will not be limited by some mathematician's reductive definition of zombie. I happen to be a reanimated astronaut, and the fact that I sometimes kill people with machetes is in no way inspired by hunger for human flesh, but merely my unthinking fealty to my insane master's directives.
posted by Astro Zombie at 12:21 PM on August 13, 2009 [17 favorites]

Let's see if I can summarize the key points:

Using the popular slow-moving zombie from movies such as Shaun of the Dead, the basic model has three variables:

S, for 'susceptibles', which means non-zombie people,
Z, for zombies, and
R, for 'removed', which means solidly dead folk.

In addition, there are transmission parameters, which are rates of occurrence:
β, which is a zombie attack,
ζ, which is zombie resurrection, and
δ, which is non-zombie death.

There is a zombie death parameter, which is α.

There is also a constant, Π, which is the birth rate.

The basic model is given by the three differential equations given in the article; this is for a situation with no latent infection, quarantine, and treatment.

The model with latent infections includes a period of 24 hours before a bitten individual in the group S joins the group Z.

The model with quarantine contains these parameters:

"-The quarantined area only contains members of the infected or zombie populations
(entering at rates κ and σ, respectively).
- There is a chance some members will try to escape, but any that tried to would be
killed before finding their ‘freedom’ (parameter γ).
- These killed individuals enter the removed class and may later become reanimated as
‘free’ zombies."

The model with treatment contains these parameters:

" - Since we have treatment, we no longer need the quarantine.
- The cure will allow zombies to return to their original human form regardless of how
they became zombies in the first place.
- Any cured zombies become susceptible again; the cure does not provide immunity."

Someone with more differential equations skillz than I have can explain the equations.
posted by kldickson at 12:26 PM on August 13, 2009 [1 favorite]

Pssh, amateurs...

If I learned anything, it's that I can defend a zombie onslaught with pea shooters and pumpkin tossers on my front lawn.
posted by Christ, what an asshole at 12:28 PM on August 13, 2009 [4 favorites]

This is really quite a nice article, actually.
posted by kldickson at 12:29 PM on August 13, 2009

leave it to the mathematicians to take all the fun out of a zombie outbreak.
posted by jadayne at 12:29 PM on August 13, 2009 [5 favorites]

I've enjoyed the agent based model approach. It gets to some much more complex behavior.
posted by a robot made out of meat at 12:30 PM on August 13, 2009

If I learned anything, it's that I can defend a zombie onslaught with pea shooters and pumpkin tossers on my front lawn.
Yeah, but the zombie dolphins, they're trouble...
posted by verb at 12:31 PM on August 13, 2009 [2 favorites]

At least they didn't go with the ridiculous "fast zombies" and contaminated blood theories of deviant movies and belief-systems. Zombies represent the ever-steady progress and inevitability of death itself (until they gang up and just want their place in society).
posted by filthy light thief at 12:31 PM on August 13, 2009 [1 favorite]

Zombies represent the ever-steady progress and inevitability of death itself (until they gang up and just want their place in society).

posted by Shepherd at 12:33 PM on August 13, 2009

Consider a spherical zombie, z....
posted by lukemeister at 12:35 PM on August 13, 2009 [17 favorites]

The one thing I always wonder is how zombies would realistically propagate. The only way the disease/curse/whatever spreads is by fluid transfer from a zombie to a human. But if a zombie gets close enough to a human to bite it, isn't the human much more likely to become a meal than a new zombie?
posted by explosion at 12:41 PM on August 13, 2009

Brings to mind the Zombie Infection Simulation.
posted by Graygorey at 12:46 PM on August 13, 2009 [5 favorites]

Best line: These equilibrium points show that, regardless of their stability, human-zombie coexistence is impossible.

So the ending of Sean of the Dead? Fake!
posted by Nanukthedog at 12:47 PM on August 13, 2009

Same with Fido.
posted by brundlefly at 12:51 PM on August 13, 2009

the fact that I sometimes kill people with machetes is in no way inspired by hunger for human flesh, but merely my unthinking fealty to my insane master's directives.

How is it that you get to use that "master's directive" line and everyone shrugs and accepts it, just because your master is a genius who probably lives in a government lab under a mountain somewhere, but when I say that the deaths were "directives" people get super bent out of shape and try to imprison me.

Is it just because my master is a voice in my head put there by an evil Siamese cat? Is that it?

Totally not fair.
posted by quin at 12:51 PM on August 13, 2009 [1 favorite]

The one thing I always wonder is how zombies would realistically propagate. The only way the disease/curse/whatever spreads is by fluid transfer from a zombie to a human. But if a zombie gets close enough to a human to bite it, isn't the human much more likely to become a meal than a new zombie?

It's a negative feedback loop: a single zombie is physically weak and unlikely to kill a person, but a mob is unstoppable. When the zombie density is low, most attacks result in the infection and eventual conversion of the victim, which in turn increases the zombie density until the zombie mobs are able to kill and consume their victims.
posted by Pyry at 12:52 PM on August 13, 2009 [8 favorites]

Bah. All the ordinary citizen needs to know is they start in the front yard, then move to the back, and then the roof. Just make sure you buy enough pots for the roof invasions.
posted by Hardcore Poser at 12:54 PM on August 13, 2009 [1 favorite]

These people aren't mathematicians. Or at least, they don't write papers like mathematicians.
posted by TypographicalError at 12:56 PM on August 13, 2009

I'm a mathematician and I'm not sure what you're talking about, Typo.
posted by escabeche at 1:04 PM on August 13, 2009 [2 favorites]

I can't tell you how happy I was to see this paper cite Brooks. Joy!
posted by butterstick at 1:17 PM on August 13, 2009

So... wait Pyry.. are you saying that I can improve my zombie group dynamic skills by playing Katamari Damarci?
posted by Nanukthedog at 1:17 PM on August 13, 2009

And buy the rake! It's got the best cost/benefit ever.

What effect would buying rakes (or other lwn implements) have on this model, I wonder?
posted by bonehead at 1:22 PM on August 13, 2009

*Naaah nananaa nananaaaaah naaah KatamaRRGGGRRRRRRRAAAAAAAAAAAAAARRGGG*
posted by slimepuppy at 1:22 PM on August 13, 2009 [2 favorites]

Same with Fido.

Fido is based upon the assumption that 1) we can build strong enough walls to keep out the roaming masses of zombies from inside our secure compounds, and 2) we can control them through science and technology.

I'm interested to see how the proposed (as in not-yet-created) Walking Dead show handles this. In the comics, there are secured outposts, but people usually eff things up.
posted by filthy light thief at 1:24 PM on August 13, 2009

How is it that you get to use that "master's directive" line and everyone shrugs and accepts it, just because your master is a genius who probably lives in a government lab under a mountain somewhere, but when I say that the deaths were "directives" people get super bent out of shape and try to imprison me.

Well, my master is John Carradine, and also Tura Satana, so there's that.
posted by Astro Zombie at 1:29 PM on August 13, 2009

funny -- a third of their references are video games or classic zombie movies.
posted by JohnFredra at 1:39 PM on August 13, 2009

I'm interested to see how the proposed (as in not-yet-created) Walking Dead show handles this.

Side note: A world with Fables and Walking Dead on TV is a world that trumps all others.

posted by pokermonk at 2:11 PM on August 13, 2009

I'm interested to see how the proposed (as in not-yet-created) Walking Dead show handles this.
I had no idea. Can't wait to see what Frank Darabont would do with it.
posted by now i'm piste at 2:17 PM on August 13, 2009

Yeah, yesterday I squealed like a little girl when I found out about the tv show.
posted by brundlefly at 2:19 PM on August 13, 2009

The senior software engineer at my office is, as we speak, trying to figure out how to make this paper relevant to visual effects so he can teach it at the next big lunchtime paper review.
posted by fairytale of los angeles at 2:25 PM on August 13, 2009 [3 favorites]

funny -- a third of their references are video games or classic zombie movies.

Would you prefer more references to stories of Haitian zombie-curses and zombie powder? Or are you more fond of the emaciated corpses that haunt graveyards in the Middle Ages? There are only so many sources for "research" on zombies.

There wasn't enough info around just yet to make any proper post, so I thought I'd toss it in here, for any who were interested.
posted by filthy light thief at 2:29 PM on August 13, 2009 [1 favorite]

As soon as I recover from my nerdgasm, I will be printing this out to give to my students at the first Math Club meeting of the school year.
posted by tits mcgee at 2:50 PM on August 13, 2009

The one thing I always wonder is how zombies would realistically propagate. The only way the disease/curse/whatever spreads is by fluid transfer from a zombie to a human. But if a zombie gets close enough to a human to bite it, isn't the human much more likely to become a meal than a new zombie?

It seems the usual way this works in onscreen depictions relies on your traditional shambling zombie. Zombie bites human, human madly scrambles free only to succumb to infection hours later. With 28 Days Later, we had some highly motivated zombies, so the speed of infection was cranked up to mere seconds. Alas, we then lose one of the tropes of the zombie movie: the character who has been bitten but is concealing his infection from the other characters. O tempora, o mores.

In other modern depictions of fast-moving zombies -- the 2003 Dawn of the Dead, the 2008 I Am Legend -- we get other explanations. I seem to recall that the infection in IAL did take hours to take hold, so your point would be valid there. (Of course, if that were the biggest problem that IAL had, it would be a better movie.) I cannot recall if there is any mention of this in Dawn of the Dead.

Yes, I know the critters in 28 Days Later and I Am Legend were not precisely zombies.
posted by ricochet biscuit at 2:53 PM on August 13, 2009

In the goofy Italian flick "Nightmare City" the zombies are relatively fast-moving and can use weapons. They get neatly around the problem of causing so much damage that infection spread is hampered by mainly just stabbing people in the neck and drinking their blood rather than eating them. (Also occasionally doing unfortunate things to nipples, but that's hardly something that'll negatively impact a new zombie's mobility and kill potential.)

It's also notable because of the outbreak scene where dozens of stab-crazy and gun-shooty quasi-zombies (they're really sort of the duck-billed platypi of the zombie tree of death) pour out of a rainbow-painted airplane like so many clowns out of a tiny car.
posted by Drastic at 3:02 PM on August 13, 2009 [1 favorite]

If zombies are fast moving and using weapons and are drinking blood rather than eating human flesh, are they even still zombies at that point?
posted by hippybear at 3:14 PM on August 13, 2009

My major question is this...

Did this paper use tax-payer funding? (University of Ottawa, right?)

If so....YAY!!!
posted by jkaczor at 3:26 PM on August 13, 2009 [2 favorites]

Near the beginning, the article refers to "regular, impulsive reductions in the number of zombies". Such a charming way of describing survivors hysterically swinging sharp objects, bits of wood and anything else at hand while hordes of undead pursue them. It's as sterile a term as 'collateral damage', though obviously not as repugnant (because, you know, it isn't real except in my nightmares. Of which there are many. *Twitches*)
posted by marmaduke_yaverland at 3:29 PM on August 13, 2009

Is it just because my master is a voice in my head put there by an evil Siamese cat? Is that it?

Totally not fair.

I can totally take care of this problem for you, quin.

(I have a related plan for handling Tura Satana. You're on your own with Mr. Carradine, AZ.)
posted by kittens for breakfast at 3:47 PM on August 13, 2009 [1 favorite]

I don't completely agree with their model. They state that you can "kill" a zombie (e.g. chop its head or destroy its brain), but then put it back in the R class from which zombies can always spontaneously rise again.

In such a model, even your cremated thrice-removed great-aunt could come back as a brain-less cloud of zombie ashes, kill you and eat your brains! So, of course they're going to find that the only stable equilibrium, if no cure is available, is a zombie apocalypse.
posted by Fruny at 3:49 PM on August 13, 2009

This SZR model is a fantastic update of the standard SIR model of infectious disease.
posted by Schismatic at 4:03 PM on August 13, 2009

If zombies are fast moving and using weapons and are drinking blood rather than eating human flesh, are they even still zombies at that point?

I have sympathy for the purist school of zombie taxonomy, but ultimately it's just a classification model; broadly useful but with exceptions and gray areas. There are species and subspecies that confound it. Much like the platypus, that's not really the platypus' fault that there are points where the classification scheme breaks down. Some species have a greater faculty for tool use than others; some species will only produce tool-using outliers like [i]Undeadicus Zombie Romerii[/i] (note that this strain, commonly cited as the only true zombies, has also produced occasional speedy mobility outliers, too) Bub and Screaming Gas Station Guy. Some zombies will avoid water; others will fight a shark to a standstill (much to the shark's befuddlement). Some zombies hunt by sight, others are blind; the blind ones have been known to use swords and ride on horseback in certain sightings. Some are faster, some slower, and some rare ones have been known to teleport.
posted by Drastic at 4:04 PM on August 13, 2009 [3 favorites]

jkaczor, it looks like it. From the paper itself:
RJS? is supported by an NSERC Discovery grant, an Ontario Early Researcher Award and
funding from MITACS.

posted by fairytale of los angeles at 4:11 PM on August 13, 2009 [2 favorites]

I have sympathy for the purist school of zombie taxonomy, but ultimately it's just a classification model; broadly useful but with exceptions and gray areas. There are species and subspecies that confound it.

Very true. For instance, the Chinese version hops around, instead of staggering.
posted by brundlefly at 4:20 PM on August 13, 2009

I think the real question here is, where can I find a copy of Up With Dead People?
posted by serazin at 6:10 PM on August 13, 2009 [1 favorite]

Dear gods of knowledge: I LOVE BEING AN ACADEMIC!
posted by strixus at 6:45 PM on August 13, 2009

I think the real question here is, where can I find a copy of Up With Dead People?

Oh, I read that, and went immediately to the link, hoping and praying that it was a touring musical group who sang neo-hippie songs about how we all need to get along and how great the world would be if we could just hold hands and sing.

I mean, seriously. Watch this and picture what it all COULD really be!
posted by hippybear at 7:02 PM on August 13, 2009

Thank-you "fairytale of los angeles", admittedly my reading comprehension and attention goes down in direct proportion to the number of equations contained within the source material.

But this is still awesome and frankly worth a few bucks - for even bigger grant money they really should have replaced "Zombie" with "H1N1".
posted by jkaczor at 7:28 PM on August 13, 2009

fairytale of los angeles, I've seen a presentation on how Zombie Freeze Tag is a decent way of looking at Diffusion Limited Aggregation, of which you can provide pretty pictures.
posted by Pronoiac at 7:39 PM on August 13, 2009

If a zombie outbreak ever happens, are you guys aware how many citations this paper will get? Kudos to the authors.
posted by caelumluna at 8:09 PM on August 13, 2009 [1 favorite]

I'm always amazed at how much thought people will put into things or concepts that don't actually exist.

See, also: religion.
posted by Avenger at 8:25 PM on August 13, 2009 [1 favorite]

Consider a spherical zombie, z

Can we assume it is frictionless?
posted by flaterik at 1:15 AM on August 14, 2009 [1 favorite]

The one thing I always wonder is how zombies would realistically propagate.

I proposed that once to a group of people and we didn't get far with it.
The slow moving zombies are probably easy to get away from and thus have a larger transference of undeadedness. With fast moving zombies, not so much, you are going to end up as chow. A large deciding factor would come down to the reanimation time lag. Is it instantaneous upon death? What about the fact that zombies don't eat other zombies, so do they stop eating when something dies? Is there a difference between dead and undead?

I have sympathy for the purist school of zombie taxonomy, but ultimately it's just a classification model; broadly useful but with exceptions and gray areas. There are species and subspecies that confound it.

How you're separating the vampires from the zombies?
posted by P.o.B. at 2:03 AM on August 14, 2009

Zombie metabolism doesn't produce its own angst and general self-absorbed emo-ness. (It's similar to how human metabolism doesn't make its own vitamin C, only they don't actually have a metabolic use for it. Angst or the vitamin.) Notice that the only zombies who wallow in woe-is-me angst are the freshly-turned; they're running off of human angst reserves but don't replenish it when they run out. (There are probably outliers that some specialists can identify; unlife is a spectrum, not a series of discrete boxes.)

The common vampire, though, do generate their own angst. Its exact metabolic role hasn't really been determined, but I'm inclined to believe the research that says its something of a waste product. Stereotypically gothy acting out, gloominess, organ-playing, ritualized "romantic" pursuits of primarily young women and girls, etc., are effectively the vampire excreting angstwaste.

But again, unlife is a spectrum. Note that earlier breeds of vampires are actually fairly close to zombies in how they're described; in modern times the hyperangst producing species has undergone more of a population explosion.
posted by Drastic at 8:30 AM on August 14, 2009 [2 favorites]

It's been several years since I stopped playing Urban Dead, and I thought I was cured of my addiction to the zombie genre. Then a few days ago I started playing the Dead Frontier open beta, and I'm now completely off the wagon. I just wish you could play as a zombie like in UD.
posted by Thoughtcrime at 9:12 AM on August 14, 2009 [1 favorite]

Let's not go crazy. Twilight is a blip (blight?) on the history of vampire mythos, but if you want to use that as a signifier I'm sure you can find emo/goth zombies at the mall looking for even sadder & darker makeup.
If we are going to say zombies are more than rotting corpses that feed on living flesh then it's harder to differentiate between an undead staring at you while your asleep because (s)he/it is doing it our of forbidden love or just plain hungry.
posted by P.o.B. at 12:12 PM on August 14, 2009

My inner teenager had multiple giggly nerdgasms just skimming that PDF: It is too excellent and too retarded.

BBRRRRRRRRAAAAAINNNSSSS!!!

*Rolls eyes into back of head and shuffles away....*
posted by Skygazer at 12:32 PM on August 14, 2009

I know this is an old thread, but I thought fire at the whole "How would the slow zombies get everyone" question.

Easy, people are social animals who, when given the choice and lacking foreknowledge, will aid anyone who looks injured, even bloody moaning injured people. Not all people, but if we weren't so social, our tightly packed cities would be easy pickings for even the one-legged Romero-style walking dead. All it takes is more than three infected to roam free (and unknown to the general public) and you have an easy-going strolling disaster on your hands.

I'd bet that social immun-responses like twitter and facebook would spread the "watch out for bloody nonsense" news fairly quickly.
posted by NiteMayr at 3:00 PM on August 16, 2009

HAI IM PLAYIN ZOMBEEZ DOUNTOUN

JOOOOOIN UUUUUS

BRING MOR BRAAAAINZ

posted by Pronoiac at 4:04 PM on August 16, 2009

I came across this on the BBC website whilst at work today. "Wow!" I said to my team-mates, "this is fantastic! Someone's published a scientific paper about zombies and how they'd eradicate humanity."

"What's a zombie?" said three people, simultaneously.

Humanity is safe, they'd survive.
posted by essexjan at 10:43 AM on August 18, 2009

Duck season! Wabbit season! Zombie season!

And because I must pimp Peter Watts (another Canuck scientist who fools around with things Man Was Not Meant To Know) whenever possible: Vampires: Biology and Evolution.
posted by maudlin at 5:26 PM on August 18, 2009

Nice catch maudlin; if you hadn't posted that one I sure would've. Any slideshow with the title 'Taming Yesterday's Nightmares for a Better Tomorrow' has my vote.

As for zombie variation, has anyone stumbled across All Flesh Must Be Eaten? It's a zombie survival RPG. Unfortunately, some of the finer mechanics are missing, and many others are scattered throughout the text, but the zombe creation rules are fantastic. What, how much, and how often a zombie must eat, weak points (head, spine/neck, heart), origin (radiation, divine wrath, aliens), movement and infection rates.

Because I'm weak like that, I think I'll have a peek at the equation and see if I can add some supplimentary modules to account for the possible variations.
posted by LD Feral at 10:19 AM on September 2, 2009

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