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The bifurcations in a four-variable ODE model of an SIS type epidemic on an\udadaptive network are studied. The model describes the propagation of the epidemic on\uda network where links (or edges) of different type (i.e.\udSI;II\udand\udSS\ud) can be activated\udor deleted according to a simple rule consisting of random link activation and deletion.\udIn the case when\udII\udlinks cannot be neither deleted nor created it is proved that\udthe system can have at most three steady states with the trivial, disease-free steady\udstate being one of them. It is shown that a stable endemic steady state can appear\udthrough a transcritical bifurcation, or a stable and an unstable endemic steady state\udarise as a result of saddle-node bifurcation. Moreover, at the endemic steady state a\udHopf bifurcation may occur giving rise to stable oscillation. The bifurcation curves in\udthe parameter space are determined analytically using the parametric representation\udmethod. For certain parameter regimes or bifurcation types, analytical results based on\udthe ODE model show good agreement when compared to results based on individual-\udbased network simulations. When agreement between the two modelling approaches\udholds, the ODE-based model provides a faster and more reliable tool that can be used\udto explore full spectrum of model behaviour

Detailed study of bifurcations in an epidemic model on a dynamic network