S04E16: The Cohabitation Formulation

There are times when television can be a public service.  On tonight’s whiteboards, the heroes of The Big Bang Theory have derived what to do in case our civilization  is attacked by zombies.

A few weeks ago, during the taping of  the differential equations episode, one of the Warner Brothers executives asked me, “What good are differential equations?”

So my theoretical physicist guest and I  eagerly began a discourse on all the wonderful mathematics of differential equations, but we could tell he was somehow unconvinced.  But then we told him about a recent well-known academic paper:  WHEN ZOMBIES ATTACK!:  MATHEMATICAL MODELLING OF AN OUTBREAK IN ZOMBIE INFECTION and he was on board.

Canadian mathematicians published a mathematical tretise on zombie population growth in the journal Infectious Disease Modelling Research Progress

Differential equations  describe how quickly things change, and how the rates of change affect other processes.  If you want to be really fancy, even the rates of rates of change can be described as well.  If you say, “The world’s population grows by 2% every year”, you have just stated a differential equation.  The solution to that one, unfortunately,  is the same function that describes compound interest.

In the case of zombies, the paper approaches the problem with all the necessary variables:  “zombies” (Z);  humans who are not yet zombies  (H); and zombies who have been neutralized — by some mechanism detailed in “Night of the Living Dead” — whom they called “removed” (R).  (Humans that die of natural causes, are also considered “removed”. How lucky is that during a zombie apocalypse?).

The role of differential equations comes into play when you consider the critical outcome: the rate of zombie increase or decrease.   For example, in the authors’ model, the number of zombies can increase by resurrection of “removed” humans or zombies.  It is just a rate proportional to the population of removed humans and zombies, so  call it +constant*R where R is the number of dead bodies around.  The “+” sign denotes an increase and the constant depends on the movie.   Because this term affects a rate, we are constructing a differential equation.

Lucky for us, the number of zombies can also decrease by removing their heads or other unpleasantness.  The probability that this happens depends now on two populations though, not one.   For example, if there were zero humans (H)  to neutralize the zombies, this  rate would be zero.  Likewise if there were no zombies (Z) to neutralize, the rate would be zero too.  The answer, the  product of H and Z, behaves just this way.  The authors describe this mathematically as –constant*H*Z, where the “-” sign corresponds to a decrease in zombies.

Unfortunately a zombie meeting a susceptible human doesn’t always work out so well.  A zombie might win and turn the human  into a zombie.  So the authors add another term like the one above but now with a plus sign: +constant*H*Z.  The “+” sign indicates an increase in zombies.

In "Zombie College" (a series of 12 animated shorts written by Big Bang Theory writer, Eric Kaplan) the constant c is explored.

(Watch Zombie College)

So you are ready to put these three parts together to form  the basic equation.   The net result of the three processes above is a rate of change of Zombies, Z’ which is just a sum of the last three paragraphs.   Calling the constants a, b and c, we have the rate of change of zombies described by

Z’= +a*R  – b*H*Z + c*H*Z.

(The variables on the board follow the paper: using S, for “susceptibles” instead of H for “human”.  But I thought H was clearer and it’s my blog.)

The authors add a few other details, such as birth rate, to find the formulas for the rate of change of humans, H’, and removed people R’.   The three equations’ variables depend on each other so these become  especially nasty kinds of differential equations, called coupled differential equations.  Undaunted, the mathematicians sharpen their pencils and follow the time honored tools for solving such coupled differential equations.  The net result is…

Everyone becomes a zombie.

(Technically, if humans  try to neutralize all the zombies everywhere at the same time, we have a fighting chance, but the authors consider the necessary cooperation unlikely.)

Full disclosure:  There is one thing I don’t understand about their model.  The authors allow zombies destroyed by humans to be counted in R.   I am no zombie movie expert, but I’d figure once you remove the head of a zombie, or even the head of  a non un-dead dead human, they cannot turn into a zombie.   (Please, someone with zombie cred pipe up in the comments.)   It would be easy enough to change the model to account for this though.  That is the beauty of differential equations.    Maybe their grim conclusion would be averted.

Despite the many scenarios envisioned, the outlook is always bleak, as summarized in the authors’ conclusions:

…an outbreak of zombies will result in the collapse of civilisation, with every human infected, or dead. This is because human births and deaths will provide the undead with a limitless supply of new bodies to infect, resurrect and convert. Thus, if zombies arrive, we must act quickly and decisively to eradicate them before they eradicate us.

While the example may seem frivolous, the mathematics of the differential equations are real.  Conclusions drawn from this study have impact on other infectious diseases with latent manifestation, presumably helping us address infections such as HIV.   For more about how mathematics can help you survive the zombie apocalypse,  or battle persistent disease in underdeveloped countries, see  my friend Jennifer Ouellette’s handy new book.

Want to solve some differential equations yourself?   I’m still waiting for someone to program the iPhone differential equation handwriting recognition and equation solver app.  You’ve had a few weeks now people.