We come from the future
We come from the future

# 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.

Of course, this is a short-term survival strategy. Eventually, the zombies will reach your fortified location and increase in numbers as they expand their population by chomping on stragglers. At that point, your only option is to hope your supplies last long enough, before the military or Brad Pitt launch a successful counterattack.

In the event that the zombies penetrate your fortress, the mathematicians offer an important piece of advice. You need to replicate the conditions of the initial outbreak by finding ways to slow them down while you put as much distance between you and them as possible. "Thus an effective fortification should have plenty of obstructions that a human could navigate but a decaying zombie, who may not be so athletic, would find challenging."

### DISCUSSION

By
Ken

a few things that always bothered me, even after getting past the dead bodies don't survive very long bit: 1)If zombies eat brains, and the only way to kill a zombie is to destroy it's brain... then doesn't that mean victims of zombies just can't become zombies due to their brain being destroyed when they are killed.

2) why do zombies need to eat brains, or people at all if they can't actually die from starvation? Seriously if you are an immortal undead, then you don't actually need to eat to live/unlive.

3) couldn't zombies feed off of each other. as the only way to kill a zombie is to destroy the brain and they eat brains, doesn't that mean that every zombie is a walking meal to every other zombie.

4) if zombie blood or bodily fluids infect other creatures and make them zombies, shouldn't we worry about undead mosquitoes and bed bugs, and fleas. The try to feed on a movie zombie target, end up infected, and then swarm living beings burrowing through eyes ears mouth and nose to try and get brains. You can't bug spray them to death, and simply landing on you and biting infects you, while trying to squish them simply exposes you to the risk of them getting a bit in.

there was a nother one I had, but it escapes me right now.