  Approximating the time to extinction of infection is an important problem in
infection modelling. A variety of different approaches have been proposed in
the literature. We study the performance of a number of such methods, and
characterize their performance in terms of simplicity, accuracy, and
generality. To this end, we consider first the classic stochastic
susceptible-infected-susceptible (SIS) model, and then a multi-dimensional
generalization of this which allows for Erlang distributed infectious periods.
We find that (i) for a below-threshold infection initiated by a small number of
infected individuals, approximation via a linear branching process works well;
(ii) for an above-threshold infection initiated at endemic equilibrium, methods
from Hamiltonian statistical mechanics yield correct asymptotic behaviour as
population size becomes large; (iii) the widely-used Ornstein-Uhlenbeck
diffusion approximation gives a very poor approximation, but may retain some
value for qualitative comparisons in certain cases; (iv) a more detailed
diffusion approximation can give good numerical approximation in certain
circumstances, but does not provide correct large population asymptotic
behaviour, and cannot be relied upon without some form of external validation
(eg simulation studies).
