Modeling Zombie Outbreaks

The math doesn’t look good: “When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection.”

An outbreak of zombies infecting humans is likely to be disastrous, unless extremely aggressive tactics are employed against the undead. While aggressive quarantine may eradicate the infection, this is unlikely to happen in practice. A cure would only result in some humans surviving the outbreak, although they will still coexist with zombies. Only sufficiently frequent attacks, with increasing force, will result in eradication, assuming the available resources can be mustered in time.

Furthermore, these results assumed that the timescale of the outbreak was short, so that the natural birth and death rates could be ignored. If the timescale of the outbreak increases, then the result is the doomsday scenario: an outbreak of zombies will result in the collapse of civilisation, with every human infected, or dead. This is because human births and deaths will provide the undead with a limitless supply of new bodies to infect, resurrect and convert. Thus, if zombies arrive, we must act quickly and decisively to eradicate them before they eradicate us.

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, or diseases with a dormant infection.

This is, perhaps unsurprisingly, the first mathematical analysis of an outbreak of zombie infection. While the scenarios considered are obviously not realistic, it is nevertheless instructive to develop mathematical models for an unusual outbreak. This demonstrates the flexibility of mathematical modelling and shows how modelling can respond to a wide variety of challenges in ‘biology’.

In summary, a zombie outbreak is likely to lead to the collapse of civilisation, unless it is dealt with quickly. While aggressive quarantine may contain the epidemic, or a cure may lead to coexistence of humans and zombies, the most effective way to contain the rise of the undead is to hit hard and hit often. As seen in the movies, it is imperative that zombies are dealt with quickly, or else we are all in a great deal of trouble.

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Posted on August 24, 2009 at 5:57 AM • 47 Comments

Comments

Paul • August 24, 2009 6:50 AM

(Apologies if this is the point and it’s obvious to everyone else)

There’s an obvious variant on this substituting “organised religion” for “zombie”. The timescales appear to be longer, but the case for hitting it hard and hitting it often seems clear.

Josh W • August 24, 2009 7:39 AM

The variant of zombie they use is perhaps over-cautious, assuming that “dead” zombies can be reanimated by another zombie. This includes headless or brainless zombie corpses. In a scenario like that, of course there can be no victory of living humans. But if pop culture has taught me anything, a shotgun blast to the head or a machete through the neck will permanantly kill the zombie and render him ineffective. According to the research paper, he is simply returned to the “removed” pool, which can be reinfected. There is no “completely removed from the game” category, which would consist of headless zombie corpses, or incinerated human bodies, or any other total removal.

Of course, my math-fu isn’t good enough to correct the authors using their language, so I just have to make comments like this here!

A Nonny Bunny • August 24, 2009 8:25 AM

Their model ignores spatial distribution of people and zombies. If you allow for that, it becomes much easier to quarantine; or just firebomb the infected area.
It would be much better to use population simulations to model the spread of zombies (and possibly disease in general).

jim • August 24, 2009 8:26 AM

@Josh W: I agree. I would say that an encounter between a human and a zombie could have 4 possible outcomes: 1) human escapes from zombie without injury to either; 2) human is zombified; 3) zombie is disabled but not permanently destroyed; 4) zombie is permanently destroyed. The paper addresses only cases 2 and 3, although you could say that 1 is implicitly handled since it is equivalent to the human and zombie never having met in the first place. I think that if you include case 4 the outlook for the human race looks a bit rosier, although I haven’t run the simulation myself.

hhh • August 24, 2009 8:37 AM

Zombies are dead, sort of, they don’t need food, so why the competion? I do believe that we should segregate them, perhaps on an island, like england

Josh W • August 24, 2009 8:50 AM

They also don’t need air. In “World War Z”, zombies were known to walk across the bottom of the ocean (or float across, given enough bouyancy), so isolating them on an island wouldn’t work, unless you had guards posted to keep them there… but why imprison, when you could just kill? (re-kill)

qwertyuiop • August 24, 2009 8:53 AM

@hhh

ermmm… England isn’t an island. However, it’s a component part of Great Britain which is.

David • August 24, 2009 9:04 AM

The treatment of the case with a cure is also naive. It wouldn’t be a matter of some percent of zombies becoming human in a given time interval, but the result of human-zombie encounters, perhaps depending on the relative size of human and zombie populations.

So, add a further status, Destroyed, as Josh W suggests. Change some of the Susceptible-to-Removeds to Susceptible-to-Destroyeds (since human funeral customs would rapidly adapt to not leave corpses vulnerable to zombification) and change most of the Zombie-to-Removeds to Zombie-to-Destroyeds, since in event of a victory the human would destroy the zombie past rezombification.

From a security point of view, the analysis is faulty, since it posits a new security threat (Zombie Attack!!!) without any change of behavior among the threatened population. It’s similar to an assumption that getting everybody to lock their doors would eliminate burglary, or that cryptography will automatically make all computers and communications secure.

Clive Robinson • August 24, 2009 9:08 AM

All jokes aside there is an important paragraph in what Bruce has quoted,

“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, or diseases with a dormant infection.”

The case they have forgotten to mention is “electronic life” such as an infected PC etc….

So I shall sit down with a cup of strong Brownian motion generator (AKA Tea) and the paper and have a think 😉

JD Bertron • August 24, 2009 9:21 AM

As pointed by Paul above, the study is interesting and has not so obvious applications to the study (and containment strategies) of contagious phenomena such as religious extremism and violent gang affiliation (such as MS 13), where prison sentences can be compared to the respawning period. Models like these can complement those from physics (Stefan problems) and from social sciences (Game theory) to better understand and reduce the spread of these problems. They can also help understand and foster the success of attributes such as those of democracy in places that have no real predisposition for them.

Canucks • August 24, 2009 9:45 AM

This is the type of research that results when you live in a place that has long, cold Winters.

Richard • August 24, 2009 10:08 AM

This is of course an interesting analysis of the effect of ever increasing copyright terms on culture.

The zombies in this case map onto the increasing number of orphan works – works by dead authors that cannot be used to create derivative works because no one know who holds the copyright. The living are current creators (or maybe new works). But of course as the current creators die and the copyright holders become difficult to trace then their works become zombies – clogging up culture.

Finite copyright terms mean that the zombies eventually die (currently 70 years after the death of the author) – but this keeps getting extended so the scenario of the dead swamping the living – making it impossible to create new works – is a plausible one….

Jason • August 24, 2009 10:18 AM

Josh W is right that by not allowing zombies to be totally destroyed, the paper is artificially guaranteeing a zombie apocalypse.

The paper also mentions a “doomsday scenario”, in which a zombie outbreak inevitably leads to the collapse of civilization. This is proven by arguing that given a constant birth rate, the sum total of living people plus zombies plus corpses increases forever, going to infinity. Since the number of living people can’t become infinite, the paper argues, either zombies or corpses must be ever increasing.

Well, of course the number of corpses increases without bound … that would be true even if there were no zombie invasion at all! The “doomsday scenario” is nothing more dangerous than a rehash of a quote by John Maynard Keynes:

“In the long run, we are all dead.”

Bernie • August 24, 2009 10:42 AM

What does Bruce mean by “The math doesn’t look good”? Does he mean “The mathematical techniques used don’t look good (ie, unsound)”? Or “The results of the (sound) math don’t look good”?

MarkP • August 24, 2009 11:14 AM

It also doesn’t analyze the Redeker plan from WWZ. That was the solution that worked for most of the world.

Marcus • August 24, 2009 11:22 AM

@Bernie – maybe he means that the maths doesn’t look good for the outcome of the human race

Davi Ottenheimer • August 24, 2009 12:09 PM

“The key difference between the models presented here and other models of infectious disease is that the dead can come back to life.”

If you substitute “continue to infect” or “spread” with “come back to life” then the models converge.

Clive Robinson • August 24, 2009 12:49 PM

@ Davi Ottenheimer,

‘If you substitute “continue to infect” or “spread” with “come back to life” then the models converge.’

Hmm maybe not it depends on what you mean by “come back to life”

You could be dead but highly infectious. However infection would only occur if somebody came into contact with your body.

So if it is known where your body is simple avoidence will stop the spread of infection.

However if you where reanimated then you can chose where to go to or be directed to go somewhere. Thus avoidance the place you fell is nolonger a working stratagie to prevent the spread of infection.

Therfore if you where malicious (and lets face it as a zombie you would have reason to have a grevance 😉 you would deliberatly seek out the uninfected which a dead body cannot do.

Maupassant • August 24, 2009 12:53 PM

Actually, looks kind of like a model for prion diseases. Like zombies, the ‘infected’ protein is not removed from the pool, and can in turn infect others.

Clive Robinson • August 24, 2009 1:20 PM

@ qwertyuiop,

“ermmm… England isn’t an island. However, it’s a component part of Great Britain which is.”

Both Hadrian and Offer very nearly succeded in segregating the English from the Scots and Welsh respectivly.

What stoped it being effective was “trade” and the taxation arising from it.

Oh and atleast two parts of England are in effect islands by themselves the most notable being “The isle of Thanet” which is a large part of the County of Kent.

Then of course there is the Irish question…

Officialy it is,

“The United Kingdom of Great Britain and Northern Ireland”

So if,

GB = E+W+S

What is the UK,

UK = GB = E+W+S

Or,

UK = (GB + NI) = E+W+S+NI

However the joys of the English language know no bounds 😉

mcb • August 24, 2009 1:33 PM

Sorry, but my eyes glazed on the maths. How do the recent “cinematic” trends toward instantaneous infection, fast zombies, or smarter zombies change the curve?

mcb • August 24, 2009 1:44 PM

@ Clive Robinson

“Officially it is, ‘The United Kingdom of Great Britain and Northern Ireland'”

Or as we call it, England.

How come ya’ll got so dang many zombies anyway?

Clive Robinson • August 24, 2009 3:28 PM

@ mcb,

“How come ya’ll got so dang many zombies anyway?”

Ah well you’ve heard of the “British Stiff upper lip” well that’s where the rot started in the Victorian era 😉

This progressed through the “Upper Classes” and their “Gentelmans Clubs” through two world wars.

It spread like an epademic through the “middle clases” during and after WWII and became a pandemic via various B blood fests from the likes of Ealing Studios and Hammer etc in the 1970’s.

And now it’s mutated in places like Essex and has a new virulant form which results in watching reality TV and wearing certain types of clothing to hide the facial disfigurment including hoods and baseball caps with extended visors.

Further the male bodies appear to be disapearing as their clothing hangs in lose drapes the trousers desperatly belted below the buttocks. The females however apear to have the oposit problem with acres of palid or burnt bloated flesh floping unrestraind out of the tops of garments that where designed for those of less corpulant measure.

You would be forgiven for thinking that the females are some how feasting of the bodies of the males, they certainly apear to have drained their ability to stand upright as their shoulders hunch and their shambling gait becomes much as you would expect from an aging tramp (hobo/bum).

These desperatly pitiful creatures are found at night performing strange rituals in and around city centers such as Notingham. They articualte strange ferral noises and emit various vile and noxious substances from various bodily orifices.

Such is the problem that it has been given a series of new names such as “Chav” and “ouff culture”.

Thomas • August 24, 2009 3:52 PM

“… we must act quickly and decisively to eradicate them before they eradicate us.”

The same old “us vs. them” mantra.

“they” are different, therefore we must fear them, avoid them, eradicate them.

Has anyone ever actually taken the time to communicate with one of “them”?

Can’t we all just get along?

pfogg • August 24, 2009 3:53 PM

@mcb

“How do the recent “cinematic” trends toward instantaneous infection, fast zombies, or smarter zombies change the curve?”

From the article: “… These zombies can move faster, are more independent and much smarter than their classical counterparts. While we are trying to be as broad as possible in modelling zombies — especially since there are so many varieties — we have decided not to consider these individuals.”

My concern is that they’ve left a few infinities just lying around. ‘No permanent removal’ (as noted above) violates the movie paradigm. Also, while it normally isn’t covered in movies, wouldn’t non-magical zombies eventually just wear out, if deprived of brains? Seems like they should dehydrate fairly quickly with all that dripping and oozing going on. Even if they don’t have a clear maximum lifespan, they should have a half-live due to various random sources of misadventure.

Harrow • August 24, 2009 4:08 PM

“…an outbreak of zombies will result in the collapse of civilisation, with every human infected, or dead…In summary, a zombie outbreak is likely to lead to the collapse of civilisation…”

My observations of the human condition lead me to suspect that this has already happened — perhaps several times, with each newly infected population being more shambling and apathetic than the last.

-Harrow.

Clive Robinso • August 24, 2009 5:06 PM

Just one thing did anybody else read the first sentence?

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

This alone says they cannot actually take over completely as their food supply would cease to exist.

So one survival strategy for the human race is just to “starve them out”.

Andrew • August 24, 2009 6:18 PM

Clearly many emergency operations and business continuity plans need to be updated to face this new and growing memetic threat.

Filias Cupio • August 24, 2009 9:20 PM

This looks to me very much like a student project that got published in a non-peer-reviewed conference proceedings or similar. Even if a reviewer had no problems with the subject matter, it would be rejected on quality grounds.

Jason has already noted the very major flaw that I spotted: they erroneously conclude that any model with non-zero human birthrate leads to a ‘doomsday’, and examine such models no further. By doing so, they eliminated any chance of finding stable coexistence solutions.

They develop some equations, but do very little exploring of the parameter space. For example, if the parameter for the rate at which dead bodies become zombies is set to zero (i.e. only living people can be turned into zombies) what effect does this have?

They also don’t model the possibility that infected people will be killed by themselves or their friends, to prevent them becoming zombified. This is a pretty standard feature of primary sources relating to zombie invasion: “Promise you’ll shoot me if they bite me.”

AMcguinn • August 25, 2009 12:38 AM

Yes, the model is very poor. I can show a human-zombie equilibrium if you assume (a) Zombies which are killed are too destroyed to ever become zombies again, and (b) Corpses rot away over time.

I’m still working out how big a shock to the equilibrium is needed to wipe out humanity – I’ll stick it all on Anomaly UK when it’s finished (before the weekend I hope).

Erik Nørgaard • August 25, 2009 11:19 AM

Why not try model the terrorist population? Different events cause positive and negative feedback:

It is of course negative feedback when a terrorist is killed, but depending on how, this may encourage new wannabees with a total positive effect.

Then just enter the numbers into the equations and we have solved the al Qaeda problem…

Jake • August 25, 2009 12:30 PM

“an outbreak of zombies will result in the collapse of civilisation, with every human infected, or dead.”

Only the people in Madagascar will survive!

BF Skinner • August 26, 2009 6:15 AM

Also doesn’t allow for changes in culture.

The zombie legion doesn’t have to be replenished by the new human dead.

At some point people realze their newly dead don’t stay dead and will respond by cremation or more traditional “stay dead” techniques used on vampires (decapitation, not the lemon in the mouth thing).

Why are zombies only human? What could end life as we know would be zombie animals. More units to spread contaigon, more mass across the face of the planet.

sidelobe • August 26, 2009 6:57 AM

It seems to me that this model should be similar to that of an STD, especially one that doesn’t kill the host for an extended time. HIV and HPV come to mind right away, but it needn’t even really be a disease. Anything that spreads from person to person, altering them in some way (Virus), but not killing them, should behave like this.

I envision a scenario where some well-intentioned but quite mad scientist releases a virus intended to make positive changes in people. The short story “The Giving Plague”, by David Brin, is an example.

How many times has this happened to humanity already?

MysticKnightoftheSea • August 27, 2009 5:56 AM

@ clive

What! No Artificial Tea Substitute?

-NutriMat

(really MKotS) (I seem to have misplaced my Bambulweenie Sub-Meson brain anyway)

Clive Robinson • August 27, 2009 9:23 AM

@ MysticKnightoftheSea,

“What! No Artificial Tea Substitute?”

No I have been told for my health I should not drink either black tea or green tea but white tea with lemon juice.

Apparently white tea is made with the bud and two leaves of the tea plant tip and is not fermented, and apparently unlike milk the lemon juice aids the transfer of useful chemicals such as anti-oxidants through the digestive wall.

I have no idea if it is true or not but as it is not made with boiling water (about 80C instead) and brews for longer it has a much more delicate fruity taste that compliments the lemon juice and is actual quite refreshing to taste.

However “white tea” appears to only be available through Chinese outlets and lets just say you would think for the price the leaves where plated in real silver, not the silvery looking down.

mcb • August 27, 2009 10:35 AM

@ BF Skinner

“Why are zombies only human? What could end life as we know would be zombie animals. More units to spread contaigon, more mass across the face of the planet.”

If everything that dies becomes a zombie then the millions of bacteria that die in our gut every day ought to infect everyperson on the planet in a couple hours…that’s so not fun it’s a non-starter.

If higher brain function is required for zombification then maybe we get zombie primates, dogs, elephants, cetacea, cephalopods, octopi, AND squids…?!!

BWB • August 28, 2009 1:23 AM

A curious study.

The supplied graph for the base case does not match the calculation results using the supplied parameters. I set up the given equations with the supplied parameters (alpha = 0.005, beta = 0.0095, delta = 0.0001, zeta = 0.0001). Adjusting the size of the time step does not help.

Modifying the base case by adding an equation for unrevivable death inflicted on zombies by susceptibles will not prevent human extinction so long as the rate constant of zombie attack on susceptibles (beta = 0.0095) exceeds the rate constant of susceptibles permanently disabling zombies (alpha = 0.005).

Further modifying the base case by preventing corpses, that have suffered natural death, from becoming zombies will prevent zombies from coming into existence. However, in this case, setting the initial population of zombies above zero and using the values of beta and alpha given above will lead to human extinction. Once again, the ratio of beta to alpha is the key.

In the final analysis, prevention of reanimation combined with the ability of susceptibles to destroy zombies faster than zombies can infect susceptibles will be a successful strategy for human survival – hardly a surprise.

The models would have been more interesting if the birth and death rate terms were included in the susceptibles equation. An extension of that equation to handle the effects of population age distribution on birth rate, death rate, and zombie killing potential would really have been cool.

Tequima • September 3, 2009 5:48 AM

You musn’t forget that there will be zombies who will maliciously seek out fresh sources of humans to infect. With even a limited amount of self-will, you only have to look at the behaviour of “griefers” in an online-gaming scenario, such as the World of Warcraft “Corrupted Blood” 2005 incident http://www.securityfocus.com/news/11330
or the pre-Wrath of the Lich King zombie invasion http://www.wow.com/2008/10/29/times-online-examines-wows-zombies-and-the-connection-to-real-l/
Though the “risk” was negligible, people did, once infected and “nothing to loose”, set out to deliberately infect others and find new ways of doing it.
It’s similar to PC viral infections “because I can”.

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