Sunday, November 18, 2012

Is the SIR model applicable for Zombie outbreaks?

I have been playing with SimZombie and thinking about the paper "When Zombies attack! Mathmatical modelling of an outbreak of Zombie Infection" (Availible here).  Both are based on the SIR model using a synchronous latice from epidemiology and I think the model is way too simple to be useful.

My problem is that humans in a population do not act like cells or cattle.  There are plenty of similarities but its the differences that break the model fundamentally. However, like all thought experiments on imaginary ailments... this is simply a case of my imagination against someone elses...

The problem is that the variables in the SIR model is very simplistic.

Transition from the S to the I state is inevitable goverened only by some probability at each possible infection event.  Similarly transition from the I to the R state is also inevitable, again regulated by some probability.

The argument being that the complexity of the system is stripped down and the value of the model is in the understanding of the outcomes given different starting conditions of the variables. 

However, I would contend that stripping the population down to such an abstraction, produces a model that is pointlessly simple and, to be honest, boring.  I mean, if you are going to spend the time to model an imaginary disease, you may as well enjoy it.

So where to begin?  Well let's start by extenting the SIR model a little.

Lets start by looking at a more interesting number of compartments in the model.

S - Succeptable ( This is the "living" population in most scenarios)
B - Bitten (Early stage Infection, but not mentally compromised ...yet)
Z - Zombie (Fully infected classical Zombie. )
D - Dead (This is permaently dead... not undead-dead.  This is game-over)

Finally, the most important divergence from the SIR model is the concept that the population is passive. I propose that we add another compartment.

H - Zombie Hunter

I propose that we accept that a certain proportion of the population are not going to be particularly "passive" and will take some steps to at least defend themselves, while others will go to the extreem of purposefully hunting down Zombies.

One of the interesting things is this proposed model is how these hunters will behave over time.  If we consider that someone learning to hunt, should improve with experience (learn) then they should become more effective given time.  While the other side of this coin may be a resource problem. If they run out of energy or ammo... then they may become less effective over time. 

We can still map out the state transition diagram. I think its pretty obvious.
S->B   (Got bitten)
S->D   (Got dead)
S->H   (Got angry)

B->Z   (Full blown infection)
B->D   (Got dead. Self inflicted, friends, treage )

H->S   (Calmed down or out of ammo)
H->B   (Got bitten)
H->D   (Got dead)

Z->D   (Got dead)

The question is always... "is there a cure"?  If there is a state transition from Z to S then does it grant immunity or is the individual again succeptable, if not, there would be a true "R" state with immunity.

Anyway, lets have a look at some of the potential variables for the different compartments.

S Compartment
* Movement rate
* Intelligence
* Clustering and Cooperation
* Communication

B Compartment
* Time to full blown infection
* Knowledge of process
* Willingness to self-terminate
* Willingness to hunt full blown Zombies

Z Compartment
* Movment rate of a Zombie
* Intelligence of the Zombie
* "Hunger" rate of the Zombie
* Infection from contact/bite
* Shelf-life.  (How long can a Zombie last without food? Is a Zombie Omnivourous? Does a Zombie die of old age? Can a Zombie die from other things? Does a Zombie "fall apart" from wear and tear?)

D Compartment
Apart from being really really dead, are these agents:

* Still infectious?
* A short or long term food source for the Zombies
* Going to drive the behaviour of either the S, B, Z or H compartments

H Compartment
* Experience
* Resources
* Movment
* Intelligence
* Cooperation
* Reproduction ( can a H turn an S into a H?)

Now this should make for an interesting model. 

I think the general reason that Zombies are so fearsome in all the scenarios is that they are implicitly or explicity given immortality and a reasonable amount of intelligence but not enough to be able to survive.  It would be pointless to debate these issues in the context of arguing about an infection model because the assumption is that Zombies have some sort of human seeking drive that makes living people their only food source.  In the event that this was true, the logical extensions of become fairly idiotic.  Are Zombies cold blooded?  If not, how do they maintain their energy needs?  Anyway... at some point they will run out of energy and then what? If they can eat something else... then why bother with humans who can be a bit hard to catch? As I said, unless you accept some of these assumptions, the whole game falls apart.


* Zombie are essentially "Immortal" and do not "die" from anything like disease, decomposition or wear-and-tear.
* Zombies are fixated on fresh humans ( cause the script writer told them too...)
* Zombies are fairly dumb, cannot communicate directly and engage in only basic food aquisition and movment. Essentially they act "alone".
* There is no cure... or the infection process is so destructive there is "nothing left tosave"... so for practical purposes its a one way trip.

The assumption that Zombies are Immortal is problematic because it creates a cascade effect of an increasingly large population of Zombies.  In the event that this is "valid" within the scenario, then it does present a challenge for the uninfected.  In the event that there is some "decay" model for the Zombies then there would need to be some effect of time on the population of the Zombies.  In this case the obvious strategy for the uninfected is to quarantine themselves or the infected and wait it out.  The fact that most of the popular fiction around Zombies tends to work in a timeframe where any decay model is not a significant factor means that this is not well explored. (28 weeks Later looked at this "wait it out" model and explored the concept of latent infection "flaring up" again.)

The assumption that Zombies are fixated on fresh meat presents a few problems.  While it would probably be fairly easy to access in the early days of an "outbreak" due to the suprise factor. I think that as soon as the knowledge about the outbreak was "released" the availible population of food would disperse and be more cautious.  This would reduce the growth rate of the Zombie population simply because the availible population of uninfected were aware and attempting to avoid the Zombies.  The other problem is that if a mob of Zombies catch up with a food source, they will probably consume it rather than simply infecting it.  So the rate at which contact with Zombies leads to infection rather than death would probably be related to the densitie of Zombies in the immediate area. The other question is how long does it take for a corpse to become unattactive to Zombies as a food source?  Would there be a period where a freshly infected Zombie was still "tasty" enough that other Zombies would try to "snack" on each other?  In this case, would they try to defend themselves?  I which case can you get Zombies to fight each other?  Will they fight to the death?  Are Zombies smart enough to distinguish between a Zombie that is covered in fresh blood from feeding vs one that is still fresh enough to snack on?  This does suggest some fairly advanced reasoning and sensing ability.    

The assumption that Zombies act alone is interesting. In most popular culture Zombie outbreaks Zombie populations display emergent flocking behaviour which makes the danger from a single Zombie much more dangerous.  For this behaviour to function, Zombies must be able to perceive each other and attribute meaning to others behaviour.  This could function simply by Zombies "walking" purposfully when they are on the trail of food, and walking differently when they are "wandering".  So any Zombie in "wander" mode who sees another Zombie walking in "pursuit" mode will perceive that they are heading toward food and join in.  The key being that the zombie joining the "pursuit" must then change their behaviour to "pursuit" mode as well.

This "model" creates an indirect system of emergent communication within the population of Zombies.   The result is that Zombies follow other Zombies and engage in "mobbing" behaviour of "food sources".   The question is why a Zombie, who can figure out the difference between another Zombie randomly walking and the same Zombie walking toward food does not use their perceptions for other cognitive functions... just imagine Zombies with Neural Plasticity....

Anyway, the obvious problem with a flocking algorithm that is based on its members following each other is that potentially it can create a "whirlpool" of Zombies following Zombies following the first Zombies.  This whirlpool could potentially grow to suck in all the Zombies in the world.  Assuming that none of the un-infected are stupid enough to disturb the whirlpool, this would keep all the Zombies occupied until they "timed out". 

As I mentioned above, if you follow some of these assumptions they become silly. 


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