 Solution

Let $A_2(t)={\int }_0^{t}1\{X(s)=2\}ds$ be the occupation time of a continuous-time Markov chain in the symptoms state. Tonelli theorem permits exchanging the nonnegative integral and expectation. Starting from state 1, put $q=\lambda +\mu >0$. ThensoFor small $t$ the answer is $\lambda t^2/2+O(t^3)$, as expected when onset must first occur. It is between zero and $t$. If $\lambda =\mu =0$, no transition can occur and the answer is zero; the displayed quotient is interpreted separately in that degenerate case.