  The standard pair approximation equations (PA) for the
Susceptible-Infective-Recovered-Susceptible (SIRS) model of infection spread on
a network of homogeneous degree $k$ predict a thin phase of sustained
oscillations for parameter values that correspond to diseases that confer long
lasting immunity. Here we present a study of the dependence of this oscillatory
phase on the parameter $k$ and of its relevance to understand the behaviour of
simulations on networks. For $k=4$, we compare the phase diagram of the PA
model with the results of simulations on regular random graphs (RRG) of the
same degree. We show that for parameter values in the oscillatory phase, and
even for large system sizes, the simulations either die out or exhibit damped
oscillations, depending on the initial conditions. This failure of the standard
PA model to capture the qualitative behaviour of the simulations on large RRGs
is currently being investigated.
