  We investigate the expected time to extinction in the
susceptible-infectious-susceptible (SIS) model of disease spreading. Rather
than using stochastic simulations, or asymptotic calculations in network
models, we solve the extinction time exactly for all connected graphs with
three to eight vertices. This approach enables us to discover descriptive
relations that would be impossible with stochastic simulations. It also helps
us discovering graphs and configurations of S and I with anomalous behaviors
with respect to disease spreading. We find that for large transmission rates
the extinction time is independent of the configurations, just dependent on the
graph. In this limit, the number of vertices and edges determine the extinction
time very accurately (deviations primarily coming from the fluctuations in
degrees). We find that the rankings of configurations with respect to
extinction times at low and high transmission rates are correlated at low
prevalences and negatively correlated for high prevalences. The most important
structural factor determining this ranking is the degrees of the infectious
vertices.
