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# The Equation That Can Help Predict Zombie Migration Patterns

The scenario: News reports say that there's been a zombie outbreak at the city hospital. The undead are pouring into the streets, biting hapless citizens. How much time do you have to gather supplies, get out of town and set up a fortified site, before the horde catches up with you? As always, there's a way to calculate this, and it comes from physics.

There have been quite a few mathematical models of zombie infection in recent years — partially for fun, but also to test out methodologies that could be modified and applied to actual, non-zombie outbreaks.

But, Thomas Woolley, at Oxford University's Mathematical Institute, believes these earlier studies share a common flaw: they looked at time- and population-dependent interactions between humans and the undead, without considering geography. "Zombies do not move homogeneously," he explains in a recently published paper, in which he is the lead author. "This allows zombies to shuffle around, giving a much more realistic picture of an invasion."

Woolley and his colleagues also take issue with models that assume zombies and humans are well mixed, meaning that zombies can be found everywhere there are humans:

Realistically, the initial horde of zombies will be localized to areas containing dead humans, such as cemeteries and hospitals. In addition, because humans and zombies are not initially separated, humans are not able to run and hide in order to try and preserve themselves. It is a well-documented fact that zombies are deadly but slow-moving. Due to their slow movements, it is quite possible that, given sufficient warning, we would be able to outrun the zombies and produce a defensible blockade where humans could live safely. In order to do so, it would be useful to know just how long an infestation of zombies would take to reach our defenses; this would give us an estimate of how long we would have to scavenge for supplies and weaponry in order that we may protect ourselves from these oncoming, undead predators.

The movements of the undead are commonly described as small irregular steps. That immediately reminded the authors of a principle in physics known as diffusion — or more specifically, the "random walk."

Diffusion describes the spread of particles through random motion from regions of higher concentration to regions of lower concentration. Although the concept originated in physics, it has applications in multiple fields of study, including biology and chemistry.

In molecular diffusion, the moving entities are small molecules that are self-propelled by thermal energy. They move at random because they frequently collide, and as they do so, they become less concentrated in a single area.

So, for example, we have this simple sketch (above). An initial group of zombies is placed in the top left corner, close together, their initial directions denoted by the small black arrows. After a time, their random motion will cause them to spread themselves out over the domain.

Applying diffusion equations to zombies allowed the researchers to estimate the density of zombies at various points in place and time. This figure (below) shows the time in minutes until the first zombies arrive at your location, for various rates of diffusion and distances. If, for example, the initial zombie outbreak is 90 meters away and they have a diffusion rate of 100m2/min, then they'll catch up with you in around 26 minutes.

Mathematically, this also demonstrates why we should flee from zombies, instead of doing something foolhardy like standing our ground and fighting back the undead:

If we were to double the distance between ourselves and the zombies, then the time for the zombies to reach us would approximately quadruple. However, if we were to slow the zombies down by half, then the time taken would only double. Since we want to delay interaction with the zombies for as long as possible… it is much better to expend energy traveling away from the zombies than it is to try and slow them down. Without some form of projectile weaponry or chainsaw, killing zombies is particularly difficult, as they do not stop until their brains are destroyed.