  We obtain tight thresholds for bond percolation on one-dimensional
small-world graphs, and apply such results to obtain tight thresholds for the
\emph{Independent Cascade} process and the \emph{Reed-Frost} process in such
graphs. These are the first fully rigorous results establishing a phase
transition for bond percolation and SIR epidemic processes in small-world
graphs. Although one-dimensional small-world graphs are an idealized and
unrealistic network model, a number of realistic qualitative epidemiological
phenomena emerge from our analysis, including the epidemic spread through a
sequence of local outbreaks, the danger posed by random connections, and the
effect of super-spreader events.
