# The zombie apocalypse

Zombies are a staple of modern popular culture, and appear in a variety of forms, including the traditional slow-moving, unintelligent zombie hordes and less common fast-moving - and perhaps intelligent - zombies. The common theme in media featuring zombies is the the zombie infection, which may affect either the living or the deceased, or both. It is generally agreed that this infection is passed from zombie to non-zombie by means of a scratch or bite, and infection always leads to a transformation into a zombie. (In some more modern interpretations, this transformation may be reversible.)

The spread of the zombie infection is an interesting problem to model mathematically. There are many factors to consider: the chances of a non-zombie becoming infected in an interaction with a zombie; the rate at which interactions between non-zombies and zombies occur; the spread the zombie horde from place to place; and the “critical mass” of the zombie horde at which point there is insufficient food for all zombies. We can model parts of this problem in isolation, with some simplifying assumptions.

For example, we might consider the following scenario. Suppose that the world is divided into nine “square” regions, numbered one to nine in the usual “keypad” arrangement, and that the zombie infection first arises in region five (in the centre of the arrangement). Then the zombie population across all nine regions can be modelled using a diffusion model.

We might also examine spread of the zombie infection from a probabilistic point of view, where we instead model the spread by assigning a probability that a non-zombie will become infected during an encounter with a zombie. At each point in time there will be interactions between existing zombies and non-infected people, and at each of these interactions, there is a chance that the non-infected person will become infected. This can be pictured as a tree, where each branch represents a different zombie, and each branch splits in two if the zombie infects a person at a given time. Mathematically, this is a branching process.

A zombie apocalypse is just one of many scenarios that can be modeled using a branching process, even if it is (most likely) fictitious. Indeed, epidemic modeling using branching processes, and more general stochastic processes, is an active field of research and provides a useful tool for predicting the evolution of an infection.

I have not studied probability formally since I was an undergraduate, and since then I have acquired a much more powerful arsenal of mathematical tools, and many concepts that once seemed impenetrable are now much more clear. I decided that it was time for me to refresh my knowledge of probability theory with my new-found knowledge and endeavor to understand some of the nuances of the theory.

This refresher is motivated by the surprising appearance of probabilistic aspects in operator theory. This gives me an excellent excuse for spending time searching for mathematical papers on the zombie apocalypse, and “researching” their role in popular culture, although I feel the latter would have happened either way.