The Joss Whedon model and beating the disease

Looks like the disease has passed and we are back to work! Being ill actually made us think about the hypothetical situation in which vampires were real and humans would have decided to develop effective measures to fight this (vampiric) disease. What would have happened?

Well, one possible outcome was portrayed very well in a film called “Daybreakers”, a 2009 science-fiction horror directed by Spierig brothers. In the film some strange disease (a plague) has turned most of the Earth’s population into vampires. Humans lost their war on vampiric disease and became outlaws and human blood (since no substitute was found – ah, where are you, True blood? :)) became scarce (which made it a very desirable commodity – remember the paradox of value (diamonds-water paradox) from your Microeconomics class?). Vampires began to hunt humans and harvested them for blood but almost depleted the resources (we, humans, always do – whether we are turned into vampires or not, it seems). Luckily, the salvation came with one former ragged vampire who calls himself Elvis and who was able to turn himself back into human form after the car crash (we see the Hand of God here, do you?).

Have you ever wondered what would happen if humans turned against vampires as they often do against diseases and succeeded? The concept is not new. The creator of “Buffy the Vampire Slayer” (TV series), Joss Whedon (and its spin-off “Angel”), 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”). 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 stake, a crucifix or a variety of other methods are pretty efficient. The model reveals how unstable the equilibrium between humans and vampires 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

where 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 (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).

Although the model 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.
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 peacefully.

Flu season, vampires and diseases

Down with a flu :( Not a very pleasant thing to encounter, that is for sure. We are sure you all know that feeling when you feel terrible, have to stay at home, drink huge quantities of hot tea and you seem to hate everyone and everything (including yourself) for getting sick.

The disease (and watching a superb, all-stars (ah, Gwyneth Paltrow dies too soon but keep coming back throughout the film :)) medical thriller “Contagion” recently) made us to sit down and write some considerations about human population and diseases. Remember vampires from our previous posts? Well, they too can be regarded as a form of a disease (well, sort of). See our recent paper about that to find that out.

Remember how not long ago the world media announced that the human population reached the 7 billion mark? Although already quite high, we think that this figure could (and possible should) have been much higher. Our calculations of projections for human population growth show that there is some hidden factor that comes into the picture and prevents human population from growing exponentially. Tossing wars and natural disaster aside (however gruesome and unfortunate, they cannot be blamed for as major responsible factors), all evidence suggests that there must be infectious diseases that bear the whole burden of responsibility. Throughout its known history, humanity has lost about million people to various infectious diseases and it is still counting.

According to Worldometers, 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. And finally, on Halloween 2011 mankind reached its 7th billion, the process that took just 12 years.

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 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 and by the beginning of 2012 will reach 7 billion people. 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). 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.

About 10 thousand years ago the population of Earth was about four million people. A simple calculation of the annual growth rate of Earth’s population in accordance with this dynamics yields the number 0.075%. This is 15 times less than the average population growth in 2010. It is obvious that there is some hidden factor preventing human population from the explosive growth. Amongst such factors one can come up with wars, famines, natural disasters and diseases. Even though wards, famines and disasters can no doubt be gruesome and bloody, there are infectious diseases that the world accounts the most victims for. Let us look closely at the history of some known infectious diseases to support our argument with facts and figures.

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 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 take a good look at them. Viruses are inert and harmless in isolation but when put in action they react and multiply quickly. There 5 thousand types of viruses that are known to science: from flu and cold to smallpox, Ebola, polio and HIW. 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”. WWI killed 21 million people in 4 years. Swine flu did the same in its first 4 months. According to Bill Bryson most 80% of American causalities in WWI came from Spanish flu (in some units the mortality was around 80%). Swine flu arose as a normal 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 (their suffering lasted 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 toll was 220 000 with the similar numbers of deaths in France and Germany. Some estimates put the world toll from Spanish flu at between 20 and 100 million (due to the poor statistics from the Third World).

Therefore, the role of infectious disease is devastating for human population growth. Many sources show that most of the infections diseases might put human population in great danger. Infections spread violently like vampires in our model. If scenarios shown in most of vampire films were real, pretty soon the world would have been taken by vampires (unless some conditions outlined in our paper on a symbiosis of humans and vampires would be established allowing humans and vampires to co-exist peacefully). The only answer was (and still is) 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 according to the IUCN definition).

The role of diseases in human population dynamics and growth cannot be underestimated. Slower growth of human population than the one predicted by our mathematical models might be attributed to the existence of diseases that act as a hidden factor in human population spread over the globe. These diseases, however, might be a good thing because they keep the world from overpopulation and therefore almost sure extinction.

So, be careful and do not catch anything! Hope our disease is not the one from “Contagion” so our recent paper about human population and diseases is not our last one :) We will keep you posted!

Draculanomics

We 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. Our 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 OK about living with an idea that their national hero and a brave freedom fighter against Turks was a vampire. How 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 ("The Lost Symbol") in our globalized world, right? Tepes 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. Take a close look at those souvenirs originating in Romania. Clumsy and kitschy as they are, they are still an attempt to market what has once been sacred. The fridge magnet represents Bran castle and the image of Vlad III "the Impaler". The two keyrings bear the engravings saying "Dracula, prince of Transylvania" and a mention of Vlad Tepes'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 would know Dracula's tomb is actually in Bulgaria). In fact, Vlad Tepes (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 maneuvers 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 could have beaten up Disneyland!

In short, Draculanomics is a new and booming window of opportunity! If you are an entrepreneur (or want to be one) we strongly advise you to consider taking up this niche - you might make yourself lots of money!

Dracula in Romania (and beyond)

Greetings from a conference in Romania! Unfortunately, the conference was not on vampires this time. We, scientists, have to meet and talk about mundane problems from time to time. Besides, Romania can be a normal country too!

What we would like to tell you today is the story of Dracula. Dracula, as we know him, was the center 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 from The Guardian here.

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 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 20 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 fellow 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 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.

Let us analyze the events described in “Dracula” (and Stephen King’s “’Salem’s Lot”, which provides a very similar scenario but is set in New England). The interactions between vampires and humans are portrayed 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” (or in “’Salem’s Lot”) were real. How would things evolve given 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). In 1897, the world population was about 1 650 million people.

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 (Jonathan Harker and Abraham van Helsing could not be, by all means, considered very efficient vampire slayers). In order to solve this, the Cauchy problem is applied. 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), we get to the predator-prey model which is diminished to a simple problem of an epidemic outbreak

The solution to this problem is the following: the human population is drastically reduced by 80% by the 165th day from the moment when the first vampire arrives. This actually means on the 165th day of Count Dracula’s arrival to England the human population reached its critical value and practically became extinct (following the definitions of “Critically Endangered species” by the International Union for Conservation of Nature). At that precise moment, the world would have been inhabited by 1 384 million vampires and 266 million people.

Let us observe the speed with which vampire population grows. The growth of vampire population is extreme: at first, the number of vampires jumps up abruptly, but then slows down and declines. We can determine the moment of time when the speed of vampire population’s growth reaches its maximal values. It is the 153rd day and the number of vampires is the highest (825 million vampires and 286 million of newly turned vampires).

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. Therefore, we have to conclude that 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 described by Bram Stoker seems highly unrealistic.

Sorry everyone! Dracula just does not work! Fortunately, there are other interesting things that do work. Follow our blog and learn more about supernatural economics!

True blood and vampire-like bacteria

Good day everyone,

There are a couple of things from the scientific world we would like to share with you. Science means progress and progress means that even the most fantastic things might be once explained and put into practice.

The first thing is that we might be just a step away from vampires revealing themselves to us (at least in accordance with the logic of Charlaine Harris’s “Sookie Stockhouse series” and “True blood” TV series that is inspired by it). The news is that according to researchers from Edinburgh and Bristol Universities who for the first time made thousands of millions of red blood cells from stem cells we might be able to produce industrial scale quantities of blood in about two years from now.

Are you excited? Want to meet your own Bill Compton? Wonder how the “True blood” tastes? According to Bill, it was nothing special – your actual human blood is better.

The second thing is that according to “Discover” magazine, scientists found a strange micro-organism (bacteria, in fact) that behaves like a vampire. Read the article for yourself and think about it in that way: if there are vampires at this primitive, small-scale level, why cannot they be at the macro-level, e.g. living amongst us? Science seems to come with more and more facts supporting this idea!