EMILY WELKINS

Till the last drop (illustrated)! Solutions for the peaceful co-existence of vampires

and humans based on the models derived from fiction literature, comic books and films

2012

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Copyright © Emily Welkins, 2012

www.emilywelkins.com

All rights reserved, no part of this book may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior written permission of the author.

2nd special illustrated edition

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TABLE OF CONTENTS

INTRODUCTION: WHAT THIS BOOK IS REALLY ABOUT ................................................... 5 DYNAMICS OF GROWTH IN HUMAN POPULATION: HIDDEN FACTORS? .............. 15

INFECTIOUS DISEASES AND THEIR IMPACT ON HUMAN POPULATION ........................................... 23 VAMPIRES: THE HIDDEN FACTOR? ....................... 29 DRACULA, THE MOST FAMOUS VAMPIRE OF ALL TIMES................................................................... 32

VAMPIRES IN THE MODEL OF HUMAN POPULATION GROWTH: PREDATOR-PREY MODEL .................................................................. 38 VAMPIRES’ AND HUMANS’ CO-EXISTENCE IN THE REAL WORLD BASED ON POPULAR LITERATURE, COMIC BOOKS AND FILMS 48

SCENARIO 1: THE STOKER-KING MODEL ............. 50 SCENARIO 2: THE RICE MODEL ............................ 62 SCENARIO 3: THE HARRIS-MEYER-KOSTOVA MODEL ................................................................. 69 SCENARIO 4: THE WHEDON MODEL ..................... 81 SCENARIO 5: THE BLADE MODEL ........................ 88

CONCLUSIONS .................................................. 100 PROLOGUE ........................................................ 104 REFERENCES .................................................... 107

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Emily Welkins

Till the last drop (illustrated)!

Solutions for the peaceful co-existence of vampires and humans based on the models derived from

fiction literature, comic books and films

For other works by this author please visit:

www.emilywelkins.com

5

1

Introduction: what this book is really about This is not just another book about vampires. This book is a multidisciplinary study that draws on a variety of fields including mathematics, biology, demography, ethnography and history. The main question this book tries to answer is: If vampires were real and lived amongst us, would their existence be possible from the scientific point of view? I present a new approach to modeling inter-temporal interactions between vampires and humans based on several types of vampire

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behavior described in popular fiction literature, comic books, films and TV series. And not just some books and films! I use the sources you surely know very well. I draw several scenarios of vampire-human co-existence and use mathematical models to test whether vampires could have existed amongst us today and under what provisions. Although mathematical principles enabled us to doubt the realism of many human and vampire encounters described in the literature, comic books and films, several sources provide what might be an acceptable description of the situation in which vampires and humans co-exist in a world that is very similar to the one we live in. This book tackles a ridiculous subject using serious mathematical tools. I analyse what I call “The Stoker-King model” (based on Bram Stoker’s “Dracula” and Stephen King’s “’Salem’s Lot”), “The Rice model” (after Anne Rice’s “Vampire Chronicles”), “The Harris-Meyer-Kostova model” (based on Charlaine Harris’s “Southern Vampire

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Series”, Stephenie Meyer’s “Twilight saga” and Elizabeth Kostova’s “The Historian”, “The Whedon model” (based on Joss Whedon’s “Buffy the Vampire Slayer” TV series) and “The Blade model” (based on Marvel Comics’ “Blade”). Therefore, the books provides a serious analysis of a ridiculous subject and is similar, in its nature, to the article entitled “The Theory of Interstellar Trade” written by an American economist and Nobel Prize Winner Paul Krugman in 1978, when he was a young, unknown Assistant Professor caught up in academic rat race. The article was not published until 2011, when the journal Economic Inquiry finally accepted it. I am not the only scientist dealing with the vampirical theme. Quite often, other colleagues turn to this topic in their own research under the influence of popular films (they are so irresistible, we all know that) and literature – after all, it is hard to refrain from using your analytical mind in solving the puzzles of the supernatural.

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For instance, the vampire metaphor in economics dates all the way back to the works of Karl Marx and his followers. In fact, Karl Marx wrote a lot about vampires. For instance, it was calculated that Marx used the vampire metaphor at least three times in “Capital”. For example, in one of the cases Marx describes British industry as “vampire-like” which “could but live by sucking blood and children’s blood too”. Marx’s colleague and follower Frederick Engels also uses the vampire metaphor in his works and public addresses. In one of his works entitled “The Condition of the Working Class in England“, Engels identifies and blames the “vampire property-holding class” as the source of the social troubles. Marx’s and Engel’s perception of vampires corresponds to the recent Hollywood film Abraham Lincoln: Vampire Hunter directed by Timur Bekmambetov. In the film, based on the eponymous book by Seth Grahame-Smith, the 16th president of the United States

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leads a secret life of a vampire hunter searching and destroying “bad” vampires, while he is assisted by “good” vampires.

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According to the film, “bad” vampires were chased out of their homes in Europe after the unveiling of bloody atrocities of Elizabeth Bathory and decided for America as their new home. They supported slave trade and started the Civil War to conquer the North and enslave all American population: black and white. In his secret diary, Abraham Lincoln writes that those vampires are “virtually everywhere” - layers, bankers, shop owners, in short the petit bourgeoisie accused by Marx and Engels of blood-sucking. Marx described vampires’ habits, their greediness and their lounging for blood in such a detail that in many cases it crossed the boundaries of the mere metaphor. Although many researchers perceive Marx’s vampires as metaphoric abstract bourgeois bloodsuckers feeding on working people, his knowledge of vampires is very peculiar. In one particular case, when describing Wallachian peasants performing forced labour for their boyars, Marx refers to one specific “boyar” who was “drunk with

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victory” and who might have been no one but Wallachian prince Vlad (called “The Impaler”) – or Count Dracula himself! All this is very interesting because the best-known novel of vampiric genre, Bram Stoker’s Dracula, did not see the daylight until 1897, some 14 years after Marx’s death. Surely, one can place the Marx’s metaphor in the wider context of nineteenth-century gothic and horror stories which were abundant these days, and of which Marx was a huge fan. On the other hand, one might assume that some of the vampire legends were true and Marx and his contemporaries were aware of that. Academic research on vampires was also done by the two mathematicians from Technische Universitat Wien in Vienna, Austria. Richard Hartl and Alexander Mehlmann published several articles explaining, among all, the problem of renewable resources (us humans!), optimal blood consumption and cyclical bloodsucking behaviour for vampires. In the

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1992 paper published in the Journal of Optimization Theory and Application, the authors present a theoretical model explaining what they call “cycles of fear”. Their model provides an alternative approach to traditional demographic oscillations associated with studying interactions between vampires and humans (they use a controlled Lotka-Volterra type of system and introduce an optimal choice of consumption). Although quite mathematical and full of formulas that might be too challenging for mainstream vampire fans, the article does not lack a certain sense of humour (for example they state that the use of optimal point control theory (the one that applies Pontryagin’s principle requiring the derivation of a shadow price for vampires) is questionable because vampires cast no shadow). Most recently, an article by Professor Costas Efthimiou and Dr Sohang Gandhi, theoretical physicists from the University of Central Florida, entitled “Cinema Fiction Vs

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Physical Reality: Ghosts, Vampires and Zombies” appeared in Sceptical Enquirer magazine in 2007. The authors concluded that “whomever devised the vampire legend had failed his college algebra and philosophy courses”. Whether explained through mathematics or not, most academic articles on vampires do not deny the existence of vampires as such. It is just all about optimal strategies and behaviour for vampires who need to take care and do not expose themselves too much to humans. Further down these pages, I am going to show that my research also provides grounds for the existence of vampires among us. And, quite interestingly, some writers and film-makers know a great deal about it and describe the vampiric behaviour and optimal strategies with stunning precision. Do they know any vampires? Is this why they can tell us so much about them?

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This book will amuse vampire fans and academics alike. But most of all, it will show you that science is not boring at all.

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Dynamics of growth in human population: hidden factors? According to Worldometers (a part of the real time world statistics project), human population has grown considerably in the last two thousand years. The industrial revolution became a milestone for the giant leap in the size of world’s population. By 1800, the population of Earth had reached 1 billion. It had added another billion within 130 years (1930) and the 3rd billion within the next 30 years (1959), 4th billion within 15 years (1974), 5th billion within 13 years (1987), and 6th billion within 12 years (1999). For just 40 years, from 1959 until 1999, the world’s population has doubled from three billion to six billion. By many

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accounts, by the end of 2011 mankind was to reach its 7th billion, the process taking just 12 years. It happened sooner than expected, though. On Halloween 2011 the world media announced that the human population reached the 7 billion mark. Today, the world’s population is growing at a rate of about 1.15% per year. The rate of growth reached its peak at the end of the 1960s, when it reached the level of 2%. The rate of population growth is due to decrease in the next few decades. However, the average annual change in population is estimated to be at a rate of more than 77 million people. It is a widely-accepted concern today that the world’s population will set itself at slightly above 10 billion after 2200. As seen from the examples above, the world’s population grows at a high rate. The Malthusian law (an exponential growth model first explained in the book “An Essay on the Principles of Population” by Robert Thomas Malthus, an English scholar who

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lived in the late 18th and early 19th century) states that small populations typically grow exponentially (especially in the absence of threatening natural enemies). Applying this law to the human population would mean that mankind that lives in a comfortable built environment it has created for itself and is not threatened by any natural enemies will grow progressively and abundantly. Assume that the world’s population is to follow the exponential growth rate x (t), and by the beginning of 2012 (x1) will reach 7 billion people (t1). This dynamics can be expressed by the following differential equation:

kxdtdx

= , (1)

where k represents the coefficient of the population growth. Using the method of division of variables we would arrive to the following solution:

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)(0

0)( ttkextx −= , (2)

where x0 is the total volume of population at the initial time period t0. It seems logical to assume that the exponential phase in the growth of our planet’s population started at the moment the first civilizations formed themselves (i.e. mankind stepped onto the certain level of socialization that allowed for the reproduction of the human species regardless the caprices of nature). According to The Cambridge Ancient History Compilation, it was scientifically proven that the first civilizations on Earth were those dating back to around 8000 B.C. (e.g. Egyptian, Sumerian, Assyrian, Babylonian, Helenian, Minoan, Indian and Chinese civilizations). Quite curiously, only Indian and Chinese civilizations have remained in existence until today, all others went extinct. This might lead to the conclusion that there was indeed a factor hampering the exponential growth of the human population throughout the ages.

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According to Atlas of World Population History, 10 thousand years ago the population of Earth was about four million people. The initial conditions can be formalized as the following: t0 = - 8000, x0 = 4 million people. The coefficient of the world’s population growth is equal to:

[ ] 4

01

01 14 6.71l n)/l n ( −⋅=+=−

= r yTttxxk , (3)

where Try is the annual growth rate of population. A simple calculation of the annual growth rate of Earth’s population in accordance with this dynamics yields the number 0.075%. This is fifteen times less than the average population growth in 2010, according to Sergei Kapitsa, famous Russian physicist and demographer. Chart 2 that follows depicts the exponential growth model of world's population until 2012.

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Chart 1: Exponential growth model of world’s population from the 8000 B.C. until 2012

7988− 6988− 5988− 4988− 3988− 2988− 1988− 988− 12 1012 20120

1400

2800

4200

5600

7000People

Time (in years)

Pop

ulat

ion

Val

ue (

in m

lns)

Chart 2 depicts the logarithmic scale of the exponential dynamics and the actual dynamics built using the values of population growth starting from 8000 B.C. It is obvious that there is some hidden factor preventing human population from the explosive growth.

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Chart 2: Logarithmic scale of Earth’s population growth (upper line – exponential model of population growth, bottom line – actual model built using the values from Atlas of World Population History)

7988− 6988− 5988− 4988− 3988− 2988− 1988− 988− 12 1012 20121

10

100

1000

1 104×People (exp model)People (data)

Time (in years)

Popu

latio

n V

alue

(log

sca

le)

Amongst such factors one can come up with are wars, famines, natural disasters and disease epidemics. According to Jared Diamond, an American scholar and an author of popular science books, even though wars, famines and disasters can no doubt be gruesome and bloody, there are infectious diseases that the world accounts

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the most victims for. Let us look closely at the history of some known infectious diseases to support our argument with facts and figures.

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Infectious diseases and their impact on human population Human history is full of infectious diseases that would leave tens of thousands and more dead. The diseases were caused either by bacterial infections or by the viral ones and humans can call themselves lucky for not dealing with the most effective killers – the most efficient diseases usually kill their hosts too quickly. And if the disease eliminates a human before it can move on, it will die out itself. According to Jared Diamond, human history is full of diseases that once caused terrifying epidemics but then disappeared without a trace. The so-called “English sweating sickness” that raged in 1495-1552 and killed tens of thousands but then disappeared without a trace might be one of the examples. Another example might be the “sleeping sickness” (which became known

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as “Encefalitis letargica”) which appeared in 1916 and quickly spread up in Europe and America. In 10 years the disease killed 10 million people but then vanished completely. While most of the diseases caused by bacteria can be cured with antibiotics, viral diseases prove to be more dangerous and unpredictable. Viruses are so small that only the invention of electronic microscope in 1943 allowed scientist to discover them. Viruses are inert and harmless in isolation but when put in action they react and multiply quickly. There are five thousand types of viruses that are known to science: from flu and cold to smallpox, Ebola, polio and HIV. They prove to be very dangerous human killers: smallpox alone killed 300 million people on Earth in the 20th century. The worst epidemic in history is often called “The Great Swine Flu Epidemic” or the “The Spanish flu Epidemic”. While WWI killed 21 million people in 4 years, swine flu managed to do the same in 4 months. According to Bill Bryson, another famous

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author of popular science books (for example “A Short History of Nearly Everything” that won him a prestigious Aventis Prize in 2004), most 80% of American causalities in WWI came from Spanish flu (in some units the mortality was around 80%). Swine flu appeared as a normal seasonal flu in a spring of 1918 but mutated into something more severe. A smaller proportion of victims suffered only mild symptoms but the rest became very ill and quickly succumbed (lasting from several hours to several days). The first deaths in the U.S. were amongst sailors in Boston in August 1918 but the epidemic quickly spread throughout to the whole country. Between the autumn of 1918 and spring 1919, 549 152 people in total died in the U.S. In Britain, the death toll was 220 000 with the similar numbers of deaths in other European countries. Some estimates put the world toll from Spanish flu at between 20 and 100 million (due to the poor statistics from the Third World).

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Therefore, the role of infectious disease is devastating for human population growth. Books on epidemic models show that most of the infections diseases might put human population in great danger. Just recall the recent medical thrilled “Contagion” which portrayed a hypothetical disease (similar to swine flu) swiping over the world and killing millions before the serum is developed. Infectious diseases spread violently like zombies (or vampires) from the popular books and horror movies. Just to recall Romero’s timeless classic “Night of the Living Dead” from 1968 or most recent Boyle’s “28 days later” (or its sequel “28 weeks later”), a British parody “Shawn of the Dead”, and of course “Resident Evil” and its sequels or many versions of “Dracula” or most recent movies like “Blade”, “Buffy the Vampire Slayer” or “True Blood”. If scenarios shown in most of zombie and vampire films were real, very soon the world would have been taken by them.

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Mathematicians from the University of Ottawa led by Professor Robert Smith! claim that zombies would have eventually taken over the world unless quick and aggressive attacks are made. The progression of zombie infection is fast and unless isolated,

quarantined and killed, very soon everyone will become a zombie (and then dies, as far as there is nothing left to feed on). The same would hold true for the infectious diseases.

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Quarantine, isolation and (if available) vaccination are a must before most of the population is whipped out by a disease (or humanity becomes an endangered spices listed in so-called “Red list” according to the definition used by the International Union for Conservation of Nature). Vampiric infection can also be regarded as a form of infectious disease. Since 1980s such topics as behaviour of vampires, economic significance of vampirism and optimal bloodsucking strategies (e.g. preventing the depletion of renewable human resources) have found their way into the research literature, becoming an inspiration for several academic articles.

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Vampires: the hidden factor? Vampires are man’s natural predators vividly described in legends and folklore. For those of the readers who consider themselves non-believers, vampires can actually be perceived as some form of infectious (blood-transmitted) disease. The word “vampire” is considered to come from the Hungarian language where it is spelled “vampir”. In Slavic languages, the word “vampire” exists in a quite similar form in Russian, Polish, Czech, Serbian and Bulgarian languages. It might be that this word comes from the old Greek root “pi” (which means “to drink”). William Ralston (1872) points out that the word “vampire” might have came from the Lithuanian (Lithuanian is the oldest of all Baltic languages) words “wempti” which means “to drink” and/or “wampiti” which means “purple”.

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The first myths and legends about vampires have probably existed since the dawn of human history. In the 19th century, ancient Mesopotamian texts dating back to 4000 B.C. were translated into English by the prominent British archaeologist and assyriologist Reginald Campbell Thompson revealing some mentioning of “seven spirits” that are very much like the description of vampires as we think of them today. What all myths and legends about vampires agree upon is that vampires are immortal and need to drink human blood to survive and to feed. Some vampires can be very old but all of them possess physical strength surpassing that of any human. Vampires typically do not cast a shadow, do not reflect in the mirror and are afraid of the sunlight, garlic and sacred objects (crucifixes, rosary beads). Vampires cannot enter human dwellings without invitation and cannot visit holy places (churches, shrines). According to most of the legends, vampires can be killed by stakes, silver bullets or by simply cutting off their heads.

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The fact that vampires constituted a threat to humans throughout the history of mankind (whether this threat was real or imaginary one) can be illustrated by the examples of recent archaeological findings at ancient burial sites where some human remains showed signs of being staked, strapped or gagged with a stone, a typical way to slay the vampire as the legends have it.

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Dracula, the most famous vampire of all times The legend of Dracula is quite interesting and amusing in itself. Dracula, as we know him, was the centre figure of Bram Stoker’s 1897 eponymous novel. According to the book, he lived in a castle in Borgo pass, so today thousands of tourists go to castle Bran near Brasov, Romania to breathe in the terrifying atmosphere and (perhaps) to see the Count with their own eyes. In fact, the castle has nothing to do with the novel (or with Count Dracula) and is only being marketed by Romanian tourist agencies to gullible tourists. You can read an interesting and funny encounter of a search for Dracula described by Tanya Gold at “The Guardian” on the 30th of October 2010 and entitled “Scary Halloween? Don’t count on it: on Dracula’s Trail in Romania”.

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However, do not try to ask your Romanian friends about “Dracula”. Dracula is a fictional figure and after several minutes of discussions you will settle the dispute in by talking about Vlad III, Prince of Wallachia (1431-1476) who was known as “Dracula” (meaning “son of the Dragon”). Vlad’s father was a member of the Order of the Dragon (Dracul) which gave him this nickname. Vlad was a cruel and gruesome ruler but he managed to fight the Ottoman Turks well and was praised by both his people and by his allies. He became known for impaling his enemies, a habit that also earned him the nickname “Vlad the Impaler” (“Vlad Ţepeş” in Romanian). Allegedly, Bram Stoker stumbled upon Vlad while studying Romanian and Hungarian history and used the name for his main character (the initial idea was “Count Wampyr”). In reality, however, Vlad III does not have anything to do with vampires. On the contrary, he is a respected historical figure and a national hero. In 1997 Romania printed stamps with his portrait and in 2010

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National Bank of Romania minted commemorative coins dedicated to Vlad on the occasion of celebrating 550 years since the first mention in writing of Bucharest, under his rule, on 20th of September 1459. Bram Stoker was rather basing his plot on so-called “vampire craze’' that took place in the 1720s and 1730s in a part of Serbia that was temporarily attached to the Habsburg monarchy after the Treaty of Passarowitz (1718). Two peasants, Petar Blagojevic and Arnaut Pavle, who died suddenly and without any obvious reason and were reported being seen after their deaths, allegedly caused several other mysterious deaths of their fell ow villagers in the settlements of Kisiljevo and Medveda. The Austrian authorities were called in and the whole affair culminated in an exhumation of suspected vampires, cutting off and burning of their heads and bodies. The whole story which was most likely caused by the poor understanding of infectious diseases and knowledge of the

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decomposition of human body, was vividly described in official reports of that time and attracted Stoker’s attention while his research in the British Library. I have always thought that Romania should market Count Dracula better! Not only that most of the people (even the most educated ones, our colleagues, economists and mathematicians) immediately link the word "Romania" with the word "Dracula", the long-deceased (and perhaps still undead) "prince of Transylvania" remains the country's best trademark. Just think for a moment: what would be your first association if someone mentions Romania? And it is not just among us, Central and Eastern Europeans. My Basque colleague has confirmed that everyone mentioned Dracula when she claimed she was going to a conference in Romania. It is apparent that Romanians are not comfortable about living with an idea that their national hero and a brave freedom fighter against Turks was a vampire. How

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would you feel if someone asked you whether George Washington was a mason, a magician and a Templar who possessed the secrets powerful enough to enslave the world (we all read Dan Brown’s "The Lost Symbol" in our globalized world, right?)? Ţepeş really did lots of good for the Romanian struggle for independence and Romanians are proud about their country, their roots, culture and national heroes. Nevertheless, there are apparently some entrepreneurial individuals who would not have any scruples in exploiting Dracula's name. Although Dracula-themed souvenirs originating in Romania are often clumsy and kitschy, they are still an attempt to market what has once been sacred. One can get fridge magnet representing Bran castle and the image of Vlad III "the Impaler" or the keyrings bearing the engravings saying "Dracula, prince of Transylvania" and a mention of Vlad Ţepeş's "official" burial place at a monastery, on an island on lake Snagov outside Bucharest (but if you read Elizabeth Kostova's "The Historian", you

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would know Dracula's tomb is actually in Bulgaria). In fact, Vlad Ţepeş (or Dracula, if you like) has nothing to do with Bran castle. The place is just said to be connected to the Transylvanian prince, although he barely passed the castle on his manoeuvres in Romania. But nowadays all tourists want to go to Bran and take their photo with "Dracula castle". Local souvenir industry slowly adjusts to the demand. Few T-shirts, pots, some postcards - the "Dracula" brand (that is surely worth hundreds of millions) is not exploited to the full. They even wanted to build a Dracula theme park but gave up the idea. Too bad - it has all the potential to be become more famous than Disneyland. In short, Draculanomics is a new and booming window of opportunity. If you are an entrepreneur (or want to be one) I would strongly advise you to consider taking up this niche.

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3

Vampires in the model of human population growth: predator-prey model Consider introducing vampires into the model of population growth presented in equation (1). The vampire population is denoted by the function )(ty , 10 =y . Vampires act as natural predators for humans. The human population dynamics can therefore be presented as the following function:

yxvk xd td x )(−= , (4)

where the equation )(xv is the rate at which humans are killed by vampires.

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Assume that the number of any vampire’s victims is growing proportionally. Thence, the function )(xv can be presented as the following:

xaxv ⋅=)( , (5) where a>0 is the coefficient of the human’s lethal interaction with a vampire (a human is either killed by a vampire or is turned into a vampire). As a result, the differential equation describing the growth rate of human population can be written as the following:

)( a ykxd td x

−= (6)

Assume the dynamics of vampire’s population change to be )(ty . The growth of vampire population will be determined by the quality and quantity of interactions with

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humans. After selecting its victim, any vampire can kill it by simply draining its blood, turning it into a new vampire or feeding on it but leaving it to live. Let us also introduce vampire slayers into the model. The slayers regulate the population of vampires by periodically killing vampires. The equation will then be modified to look like as the following:

c yb a x yd td y

−= , (7)

where 10 ≤< b is the coefficient reflecting the rate with which humans are turned into vampires, 0≥с is the coefficient of lethal outcome of the interaction between a vampire and vampire slayer. Consider a Lotka-Volterra system (a pair of first-order, non-linear differential equations used to describe the dynamics of biological systems and named after the early-20th

41

century mathematicians Alfred Lotke and Vito Loterra ). This system is classified as a “predator-prey” type model:

−=

−=

)(

)(

cb a xyd td y

a ykxd td x

(8)

The system allows for the stationary solution, meaning that there is a pair of solutions for the system that creates a state when human and vampire populations can co-exist in time without any change in numbers. In order to find the solutions for these two populations, sx and sy , we have to solve the following system putting it equal to zero:

=−=−

0)(0)(

cb a xya ykx

s

s (9)

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As a result, in the stationary case the initial system breaks down into two independent equations yielding the following parameters:

=

ak

b acyx ss ),( (10)

It is obvious from a stationary case that the size of human population is determined by the effectiveness of slaying vampires by vampire hunters c and the number of cases when the humans are turned into vampiresba . The size of vampire population depends on the growth rate of human population k and vampires’ thirst for human blood a. The stationary solution shows that when vampires are capable of restraining their blood thirst, the size of both populations can be rather high in mutual co-existence. The system is held in balance by the existence of vampire slayers.

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The model described in equation (8) represents a system of ordinary differential equations which can be solved by using iterative numerical methods. The most widely-used ones are a family of the Range-Kutta methods that represent the modified and corrected Euler’s method with a higher degree of precision. The time-step algorithm includes the integration of differential equation from the initial to the final condition and computing the value of equation at the next step through the previous one:

∫+

+=+

1

),(1

i

i

t

tii dytgyy (11)

The basic idea of the Runge-Kutta algorithms lies in substituting the function ),( ytg that depends on the unknown function )(ty by certain approximation. The more precise the approximation of the

44

integral, the more accurately one can determine 1+iy .Integrals can be approximated using either the rectangle method (2nd degree of precision) or Simpson’s rule of numerical approximation of definite integrals (3rd degree of precision). The price one has to pay for the higher degree of precision would be the necessity to get the approximation of the integral in three points. Running numerous experimental calculations it was established that the best ratio of precision and the volume of calculations is yielded by the fourth-order Runge-Kutta method. The formulae of calculations using the fourth-order Runge-Kutta method for the system described in (5) are presented below:

[ ]43211 2261 kkkkxx ii ++++=+ (12)

[ ]43211 2261 mmmmyy ii ++++=+

45

tyxtfk iii ∆= ),,(1 tyxtgm iii ∆= ),,(1

tmykxttfk iii ∆++∆

+= )2

,2

,2

( 112

tmykxttgm iii ∆++∆

+= )2

,2

,2

( 112

tmykxttfk iii ∆++∆

+= )2

,2

,2

( 223

tmykxttgm iii ∆++∆

+= )2

,2

,2

( 223

tmykxttfk iii ∆++∆+= ),,( 334

tmykxttgm iii ∆++∆+= ),,( 334 )(),,( iiiii a ykxyxtf −=

)(),,( cb a xyyxtg iiiii −=

We employ the fourth-order Runge-Kutta method in Mathcad calculation environment using the function ),,,,( DnbaVr k f i x e d in order to solve the system described above.

46

The function has five arguments: V - the vector of initial values of the functions (border conditions), [ ]ba, - coordinates of the beginning and the end of the computation interval, n - the number of the network segments, D - the vector of the first derivatives of the system. The function

()rkfixed yields a matrix consisting of )1( +n rows and three columns.

Figure 1: Algorithm of the Runge-Kutta method for predator-prey model in Mathcad computational environment

R e s),1 0 0 0 0,,,(R e s

)()(

),(

00

:)0,0(R K01

10

Dt ft sx y sr k f i x e dcx y sabx y s

x y sakx y sx y stD

yx

x y s

yx

←

−⋅⋅⋅

⋅−⋅←

←

=

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The first column contains the values of the network coordinates, the second and the third contain the solutions by the Runge-Kutta method in the nodes of the network for the functions )(tx and )(ty . The algorithm that uses the function for solving the system in (8) is shown at Figure 1.

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4

Vampires’ and humans’ co-existence in the real world based on popular literature, comic books and films Starting from Bram Stoker’s “Dracula”, the theme of vampirism has been widely exploited by many authors: Anne Rice, Stephen King, Stephenie Meyer, Elizabeth Kostova or Charlaine Harris, just to name a few. In addition to that, vampires often appear in comic books and films and TV series based on these books (e.g. “Blade” or “Buffy the Vampire Slayer”). I carefully reviewed popular literature, comic books, and films on vampires and identified five types of scenarios describing vampire-human interactions. These scenarios were used to draw models of vampire-human

49

confrontation using the predator-prey model described and defined above.

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Scenario 1: The Stoker-King model Bram Stoker’s “Dracula” and Stephen King’s “’Salem’s Lot” describe interactions between vampires and humans in the following way: a vampire selects a human victim and gets into its proximity (it typically happens after dark and the vampire needs the victim to invite her/him in). Often the vampire does not require permission to enter the victim’s premises and attacks the sleeping victim. The vampire bites the victim and drinks the victim’s blood, then returns to feed for 4-5 consecutive days, whereupon the victim dies, is buried and rises to become another vampire (unless a wooden stake is put through its heart). Vampires usually need to feed every day, so more and more human beings are constantly turned into vampires. Assume the events described in “Dracula” were real. How would things evolve given

51

the Stoker-King model dynamics described in both sources? Let us take 1897 as the starting point (i.e. the year Stoker’s novel was first published). According to the United Nations statistical compendium, in 1897 the world population was about 1 650 million people.

52

The initial conditions of the Stoker-King model are the following: 1 vampire, 1 650 million people, there are no organized groups of vampire slayers. The model can be presented in a form of a diagram (see Diagram 1). Diagram 1: The Stoker-King model

where H denotes humans and V denotes vampires, H0 is the initial state of human population, v0 is the initial state of vampire population and the aHV describes an interaction between a human and a vampire (with a as the coefficient of a lethal outcome for vampire-human interaction for humans). Let us calibrate the parameters of this specific case of predator-prey model. The

53

calculation period is set at 1 year with a step of 5 days ( 7 30=t ). The coefficient of human population growth k for the given period is very small and can be neglected, therefore 0=k . The coefficient of lethal outcome for humans interacting with vampires can be calculated according to the scenario presented in the Stoker-King model

tqyty 00 )( = , where 10 =y , q=2. The probability of a human being turned into a vampire is very high, thence 1=b . Jonathan Harker and Abraham van Helsing could not be, by all means, considered very efficient vampire slayers, therefore we can put 0=c . The resulting simplified model is presented in a form of the following Cauchy problem:

=⋅=

=

−=

1)0(1065.1)0( 9

yx

axydtdy

axydtdx

(13)

54

Due to the fact that the total sum of humans and vampires does not change in time (human population does not grow and humans gradually become vampires), the predator-prey model is diminished to a simple problem of an epidemic outbreak.

It can be assumed that for any moment t there holds an equality 00)()( yxtytx +=+ , where 9

0 1 06 5.1 ⋅=x . The system of differential equations can be presented in a form of a single differential equation:

[ ])()()()( 00 tyxyta ytyta xd td y

−+== (14)

with the initial condition 1)0( 0 == yy . This differential equation belongs to the class of logistic equations (e.g. the Verhulst equation that describes the growth of population). Let us solve the Cauchy problem for this equation:

55

[ ] axyyxy

d yy

d ya d tyyxy

d y )(= > 000000

+=−+

+=−+ ⇒

)) (()l n (l n)l n ()l n ( 000000000 ttxyayxyyyxyy −++−+−=−+− ⇒

)) ((l nl n 0000

0

00

ttxyaxy

yxyy

−++

=

−+ ⇒

)) ((

0

0

00

000 ttxyaexy

yxyy −+=

−+ By solving the problem above we get the following equation:

)) ((00

000000

)()( ttxyexyyyxty −+−+

+= α

, 0≥t (15) The equation clearly shows that with passing time the number of vampires grows and very soon there are no humans left:

00)(l i m yxtyt

+=→ ∞

The solution to this problem is presented below (Chart 4). It is clearly visible that the human population is drastically reduced by 80% by the 165th day from the moment when the first vampire arrives. This means

56

that the human population reaches its critical value and practically becomes extinct (following the definitions of “Critically Endangered species” by the International Union for Conservation of Nature). At that precise moment, the world will be inhabited by 1384 million vampires and 266 million people.

Chart 4: The change in the numbers of humans (left line) and vampires (right line) in time (1 step = 5 days) in the Stoker-King model

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 730

3 108×

6 108×

9 108×

1.2 109×

1.5 109×

1.8 109×PeopleVampiresEndangeredProgression

Time (step - 5 days)

Pop

ulat

ion

Val

ue

57

Let us observe the speed with which vampire population grows. In order to do that, an analysis of the following magnitude should

be carried out: dtdy

.

2)) ((00

)) ((00

200

)()(

000

000

ttxya

ttxya

exyexyxya

d td y

−+−

−+−

++

= (16)

The results are shown on Chart 5 that follows: Chart 5: The change in vampires’ growth dynamics (1 step = 5 days) in Stoker-King model

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73

1 108×

2 108×

3 108×

Time (step - 5 days)

Popu

lation

Gro

wth

58

Chart 5 clearly shows that the growth of vampire population is extreme: at first, the number of vampires jumps up abruptly, but then slows down and declines. In order to determine the moment of time when the speed of vampire population’s growth reaches its maximal values, we need to take a look at the following magnitude:

[ ][ ]3)) ((

00

)) ((00

)) ((00

300

2

2

2

000

000000)(ttxya

ttxyattxya

exyexyexyxya

d tyd

−+−

−+−−+−

+

−+−=

=

(17) Chart 6 shows that the speed of vampire population’s growth accelerates until the point denoted by maxt and then slows down.

59

Chart 6: The change of speed of growth for vampire’s population (1 step = 5 days) in the Stoker-King model

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 731− 108×

0

1 108×

2 108×

3 108×GrowthChange in Growth

Max Growth

Time (step - 5 days)

Cha

nge

in P

opul

atio

n G

row

th

The maximal growth of the number of vampires (infected humans) will be observed in a moment of time maxt :

000

00m a x )(

)/l n ( txyayxt +

+= , (18)

where 1 5 3m a x =t is the day (153rd day) when the number of vampires is the highest,

8 2)( m a x =tx million is the number of

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vampires in a moment of time maxt , 2 8)(' m a x =tx million is the number of newly

turned vampires in day maxt . Chart 7 shows the phase diagram of both populations. It is apparent that the increase in one population (vampires) inevitably leads to the decrease in another (humans). When the number of vampires reaches the number of human population, the humans disappear altogether. The presence of vampires in the Stoker–King model brings the mankind to the brink of extinction.

Chart 7: Phase diagram of vampire ( 2z ) and human ( 1z ) populations in the Stoker-King model.

0 5 108× 1 109× 1.5 109× 2 109×0

5 108×

1 109×

1.5 109×

2 109×

Vampires

Peop

le

61

The Stoker-King model describes the “explosive” growth of vampire population. Within the two months of Dracula’s arrival to England (or Kurt Barlow’s arrival to New England), there would have been 4 thousand vampires in operation. The model analyzed in this scenario is very similar to an epidemic outbreak caused by a deadly virus (e.g. Ebola or SARS). According to the Stoker-King model, vampires need just half a year to take up man’s place in nature. Therefore, the co-existence of humans and vampires seems highly unrealistic.

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Scenario 2: The Rice model Anne Rice's “Vampire Chronicles” describes the world with vampires, where vampires still need to feed on human beings (like in the Stoker-King model) but do so discretely. The vampire can attack a human being, feed on it and leave it to live. In some cases (if they are too hungry), vampires kill their victims by draining their blood. The vampire cannot easily turn the human into another vampire (in order to do so, the victim’s permission needs to be gained, it needs to drink some of vampire’s blood and the whole process is painful for both of them and takes several days, so it happens very rarely). Vampires do not need to feed every day: some blood once a week or so is enough to survive. Assume the events described in “Vampire Chronicles” were real. How would things evolve given the Rice model dynamics

63

described in her literary works? Let us take 1791 as a starting point (a year Lestat made

64

Lui a vampire). In 1791, the world population was about 982 million people. The initial conditions of the Rice model are the following: 2 vampires, 982 million people, there are no organized groups of vampire slayers. The model can be presented in a form of a diagram (Diagram 2). Diagram 2: The Rice model

where H denotes humans and V denotes vampires, H0 is the initial state of human population, kH denotes the exponential growth of human population, v0 is the initial state of vampire population, and aHV and baHV both describe interactions between a

65

human and a vampire (with a as the coefficient of a lethal outcome for vampire-human interaction for humans and b as the coefficient describing the rate with which humans are turned into vampires). Let us calibrate the parameters of this specific predator-prey model. The calculation period is set to 100 years with a step of 7 days ( 5 2 00=t ). The coefficient of human population growth is calculated as

01

01 )/l n (ttxxk

−= where 15201 =x million people

at a moment of time 18911 =t , 9820 =x million people at 17910 =t . Humans do not necessarily die or become vampires after their encounter with vampires, thence the coefficient of lethal outcome a will be considerably lower than in the Stoker-King model and is therefore denoted as a⋅1.0 . The probability of a human turned into a vampire is quite low and can be denoted as 1.0=b . There are no efficient groups of vampire slayers, therefore we can put 0=c .

66

Chart 8: The change in the numbers of humans (top line) and vampires (bottom line) in time (1 step = 7 days) in the Rice model.

0 520 1040 1560 2080 2600 3120 3640 4160 4680 52000

2 108×

4 108×

6 108×

8 108×

1 109×

1.2 109×PeopleVampiresEndangered

Time (in weeks)

Popu

latio

n V

alue

The resulting simplified model is presented in a form of the following Cauchy problem:

=⋅=

=

−=

2)0(10982)0(

)(

6

yx

baxydtdy

aykxdtdx

(19)

67

The following system is solved using the Range-Kutta method. The results are presented in a graphical form on Chart 8 above.

Chart 9: Phase diagram of vampire ( 2z ) and human ( 1z ) populations in the Rice model.

0 5 107× 1 108× 1.5 108×0

5 108×

1 109×

1.5 109×

Vampires

Peop

le

It is apparent that in spite of the presence of vampires, the human population in the Rice model grows in the beginning. However, when the number of vampires reaches its critical mass, the human population starts to

68

shrink and after 48.7 years is almost extinct. The number of vampires at this moment is equal to 100 million. Chart 9 above depicts the phase diagram of the system. The chart shows a clear pattern: when the vampire population is small, the human population is growing at its natural rate of reproduction. However, when the number of vampires starts to rise, the human population is diminishing proportionally to the increase in vampire population. When compared to the Stoker-King model, the Rice model merely delays the total extinction of mankind. It is not possible to find equilibrium or stationary solution for this system. According to the Rice model, the co-existence of humans and vampires is possible for a short period of time. However, as time passes, all humans will be extinct or turned into vampires. Therefore, the co-existence of humans and vampires described by Anne Rice also seems unrealistic.

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Scenario 3: The Harris-Meyer-Kostova model Stephenie Meyer’s “Twilight series”, Charlaine Harris’s “Sookie Stockhouse (Southern Vampire) series”, “True Blood” (TV series) and Elizabeth Kostova’s “The Historian” show the world where vampires peacefully co-exist with humans. In Stephenie Meyer’s “Twilight series” vampires can tolerate the sunlight, interact with humans (even fall in love with them) and drink animals’ blood to survive. Of course, they have to live in secrecy and pretend to be human beings. In “True Blood” TV series, however, a world is shown where vampires and humans live side-by-side and are aware of each other. Vampires can buy synthetic blood of different blood types that is sold in bottles and can be bought in every grocery store, bar or gas station. They cannot walk during daytime, so they usually come

70

out at night. Humans also find use of vampires’ essence – vampires’ blood (called

71

“V”) is a powerful hallucinogenic drug that is sought by humans and traded on the black market (sometimes humans capture vampires with the help of silver chains or harnesses and then kill them by draining their blood). Some humans seek sex with vampires (vampires are stronger and faster than humans and can provide superb erotic experience). There is a possibility to turn a human being into a vampire, but it takes time and effort (as described in the Rice model). In Elizabeth Kostova’s novel “The Historian”, vampires are rare although real and do not reveal themselves to humans too often. Their food ratios are limited and they spend lots of time brooding in their well-hidden tombs. “Sookie Stackhouse (Southern Vampire) Series” by Charlaine Harris comes with an interesting concept of vampires “coming out” in the 2000s: vampires have ultimately decided to reveal themselves to humans (a concept totally unacceptable in the works of

72

Stephenie Meyer) and co-exist with them peacefully exerting their citizens’ rights. Assume that at the time of the events described in the first book of the series, “Dead Until Dark” (2001), the world’s vampire hypothetical population was around five million (the population of the state of Louisiana in 2001). Diagram 3: The Harris-Meyer-Kostova model

The initial conditions of the Harris-Meyer-Kostova model are the following: five million vampires, 6 159 million people, there are organized groups of vampire “drainers”.

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The model can be presented in a form of a diagram (Diagram 3). H denotes humans, V denotes vampires and VS denotes vampire slayers. H0 is the initial state of human population, kH denotes the exponential growth of human population, v0 is the initial state of vampire population, aHV and baHV both describe interactions between a human and a vampire (with a as the coefficient of a lethal outcome for vampire-human interaction for humans and b as the coefficient describing the rate with which humans are turned into vampires) and cV denotes the death rate for vampires. Let us calibrate the parameters of this specific case of predator-prey model. The calculation period is set at 100 years with a step of 1 year ( 2 1 02 0 0 1=t ). The coefficient of human population growth is

calculated as 01

01 )/l n (ttxxk

−= where 7 0 0 01 =x

million people at a moment of time 20121 =t , 6 1 5 00 =x million of people at time 20010 =t .

Humans almost always come out alive from

74

their encounters with vampires, hence the coefficient of lethal outcome a will be low and is denoted by a⋅01.0 . The probability of a human being turned into a vampire is similar to the on in the Rice model and equals to 1.0=b . There are numerous groups of vampire “drainers” (although the number of drained vampires is relatively low and would not lead to their total extinction), therefore we can put 0>c (c is calculated similarly to the coefficient k). The resulting model is presented in the initial set-up of predator-prey framework:

⋅=⋅=

−=

−=

6

6

105)0(106150)0(

)(

)(

yx

cbaxydtdy

aykxdtdx

(20)

75

Chart 10: Chart with stationary solution presented on a logarithmic scale for the vampire ( sy ) and human ( sx ) populations in the Harris-Meyer-Kostova model.

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 20121 106×

1 107×

1 108×

1 109×

1 1010×PeopleVampires

Time (in years)

Popu

latio

n V

alue

(log

sca

le)

The model allows for a stationary solution: there are system parameters ),( ss yx that would stabilize the populations of humans and vampires in time. In order to find the stabilized populations of both spices,

sx and sy , an equality described in (10) might be employed:

76

( )87 7 0 4),( =ss yx million individuals. Chart 10 above shows the stationary solution presented on a logarithmic scale. This stationary solution for 2001 cannot be found with the chosen population growth coefficient k and can be reached applying some conditions only after 2012. The deviations in the number of people and vampires from the stationary state at the initial period of time are quite small which points at the fact that the system might be stable and auto-cyclical (Chart 11, Chart 12). It is apparent from both charts (Chart 11, Chart 12) that the human population will be growing until 2046 when it reaches its peak of 9.6 billion people, whereupon it will be declining until 2065 until it reaches its bottom at 6.12 billion people. This process will repeat itself continuously1.

1 According to the UN Population prognoses, the population of Earth will reach 9.1 billion people by 2050 which proves the rational in my results.

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Chart 11: The change in the number of humans in the Harris-Meyer-Kostova model (cyclical nature)

2001 2011 2021 2031 2041 2051 2061 2071 2081 2091 21010

2 109×

4 109×

6 109×

8 109×

1 1010×

1.2 1010×

1.4 1010×

Time (in years)

Popula

tion V

alue

The vampire population will be declining until 2023 when it reaches its minimum of 289 thousand vampires, whereupon it will be growing until 2055 until it reaches its peak at 397 million vampires. This process will repeat itself continuously.

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Chart 12: The change in the number of vampires in the Harris-Meyer-Kostova model (cyclical nature)

2001 2011 2021 2031 2041 2051 2061 2071 2081 2091 21010

1 107×

2 107×

3 107×

4 107×

Time (in years)

Pop

ulat

ion

Val

ue

Chart 13 shows the phase diagram of the cyclical system of human-vampire co-existence.

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Chart 13: Phase diagram of vampire ( 2z ) and human ( 1z ) populations in the Harris-Meyer-Kostova model

0 1 107× 2 107× 3 107× 4 107×6 109×

7 109×

8 109×

9 109×

1 1010×

Vampires

Peo

ple

Under certain conditions, the Harris-Meyer-Kostova model seems plausible and allows for the existence of vampires in our world. Peaceful co-existence of two spices is a reality. However, this symbiosis is very fragile and whenever the growth rate of human population slows down, the blood

80

thirst of vampires accelerates, or vampire drainers become too greedy, the whole system lies in ruins with just one population remaining.

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Scenario 4: The Whedon model The creator of “Buffy the Vampire Slayer” (TV series), Joss Whedon, presents the most simplistic, yet the most dreadful “doomsday” scenario of vampire-human interaction (similar to Zombie infection outbreak in other fiction movies, such as “28 days later” or “Resident Evil” that is very well described in the article by Robert Smith! and his colleagues from the University of Ottawa). The vampire bites its victim who (in a very short period of time) rises as another undead vampire and, in turn, bites another human victim, and so on. Luckily enough for humans, the world is populated by an unknown (but considerably large) number of vampire slayers (with a girl named Buffy Summers being their most remarkable representative). Killing a vampire according to the Whedon model is relatively easy: using a wooden

82

stake, a crucifix, or a variety of other methods that prove to be quite efficient.

The Whedon model is a modified version of the Harris-Meyer-Kostova model. In

83

addition to the original model, it uses the higher coefficient of vampire-slaying effectiveness, c. The model reveals how unstable the equilibrium reached in Meyer-Harris-Kostova model might be. The initial conditions of the Whedon model are the following: five million vampires, 6 159 million people, there are organized groups of zealot vampire slayers (who kill vampires not just for profit but out of persuasion and in an altruistic attempt to destroy the Evil and save mankind). The model can be presented in a form of a diagram (Diagram 4). Diagram 4: The Whedon model

where H denotes humans, V denotes vampires and VS denotes vampire slayers.

84

H0 is the initial state of human population, kH denotes the exponential growth of human population, v0 is the initial state of vampire population, aHV and baHV both describe interactions between a human and a vampire (with a as the coefficient of a lethal outcome for vampire-human interaction for humans (which is higher this time) and b as the coefficient describing the rate with which humans are turned into vampires) and cV denotes the death rate for vampires (with a much more higher c). Let us calibrate the parameters of the model. The calculation period is set at 10 years with a step of 1 year ( 2 1 02 0 0 1=t ). The coefficient of human population’s growth is

calculated as 01

01 )/l n (ttxxk

−= where 7 0 0 01 =x

million of people at a moment of time 20121 =t , 6 1 5 00 =x million of people at time

20010 =t . Humans almost always die after their encounters with vampires (are turned into new vampires), the coefficient of lethal outcome is high and it is denoted by a . The

85

probability of a human turned into a vampire is 1.0=b . There are numerous groups of vampire slayers, therefore we put c⋅10 .

The resulting model is presented on Charts 14 and 15. Chart 14: Changes in human population in the Whedon model (disbalance, human population recovers)

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 20120

1 109×

2 109×

3 109×

4 109×

5 109×

6 109×

7 109×

Time (in years)

Popu

latio

n V

alue

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Chart 15: Changes in vampire population in the Whedon model (disbalance, vampires are exterminated)

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 20120

1 107×

2 107×

3 107×

4 107×

Time (in years)

Popu

latio

n V

alue

Although the model described in (8) allows for the stationary solution, the initial conditions of the problem lead to the disbalance in the system. The Whedon model is too unstable to be realistic. Vampires and humans cannot co-exist for a long period of time because human vampire slayers (e.g. famous Miss Summers) exterminate all vampires entirely. The human population recovers from the damage caused to it by vampires and continues to grow steadily.

87

Even though the Whedon model’s structure theoretically allows for co-existence of humans and vampires, the laborious vampire slayers contribute to putting the system out of balance by killing all vampires. This outcome is predetermined by the initial parameters of the Whedon model (vampires constantly need to feed, cannot effectively control their blood thirst and attack humans whenever possible). A very important feature of the Whedon model is for the vampire slayers to exterminate all vampires before they do an irreversible damage to the human population and it will fail to return to its initial state and continue to grow steadily and peacefully.

88

Scenario 5: The Blade model The last scenario is a peculiar case of the world inhabited by humans and vampires described in “Blade” comic book series which became an inspiration to the popular film trilogy and TV series (hereinafter referred to as “the Blade model”). The Blade model shows the world where some vampires developed so-called “Reaper virus” making them some sort of “super-vampires”. They are stronger and faster than “normal” vampires and can sustain sunlight and usual weapons (garlic, silver). “Super-vampires” need to feed on other “normal” vampires and are not interested in human beings. The feeding takes place once in about 5-7 days (when “super-vampires” become thirsty). “Super-vampire’s” victim usually becomes another “super-vampire”.

89

“Normal” vampires still feed on human beings (typically once in 5-7 days) and usually kill them after feeding. Humans can be turned into new “normal” vampires but it takes time and effort. The initial conditions of the Blade model are the following: five million vampires, 6 159

90

million people, 1 super-vampire, there are organized groups of both super-vampire slayers and vampire slayers (Diagrams 5a and 5b). Diagram 5a: The Blade model: h > 0

Diagram 5b: The Blade model: h = 0

where H denotes humans, V denotes vampires, VS denotes vampire slayers, SV denotes super-vampires and B denotes a super-vampire slayer (Blade). H0 is the

91

initial state of human population, kH denotes the exponential growth of human population, v0 is the initial state of vampire population, aHV and baHV both describe interactions between a human and a vampire (with a as the coefficient of a lethal outcome for vampire-human interaction for humans and b as the coefficient describing the rate with which humans are turned into vampires), cV denotes the death rate for vampires. SV0 is the initial state of super-vampire population, cVSV and dcVSV both describe interactions between vampires and super-vampires with c being the coefficient of the lethal outcome for vampires and d being the coefficient of turn rate for vampires. hSV denotes the lethal outcome for super-vampires when meeting the super-vampire slayer. In this specific case, the Blade model is somewhat different from the predator-prey model defined in (8) due to the new population represented by super-vampires. The model that describes the interaction of three populations (humans, vampires and super-vampires) looks like the following:

92

=⋅=

⋅=

−=

−=

−=

1)0(105)0(

106150)0(

)(

)(

)(

6

6

zy

x

hdcyzdtdz

czbaxydtdy

aykxdtdx

(21)

Let us calibrate the parameters of the model. The calculation period is set to 500 years with a step of 7 days ( 2 6 00=t ).The coefficient of human population’s growth is

calculated as 01

01 )/l n (ttxxk

−= , where 7 0 0 01 =x

million people at a moment of time of 20121 =t , 6 1 5 00 =x million people at a

moment of time of 20010 =t . Humans almost always die after their encounters with vampires (are turned into new vampires), the coefficient of lethal outcome is high and it is denoted by a . The probability of a human being turned into a vampire is rather low and equals to b=0.001. There are numerous groups of super-vampire

93

slayers, therefore we use considerably high coefficient of c. Almost every vampire becomes a super-vampire after an unfortunate encounter with the latter, therefore 1=d . In order to solve the system of ordinary differential equations the fourth-order Runge-Kutta method is employed. This is done by building the following algorithm: Figure 2: Runge-Kutta method algorithm for the Blade model

R e s),2 6 0 0 0,,,(R e s

)()(

)(),(

000

:)0,0,0(R K D

12

201

10

Dt ft sx y sr k f i x e dhx y scdx y s

x y scx y sabx y sx y sakx y s

x y stD

zyx

x y s

zyx

←

−⋅⋅⋅⋅−⋅⋅⋅

⋅−⋅←

←

=

Following the scenario described in “Blade 2” film, here are two cases to be considered:

94

Two cases of the Blade model scenario: 1. Blade agrees to help

vampires in exterminating super-vampires. The parameter regulating the extermination rate of super-vampires is h > 0 (Diagram 5a).

2. Blade refuses to help

vampires in killing super-vampires (super-vampires are no threat for humans), h = 0 (Diagram 5b).

In the first case, the stationary solution can

be reached only if d ch

akys == , ba

czx ss = (strict

regulation of the vampire population and reaching proportionality between the population of humans and super-vampires). Slight deviations from the stationary solution will lead to cycles in all 3 populations (see Charts 16 and 17).

95

From Chart 16 it is apparent that the human population will be diminishing from the initial period of time and in 21 years (1092 weeks) will reach its minimum with the critical point of 1.23 billion people. Moreover, the human population will grow and by 2196 (10140 weeks) will reach its maximum of 8.53 billion people. This process will repeat itself with a cyclical periodicity. Chart 16: Changes in human population in the Blade model (cyclical nature)

0 2600 5200 7800 10400 13000 15600 18200 20800 23400 260000

2 109×

4 109×

6 109×

8 109×

1 1010×

Time (in weeks)

Popu

latio

n V

alue

96

Chart 17: Changes in vampire population (top line) and super-vampire population (bottom like) (cyclical nature)

0 2600 5200 7800 10400 13000 15600 18200 20800 23400 260000

2 106×

4 106×

6 106×

8 106×

1 107×VampiresSuper-Vampires

Time (in weeks)

Popu

latio

n V

alue

The changes in the populations of vampires and super-vampires are also of a cyclical nature (Chart 17). Starting from the initial point of time, the number of vampires will grow from 5 million to 9.8 million within 15 years. Further, it will decline to the point of mere 554 vampires in 2055 (2808 weeks). The population of super-vampires will grow from just one super-vampire to 7.2 million vampires in 21 years (1092 weeks) and then will decline until the super-vampires will be on a brink of extinction.

97

Chart 18: Phase diagram for the populations of vampires (Y ), humans ( X ) and super-vampires ( Z ) in the Blade model.

Chart 18 presents the phase diagram of the system with vampires, humans and super-vampires which looks like the connected curve. In case Blade refuses to help vampires in their quest to exterminate their new enemy, super-vampires, the Blade model cannot

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sustain a stable co-existence of three species. Chart 19 depicts the situation when the human population is diminishing being under continuous attacks by vampires and by 2020 (988 weeks) will reach its minimum of 1.5 billion, whereupon it will start growing steadily again. This is going to happen due to the fact that super-vampires will exterminate all vampires (Chart 20) and there will be no natural enemies for humans. By 2032 (1612 weeks) there will be no vampires left in the world of the Blade model (they all will be killed or turned into super-vampires). The super-vampire population will reach its peak of 10.4 million and will remain at its values until super-vampires die off due to food shortages (since they cannot feed on humans, due to their nature, and require vampires’ blood for feeding).

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Chart 19: The change in human population (disbalance)

0 520 1040 1560 2080 2600 3120 3640 4160 4680 52001 109×

2 109×

3 109×

4 109×

5 109×

6 109×

7 109×

Time (in weeks)

Popu

latio

n V

alue

Chart 20: The change in the vampire population (red line) and super-vampire population (brown line) (deviation)

0 520 1040 1560 2080 2600 3120 3640 4160 4680 52000

5 106×

1 107×

1.5 107×VampiresSuper-Vampires

Time (in weeks)

Popu

latio

n V

alue

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5

Conclusions

The book analysed the possibility of co-existence for human and vampire populations in five different scenarios described by the conditions narrated in popular literature, comic books, films and TV series. It appears that although vampire-human interactions would in most cases lead to great disbalances in the ecosystems, there are several cases that might actually convey plausible models of co-existence between humans and vampires. In total, five different models were defined, calibrated and analysed. The Stoker-King model (based on Bram Stoker’s “Dracula”

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and Stephen King’s “’Salem’s Lot”) described the “explosive” rate of growth in vampire population that would lead to exterminating 80% of the human population on the 165th day of the first vampire’s arrival. The scenario is similar to severe epidemic outbreaks and would lead first to the complete extinction of humans and then to the death of all vampires. The Rice model (based on Anne Rice’s “Vampire Chronicles”) would merely delay the total extinction of mankind by vampires by 48 years with respect to the first model and therefore cannot be considered as realistic. Unlike the previous two, the Harris-Meyer-Kostova model (based on Charlaine Harris’s “Southern Vampire Series”, Stephenie Meyer’s “Twilight saga” and Elizabeth Kostova’s “The Historian”) allows for the peaceful (and totally unnoticeable) existence of vampires in our world. However, the system is very fragile and some coordination is required to keep things in balance.

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The fourth model, the Whedon model (based on Joss Whedon’s “Buffy the Vampire Slayer” TV series), allows for the co-existence of humans and vampires, however in this case vampires become one of the endangered spices due to the existence of super-effective vampire slayers. Unless the slayers calm their rigour, the vampire population would be soon extinct. Finally, the fifth model, the Blade model (based on Marvel Comics’ “Blade”), presents an extension to the previous class of models introducing the super-vampires and examining the balance in the new system. According to the Blade model, all three populations are in disbalance and it largely depends on whether vampires and humans would join their forces to fight the super-vampires that vampires remain in existence or are exterminated by the super-vampires. Overall, although mathematical principles enabled me to doubt the realism of many human and vampire encounters described in the literature, comic books and films several

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sources provide what might be an acceptable description of the situation in which vampires and humans can co-exist in a world that is very similar to the one we live in.

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6

Prologue Have you ever wondered whether vampires could actually live amongst us? Although their existence is possible (at least from the scientific point of view) as I have shown in my book, one may ask why we do not see them or meet them every day. One possible explanation might be that they are well-hidden in their coffins or hide-outs and only come out at night. Another explanation might be that they are so scarce, that a chance of meeting a vampire is practically equal to zero. However, some individuals we occasionally meet do make us think that the argument is not lost and that vampires are among us. There are lots of them among economists and mathematicians (especially my

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colleagues). In particular, there is one certain gentleman I used to know and work with, a director of a respectable research Institute. I have been suspecting him for a long time now! And there are many things to back up this conclusion: i) no one has ever seen him walking in daylight, ii) most of the time he lingers in his office behind closed shutters and windows and when one is invited to meet him, he receives his guest in a dark 19th century room with frescoes ceiling dimly let by one lamp (which proves he has a style – so common for vampires (just recall Lestat, Edward Cullen or Bill Compton)), iii) he looks pretty sinister, iv) he always looks pale and never wears a tan, even though he claims he goes on vacations sometimes, v) he is often away (allegedly on business trips, but we know better – he probably goes out hunting or attends vampires councils). Look around you! If you teach or do research in some field of social or natural sciences at the University or any other research institution, there might be similar

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sinister-looking individuals among your colleagues. They always stay in the shadows; they linger behind the closed doors of their offices and never expose their pale bodies to the warm rays of the sun. Think about how many colleagues are there whom you actually never met during the daytime! You always stumble upon then when you are about to leave work and they are still there, in their offices, pretending to work, wanting to chat, inviting you to come in and talk to them. Be careful! We know better and advise you to stay away from those suspicious individuals – you might be turned into one of them!

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7

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Till the last drop (illustrated)! Solutions for the peaceful co-existence of vampires

and humans based on the models derived from fiction literature, comic books and films

Emily Welkins

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2nd special illustrated edition

116 pages

Copyright © Emily Welkins, 2012