  We study continuous-time birth-death type processes, where individuals have
independent and identically distributed lifetimes, according to a random
variable Q, with E[Q]=1, and where the birth rate if the population is
currently in state (has size) n is \alpha(n). We focus on two important
examples, namely \alpha(n)=\lambda n being a branching process, and
\alpha(n)=\lambda n(N-n)/N which corresponds to an SIS epidemic model in a
homogeneously mixing community of fixed size N. The processes are assumed to
start with a single individual, i.e. in state 1. Let T, A_n, C and S denote the
(random) time to extinction, the total time spent in state $n$, the total
number of individuals ever alive and the sum of the lifetimes of all
individuals in the birth-death process, respectively. The main results of the
paper give expressions for the expectation of all these quantities, and shows
that these expectations are insensitive to the distribution of Q. We also
derive an asymptotic expression for the expected time to extinction of the SIS
epidemic, but now starting at the endemic state, which is not independent of
the distribution of Q. The results are also applied to the household SIS
epidemic, showing that its threshold parameter R_* is insensitive to the
distribution of Q, contrary to the household SIR epidemic, for which R_* does
depend on Q.
