  For a susceptible-infectious-susceptible (SIS) infection model in a
heterogeneous population, we present simple formulae giving the leading-order
asymptotic (large population) behaviour of the mean persistence time, from an
endemic state to extinction of infection. Our model may be interpreted as
describing an infection spreading through either (i) a population with
heterogeneity in individuals' susceptibility and/or infectiousness; or (ii) a
heterogeneous directed network. Using our asymptotic formulae, we show that
such heterogeneity can only reduce (to leading order) the mean persistence time
compared to a corresponding homogeneous population, and that the greater the
degree of heterogeneity, the more quickly infection will die out.
