MATHS OF ZOMBIES

Mathematical modelling can be used to approximate many scenarios, some more “interesting” than others. Using zombies to talk about the statistics of an infectious disease spread is sure to catch the reader’s attention. This is exactly what the paper You Can Run, You Can Hide: The Epidemiology and Statistical Mechanics of Zombies by Alexander A. Alemi et al. does. The authors state that Zombies are a fun framework to study concepts of modern epidemiology: “Zombism is just that – a (fictional) disease – and so should be amenable to the same kind of analysis and study that we use to combat more traditional diseases.”

The results of a full scale simulation of a zombie pandemic in the United States are in the paper as well as a discussion about the geography-dependent survival rates. For example, the disease is supposed to spread faster in the coasts due to the higher population density while the more remote areas in the centre of the country remained zombie free (even after four months). To model the simulation the following conditions are taken into account:

  • In this simulation, zombies move but humans do not because the authors assume that in the chaos of the first days of the infection, the transportation networks shut down.
  • How fast zombies bite humans and how effectively humans kill zombies is chosen based on the work of Witkowski and Blais, which, in turn, is based on the films Night of the Living Dead (1968) and Shawn of the Dead (2004).

The SZR model divides the agents in three groups: Susceptible, Zombie, and Removed. The authors add a fourth state “Exposed” to add “realism” to the simulation of a zombie outbreak:

  • The susceptible population, S,  are the uninfected humans. They try to kill the zombies they encounter.
  • The zombies, Z, were once humans but they are now (un)dead. They seek out humans to infect.
    • In the simplified SZR model, zombies do not move and can only infect their neighbours in a two-dimensional grid.
  • The removed state, R, is the case of killed zombies.
  • The exposed state, E, represents humans that have been bitten but have not converted just yet – latent zombies.

The three possible transitions are: a human becomes exposed if he is bitten by a zombie, exposed individuals will become zombies at some constant rate (\nu), and a zombie can be killed by a human (therefore changing to the removed state).

  • The bite parameter, \beta, is the probability that a zombie bites a human if they meet.
  • In the same situation, the kill parameter, \kappa, is the probability that a human kills the zombie.

The parameters \beta and \kappa are merged in a single virulence parameter, \alpha = \frac{\kappa}{\beta}, which tells us how much better are humans at killing as compared to zombies at biting. Therefore, when \alpha is big, zombies are quickly defeated by humans and the infection does not expand far. Nevertheless, if \alpha is small, the pandemic eventually covers the whole territory. Of course, in a stochastic simulation, happy accidents may occur and the human population could wipe out all the zombies early on. Anyhow, according the films mentioned previously, zombies are a 25% more effective than human. For the simulation, authors therefore use \alpha = 0.8.

In the implementation of the model, a container with all Z-S contacts is created. An element is chosen randomly and with probability \frac{1}{1+\alpha} the human is bitten while its infected neighbour is killed with probability \frac{\alpha}{1+\alpha}. When a susceptible, S, is bitten – and therefore turned into a zombie – his neighbours are queried and the new Z-S pairs are added to the container. On the other hand, when a zombie is killed, all the Z-S pairs with him are removed.

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Figure 1. Change in the size of the human (blue), zombie (red), killed (black), and bitten (green) populations over a lapse of 112 days in the case of an outbreak in the continental United States. At the beginning, one in every million was infected.

With the parameters chosen by the authors, as we can see in the above graph, 7 days is enough time for zombies to infect about half of the population of the United States. In the video below, available online in the supplemental materials of the paper, we can see the evolution of the outbreak:

Using the results of the simulations, the authors provide a “zombie susceptibility map” that shows the probability of every cell to be infected. This way, we can see what regions are at greater risk at a given time after the apparition of the patient zero.

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Figure 2. Probability of being infected in each pixel (a) 7 and (b) 28 days after the apparition of a single infected individual randomly chosen among the US population. Brighter areas represent regions at risk while darker pixels represent smaller probabilities.

As fictional as a zombie infestation may be, a similar phenomena is not difficult to imagine. While in the zombie outbreak, humans need to kill zombies to stop the transmission of the disease, in real pandemics, some susceptible hosts are required to involve themselves with the infectious population risking their health in the process: the medical personnel. Due to the similar behaviour of modelled and real epidemics, we can say that this techniques discussed in the paper are useful to have an informed discussion about how to proceed in case of an outbreak (zombie or not):

  • In Bayesian Analysis of Epidemics – Zombies, Influenza, and other Diseases, the paper by Caitlyn Witkowski and Brian Blais, the possible success of a military intervention to annihilate zombies is discussed.
  • Zombies, unlike Ebola infected people, do us the favour of not flying so the authors of the paper haven’t taken planes into account. Nevertheless and in spite of the efforts of public health authorities, other infectious diseases might spread across a more complex lattice so the model should be able to adapt to those situations.

As discussed in the Conclusion of the paper, variants of the SZR model might be useful to model another real-life situation: the spread of ideas and opinions. Maybe some future work could create the Audience-Slanderer-Mistaken model or something along those lines? That would probably be fun to read.

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