Here is a question from my stochastic processes exam that I gave this morning:

Question: There is a zombie outbreak in Richmond. The zombie population can be modeled as a linear growth birth death process.  Each zombie independently reproduces at a rate of λ = 2/hour and is killed by resourceful Virginians at a rate of μ = 0.5/hour.  If the population started with a pack of two zombies, find the average size of the zombie population after 24 hours.

Answer: The average size of the population can be modeled using a linear growth birth death process. Let Ei denote the expected size of the zombie population after 24 hours given that there are initially i zombies. Then Ei = i * E1.

The expected size of the zombie population is given by

Ei = ie^((λ-μ)t) = 2*e^36 after t=24 hours. That is a lot of zombies!

Comments: Is this model appropriate? Well, let’s take a look at the assumptions. They key assumption here is that there are an unlimited number of humans that can serve as zombie fodder. Clearly there are not  e^26 humans on the planet. More generally, as the zombie population grows, they will eventually run out of food (humans!), so their growth will slow down. That is, the linear growth assumption doesn’t make sense. It might work for the first hour or two, however. Likewise, the linear death assumption is not realistic, because after the zombie population explodes, there are no more resourceful Virginians left to kill zombies. Even when there are Virginians around, the rate that they can kill the zombies is probably not proportional to the size of the zombie population. But the students didn’t need to assess model realism for full credit.

Are you prepared for a zombie apocalypse?  The CDC is apparently very concerned about this: they recently issued some advice for a zombie outbreak.  Some of their advice includes identifying optimal evacuation routes and quarantine plans. It sounds like operations research could play a critical role in surviving a zombie apocalypse and propagating the human species. I’m game. Are you?

Unlike the vampire threat, I take zombie threats very, very seriously. Zombie outbreaks in movies and films consistently exhibit exponential growth, which one would expect if they modeled a zombie outbreak using mathematical tools such as birth-death models or differential equations.  This consistency with mathematical modeling obviously means that zombie apocalypse is a real threat, doesn’t it?

A similarly concerned student gave me a book chapter (see the reference below) that appeared in an academic book about infectious diseases that is helpful for preparing for a zombie attack.  They set up a series of differential equations to model a zombie outbreak and then determine optimal strategies for responding.  Their model consists of three sub-populations:

• Susceptible (S): humans,
• Zombies (Z): this is self-explanatory, and
• Removed (R): dead humans.

Humans can die (i.e., they are removed) through natural causes (with rate d).  The removed humans can become zombies with rate z, and suscpetibles can become zombies through a zombie encounter with rate b.  Zombies die with rate a. Since zombies only crave human flesh, other species do not need to be considered.  The birth rate is a constant P.  Therefore, the differential equations modeling these interactions are

S’ = P – bSZ – dS

Z’ = bSZ + zR – aSZ

R’ = dS + aSZ – zR

The key difference between this model and other infectious disease models is that the dead can be resurrected.  The authors identify the conditions under which the human species is wiped out(!)  However, they show that quick, aggressive attacks can stave off a zombie apocalypse.

* P. Munz, I. Hudea, J. Imad, and R.J. Smith, 2009. “When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection.” In Infectious disease modelling research progress, Nova Science Publishers, Inc., p. 133 – 150.

Strangely enough, this book chapter was supported by several grants, including a NSERV Discovery grant, an Ontario Early Researcher Award, and funding from MITACS.  With its low population density, you’d think Canada would be less susceptible to a zombie attack.

How have you prepared for a zombie attack?

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