Yesterday I updated and expanded a long-ago post of mine called "Mmmm...Brains!: Using Mathematics To Save Us On Z-Day."This post summarized a book chapter in 2009 by Philip Munz, Ioan Hudea, Jo Imad, and Robert Smith? called "When Zombies Attack!: Mathematical Modeling of an Outbreak of Zombie Infection." This work was interesting because it combined basic biological assumptions and epidemic modeling with the rise and spread of zombies. Now, Caitlyn Witkowski of Bryant University and Brian Blais of Brown University have written a paper, which was published on the arXiv pre-print server, that extends the Munz et. al. (2009) work and then applies the methods to influenza dynamics.
Let's start with stochastic vs. deterministic models. Stochastic models are all about random variables and chance variations. They estimate probability distributions and outcomes by allowing for random variation over time, and they are good for small populations. Deterministic models assign individuals to subgroups or categories. They work well for large populations, assuming the size of each category can be calculated using only the history used to develop the model. This deterministic type of model is where we'll focus as it is heavily used in modeling disease dynamics. The categories used in a model each represent a specific stage of an epidemic with letters used in equations to represent each. The two types of disease models we'll focus on are the SIR model and the SEIR model. The SIR model is mathematically simpler and follows the flows of people between three states: susceptible (S), infected (I), and recovered/resistant (R). The SEIR model adds a fourth state, exposed (E), representing an infected individuals who are not yet symptomatic or infectious. This addition of a latency period is the primary difference between the models. Additional parameters are then added including the contact rate or transmission parameter (beta), removal of infection rate (alpha), rate from exposed to infected (sigma), rate of infected to recovered/resistant (gamma), and natural mortality rate (mu).
In order to create their model, Witkowski and Blas first needed data. As we currently have no data on actual zombies (that we know of), they gained insight into zombie dynamics by binge watching zombie films and television shows. From this they found that virtually all zombie movies fall into one of two forms that can each be represented by a particular film, either Night of the Living Dead (1968) or Shaun of the Dead (2004).
The Night of the Living Dead (which I'll abbreviate NotLD) category includes the following observations:
- "Anyone who dies becomes a zombie, regardless of contact with one.
- Because contact with a zombie is likely to lead to death, the interaction between the two subpopulations of susceptibles and zombies is signi cant.
- This interaction between susceptibles and zombies results in a temporarily removed subpop- ulation before members of that population become zombies.
- The only way in which a zombie can be permanently removed is by destroying the brain or burning the body."
With this knowledge, they used the SIR-type models and applied Bayesian parameter estimations. They then applied Markov Chain Monte Carlo (MCMC) techniques to estimate the posterior probabilities of the parameters. This allowed them to provide both the best estimates and their uncertainty. In both movie categories, they were able to estimate the initial susceptible population by using simple approximation, estimated zombie numbers from scenes with a field of view approximately 50 meter squared area, and estimate overall time values from visual cues (clocks, sun, etc.). They then used the exact same techniques to analyze real-world data on influenza using Google Trend data.
Witkowski and Blas found their models to be a significant improvement over the models in Munz et. al. (2009) in that their model structure had a closer match to the zombie system. They found that when they removed Munz et. al.'s "recycling" parameter (humans in the removed class can resurrect and become a zombie) the stability of the system changed significantly. This showed that the zombies can be completely removed as long as they are removed faster than they are created. They found the rates of infection and removal (beta and alpha, respectively) to be nearly a factor of two smaller for SotD than for NotLD. NotLD also has a higher value for the rate of exposed becoming fully infected than does SotD. NotLD also shows an interesting joint distribution patterns in that higher values of the infection rate (beta) require a higher rate of removal of infection (alpha), and no relationship between the removal rate of infection and the rate of exposed becoming fully infected (sigma). In both the NotLD and the SotD categories military intervention saves human civilization, the strength of this military attack and time at which it occurs being the two major factors determining the success. A quick, early intervention expectedly annihilates the zombie horde, but a later intervention is essentially a wasted effort. They explored this scenario with SotD. They assumed that the military intervention seen at the end of the film increased the alpha parameter ten-fold which effectively eliminates the zombies, restoring civilization. However, their calculations suggest that if the film had lasted 30 minutes longer, the same intervention strength would have lead to a doomsday scenario. So this model does agree with Munz et. al. that timing is everything.
Caitlyn Witkowski, & Brian Blais (2013). Bayesian Analysis of Epidemics - Zombies, Infuenza, and other Diseases arXiv.org, 1-16
Learn more about SIR-type disease models:
Murali Haran's presentation slides "An introduction to models for disease dynamics"
Hans Nesse's description and simulator his page "Global Health - SEIR Model"
Jeffrey Moehlis tutorial page "An SEIR model"
Ottar Bjørnstad's 2005 paper "SEIR models"
Some other blog write-ups:
Geekosystem's article "Mathematicians Wrote a Paper on How the Zombie Apocalypse Won’t Kill Us All, Made Us Grateful for Math"
Medium's article "Mathematical Model of Zombie Epidemics Reveals Two Types of Living-Dead Infections"
(image via IMDB)