finding the limiting fractions of dead, sick and well individuals in the population Suppose that there is an epidemic in which every month half of those who are well become sick (the other half stay well), and a quarter of those who are sick die (the remaining three quarters stay sick). Find the limiting fractions of dead, sick and well individuals in the population as the number of months ahead goes to infinity. I am not quite sure how to set it up but so far I have: Let $x$ be the population. The limiting fraction for those who die is $\lim_{n\rightarrow \infty } \left ( x\times \frac{x}{2} \times \frac{x}{4} \right )^{n} = \lim_{n\rightarrow \infty } \frac{x^{3n}}{8^{n}}$. The limiting fraction for those who are sick is $lim_{n\rightarrow \infty } \left ( x\times \frac{x}{2} \right )^{n} = \lim_{n\rightarrow \infty } \frac{x^{2n}}{2^{n}}$. The limiting fraction for those who are well is the same as the limting fraction for those who are sick. Am I on the right track?

Let $W_n$ = the fraction of the total population that is well at month $n$, $S_n$ = the fraction that is sick, and $D_n$ = the fraction that is dead. So for any month, these three represent the entire population: $W_n + S_n + D_n = 1$. We are given that $W_{n+1} = \frac{W_{n}}{2}$, $S_{n+1} =\frac{3}{4}S_n + \frac{W_{n}}{2}$ and $D_{n+1} = D_n + \frac{S_n}{4}$, and you can easily confirm that $W_{n+1} + S_{n+1} + D_{n+1}$ is still 1.

Taking the limits as $n \to \infty$ of both sides of the three $n+1$ equations gives a fairly easy way of finding the limiting fractions (though this doesn't prove that the limits exist, but it sounds like you haven't been asked to do that).