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We consider extinction times for a class of birth-death processes commonlyfound in applications, where there is a control parameter which determineswhether the population quickly becomes extinct, or rather persists for a longtime. We give an exact expression for the discrete case and its asymptoticexpansion for large values of the population. We have results below thethreshold, at the threshold, and above the threshold (where there is aquasi-stationary state and the extinction time is very long.) We show that theFokker-Planck approximation is valid only quite near the threshold. We compareour analytical results to numerical simulations for the SIS epidemic model,which is in the class that we treat. This is an interesting example of thedelicate relationship between discrete and continuum treatments of the sameproblem.

Extinction Times for Birth-Death Processes: Exact Results, Continuum Asymptotics, and the Failure of the Fokker--Planck Approximation