Thursday, April 1, 2010

Mmmm...Brains!: Using Mathematics To Save Us On Z-Day

Updated and Expanded: 1-14-2014

In the science of the undead it's publish or perish, and rise to publish again. Make no mistake, the threat of zombies is real and Z-day is closer than you think. If caught unprepared you may wake up with a flesh-hungry, reanimated corpse on your doorstep.

But, realistically, can we use what we already know to examine a zombie outbreak?

Philip Munz, Ioan Hudea, Jo Imad, and Robert Smith? (yes, there really is a question mark in his name) wrote a book chapter in 2009 called "When Zombies Attack!: Mathematical Modeling of an Outbreak of Zombie Infection."  This article models a zombie attack, using basic biological assumptions and equations currently used to examine the spread of infections such as HIV, malaria, and HPV. It also takes into account what we know (or think we know) about zombies: they are cannibalistic, they move in small, irregular steps (using the popular slow-zombie rather than the newer fast-zombie), they show signs of physical decomposition, and their bite changes a non-zombie into a zombie.

Munz et al.'s basic model considered three basic classes: susceptible, zombie, and removed. Susceptibles become deceased through natural, non-zombie-related causes but can become zombies through bite/blood transmission. As it suggests, the removed are those who have died via attack or natural causes. So, assuming a short time period where the birth rate is constant, zombies can only come from the resurrected or susceptibles, and zombies move to the removed class when they are defeated. Using these variables he was able to put together a simple model. Then things got more complicated, as models tend to do.

The authors took his basic model and ran it through different scenarios, adding parameters as needed. The first factors he looked at were mass-action incidence and random contact. These basic additions show a disease-free equilibrium to be unstable and a human-zombie coexistence to be impossible. Next, they revised the model to include a latent class of infected individuals which shows that although the zombies will still take over, it will take approximately twice as long. The next version of the model was run with a partial quarantine of zombies then there will be a slight delay in the time to eradication but, ultimately, the zombies still get us. Then the model was run assuming a cure has been found that allows a zombie to return to human form but does not offer immunity. In this case, humans are not eradicated, but only exist in low numbers. Better but still not great. Finally, they attempted to control the zombie population by strategically destroying them as Max Brooks suggests in his book World War Z - An Oral History of the Zombie War. This scenario assumes that it would be difficult to have the resources and coordination needed and so more than one attack would be needed resulting in an impulsive effect. This model found that after 2.5 days, 25 percent of the zombies destroyed; after 5 days, 50 percent; after 7.5 days, 75 percent; and after 10 days, 100 percent of zombies were destroyed. It is important to note that the time scale of this model is short. If the time scale of the outbreak increases then you get a doomsday scenario with a complete collapse of civilization, every human infected or dead. Essentially, if we can contain the outbreak initially and quickly then we can save our own asses (literally).

Next, consider Daniel Lakeland's Improved Zombie Dynamics model over at his Models of Reality blog. He builds on the above model, taking into account several factors that Smith?'s group did not include. Lakeland's model removes the short timescale, allowing the human population to grow at approximately 4.5 percent per year growth through a birth rate of 6.5 percent and a death rate of 2 percent. He also tackles the classification categories. Once a zombie is categorized as removed it can not be reanimated and therefore a special class of fully removed zombies should be required. Also in this fully removed category should go those human who died of natural causes too long ago to be zombified (basically, skeletons). Resurrection from the dead is assumed to be a relatively rare event, 1 percent per year, and rotting of the dead to be much faster, perhaps 5 percent per dayAny good model should also take human experience into account. Surviving humans have learned how to fight zombies and avoid zombification.  Initially, the probability of a human winning a zombie fight is relatively low, about 0.1 percent, but then again the percentage of zombies starts out low too, about 1 in 10,000. But the zombie-killing learning parameter is quite large. Elite zombie killers, in particular, serve an important role in that they are most efficient, rapid, and effective in zombie eradication. However, this skill decays at approximately 1 percent per day in the absence of education. The lack of skill and readiness makes all the difference and causes the probability of a human victory to decline rapidly to zero. It is this, the education parameter, that Lakeland's model found to be most important. Reasonably large numbers of people should be at the very least vigilant with an elite force (perhaps 4/10,000 people) there to drop some zombies. So watch some zombie movies, ya'll.

ResearchBlogging.orgPhilip Munz, Ioan Hudea, Joe Imad, & Robert J. Smith? (2009). When zombies attack!: Mathematical modelling of an outbreak of zombie infection Infectious Disease Modelling Research Progress, 133-150

Lakeland, D. (2010). Improved zombie dynamics. Models of Reality blog, 1 March.

Robert Smith?'s articles:
"A report on the zombie outbreak of 2009: how mathematics can save us (no, really)"
"What can Zombies Teach us about Mathematics?"

NPR interview with Robert Smith?: "Who Will Win In Human, Zombie War?"
CBC News Story: "Zombie Math"

Some other things to consider:

This unpublished paper by Andrew Gelman on a way to study zombies indirectly using surveys that don't risk the interviewers: "'How many zombies do you know?' Using indirect survey methods to measure alien attacks and outbreaks of the undead"

Blake Messer over at The Tortoise's Lens blog creates a model where looks at the Munz model and he takes into account that the humans who survive do so for a reason (they are stronger, faster, smarter, etc.) and the distribution of these people in a two dimensional landscape. Read his post called "Agent-Based Computational Model of Humanity's Prospects for Post Zombie Outbreak Survival"

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