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a higher degree and are connected to other susceptible individuals a positive degree
correlation appears if the whole population is considered.
An even stronger eect of state-sensitive rewiring can be observed if rewiring
and epidemic spreading take place on the same time-scale. The right column of
Fig. 18.2 shows that the degree distribution becomes very wide (note the dierent
scale) and a strong positive (assortative) degree correlation appears. To understand
this consider the following: In the case of fast rewiring, the degree of susceptible
nodes increased because of links that were rewired into the susceptible population.
However, this rewiring had to stop once all SI-links had been converted into SS-
links eectively limiting the width of the degree distribution. If both epidemic and
topological dynamics take place on the same timescale either the disease will go
extinct or rewiring will continue indenitely. In the latter case the degree of a
susceptible node growth linearly until it eventually becomes infected, giving rise to
a wide degree distribution.
It is interesting to note that there are several parallels between the adaptive SIS
model studied here and models of opinion formation on adaptive networks (Do and
Gross, 2009). Also in opinion formation the strongest impact of the dynamics on
the topology is observed if topological evolution and local dynamics take place on
approximately the same time scale (Gil and Zanette, 2006; Holme and Newman,
2007). In this case the appearance of a broad degree distribution can be linked
to a single continuous phase transition. In case of the adaptive SIS model one
could suspect the corresponding phase transition is actually the epidemic threshold.
However, as I will show in the next section the situation is slightly more complex.
Finally, note that the shape of the degree distribution is reminiscent of the
one observed by Ito and Kaneko in an adaptive network of coupled oscillators (Ito
and Kaneko, 2002). However, while Ito and Kaneko start with a homogeneous
population that spontaneously divides into two distinct topological classes, it clear
that in the adaptive SIS model these classes correspond to susceptible and infected
nodes. Hence the emergence of the bimodal degree distribution is less surprising in
the present case.
18.4. Extracting Dynamics by Moment Closure
Let us now study the dynamics of the adaptive SIS model with the tools of nonlinear
dynamics. For this purpose we need to derive a low-dimensional emergent-level
description of the system. A convenient tool to achieve this is the so-called moment-
closure approximation which is frequently used in epidemiology (Keeling et al., 1997;
Parham et al., 2008; Peyrard et al., 2008).
18.4.1. Basic moment-expansion of the model
Based on previous section it is reasonable to assume that the state of the system
can be characterized by the density of infected nodes, I; the density of SS-Links per
