We have studied in detail an epidemiological model on a small-world network as constructed by Watts and Strogats (1998). The model was first studied by Kuperman and Abramson (2001). They found that as the parameter which defines the network is changed towards random network, the dynamical system bifurcates from an endemic situation towards a cyclic epidemies. This result suggests the importance of the network topology in the transmission and outbreak of cyclic epidemies.
Our results show that the governing transition is a Hopf bifurcation, and we show how the transition behaves varying the topology, the infectious time scale, and the recovery time scale.
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