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.

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.

(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.

February 17, 2011 at 8:26 pm |

““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.”

I need to point out that this particular differential equation didn’t hold true. World Population Rate of Change actually decreases .2%/year, and will peak about 2050 before suddenly reversing as all the baby boomers die off.

February 17, 2011 at 10:15 pm |

That’s what people hope. I’ve seen this statement in a UN document about population as well. However, this projection is an extrapolation. It is a simple linear fit to the argument of the exponential. A linear extrapolation is a thin straw to grasp.

Meanwhile our whole economic system is based on an assumption of exponential growth. I’m no economist, but it appears to me that without exponential growth we either fail or need to change our system in ways yet to be discovered. Or with exponential growth. we eventually fail even worse.

Of course such leveling off of the exponential must occur at some point. That is true or we will have to stand on each others’ heads. Before that happens, plagues and wars should reduce the growth rate. The question I think you are raising is:Will our future see a smooth leveling that somehow we learn to adjust to, or will it be a sudden miserable collapse. You can choose assumptions that lead to a differential equation that describes either case.

February 22, 2011 at 11:38 pm

Economics does not require an exponential population growth. A targeted 2% inflation rate is to ease downward pressure on sticky prices. It’s easier to raise the inflation rate than it is to cut salaries. The current crises is what a world without targeted inflation would look like all the time. Arguments that population inversion was the cause of Japan’s “Lost Decade” are not taken seriously from as broad a range as Milton Friedman to Paul Krugman. And the current Austrians to the extreme reject pretty much all of economics as a science. They have become economics’ “Intelligent Design.”

February 18, 2011 at 8:58 am |

Nature has a habit of restoring ecological imbalances. The Black Plague of Europe being one example. Nowadays, with our advances in medical care may limit what Nature can throw at us. However, the recent events in the Middle East should reminds that come what may, War, one of the Four Horsemen, is always in play.

February 19, 2011 at 12:04 am |

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February 19, 2011 at 1:19 am |

Don’t worry! Innovation is increasing faster than population. :)

People think that the economy is in trouble. It’s actually never been better: automation has made productivity almost astronomical. With the push of a button, you can churn out endless quantities of goods. There is no manual task a machine cannot surpass, or at least equal, a person doing the same work. E-book readers will save entire forests, but cut swathes in forestry jobs. (as a Canadian, I’m feeling mixed about that!) The same thing is happening for white collar work. Where you once needed 10 accountants, 1 accountant with Excel will do.

Even programming is subject to this: Java is a program that simplifies the task of telling a computer what to do. The “offshore to India” is only part of the story.

The next step is “vague” orders, and is very difficult to do. But the Watson cpu on the last Jeopardy show did exactly that. Computers programming computers is the next step. (and according to futurists also the final step) What Kurzwell calls a singularity, I would call a spark. And who knows what new Universe we’ll see then?

Anyway, as long as your family name is Walmart, Gates, or any other “factory” owner, things are quite good at the moment… however, people talking of “balancing budgets” while you sit on an ever increasing mountain of gold is a somewhat uneasy feeling, I’m sure.

Simply stated, we NEED people (or zombies) less and less to do things for us. The question now becomes: do we WANT people around? Well… personally, I always enjoy talking and meeting people (it’s quite exhausting however!)

A sticky point is petroleum. The ever increasing turmoil in the middle east is a red flag of this. How can such supposedly wealthy countries be so troubled? How can dictator/hierarchies with 40 years of stability simply crumble? Because they’ve literally run out of gas and can no longer buy peace. The rate of oil extraction has hit a vertex, and is now decreasing. Egyptian oil workers are just the 1st ones let go. But… where was the bloodbath? The supply wars? The zombies?

Did… we just see an Egyptian revolution who’s theme was basically innovation and solving problems!? WE DID!

We need alternatives to oil, sooner rather than later… and guess what’s the fuel of innovation? Need? Do I need a cell phone app to solve quations for me? No. Do I need to read big physics book? No. Do I need a diploma for a job? Well, a robot will be doing it, so that’s moot! But what then? Stay in bed and live on welfare? That’s a minimum!

So… what’s the maximum? What do I WANT to do if I don’t need anything? What does any Sheldon want to do!? Go out into the world! See some physics! Talk about it! Get tired, go home and think about it! (and do laundry on saturday…)

Now we just need 6.7 billion people to do the same thing! Think of the spark that would make! Or did I just get carried away?

February 21, 2011 at 10:41 am |

I´m a bit disappointed Dr. S. Zombies were nowhere to be seen, or heard, on this episode. When I finally saw it, it was a bit of an anticlimax (nice episode, though)

Back to the paper. I too was puzzlez as to why count the removed-as-hell zombies. You shoot them in the head, they´re not coming in the next movie, right? Also, the implicit hypothesis that zombies are inmortal is a bit odd. Just manage to survive for a while, and they should die of exhaustion or hunger, á la 28 Days Later; or they will stop being that exquisite for dinner, eat each other, and wipe themselves out.

Hope somebody adds those new scenarios to the original paper, and turn into Android. Now THAT is an app I´d love to have!

April 8, 2011 at 2:53 pm |

So Warner Brothers call physicians. Of course! I had my suspicions… I’ll include your post at the one I’m doing about the series.

Thank you for sharing.