Thursday, 17 November 2011

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.