  We consider a Susceptible-Infective-Recovered (SIR) model, where the
mechanism for the renewal of susceptibles is demographic, on a ring with next
nearest neighbour interactions, and a family of correlated pair approximations
(CPA), parametrized by a measure of the relative contributions of loops and
open triplets of the sites involved in the infection process. We have found
that the phase diagram of the CPA, at fixed coordination number, changes
qualitatively as the relative weight of the loops increases, from the phase
diagram of the uncorrelated pair approximation to phase diagrams typical of
one-dimensional systems. In addition, we have performed computer simulations of
the same model and shown that while the CPA with a constant correlation
parameter cannot describe the global behaviour of the model, a reasonable
description of the endemic equilibria as well as of the phase diagram may be
obtained by allowing the parameter to depend on the demographic rate.
