  This paper is concerned with a stochastic model for the spread of an SEIR
(susceptible -> exposed (=latent) -> infective -> removed) epidemic with a
contact tracing scheme, in which removed individuals may name some of their
infectious contacts, who are then removed if they have not been already after
some tracing delay. The epidemic is analysed via an approximating, modified
birth-death process, for which a type-reproduction number is derived in terms
of unnamed individuals, that is shown to be infinite when the contact rate is
sufficiently large. We obtain explicit results under the assumption of either
constant or exponentially distributed infectious periods, including the
epidemic extinction probability in the former case. Numerical illustrations
show that, while the distributions of latent periods and delays have an effect
on the spread of the epidemic, the assumption of whether the delays experienced
by individuals infected by the same individual are of the same or independent
length makes little difference.
