stochastic processes exam question: find the size of a zombie population during an outbreak

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.

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