  In the Susceptible-Infectious-Recovered (SIR) model of disease spreading, the
time to extinction of the epidemics happens at an intermediate value of the
per-contact transmission probability. Too contagious infections burn out fast
in the population. Infections that are not contagious enough die out before
they spread to a large fraction of people. We characterize how the maximal
extinction time in SIR simulations on networks depend on the network structure.
For example we find that the average distances in isolated components, weighted
by the component size, is a good predictor of the maximal time to extinction.
Furthermore, the transmission probability giving the longest outbreaks is
larger than, but otherwise seemingly independent of, the epidemic threshold.
