Mathematical Possibility of a Zombie Apocalypse
In: Infectious Disease Modelling Research Progress
ISBN 978-1-60741-347-9 c Editors: J.M. Tchuenche and C. Chiyaka, pp. 133-150 2009 Nova Science Publishers, Inc.
Chapter 4
W HEN Z OMBIES ATTACK !: M ATHEMATICAL
M ODELLING OF AN O UTBREAK OF Z OMBIE
I NFECTION
Philip Munz1 ∗ Ioan Hudea1 † Joe Imad2 ‡ Robert J. Smith?3 §
,
,
,
1
School of Mathematics and Statistics, Carleton University,
1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada
2
Department of Mathematics, The University of Ottawa,
585 King Edward Ave, Ottawa ON K1N 6N5, Canada
2
Department 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
∗
E-mail address:
E-mail address:
‡
E-mail address:
§
E-mail address:
†
pmunz@connect.carleton.ca iahudea@connect.carleton.ca jimad050@uottawa.ca
rsmith43@uottawa.ca.
ISBN 978-1-60741-347-9 c Editors: J.M. Tchuenche and C. Chiyaka, pp. 133-150 2009 Nova Science Publishers, Inc.
Chapter 4
W HEN Z OMBIES ATTACK !: M ATHEMATICAL
M ODELLING OF AN O UTBREAK OF Z OMBIE
I NFECTION
Philip Munz1 ∗ Ioan Hudea1 † Joe Imad2 ‡ Robert J. Smith?3 §
,
,
,
1
School of Mathematics and Statistics, Carleton University,
1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada
2
Department of Mathematics, The University of Ottawa,
585 King Edward Ave, Ottawa ON K1N 6N5, Canada
2
Department 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
∗
E-mail address:
E-mail address:
‡
E-mail address:
§
E-mail address:
†
pmunz@connect.carleton.ca iahudea@connect.carleton.ca jimad050@uottawa.ca
rsmith43@uottawa.ca.
References: [1] Brooks, Max, 2003 The Zombie Survival Guide - Complete Protection from the Living Dead, Three Rivers Press, pp [2] Romero, George A. (writer, director), 1968 Night of the Living Dead. [3] Davis, Wade, 1988 Passage of Darkness - The Ethnobiology of the Haitian Zombie, Simon and Schuster pp [4] Davis, Wade, 1985 The Serpent and the Rainbow, Simon and Schuster pp. 17-20, 24, 32. [5] Williams, Tony, 2003 Knight of the Living Dead - The Cinema of George A. Romero, Wallflower Press pp.12-14. [6] Capcom, Shinji Mikami (creator), 1996-2007 Resident Evil. [7] Capcom, Keiji Inafune (creator), 2006 Dead Rising. [8] Pegg, Simon (writer, creator, actor), 2002 Shaun of the Dead. [9] Boyle, Danny (director), 2003 28 Days Later. [10] Snyder, Zack (director), 2004 Dawn of the Dead. [12] Heffernan, J.M., Smith, R.J., Wahl, L.M. (2005). Perspectives on the Basic Reproductive Ratio. J R Soc Interface 2(4), 281-293. [13] van den Driessche, P., Watmough, J. (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission [14] Brooks, Max, 2006 World War Z - An Oral History of the Zombie War, Three Rivers Press. [15] Bainov, D.D. & Simeonov, P.S. Systems with Impulsive Effect. Ellis Horwood Ltd, Chichester (1989). [17] Bainov, D.D. & Simeonov, P.S. Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore (1995).