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Mathematical Modelling of Zombies

Robert Smith?
Copyright Date: 2014
Pages: 468
https://www.jstor.org/stable/j.ctt1287nxm

Table of Contents

1. Front Matter
(pp. i-vi)
2. Table of Contents
(pp. vii-xii)
3. FOREWORD: I RAN WITH A ZOMBIE
(pp. xiii-xvi)
Andrew Cartmel

One of the hallmarks of a good horror film is that you leave the cinema retaining a residual sense of unease. Suddenly, there seems to be a chill to the sunlight and the world becomes an oddly-less-comfortable place. Then you pinch yourself and remember that you aren’t actually living in a society beleaguered by perpetual onslaughts from hordes of the undead. Cue an embarrassed rush of relief.

And the only time such a sense of unease, perhaps blossoming into full-blown terror, is likely to recur is in your dreams. Or nightmares. I’ve personally lost count of the number of times...

4. INTRODUCTION: WHAT CAN ZOMBIES TEACH US ABOUT MATHEMATICS?
(pp. xvii-xx)
Robert Smith?

In 2009, I published an article I thought would amuse me and no one else: a mathematical model of zombies. The idea was to examine a hypothetical zombie apocalypse through the lens of disease modelling: that is, using the same types of differential equations I use every day to examine the spread of infections such as HIV, malaria, human papillomavirus and a variety of tropical diseases. What we found was that zombies would overwhelm a mid-sized city like Ottawa in just four days.

Much to my surprise, the media appeared to share my sense of humour. The zombie paper was the...

5. THE VIRAL SPREAD OF A ZOMBIE MEDIA STORY
(pp. 1-26)
Robert Smith?

Like any huge event, it started small. In August 2009, an online blog for a newspaper [1] and an article inNational Geographic[2] triggered a tidal wave of reports: a group of Canadian researchers had created a mathematical model of zombies [3]. The story was reported inWired[4], which acted as a hub for spreading it significantly further afield. It was picked up in Canada’sGlobe and Mail[5] and then spread toThe Toronto Star[6],The Wall Street Journal[7] andBBC News[8], where it was the number one story in the world for 24...

6. THE UNDEAD: A PLAGUE ON HUMANITY OR A POWERFUL NEW TOOL FOR EPIDEMIOLOGICAL RESEARCH?
(pp. 27-44)
Jane M. Heffernan and Derek J. Wilson

In recent years, the undead have advanced considerably, both in numbers and in their adeptness at flesh-eating and blood-sucking. Indeed, there is a growing list of cities completely overrun with zombies (e.g., London [1]; Milwaukee suburbs [2]; and New York [3]) or infested with vampires (e.g., Los Angeles [4]; Beaumont, Louisiana [5]; and Forks, Washington [6]). The viral spread of zombies through the media has already been documented. Things are looking grim …

… or are they? While some may feel a twinge of concern over these recent developments, we at the Umbrella Corporation [7] Centre for Research on the...

7. WHEN ZOMBIES ATTACK! ALTERNATE ENDING
(pp. 45-56)
Phil Munz

Authors of an earlier paper on mathematical modelling of zombies,When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infectionused impulsive differential equations to model the eradication of zombies, leaving none undead. While the paper presented an effective conclusion to the zombie catastrophe, it omitted an important idea from zombie entertainment: that many zombie works provoke sequels. This suggests that a more effective model should include cyclic tendencies between human and zombie populations in order to describe fluctuations in the various populations (that is, nearly all zombies are destroyed in the conclusion of the original work; however, they...

8. WHEN HUMANS STRIKE BACK! ADAPTIVE STRATEGIES FOR ZOMBIE ATTACKS
(pp. 57-70)
Bard Ermentrout and Kyle Ermentrout

In a recent article, Munz et al. [1] suggested a number of models for zombie attacks on human populations, likening these to disease outbreaks. In addition to the trends of a standard disease outbreak, the zombie apocalypse may show some more interesting dynamics such as multistability and oscillations, as Munz himself has shown in the zombie paper’s sequel (Chapter 3). To see this in more detail, we allow for some reaction of humans to their zombie enemies. Rather than playing a defensive role against disease, humans are capable of becoming the aggressors against the undead. The zombie makes for a...

9. INCREASING SURVIVABILITY IN A ZOMBIE EPIDEMIC
(pp. 71-92)
Ben Tippett

It is said that the world is only 70 days away from starvation [1]. By this, we mean that, given our rates of consumption and global stockpiles of food, if workers were to stop farming, fishing or gathering food, the human population would not survive beyond two months.

In the midst of a zombie pandemic, where national and municipal quarantines would grind trade to a halt, it is safe to assume that famine’s skeletal spectre would pose as much of a threat as the literal skeletal anthropophagite ghouls. For this reason, survivors would face a morbid calculus as they are...

10. HOW LONG CAN WE SURVIVE?
(pp. 93-116)
Thomas E. Woolley, Ruth E. Baker, Eamonn A. Gaffney and Philip K. Maini

Humans would not survive a zombie holocaust, or so current work suggests [1]. The fact that reanimated corpses do not stop unless their brain is destroyed, coupled with an insatiable appetite for human flesh, has proven to be a deadly combination [2]. However, the work produced by Munz et al. [1] assumes that zombies and humans are well mixed, meaning that zombies can be found everywhere there are humans. Realistically, the initial horde of zombies will be localized to areas containing dead humans, such as cemeteries and hospitals. In addition, because humans and zombies are not initially separated, humans are...

11. DEMOGRAPHICS OF ZOMBIES IN THE UNITED STATES
(pp. 117-128)
Daniel Zelterman

Zombies are an infrequently studied demographic group that is not recognized by the US Census. The recently completed decennial census (as of the time of this writing) does not include a zombie category. We will not attempt to address controversies such as whether zombie status is inherited (How many of your parents were zombies? Is the genetic trait dominant, recessive or linked to other genetic traits?), a lifestyle decision (of the rich and famous, perhaps), contagious like chicken pox or a conscious decision (like joining the National Rifle Association or the Westboro Baptist Church). A web-based search for ‘zombies’ on...

12. IS IT SAFE TO GO OUT YET? STATISTICAL INFERENCE IN A ZOMBIE OUTBREAK MODEL
(pp. 129-148)
Ben Calderhead, Mark Girolami and Desmond J. Higham

Ordinary differential equations (ODEs)—the kind that we meet in introductory calculus classes—have proved to be extremely useful tools for describing, in quantitative terms, how physical systems change over time. Some systems are so well understood that they follow widely accepted ‘laws of motion,’ such as planetary orbits, ballistic missiles, elastic springs, chemical reactions and radioactive decay. In other cases, where a range of complicated objects interact in ways that are impossible to pin down, ODEs can still be extremely effective for characterizing the main features of the system. Modelling with ODEs—that is, constructing ODEs that have some...

13. THE SOCIAL ZOMBIE: MODELLING UNDEAD OUTBREAKS ON SOCIAL NETWORKS
(pp. 149-170)
Laurent Hébert-Dufresne, Vincent Marceau, Pierre-André Noël, Antoine Allard and Louis J. Dubé

We constantly encounter networks in our everyday lives. We might be on our way to work or school (roads and public transport networks), updating a Facebook status (internet, online social networks, World Wide Web), meeting friends or getting a high-profile job (acquaintance networks), calling abroad or getting directions using a GPS (satellite networks) or simply turning on the radio (electrical and information networks). By improving our understanding of the structure of such networks, we are increasingly able to derive as much benefit as possible from the advantages that networks have to offer, while efficiently protecting ourselves from the disadvantages.

Being...

14. ZOMBIE INFECTION WARNING SYSTEM BASED ON FUZZY DECISION-MAKING
(pp. 171-190)
Micael S. Couceiro, Carlos M. Figueiredo, J. Miguel A. Luz and Michael J. Delorme

Fighting infectious disease has been part of the human experience since the beginning of recorded history. Even with the significant advances in treating and preventing disease, humans are still susceptible to many infectious diseases. Surveillance systems are integral to modern epidemiology and important for pointing out specific infectious-disease outbreaks. These systems can be described as public health warning systems and typically use routinely collected information. Warning systems are used for hazardous natural events (e.g., hurricanes, volcano eruptions or tsunamis). In contrast, little attention has been given to the development of such versatile systems for infectious-disease epidemics [2]. The goal of...

15. IS THERE A ZOMBICIDAL MANIAC NEAR YOU? YOU’D BETTER HOPE SO!
(pp. 191-208)
Nick Beeton, Alex Hoare and Brody Walker

Zombies. Ghouls. The undead. Whatever you call these creatures, it cannot be denied that the rise of the zombie is a problem that has troubled the imaginations of people throughout the world [1, 2, 3, 4]. Unfortunately, the great majority of data on zombie behaviour patterns exist merely as conjecture and speculation from people who have probably never even seen a zombie, let alone killed one. This has led to a growing gap between the readiness of the general population and that of certain private individuals, which needs to be rectified. Otherwise, in the case of a major zombie outbreak,...

16. ZOMBIES IN THE CITY: A NETLOGO MODEL
(pp. 209-232)
Jennifer Badham and Judy-anne Osborn

The entry of ‘zombie-ism’ into the disease-modelling literature came with the 2009 publication of “When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection” [1]. We refer to this original chapter for a premodern and modern mythology of zombies. We list the characteristic of zombies given in [1] that are essential to our treatment:

Slow-moving

Unintelligent (this was implied by a discussion of brain damage, rather than stated)

Prone to attacks on humans that, if successful, transform humans into zombies

Capable of being removed from the infectious population (‘killed’) by decapitation.

The original model of Munz, Hudea, Imad and...

17. AN EVOLVABLE LINEAR REPRESENTATION FOR SIMULATING GOVERNMENT POLICY IN ZOMBIE OUTBREAKS
(pp. 233-248)
Daniel Ashlock, Joseph Alexander Brown and Clinton Innes

The issue of formulating effective government policy for dealing with a zombie outbreak has never been more critical than since the zombie wars began at the start of this millennium. Government intervention has proven to be quite ineffectual during many outbreaks. Command and control breaks down in sectors with outbreaks, while civilians are uninformed or misinformed of locations for evacuations and medical supplies [1]. Furthermore, Brooks [2] gives a highly detailed account of the impacts policy initiatives have after humanity has been overrun. One horrific example is the Redeker Plan. In this plan, a phased retreat was made into largely...

18. BANELING DYNAMICS IN LEGEND OF THE SEEKER
(pp. 249-264)
Gergely Röst

Previous chapters have dealt directly with the zombie threat, with aspects including, but not limited to, spatial diffusion, government decision-making, the estimation of parameters, etc. However, in order to understand the zombie epidemic from all perspectives, it is instructive to examine the effects of similar undead creatures on humanity. Although banelings are mere fiction—unlike the zombies we all live in fear of—they are still helpful in showcasing the rise of an undead species.

In theLegend of the Seekertelevision series [1, 2], banelings are those who have died and been offered a second chance at life in...

19. THE ZOMBIE SWARM: EPIDEMICS IN THE PRESENCE OF SOCIAL ATTRACTION AND REPULSION
(pp. 265-300)
Evelyn Sander and Chad M. Topaz

The notion of a zombie has its roots in the African Kingdom of Kongo, from whence the African diaspora brought it to Haiti. In Haitian voodoo, a sorcerer (or boko) could force someone (either living or dead) to become a zombie, while the boko maintained control of the soul. In contrast, the more modern view of zombie-ism is that it does not result from a sorcerer’s control, but rather is an uncontrolled and unmodulated epidemiological effect, spread by contact like a disease. This ‘viral’ zombie-ism is depicted the classic 1968 movieNight of the Living Dead, in Michael Jackson’s 1993...

20. CONCLUSION
(pp. 301-302)
Robert Smith?

So what did we learn?

To answer this question, I’d like to return to the very roots of the current epidemic: that’s right, the original 2009 Munz zombie paper. Or, more accurately, the reason it was published in the first place.

Although publishing a mathematical modelling paper on zombies was a fun thing to do (as the contributors to this volume will surely agree), there was reason it was published in the book it was. Specifically, the editors of that book requested chapters that showcased the way in which models were constructed and requested that each article include more than...

21. CONTRIBUTORS
(pp. 303-312)
The Undead
22. AFTERWORD
(pp. 313-315)
Robert Smith?

I didn’t really mean to invent an entire mathematical subdiscipline. But I suppose no one ever does. I’d always kept my love of science fiction totally separate from my love of mathematics. But when I touched the two together briefly in 2009, the whole world went mad. And one of the nice consequences was discovering I wasn’t the only one. As you can now tell, there are a surprisingly large number of people into mathematics and zombies. And if you made it this far, presumably you’re one of them. Congratulations.

A book like this doesn’t just throw itself together. All...