  We consider epidemic extinction in finite networks with broad variation in
local connectivity. Generalizing the theory of large fluctuations to random
networks with a given degree distribution, we are able to predict the most
probable, or optimal, paths to extinction in various configurations, including
truncated power-laws. We find that paths for heterogeneous networks follow a
limiting form in which infection first decreases in low-degree nodes, which
triggers a rapid extinction in high- degree nodes, and finishes with a residual
low-degree extinction. The usefulness of the approach is further demonstrated
through optimal control strategies that leverage finite-size fluctuations.
Interestingly, we find that the optimal control is a mix of treating both high
and low-degree nodes based on large-fluctuation theoretical predictions.
