  We study epidemic spreading according to a
\emph{Susceptible-Infectious-Recovered} (for short, \emph{SIR}) network model
known as the {\em Reed-Frost} model, and we establish sharp thresholds for two
generative models of {\em one-dimensional small-world graphs}, in which graphs
are obtained by adding random edges to a cycle.
  In $3$-regular graphs obtained as the union of a cycle and a random perfect
matching, we show that there is a sharp threshold at $.5$ for the contagion
probability along edges.
  In graphs obtained as the union of a cycle and of a $\mathcal{G}_{n,c/n}$
Erd\H{o}s-R\'enyi random graph with edge probability $c/n$, we show that there
is a sharp threshold $p_c$ for the contagion probability: the value of $p_c$
turns out to be $\sqrt 2 -1\approx .41$ for the sparse case $c=1$ yielding an
expected node degree similar to the random $3$-regular graphs above.
  In both models, below the threshold we prove that the infection only affects
$\mathcal{O}(\log n)$ nodes, and that above the threshold it affects
$\Omega(n)$ nodes.
  These are the first fully rigorous results establishing a phase transition
for SIR models (and equivalent percolation problems) in small-world graphs.
Although one-dimensional small-world graphs are an idealized and unrealistic
network model, a number of realistic qualitative phenomena emerge from our
analysis, including the spread of the disease through a sequence of local
outbreaks, the danger posed by random connections, and the effect of
super-spreader events.
