Dr. McLay on Punk Rock Operations Research posted the following exam question she gave to a stochastic processes class:
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!
Math is cool. There are some assumptions that go into this solution that are addressed in her post.