How the zombies plan to take over the world

It's good to be a zombie. You get bitten, you die (OK, that’s not so great) and an hour later you are up and shuffling round with your arms in the air, ready to spread the virus yourself.

Of course, shuffling round and moaning means that you are unlikely to surprise or outrun your potential victims, so it’s hard to see how they are such a staple of horror movies. Let’s look at the mathematics of zombie infection, assuming that each zombie only gets to catch an average of one person per hour.

Although you have no choice about dying and coming back as a zombie if you are bitten, the infection rate is only one person per hour per zombie - not exactly a path to world domination.

However, if they started in New York, where there is a large pool of potential converts within easy reach, they could build up their numbers before lurching onwards into Connecticut and New Jersey.

The doomsday clock starts when our zombie bites his first victim. One hour later, we have two zombies each biting another person. An hour after that, we have four gnawing zombies.

So how long would it take to infect the 8.2 million citizens in the five boroughs of New York? Just 24 hours.

Zombie mathematics, and what it has to do with memes

Let’s introduce the concept of a 'success factor' for zombies, which is defined as the number of zombies in one generation divided by the number in the preceding generation.

We start off with one zombie, and he bites someone.

After an hour’s delay, allowing for death and resurrection, we have two zombies. They each bite one other person and an hour later we have four zombies, and so on. No-one ever stops being a zombie, so the success factor for these zombies is therefore two.

After 24 hours (23 generations, allowing for death and resurrection) we have 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 zombies, more concisely written as 223, namely 2 raised to the power of 23.

This number is, roughly speaking, 8.4 million, around the same as the population of New York.

We are ignoring saturation effects here - small numbers of survivors could hole up together and build barricades, thus slowing the spread - but the odds are not looking good for New Yorkers, are they?

What if we had started with two zombies instead of one, that would shorten things by one generation (i.e. one hour), but no more.

But if the zombies bit two people per hour on average, the success factor rises to three, and (again, ignoring saturation effects) we wipe out New York in 16 hours. If they bite four people an hour, we are down to 13 hours.

So the number of copies made per generation of zombies has a greater effect on the rate of infection than the numbers you start with (an idea we will come back to again in the next section). The modern generation of cinematic zombies can run, and so they would be better at catching victims and would therefore reproduce faster.

Let’s assume they each only bite one person at a time, but manage to do so every 30 minutes instead of every 60. After one hour (resurrection time again) and 30 minutes we have two zombies, after two hours we have four, after two-and-a-half hours we have eight, after three hours we have 16... New York is again wiped out after 23 generations, but now that corresponds to 12½ hours. If they manage to shorten the average attack time to 15 minutes, then New York is toast in just over six hours.

So the number of generations per unit time has a greater effect on the population than the number of copies per generation, which in turn is more important than the number of zombies you start with.

This is, as I sure you will have recognised, nothing more than old-fashioned exponential growth. If the success factor is equal to one, then the population is constant, and if it is less than one, then we see a decline in the infected population.

And the relevance to memetics is?

Memes follow exactly the same laws of arithmetic as our zombies when they start to spread.

They have a success factor too (which we define in A x B x C x D = Belief) and when we talk about a meme ‘going viral’ this is a shorthand for exponential growth in population.

We would expect this kind of behaviour to show up in all kinds of places, such as the spread of popular fads (as we shall see elsewhere) and the take-up of products during an advertising campaign, which is dealt with in greater detail in advertising and branding.