The Math Behind Zombie Attacks

Good news…bad news. The good news is that if there was a zombie attack, you wouldn’t have to worry about whether or not there are scientists focused on calculating all things related to zombies because there are. The bad news (& tl;dr) is that without a cure, you can run, but will only die tired.

Much of the zombie related research is based on the SZR (number susceptible, number zombied, number recovered) model which is itself based on the SIR (number susceptible, number infectious, number recovered (immune)) model that is used to compute theoretical spread of a disease. According to the model by Munz, et al. (2009),

β — the probability that a susceptible becomes a zombie when the two encounter each other
δ — the rate that a susceptible can become diseased through non-zombie means
ζ — the rate that humans are no longer humans and become zombies
α — the rate that zombies are killed

The outcome is zombies infect everyone.

The addition of latency is considered to be more “realistic” in that there is a period (up to 24 hours) from when the infected were infected to when they show symptoms of becoming a zombie.

If latency is added, zombies still take over, but it takes twice as long.

Munz, et al. takes the model a step further by adding the element of quarantine. Quarantined individuals are removed from the population and can not infect anyone else. So, the quarantine only has zombies.

If quarantine is added, then there is a delay, but everyone still turns into zombies.

At this point, you are probably thinking there is no hope, but there is if there is a cure! According to Munz, et al. it is assumed that if you have a treatment, there is no need for a quarantine. Also, it is assumed that cured zombies will be able to return to their human form, no matter how they turned into zombies. (← IMHO that is hogwash, because if the person’s body is torn apart in the process, you can’t just return to being a human with ease. My guess is that some number of the population would bleed out or die of infection because of raw wounds.) The outcome of this model shows us that if there was a cure, zombies and humans would exist together, where the zombie population would outnumber the human population, but because of the existence of a cure, the human population trucks on.

If a cure is added, then zombies and humans will live on together.

Sander and Chad (2014) took this a little further and created a swarming model of how zombies might move through a space and attack humans. According to Sander and Chad:

• zombies are attracted to humans, so they move to high human-populated areas
• humans seek to avoid zombies, so they move to low zombie-populated areas

The top row is S(usceptible), the middle row is Z(ombies), and the bottom row is R(ecovered). On the left image (two columns) are patterns resulting from models without a cure. Transient pulses occur slightly on the far left, but not in the second example. On the right image (three columns) are patterns from models with a cure. From left to right on the right image, each column has a varying velocity — a traveling pulse with constant velocity, traveling pulse with oscillating velocity, and merging and splitting pulses.

The results of the swarming model with a cure (image on the right) gives strikingly different results then without (image on the left). Also, Sander and Chad believe that many types of solutions are useful to understand for early detection and public health planning. But for you, the reader of this post, the point is to “head for the [unpopulated] hills” if there is a cure, watch the movements of zombies and adjust accordingly in order to survive.

If you found this post interesting/fun, you can read a lot more on the mathematics of zombies in Mathematical Modeling of Zombies.

REFERENCES

Munz, P., Hudea, I., Imad, J., & Smith, R. J. (2009). When zombies attack!: mathematical modelling of an outbreak of zombie infection. Infectious Disease Modelling Research Progress, 4, 133–150.

Sander, E., & Chad, M. (2014). Topaz. The zombie swarm: Epidemics in the presence of social attraction and repulsion. Mathematical Modelling of Zombies, 265.