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Fig. 18.3. Bifurcation diagram of the stationary density of the infected I
as a function of the
infection probability p for dierent values of the rewiring rate w. In each diagram I
has been
computed analytically from the ODE system (thin lines). Along the stable branches these results
have been conrmed by the explicit simulation of the full network (circles). Without rewiring only
a single continuous transition occurs at p=0.0001 (a). By contrast, with rewiring a number of
discontinuous transitions, bistability, and hysteresis loops (indicated by arrows) are observed (b),
(c), (d). Fast rewiring (c), (d) leads to the emergence of limit cycles (thick lines indicate the lower
turning point of the cycles), which have been computed numerically with the bifurcation software
AUTO. Parameter: r = 0:002. Figure reprinted from Gross et al. (2006).
bifurcation still occurs it no longer marks the persistence threshold as the stability
of the endemic state is lost in the Hopf bifurcation, before the fold bifurcation is
reached.
At higher rewiring the Hopf bifurcation becomes supercritical. While this bi-
furcation still marks the threshold for stationary persistence of the disease, it gives
rise to a stable limit cycle on which non-stationary persistence is possible at lower
infectiousness. However, the oscillatory parameter region is very narrow. Toward
lower infection rates it is bounded by a fold bifurcation of cycles, which leads to the
extinction of the disease.
The ndings from the bifurcation analysis are summarized in a two-parameter
bifurcation diagram shown in Fig. 18.4. A more detailed understanding can be
gained if one considers what happens on the network level. Indeed I will come back
to this point in the subsequent section. But, before let us discuss some possible
extensions of the moment closure approximation.
