Epidemics in small world networks
  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. 
M. M. Telo da Gama and A. Nunes