 Solution

Let $P(t)=e^{Qt}$ be the transition probability matrix of the four-state continuous-time Markov chain, and put $N=\{S,R\}$ for a negative test and $P=\{E,I\}$ for a positive test. Starting susceptible at time zero, the likelihood contribution isEquivalently, with indicator diagonal matrices $D_N,D_P$ and the susceptible basis vector $e_S$ it isThis sums over every hidden state sequence compatible with the three test results.