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

Using the kolmogorov forward/backwards equations prove:

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

They also ask me to prove for  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.