Physicists often use 'model systems', which are systems that share properties with more complex systems. We try to gain fundamental insight into the more complicated system by studying the model system. For example, an alloy with seven elements in it might share some subset of properties with an alloy made from only two elements. You might be able to learn something about the seven-element alloy from studying the two-element alloy because there are simply fewer variables. The information you learn from your model system can give you a stepping stool for trying to understand the more complex system.
Biologists also have their own types of model systems, called model organisms. Model organisms serve the same purpose - they are easier to deal with and share some of the features of more complex organisms that we don't yet understand. There is a field of mathematics called biometry, biostatistics or biometrics that combines math and biology to study everything from epidemiology to ecological forecasting. When you asked your math teacher "what am I ever going to use math for", this is the field that provides some really interesting answers. Biometrists usually focus on real systems, but a recent paper in Infection Disease Modelling Research Progress 2009 features a very underutilized model system.
In particular, the "classic zombie" as typified by Night of the Living Dead, as opposed to the more modern zombie. Modern zombies are a little smarter and move a little faster than classic zombies and thus are harder to model. As an experimentalist, I question the choice of zombies for a model system, as zombies are notoriously ill-behaved. They physically decay on very short time scales (often losing limbs without notice), they are mindless, make annoying noises and have as their sole purposes eating people or turning them into zombies. (Now that I think about it, I'm questioning whether a couple of my colleagues might fall in that category...)
This is more than just a cheap shot at getting your research on NPR. Maybe we don't have a zombie problem in real life, but we do have AIDS, river blindness, West Nile, malaria and a host of other diseases that spread as individuals make contact with either infected individuals or carriers (such as mosquitoes).
The paper was written by three students (Philip Munz, Ioan Hudea and Joe Imad) and a professor from the Department of Mathematics and Faculty of Medicine at the University of Ottawa Robert J. Smith?. No, the question mark is not a typo. It's actually part of the professor's name. Being a statistics type, he knows how many "Robert Smith"s there are in the world and thought he might distinguish himself through creative punctuation. He notes
"Not that the question mark actually solves that, but at least it differentiates me from that guy from The Cure. It's been twenty years now and sadly his career shows no sign of drying up."
One of the things I like about this paper is that is demonstrates how you start with a relatively simple model and gradually add elements to make the model more complex - and more capable of modeling reality. It also reminded me that every field has its own particular notation that makes no sense to people not in the field. I'm going to depart a little from the notation used in the paper because I personally found it confusing to represent the number of dead people by the symbol "R". But that's just me. Stat mech taught me that anything representing a number is N with a subscript.
The first model in the paper assumes three classes of beings: Humans, Zombies and Dead. I'm calling the number of humans NH, the number of zombies NZ, and the number of dead ND. We're interested not so much in the actual numbers in each category, but in how these numbers change with time. If the number of zombies gets bigger faster than the number of humans being born, we've got problems. So we don't care as much about NZ as we do about the rate of change of NZ with time. Just as speed is how fast your position changes and acceleration is how fast your speed changes, the rate of change in zombies is the net increase or decrease during some period of time.
To calculate this number, you have to understand the zombie rules. A human becomes a zombie when bitten or otherwise infected by a zombie. A dead person can also be resurrected into a zombie. Let's say we had 6 humans turned into zombies every hour and four dead people resurrected into humans every hour, so there would be 10 zombies created every hour.
Much to my surprise, you can kill zombies, so calling them the 'undead' is technically wrong. You can kill a zombie by removing their head or destroying their brain. So let's say 3 zombies per hour are killed every hour. Then the rate of change in the number of zombies is:
All that was just to calculate the rate of zombie number change, so we also have to write equations for how the number of dead and the number of humans change. To make the interconnectedness of the numbers clear, I've taken the liberty of reproducing the diagram on page 135 and putting a translation into English immediately below it.
Using this model and assuming that the infection occurs quickly (meaning we can ignore the birth and death rates because they are small during the time over which the infection occurs), the paper authors show mathematically that zombie-human coexistence is impossible. The infection will spread and zombies will infect everyone in pretty short order. Bummer.
But the reality, as they go on to point out, is that most infections are latent: that is, people don't turn into zombies immediately, but they walk around (for about 24 hours, according to The Zombie Survival Guide) looking normal even though they are doomed to become zombies. This means we have to introduce another box in the diagrams above to represent people who will, but have not yet become zombies. The researchers call this group 'Latents' which they represent by "I". (I know, I shouldn't be smirking since my discipline represents momentum by "p" and angular momentum by "L".) In addition to a fourth box, you'd have to include arrows from Human to Latent, Latent to Dead and Latent to Zombie, which gives you a system of four coupled ordinary differential equations to solve. I didn't bother drawing it because the authors show that, even with a latency period, we are doomed and zombies win in the end.
A null result is less interesting that a non-null result. If there is no way for us to defeat the zombies, do you think NPR would care? So the researchers asked themselves "what would happen if we were able to quarantine some of the Latents or Zombies?" OK, now we've got a mess because we have five boxes, some Zombies or Latents would be killed trying to escape quarrantine and they also include the possibility that some dead people would become re-animated as 'free' zombies. The results from this analysis are that stopping the zombies depends critically on getting infected people and zombies into quarantine. If you don't get enough of them into quarantine fast enough, the zombies win. Again.
Next, the researchers allow the possibility that a cure for zombie-ism is developed. The cure transforms zombies into humans, but it doesn't protect them from becoming re-zombified. If a zombie who was resurrected from the dead is cured, he becomes human again. Got all that? Well, if you don't, don't worry about it because we get only marginally better chances of survival: Humans are not entirely eradicated, but not many survive. At some point, the individual humans left are going to realize that curing the zombies is not statistically worth the chance.
Finally, the researchers consider a "brute force" method - what if we just killed as many zombies as we possibly could as fast as we possibly could? When they solve the system of equations, they find that this model provides a much higher probability of human survival, but only if you kill zombies really aggressively. The slower you kill the zombies, the higher the likelihood they win.
I love the tongue-in-cheek discussion section: "The key difference between the models presented here and other models of infectious disease is that the dead can come back to life". They do point out that there are plenty of real-world cases in which this model might apply, like modeling people's allegiance to political parties. I've got to find some way that nanomagnetism might relate to zombies, because I'd love to see whether one could get something like that into Phys Rev.
I realize that these are students, but I do have to raise a quibble with their results. Given what we know about zombies, they really should have extended their final model (the one that gives us humans a chance of survival) to include a latency period. I haven't worked through the math, but my gut-level instinct is that if you assume a 24-hour latency period, the zombies have a much better chance of winning unless we start aggressively killing off not only full-blown zombies, but people we suspect have been infected as well. What a great plot for a movie...
IMPORTANT GUY AT CDC: Well, Dr. Smith?, I don't care how right you think your model is, we're not taking action until it's been peer-reviewed in a major journal.
GENERAL: I don't pretend to understand anything having to do with math - never did get into that 'number thing', but if the numbers say we need to start killing millions of people to save the country, let's get going.
SMITH?: Um... did I explain the concept of a model system to you guys?
Really, this would be a great plot: scientists develop a method for saving civilization and it hinges on whether the survival of the world is worth killing some number of people who aren't infected? Uh, don't laugh. Ethiopia ordered all pigs in the country to be killed as an approach to prevent the H1N1 virus from spreading.) There's a good twist: let's put the zombie invasion not in Los Angeles, but in Libya or Iran - somewhere a significant number of Americans might not feel so bad about invading wiping off the map.
OK, that was a fun and fluffy piece of science - the paper got enough coverage that people are griping about why the Canadian government would fund such a thing. (For the record, it was a student project and the professor was acknowledging the agencies that fund him in general.) But there is a serious issue here about science journalism that gets back to the 'science popularizer' vs. science journalist dichotomy.
You can't argue that the research results were being reported for the sake of science, so besides being cute, what was the message here? I could come up with a couple slants on reporting on this that go beyond 'humans vs. zombies, ha ha'. For example, this is a great case study of graduate students learning about communicating science beyond their own subdiscipline. For example, from canada.com
"If you look at it in a more realistic way, zombies are about the same as any other major infectious disease — they get out and we try to eliminate them," said Joe Imad, a University of Ottawa mathematics student and one of the paper's co-authors. "Modelling zombies would be the same as modelling swine flu, with some differences for sure, but it is much more interesting to read."
But the even bigger story -- which I haven't seen in any of the stories I've found is that Smith? and his students do work with important problems. After I came up with the idea of setting my zombie vs. scientist movie in a less-developed country, I looked at some of Smith's other work. One paper about to be published in the Journal of Health Care for the Poor and Underserved points out that there are a slew of neglected diseases, such as leishmanaisis and dracunculiasis that don't get as much attention (or funding) because they: affect less-developed countries, and aren't as fatal, but do have long-term impacts on large numbers of people not so much by killing them, but by socioeconomic effect and long-term suffering. Diseases with high mortality rates attract a lot of attention (and funding), but chronic diseases can actually have a much greater impact on cost of care, lost days of work and social ostracism. These diseases are much more like the zombie scenario - where the infected don't die, they live and can transmit the disease to others.
The graduate students who contributed to the paper are unlikely to specialize in zombie studies, but if they pick up their advisor's obvious passion for using math to make the world a better place one or all of them just might be the person(s) who make a difference in the lives of people who don't have the ability to lobby NIH or Congress. And if they pick up his sense of humor as well, they'll probably have an easier time getting in the door of those institutions as well.