Given the Markov chain modelling the outbreak of measles with states $0$, $1$ and $2$ corresponding to susceptible, infected and recovered and immune, respectively. The transition matrix is given by $$ P = \begin{bmatrix} 1 - \beta & \beta & 0 \\ 0 & 1 - \gamma & \gamma \\ 0 & 0 & 1 \end{bmatrix} $$ Find the expected time until a susceptible individual becomes infected (state 0 to 1), and find the expected time until an infected individual becomes recovered (state 1 to 2).
Not really sure how you can say anything about time in the context of Markov chains? The time taken would depend entirely on how long each transition takes?
$P(X_{n+1}=1|X_n=0) = \beta, P(X_{n+1}=0|X_n=0) = 1- \beta$ from geometric distribution properties https://en.wikipedia.org/wiki/Geometric_distribution $$EX = 1/p_{success}$$ $$EX = \frac{1}{\beta}$$ this is expected number of rounds to reach state 1 from state 0
analogically $$EX = \frac{1}{\gamma}$$ to reach state 2 from state 1