BrainDen.com - Brain Teasers

## Question

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?

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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

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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.

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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.

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