How to get the equation of this and plot it?

Question

I was playing Mine Craft when I thought of this idea.

first I mention some explanation :

In this game there are creatures named villagers and also some named zombies.

If a zombie hits a villager the villager will into a zombie.

Now consider we have 10 villagers and 1 zombie and zombies can turn a villagers into a zombie every 1 second. and all of the zombies focus on a single villager until they kill it.

plot the equation. y=the number of villagers x=time in seconds

first I thought the plot must be a line with a negative slope. but it is not.

because as long as a villager turns into a zombie the speed of changing increases and the slope must increase with the time.

I have trouble finding the equation of the problem.

Answer 1

Do you mean that no matter the number of zombies the zombies can convert one villager per second, or that each zombie can convert a villager per second?

Let t be the number of seconds that have transpired, V(t) the number of villagers at t seconds, and Z(t) be the number of zombies at t seconds. Note that the number of overall people does not change so in general $V(t)+Z(t)=11$

If you mean that the zombies together can convert one villager per second, you are right and it is a linear map. At t=0 there is one zombie and 10 villagers, after each second there is one less villager and one more zombie so you get the following chart

\begin{array}{ccc} t & Z(t) & V(t) \\ 0 & 1 & 10\\ 1&2&9\\ 2&3&8\\ 3&4&7\\ 4&5&6\\ 5&6&5\\ 6&7&4\\ 7&8&3\\ 8&9&2\\ 9&10&1\\ 10&11&0 \end{array}

Thus your equations are $Z(t)=t+1$ and $V(t)=11-(t+1)=10-t$, linear equations.

However if each Zombie can convert a villager per second, you are right and the slope does change. (Note that even if they all attack one person together, this would just mean that n Zombies could convert n people in n seconds). After one second the 1 Zombie turns into 2 Zombies. After 2 seconds the 2 zombies turn into 2x2=4 zombies. You get the following chart:

\begin{array}{ccc} t & Z(t) & V(t) \\ 0 & 1 & 10\\ 1&2&9\\ 2&4&7\\ 3&8&3\\ 4&11&0\\ \end{array}

This is actually an exponential map, we have $Z(t)=2^t$ and $V(t)=11-2^t$

Also one last, note. This expression will not work for the last value in the chart. We would expect there to be 16 zombies, but they run out of people to convert.

Answer 2

Interesting question! This falls into the category of SIR(succeptible infected recovered) epidemic models, and is used by scientists to see how fast diseases can spread.

For your problem, you want to only consider SI since youre saying you remain a zombie forever and you cant recover or die.

This paper goes into some of the details and shows some plots around slide 21. They give you the general shape of the plots and then show how they apply to some popular zombie movies. Just a warning, the math a tad complicated, but if you look at the SZ plots youll get the jist of the idea, gradually (depending on how often the two species interact, how likely the disease is to spread, etc etc), all the people turn into zombies over time. The shape of the function is easy enough to understand, but is a fairly complicated function. Take a look at that paper for some more details!