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simplest framework at hand in which epidemic dynamics and topological evolution
can be combined. The purpose of these investigations was to demonstrate that the
adaptive interplay between local dynamics and topological evolution can give rise
to certain phenomena that can not be observed in static networks. The analysis has
revealed three such phenomena: (1) a signicant shift of the invasion threshold to-
ward higher infection rates; (2) the appearance of a persistence threshold below the
invasion threshold, giving rise to bistability and hysteresis; and (3) the emergence
of an oscillatory phase.
The next logical step is to search for these phenomena in more realistic models.
Indeed, since the original publication of the adaptive SIS-model in Gross et al.
(2006) subsequent works have appeared that add realism by extending the model
in several ways (Zanette, 2007; Shaw and Schwartz, 2008; Gross and Kevrekidis,
2008; Risau-Gusman and Zanette, 2008; Zanette and Risau-Gusman, 2008). In
order to understand the common themes and dierences between these works and
to extrapolate to real world situations, it is conductive to revisit the phenomena
listed above and consider them from an individual-based perspective.
As we have seen in Sec. 18.3 rewiring aects epidemic spreading in two opposing
ways. The primary eect of rewiring is two reduce the number of links through
which the epidemic can spread. But, in time rewiring can lead to the formation of
a large densely-linked cluster of susceptible nodes in which the infection can spread
rapidly once it has been invaded.
For the phenomenon (1), the increase of the invasion threshold, only the pri-
mary eect of rewiring is of importance. By denition, only dynamics that take
place at a low density of infected nodes are relevant for invasion. However, in this
limit the number of links that are rewired is also low and hence cannot lead to a
signicant build-up of connectivity in the susceptible subpopulation. At least under
our optimistic assumptions the epidemic threshold is therefore always increased by
rewiring.
In more realistic scenarios the epidemic threshold could in principle be decreased
by rewiring if rewiring led to an increased density of SI-links. In reality this can hap-
pen in several ways. Susceptible nodes may erroneously rewire SS-links to infected
nodes. Also, infected nodes may want to avoid contact with other infected nodes
and therefore rewire II-links to susceptible nodes. More relevant is probably, the
existence of additional disease states. While there are many infectious diseases that
follow SIS-dynamics, these are mostly too harmless or too easily controlled to trig-
ger a strong rewiring response in the population. Real world diseases that trigger a
stronger behavioral response often have pronounced exposed (E) or asymptomatic
(A) phases. Nodes in these phases appear to be susceptible, but have already been
in contact with the infectious agent, and will eventually progress to the infected
phase. While they are still in the E or A phase they act like susceptible nodes,
i.e., they rewire their links to other susceptible nodes, whom they can subsequently
(E-phase) or immediately (A-phase) infect.
