In: Infectious Disease Modelling Research Progress
Editors: J.M. Tchuenche and C. Chiyaka, pp. 133-150
ISBN 978-1-60741-347-9
c
2009 Nova Science Publishers, Inc.
Chapter 4
WHEN ZOMBIES ATTACK!: MATHEMATICAL
MODELLING OF AN OUTBREAK OF ZOMBIE
INFECTION
Philip Munz1, Ioan Hudea1y, Joe Imad2z, Robert J. Smith?3x
1School of Mathematics and Statistics, Carleton University,
1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada
2Department of Mathematics, The University of Ottawa,
585 King Edward Ave, Ottawa ON K1N 6N5, Canada
2Department of Mathematics and Faculty of Medicine, The University of Ottawa,
585 King Edward Ave, Ottawa ON K1N 6N5, Canada
Abstract
Zombies are a popular figure in pop culture/entertainment and they are usually
portrayed as being brought about through an outbreak or epidemic. Consequently,
we model a zombie attack, using biological assumptions based on popular zombie
movies. We introduce a basic model for zombie infection, determine equilibria and
their stability, and illustrate the outcome with numerical solutions. We then refine the
model to introduce a latent period of zombification, whereby humans are infected, but not infectious, before becoming undead. We then modify the model to include the
effects of possible quarantine or a cure. Finally, we examine the impact of regular,
impulsive reductions in the number of zombies and derive conditions under which
eradication can occur. We show that only quick, aggressive attacks can stave off the doomsday scenario: the collapse of society as zombies overtake us all.
1. Introduction
A zombie is a reanimated human corpse that feeds on living human flesh [1]. Stories
about zombies originated in the Afro-Caribbean spiritual belief system of Vodou (anglicised voodoo). These stories described people as being controlled by a powerful sorcerer. The walking dead became popular in the modern horror fiction mainly because of the success of George A. Romero’s 1968 film, Night of the Living Dead [2]. There are several possible etymologies of the word zombie. One of the possible origins is jumbie, which comes from the Carribean term for ghost. Another possible origin is the word nzambi which in Kongo means ‘spirit of a dead person’. According to the Merriam-Webster dictionary, the word zombie originates from the word zonbi, used in the Louisiana Creole or the Haitian Creole.
According to the Creole culture, a zonbi represents a person who died and was then brought to life without speech or free will. The followers of Vodou believe that a dead person can be revived by a sorcerer [3]. After being revived, the zombies remain under the control of the sorcerer because they have no will of their own. Zombi is also another name for a Voodoo snake god. It is said that the sorcerer uses a ‘zombie powder’ for the zombification. This powder contains an extremely powerful neurotoxin that temporarily paralyzes the human nervous system and it creates a state of hibernation. The main organs, such as the heart and lungs, and all of
the bodily functions, function at minimal levels during this state of hibernation. What turns these human beings into zombies is the lack of oxygen to the brain. As a result of this, they suffer from brain damage.
A popular belief in the Middle Ages was that the souls of the dead could return to earth one day and haunt the living [4]. In France, during the Middle Ages, they believed that the dead would usually awaken to avenge some sort of crime committed against them during their life. These awakened dead took the form of an emaciated corpse and they wandered around graveyards at night. The idea of the zombie also appears in several other cultures, such as China, Japan, the Pacific, India, Persia, the Arabs and the Americas.
Modern zombies (the ones illustrated in books, films and games [1, 5]) are very different from the voodoo and the folklore zombies. Modern zombies follow a standard, as set in the movie Night of the Living Dead [2]. The ghouls are portrayed as being mindless monsters who do not feel pain and who have an immense appetite for human flesh. Their aim is to kill, eat or infect people. The ‘undead’ move in small, irregular steps, and show signs of physical decomposition such as rotting flesh, discoloured eyes and open wounds.
Modern zombies are often related to an apocalypse, where civilization could collapse due to a plague of the undead. The background stories behind zombie movies, video games etc, are purposefully vague and inconsistent in explaining how the zombies came about in the first place. Some ideas include radiation (Night of the Living Dead [2]), exposure to airborne viruses (Resident Evil [6]), mutated diseases carried by various vectors (Dead Rising [7] claimed it was from bee stings of genetically altered bees). Shaun of the Dead [8] made fun of this by not allowing the viewer to determine what actually happened.
When a susceptible individual is bitten by a zombie, it leaves an open wound. The
wound created by the zombie has the zombie’s saliva in and around it. This bodily fluid mixes with the blood, thus infecting the (previously susceptible) individual.
The zombie that we chose to model was characterised best by the popular-culture zombie.
The basic assumptions help to form some guidelines as to the specific type of zombie
we seek to model (which will be presented in the next section). The model zombie is of the classical pop-culture zombie: slow moving, cannibalistic and undead. There are other ‘types’ of zombies, characterised by some movies like 28 Days Later [9] and the 2004 reWhen Zombies Attack! 135 make of Dawn of the Dead [10]. These ‘zombies’ can move faster, are more independent and much smarter than their classical counterparts. While we are trying to be as broad as possible in modelling zombies – especially since there are many varieties – we have decided
not to consider these individuals…
7. Discussion
An outbreak of zombies infecting humans is likely to be disastrous, unless extremely aggressive tactics are employed against the undead. While aggressive quarantine may eradicate the infection, this is unlikely to happen in practice. A cure would only result in some humans surviving the outbreak, although they will still coexist with zombies. Only sufficiently frequent attacks, with increasing force, will result in eradication, assuming the available resources can be mustered in time.
Furthermore, these results assumed that the timescale of the outbreak was short, so that the natural birth and death rates could be ignored. If the timescale of the outbreak increases, then the result is the doomsday scenario: an outbreak of zombies will result in the collapse of civilisation, with every human infected, or dead. This is because human births and deaths will provide the undead with a limitless supply of new bodies to infect, resurrect and convert. Thus, if zombies arrive, we must act quickly and decisively to eradicate them before they eradicate us.
The key difference between the models presented here and other models of infectious disease is that the dead can come back to life. Clearly, this is an unlikely scenario if taken literally, but possible real-life applications may include allegiance to political parties, or diseases with a dormant infection.
This is, perhaps unsurprisingly, the first mathematical analysis of an outbreak of zombie infection. While the scenarios considered are obviously not realistic, it is nevertheless instructive to develop mathematical models for an unusual outbreak. This demonstrates the flexibility of mathematical modelling and shows how modelling can respond to a wide variety of challenges in ‘biology’.
In summary, 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.
