  For many infectious diseases, a small-world network on an underlying regular
lattice is a suitable simplified model for the contact structure of the host
population. It is well known that the contact network, described in this
setting by a single parameter, the small-world parameter $p$, plays an
important role both in the short term and in the long term dynamics of epidemic
spread. We have studied the effect of the network structure on models of immune
for life diseases and found that in addition to the reduction of the effective
transmission rate, through the screening of infectives, spatial correlations
may strongly enhance the stochastic fluctuations. As a consequence, time series
of unforced Susceptible-Exposed-Infected-Recovered (SEIR) models provide
patterns of recurrent epidemics with realistic amplitudes, suggesting that
these models together with complex networks of contacts are the key ingredients
to describe the prevaccination dynamical patterns of diseases such as measles
and pertussis. We have also studied the role of the host contact strucuture in
pathogen antigenic variation, through its effect on the final outcome of an
invasion by a viral strain of a population where a very similar virus is
endemic. Similar viral strains are modelled by the same infection and
reinfection parameters, and by a given degree of cross immunity that represents
the antigenic distance between the competing strains. We have found, somewhat
surprisingly, that clustering on the network decreases the potential to sustain
pathogen diversity.
