Expected absorption time in an illness-death model ($E_1(T_D)=1/(a+b)+a/((a+b)c)$)

With initial exit rates $a$ to the advanced living state and $b$ directly to death, and death rate $c>0$ from the advanced state, the expected time to death from state 1 is $1/(a+b)+a/((a+b)c)$. The first term is initial holding time; the second weights the advanced-state holding time by the probability of reaching it.