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real world social networks (Caldarelli, 2007). But, is it safe to assume that these
properties will remain unchanged in the face of a major epidemic? The example of
SARS has shown that epidemics can induce behavioral changes that feed back to
the contact network (Omi, 2006). Including this feedback in a model gives rise to
an adaptive network: a system on which the dynamics of the nodes depends on the
network topology, while the evolution of the network topology depends on the state
of the nodes (Gross and Blasius, 2008).
While the dynamics OF networks and the dynamics ON networks have been
studied for a long time in physics (Albert and Barabasi, 2002; Dorogovtsev and
Mendes, 2003; Newman, 2003; Boccaletti et al., 2006; Newman et al., 2006), adap-
tive networks have only very recently come into focus (Gross and Blasius, 2008).
The dynamical interplay between the state and topology has been shown to give
rise to several phenomena: In particular adaptive networks have been shown to
self-organize robustly to critical states (Bornholdt and Rohlf, 2000) and exhibit the
spontaneous emergence of distinct classes of nodes (Ito and Kaneko, 2002) and com-
plex topologies (Holme and Ghoshal, 2006; Rosvall and Sneppen, 2006) based on
simple local rules. Moreover, in adaptive networks bifurcations (Gross et al., 2006)
and phase transitions (Holme and Newman, 2007) appear that involve local as well
as topological degrees of freedom.
In this chapter I discuss a simple conceptual adaptive network model of epidemic
spreading. It is shown that, even in this very simple model, adaptive feedback
leads to qualitative changes in the dynamics. The core of the chapter is formed
by results that have been previously published in Gross et al. (2006). In contrast
to the original publication the results are discussed in the context of subsequent
works (Zanette, 2007; Shaw and Schwartz, 2008; Gross and Kevrekidis, 2008; Risau-
Gusman and Zanette, 2008; Zanette and Risau-Gusman, 2008) which provide a
broader perspective.
The chapter starts in Sec. 18.2 with a, necessarily brief, introduction to epi-
demics on networks. The subsequent section, Sec. 18.3, introduces a simple model
of epidemics on adaptive networks. Furthermore, numerical results are discussed
that show the adaptive response of the contact network to the emergence of the
disease. Section 18.4 contains a detailed introduction to the moment closure ap-
proximation which is then used to study the dynamics of the adaptive network
analytically on an emergent level. In the subsequent section, Sec. 18.5, the focus
shifts back to the detailed level as we discuss the mechanisms behind the observed
dynamics in the context of subsequent investigations. The nal section, Sec. 18.6,
summarizes the results.
18.2. Epidemics on Networks
Traditionally epidemics are modeled in the mean eld limit of a well-mixed popu-
lation or by systems of partial dierential equations that account for physical space
