Using a simple sickness model i.e states 0 and 1 only. We get a death and birth process with rate $\displaystyle \lambda $ going from state 0 to state 1 and rate $\displaystyle \mu $ going from state 1 to state 0.

Using the kolmogorov forward/backwards equations prove:

$\displaystyle P_{0,0}=\frac{\mu}{\mu+\lambda}+\frac{\lambda}{\mu +\lambda}e^{-(\mu+\lambda)s} $

They also ask me to prove for $\displaystyle P_{1,0}, P_{0,1}, P_{1,1} $
But i think i can do these ones once i know how to prove for one of them.