  Population extinction is a rare event which requires overcoming an effective
barrier. We show that the extinction rate can be fragile: a small change in the
system parameters leads to an exponentially strong change of the rate, with the
barrier height depending on the parameters nonanalytically. General conditions
of the fragility are established. The fragility is found in one of the
best-known models of epidemiology, the SIS model. The analytical expressions
are compared with simulations.
