Mathematical model

In a paper titled ‘When zombies attack! Mathematical modelling of an outbreak of zombie infection’, a group of mathematicians introduced the basic model of how a zombie infection might spread. It’s similar to other basic models of infectious diseases.

The mathematicians began by interpreting the zombie scenario to include these three states of being:

  • Susceptible (S): humans who can be turned into zombies
  • Zombie (Z): zombies
  • Removed (R): humans or zombies that have been removed from the normal population by death, but may return as a future zombie

We can illustrate the relationships between these states as a flowchart, with arrows that represent possible movements from state to state (such as S → R, which is the conversion of a human into a zombie; or R → Z, which is the conversion of a corpse into an active zombie).

The basic model

Flowchart: under the basic model of zombie infection, we have three states: susceptible (S), zombie (Z) and removed (R). S can become Z (by zombification) or R (by dying); Z can become R (by death); and R can become Z (by revival).

If we want to use this model to analyse the outlook of humanity, we need to know how often the movements between these states occur, and how many zombies and susceptible people we start off with. The researchers also made a few more assumptions: that zombies do not defeat each other, and no other kinds of creatures (like zombie dogs) are involved in the spread of the disease.

Zombies win

With a latent infection time (that is, infected humans will carry the disease for some period of time before becoming a zombie) plus quarantine (infected humans and zombies can be isolated to contain the outbreak) we get the following …

The basic model with latent infection and quarantine

Flowchart: when latent infection and quarantine are added to the model, two new states are added: infected (I) and quarantined (Q). S can become I (by infection) or R; I can become Q (by quarantine), Z or R; Z can become Q (by quarantine) or R; and R can become Z.

Zombies win

It may take longer for them to succeed, but we’re still doomed. But what if we can produce a cure? If we found a way to turn zombies back into the susceptible (replacing the need for quarantine) …

The basic model with infection and a cure (but no immunity)

Flowchart: when there is a cure (but no immunity), quarantine is removed and the flowchart is simplified. S can become I or R; I can become Z or R; Z can become S (through the cure) or R; and R can become Z.

Humans survive, in low numbers

The authors of the basic model for zombie infection concluded that

A zombie outbreak is likely to lead to the collapse of civilisation, unless it is dealt with quickly. While aggressive quarantine may contain the epidemic, or a cure may lead to coexistence of humans and zombies, the most effective way to contain the rise of the undead is to hit hard and hit often. As seen in the movies, it is imperative that zombies are dealt with quickly, or else we are all in a great deal of trouble.

You may have realized that some factors are not listed in the models above. This is a common procedure of modelling—you start with basic knowledge of the situation, then add more information as it becomes available. So mathematicians from around the world built on the basic model of zombies in a book called ‘Mathematical Modelling of Zombies’, edited by one of the authors of the original article—Robert Smith?—a Canadian-based Australian mathematician whose name legally contains a question mark.

In the book “Mathematical Modelling of Zombies”, some of the additional details considered included the slow movement and low intelligence of zombies; the strong human capacity for learning; the spatial distribution of humans and zombies; capacity of biological agents to “swarm”; the effects of government policies; the potential for fortification; the effects of making all our choices in the face of uncertainty; and the effects arising from humans who are brave, cowardly or “unflappable”.

Some unexpected conclusions followed. For instance, Ben Tippett’s analysis showed that city dwellers have better long-term survival outcomes than country dwellers, but there is a risk of starvation if no one continues to produce food. The same article shows that a small well-trained militia is far more effective than a large unskilled one, partly because the large unskilled mobs eat more food without producing their own, and also because there are more opportunities for the group members to become zombies.

Australian mathematicians Judy-anne Osborn and Jennifer Badham showed that human fighting skill and capacity to increase skill with practice really matters. The initial skill and speed with which we learn can make the difference between zombies winning, humans winning, and a kind of stalemate in which packs of humans are forever chased by zombies.

Another factor that matters is the wisdom of the government. Daniel Ashlock, Joseph Alexander Brown and Clinton Innes showed that wise policy choices could swing the odds in favour of humans even when outnumbered by zombies.

In: Infectious Disease Modelling Research Progress ISBN 978-1-60741-347-9

Editors: J.M. Tchuenche and C. Chiyaka, pp. 133-150 c

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