Solution
= Solution
The hypotheses imply $a=1/5=0.2$ and $b=1/1.25=0.8$, so $a+b=1$. Conditional on the negative result at month 1, the likelihood is the probability of remaining in $S$ at month 2 and moving to $I$ by month 3. The supplied <matrix exponential> gives
$$
P_{SS}(1)=0.8+0.2e^{-1},
\qquad
P_{SI}(1)=0.2(1-e^{-1}).
$$
The <Markov property> therefore gives
$$
P_{SS}(1)P_{SI}(1)
=(0.8+0.2e^{-1})0.2(1-e^{-1})
=0.16-0.12e^{-1}-0.04e^{-2}.
$$