  We have performed individual-based lattice simulations of SIR and SEIR
dynamics to investigate both the short and long-term dynamics of childhood
epidemics. In our model, infection takes place through a combination of local
and long-range contacts, in practice generating a dynamic small-world network.
Sustained oscillations emerge with a period much larger than the duration of
infection. We found that the network topology has a strong impact on the
amplitude of oscillations and in the level of persistence. Diseases do not
spread very effectively through local contacts. This can be seen by measuring
an {\em effective} transmission rate $\beta_{\mbox {\scriptsize eff}}$ as well
as the basic reproductive rate $R_0$. These quantities are lower in the
small-world network than in an homogeneously mixed population, whereas the
average age at infection is higher.
