They’re coming to get you, Barbara!

-George A. Romero, Night of the Living Dead, 1968

Ah, you are back. You’re braver than you look, that’s good. You’ll need all the nerves you can muster here. You see, we’ve created a little problem: Our mathematical, metaphysical musings have created quite a stir. The deceased our crossing Heaven off of their plans for the hereafter, and Hell is full to bursting, and when there is no more room in Hell, well, you know what happens:

Yes pilgrim, I’m sorry to say it, but we are on the brink here. Today is what Max Brooks calls Z-Day, the beginning of the Zombie Apocalypse! That cold chill you are feeling is the natural reaction to this news. These cannibalistic cadavers are gruesome guests indeed! And look- there comes one now! What a sinister, shambling, spook! Watch as the lurching living dead approaches – it is really quite pathetic with its ponderous pace. Does it really pose such a grave threat? Well let’s find out.

We all know that the most dangerous part of a zombie is its maggot-ridden maw. A love bite from that and you will soon be shambling yourself. Yes, the bite is the zombies’ main form of propagation.  Inevitably, someone bitten by the undead becomes undead themselves. This pattern will serve as the basis of our model.

For purposes of simplicity, let us assume that the current world population of about 6.8 billion is evenly distributed across one massive, flat continent, and that each person is within one hour’s walking distance of at least one other person. In this world, a zombie can reach at least one victim an hour. Under these conditions, we will try to define a function that predicts the population of zombies at any time after the progenitor zombie has been encountered.

We will define three stages of the zombie disease: 1.) Exposure: This is when a person is bitten. 2.) Metastasis: The time during which a person is infected, but not yet a zombie. For our purposes we will assume this to be one hour. 3.) Zombie: Without exception, exposure leads to a person becoming a zombie. So zombiism progresses like this: Exposure – Metastasis – Zombie.

Now, consider the first five hours of the outbreak:

Hour One: There is 1 zombie (the progenitor zombie) and it bites one person (1 person is exposed).

Hour Two: There is 1 zombie, 1 metastatic person (the previous exposed), and 1 exposed (the progenitor has bitten another person).

Hour Three: There are 2 zombies (the progenitor zombie and the previously metastatic person), 1 metastatic person, and 2 exposed (both zombies bite people).

Hour Four: 3 Zombies, 2 metastatics, 3 exposed.

Hour Five: 5 Zombies, 3 metastatics, 5 exposed.

Hour Six: 8 Zombies, 5 metastatics, 8 exposed.

Look at the progression of the zombie population: 1,1,2,3,5,8. Do you notice a pattern? Each new term is the sum of the proceeding two: 1+1=2, 1+2=3, 2+3=5, 3+5=8… A sequence that is determined by its preceding terms is said to have a recurrence relationship, and we can use this to help us define our Zombie Function. Let the function Z(t) indicate the zombie population t hours after the appearance of the progenitor zombie, using the recurrence relationship we just observed, we define Z(t) like this: Z(t)=Z(t-1)+Z(t-2), given Z(1)=1, Z(2)=1.

Now that we know the Zombie Function, how does that help us? Well, we can use it to see if the human race has any chance of survival. That is, when and will the entire human population become zombies? To help us answer that question, here the Zombie Function values for the first fifty hours:

…So short answer: Yes, and soon. In less than 49 hours, that is, in just over two days, only the dead walk the Earth. Today being the 29th, that will make for a gruesomely, ghoulishly appropriate Halloween.

I can understand those of you who may no longer wish to read this, and would prefer to spend your remaining time huddled futilely with your friends and families, but for those that choose to stay, there is more to consider. You have probably realized that the numbers produced by the Zombie Function are indeed the Fibonacci numbers. This famous sequence (1,1,2,3,5,8,13…) naturally occurs in all manner of phenomenon from biology to finance for reasons similar to the reasons it describes zombie outbreaks. It was first studied by Indian mathematicians, most notably, Pingala, Virahanka, Gopāla, and Hemachandra. Drawing upon their work, Leonardo of Pisa, better known as Fibonacci, discussed the sequence in his 1202 book, Liber Abaci. We owe a great deal to this book, for it was this book that popularized the use of the efficient numeral system we use today (1,2,3,4,5,6,…) and our method for writing fractions, both of which Fibonacci borrowed from the Indian mathematicians. In Liber Abaci he discusses the Fibonacci sequence in an application dealing with rabbit populations. For as you know, rabbits breed like zombies, and it makes you wonder which we should truly fear more.

There was a flash of lightning and in its glare I noticed that Mr. Monroe was carrying a little bundle- a bundle with tiny glistening eyes.

-Bunnicula, James and Deborah Howe

Tune in tomorrow night, as the monster math continues

This is the third in a series of Halloween posts. If you enjoyed it, you may like the others: Vampire Numbers, Infernal Integers, The Sum of All Fears


About Webster

Transitionally, I’m a math student and tutor en route to becoming a math professor. Permanently, I’m a mathematics enthusiast. I study mathematics professionally, and as a leisure activity. At the time of writing this, I’m a generalist. I have let to reach the depth of understanding that requires specialization. Though I eagerly await that time, I do enjoy the ‘now’ and find there is bountiful food for thought at any level.
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7 Responses to Zombi(nacci)

  1. Pingback: Vampire Numbers | Spherical Cow

  2. Annabelle says:

    hahahahahha this looks like my differential equations homework, literally. we studied the rate at which diseases spread, trends spread, etc, populations grow and die under conditions, etc.

    a zombie differential equation D:

    • Webster says:

      Z'(t)= (φ^(-t) (φi^(2t) log(φ)+cos(╥) log(φ)+pi sin(╥t)))/sqrt(5), Where φ=1/2 (1+sqrt(5)), the golden ratio.

      No, really. I’ll probably discuss that later. On a lighter note, I bet my population was cooler than yours 😉

  3. Melody says:

    This is to post a comment to you Webster that now Annabelle is trying to teach me how to post and read other peoples comments

  4. Pingback: The Sum of All Fears | Spherical Cow

  5. Pingback: Fibonacci Findings | Spherical Cow

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