  We model an epidemic where the per-person infectiousness in a network of
geographic localities changes with the total number of active cases. This would
happen as people adopt more stringent non-pharmaceutical precautions when the
population has a larger number of active cases. We show that there exists a
sharp threshold such that when the curing rate for the infection is above this
threshold, the mean time for the epidemic to die out is logarithmic in the
initial infection size, whereas when the curing rate is below this threshold,
the mean time for epidemic extinction is infinite. We also show that when the
per-person infectiousness goes to zero asymptotically as a function of the
number of active cases, the mean extinction times all have the same asymptote
independent of network structure. Simulations bear out these results, while
also demonstrating that if the per-person infectiousness is large when the
epidemic size is small (i.e., the precautions are lax when the epidemic is
small and only get stringent after the epidemic has become large), it might
take a very long time for the epidemic to die out. We also provide some
analytical insight into these observations.

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