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.
$P_{0,0}=\frac{\mu}{\mu+\lambda}+\frac{\lambda}{\mu +\lambda}e^{-(\mu+\lambda)s}$
They also ask me to prove for $P_{1,0}, P_{0,1}, P_{1,1}$