Surviving the Zombie Apocalypse

This is part 2 in the maths behind zombies series. See part 1 here

We have previously looked at how the paper from mathematicians from Ottawa University discuss the mathematics behind surviving the zombie apocalypse – and how the mathematics used has many other modelling applications – for understanding the spread of disease and the diffusion of gases. In the previous post we saw how the zombie diffusion rate could be predicted by the formula:

In this equation Z(x,t) stands for the density of zombies at point x and time t. Z0 stands for the initial zombie density – where all zombies are starting at the same point (x between 0 and 1). L stands for the edge of the domain.  This is a 1 dimensional model – where zombies only travel in a straight line.  For modelling purposes, this would be  somewhat equivalent to being trapped in a 50 metre by 1 metre square fenced area – with (0,0) as the bottom left corner of the fence. L would be 50 in this case, and all zombies would initially be in the 1 metre square which went through the origin.

We saw that as the time, t gets large this equation can be approximated by:

Which means that after a long length of time our 50 metre square fenced area will have an equal density of zombies throughout. If we started with 100 zombies in our initial 1 metre square area (say emerging from a tomb), then Z0 = 100 and with L = 50 we would have an average density of 100/2 = 2 zombies per metre squared.

When will the zombies arrive?

So, say you have taken the previous post’s advice and run as far away as possible.  So, you’re at the edge of the 50 metre long fence.  The next question to ask therefore, how long before the zombies reach you? To answer this we need to solve the initial equation Z(x,t) to find t when x = 50 and Z(50,t) = 1.  We solve to find Z(50,t) = 1 because this represents the time t when there is a density of 1 zombie at distance 50 metres from the origin.  In other words when a zombie is standing where you are now!  Solving this would be pretty tough, so we do what mathematicians like to do, and take an approximation.  This approximate solution for t is given by:

where L is the distance we’re standing away (50 metres in this case) and D is the diffusion rate.  D can be altered to affect the speed of the zombies.  In the study they set D as 100 – which is claimed to be consistent with a slow, shuffling zombie walk.  Therefore the time the zombies will take to arrive is approximately t = 0.32(50)2/100 = 8 minutes. If we are a slightly further distance away (say we are trapped along a 100 metre fence) then the zombies will arrive in approximately t = 0.32(100)2/100 = 32 minutes.

Fight or flight?

Fighting (say by lobbing missiles at the oncoming hordes) would slow the diffusion rate D, but would probably be less effective than running – as the time is rapidly increased by the L2 factor.  Let’s look at a scenario to compare:

You are 20 metres from the zombies.  You can decide to spend 1 minute running an extra 30 metres away (you’re not in good shape) to the edge of the fence (no rocks here) or can spend your time lobbing rocks with your home-made catapult to slow the advance.  Which scenario makes more sense?

Scenario 1

You get to the edge of the fence in 1 minute.  The zombies will get to the edge of the fence in t = 0.32(50)2/100 = 8 minutes.  You therefore have an additional 7 minutes to sit down, relax, and enjoy your last few moments before the zombies arrive.

Scenario 2

You successfully manage to slow the diffusion rate to D = 50 as the zombies are slowed by your sharp-shooting.  The zombies will arrive in 0.32(20)2/50 = 2.6 minutes.  If only you’d paid more attention in maths class.

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