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Another phase that often appears in epidemic models is the R-phase, which
denotes either recovered individuals that have acquired (possibly temporary) im-
munity or removed nodes that have died from the disease. However, in either case
R-nodes should not aect the invasion threshold, as their density vanishes in the
relevant limit.
The impact of additional epidemic states and rewiring rules was investigated in
Shaw and Schwartz (2008); Risau-Gusman and Zanette (2008); Zanette and Risau-
Gusman (2008). While it was found that rewiring can have detrimental eects
under certain conditions these were relatively mild.
Phenomenon (two), the emergence of the persistence threshold is clearly re-
lated to the appearance of highly connected susceptible nodes; Below the invasion
threshold, an infectious disease can persist on the networks if a large fraction of
the population is already infected. In this case SI-links are rewired into a relatively
small population of susceptible nodes. The number of SS-links per susceptible node
increases correspondingly. Therefore many SI-links are created in every infection
event, which almost compensates the SI-links lost due to rewiring. One could say
that, if the susceptible subpopulation is too small, rewiring becomes ineective as
it is unlikely that the rewired link will remain out of reach of the disease for a
signicant time.
The mechanism described above is robust as long as links are rewired and not just
cut. One could argue that few real world diseases reach suciently high prevalence
for the eect to be of relevance. However, in particular the new emerging diseases
on which our main focus lies are known to reach locally high densities of infected
(Karlen, 1995).
The third phenomenon, the existence of an oscillatory phase, is caused by the
interplay of the two opposing eects of rewiring; First, the number of SI-links are
reduced by rewiring. However, even as the density of infected nodes decreases a
densely linked cluster of susceptible nodes is build up. Eventually the epidemic
invades this cluster which causes a sharp increase in the density of infected nodes,
thus completing the cycle. What appears as a smooth oscillation on the level of the
ODE is therefore really a series of outbreaks and avalanches if considered on the
detailed level.
At rst glance it seems surprising that in the oscillatory phase the primary
epidemic-suppressing eect of rewiring governs the dynamics when the density of
infected is high, while the secondary epidemic-promoting eect rules the increase
that sets in at the low point. However, as it is often the case in autonomous
oscillations, the cyclic behavior becomes possible due to the dierences in the time
scales on which the two eects act; The primary reduction of SI-links is immediately
eective, but the build-up of links in the susceptible subpopulation requires some
time before it can aect the dynamics. Nevertheless the mechanism that causes the
oscillatory behavior is relatively fragile. If the density of infected at the low point
of the cycle becomes too low the epidemic-promoting eect of rewiring is too weak
