4 replies to this topic

[Yoruichi-san]

Two of my favorite things ;P

 

The zombie apoc has started!  But thanks to the new Zombie Preparedness Plan put forth by the Pentagon, humans have a 50% chance of leaving an encounter with a zombie unscathed.  Assume we are at the beginning of the outbreak, and assume the number of encounters b/w zombies and humans is proportional to the product of their proportions.

 

Some studies done have shown that an average person encounters ~13 other people a day.  Assuming this holds, how long until half the population is munching on brains?

 

[k-man]

Y-san, can you clarify the "number of encounters b/w zombies and humans is proportional to the product of their proportions", please?

[Prime]

Spoiler for Not that long

Assuming a zombie is an average person, he/she encounters 13 other persons a day converting 6.5 of them into zombuism.
It would be a simple geometric series, except as the number of zombies grows, more and more of them encounter each other.

Edited by Prime, 23 February 2013 - 12:39 AM.

[Yoruichi-san]

Y-san, can you clarify the "number of encounters b/w zombies and humans is proportional to the product of their proportions", please?

 

If X is the proportion of the population that is human and Y is the proportion of the population that are zombies, then the number of encounters b/w zombies and humans is proportional to X*Y.  It's a common assumption in simple epidemic modeling.

 

Assume the 13 encounters will hold and include zombie/zombie, zombie/human, and human/human encounters.

[ThunderCloud]

Spoiler for I think...

After [(2/13) * ln(Population - 1)] days, half the population will be zombies. Reasoning: I assume that each person (human or zombie) encounters 13 other people every day. On average, the proportion of those people who are not already zombies is [(P - z) / P] (where P is the total population of humans and zombies, and z is the zombie count), and 50% of them will be transformed upon contact. I also assume that the total population P is constant (nobody dies, they just become zombified). Then we have dz/dt = (13/2)*z*((P - z) / P). That is, the rate of change of zombie population is proportional to both the zombie count and the proportion of non-zombie humans each zombie encounters. With a little manipulation, this becomes integrable, and gives (P - z) / z = e^((-13/2)t + C). Assuming there is only one zombie at the outset, C = ln(P - 1). Solving for when (P - z) / z = 1 (i.e., there are as many humans as zombies) gives the above result.