  We study a variant of the classical Markovian logistic SIS epidemic model on
a complete graph, which has the additional feature that healthy individuals can
become infected without contacting an infected member of the population. This
additional ``self-infection'' is used to model situations where there is an
unknown source of infection or an external disease reservoir, such as an animal
carrier population. In contrast to the classical logistic SIS epidemic model,
the version with self-infection has a non-degenerate stationary distribution,
and we show that it exhibits the cutoff phenomenon, which is a sharp transition
in time from one to zero of the total variation distance to stationarity. While
this result is interesting in its own right, an additional contribution of our
work is that the proof illustrates a recently formalised methodology of
Barbour, Brightwell and Luczak, which can be used to show cutoff via a
combination of concentration of measure inequalities and coupling techniques.
