  Much of the research on the behavior of the SIS model on networks has
concerned the infinite size limit; in particular the phase transition between a
state where outbreaks can reach a finite fraction of the population, and a
state where only a finite number would be infected. For finite networks, there
is also a dynamic transition---the immortality transition---when the
per-contact transmission probability $\lambda$ reaches one. If $\lambda < 1$,
the probability that an outbreak will survive by an observation time $t$ tends
to zero as $t \rightarrow \infty$; if $\lambda = 1$, this probability is one.
We show that treating $\lambda = 1$ as a critical point predicts the
$\lambda$-dependence of the survival probability also for more moderate
$\lambda$-values. The exponent, however, depends on the underlying network.
This fact could, by measuring how a vertex' deletion changes the exponent, be
used to evaluate the role of a vertex in the outbreak. Our work also confirms
an extremely clear separation between the early die-off (from the outbreak
failing to take hold in the population) and the later extinctions
(corresponding to rare stochastic events of several consecutive transmission
events failing to occur).
