  Motivated by the study of controlling (curing) epidemics, we consider the
spread of an SI process on a known graph, where we have a limited budget to use
to transition infected nodes back to the susceptible state (i.e., to cure
nodes). Recent work has demonstrated that under perfect and instantaneous
information (which nodes are/are not infected), the budget required for curing
a graph precisely depends on a combinatorial property called the CutWidth. We
show that this assumption is in fact necessary: even a minor degradation of
perfect information, e.g., a diagnostic test that is 99% accurate, drastically
alters the landscape. Infections that could previously be cured in sublinear
time now may require exponential time, or orderwise larger budget to cure. The
crux of the issue comes down to a tension not present in the full information
case: if a node is suspected (but not certain) to be infected, do we risk
wasting our budget to try to cure an uninfected node, or increase our certainty
by longer observation, at the risk that the infection spreads further? Our
results present fundamental, algorithm-independent bounds that tradeoff budget
required vs. uncertainty.
