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a random-graph-like approximation. It is not a high-degree approximation as is
sometimes held in the literature and therefore holds also in sparse networks.
Setting = 1 we can approximate the density of triplets by [ISI] = [SI]
2
=S,
and following a similar reasoning [SSI] = 2[SS][SI]=S. Substituting these relations
into the balance equations we obtain a closed system of dierential equations
d
dt
I = p[SI]rI ;
(18.5)
dt
[SS] = (r + w)[SI]2p[SI]
[SS]
d
;
(18.6)
S
dt
[II] = p[SI](1 +
[SI]
d
)2r[II] :
(18.7)
S
18.4.2. Dynamics of the adaptive SIS model
Systems of ordinary dierential equations (ODEs) can be studied by the standard
tools of dynamical systems theory. In Gross et al. (2006) the stationary solutions of
these equations and their stability are computed analytically. This analysis reveals
several transitions in which the dynamics changes qualitatively. In the context of
the low-dimensional emergent-level ODEs these transitions appear as bifurcations,
while in the context of the individual-based detailed-level simulations they corre-
spond to phase transitions. Note, that even the equilibrium solutions of the ODE
system correspond in general to highly dynamic states on the detailed level, in which
individual nodes undergo infection and recovery and links are continuously rewired.
In Fig. 18.3 results of the analytical investigation of the system of ODEs are
shown in comparison to results from individual-based simulation of the full network.
Without rewiring, there is only a single, continuous dynamical transition, which
occurs at the epidemic threshold and corresponds to a transcritical bifurcation. As
the rewiring is switched on, this threshold increases. The epidemic threshold still
marks the critical value of infectiousness, p, for the invasion of the disease. However,
another lower threshold, marked by a fold bifurcation point, appears. Above this
threshold an epidemic that is already established in the network can persist (endemic
state). In the following we distinguish between the invasion threshold and the
persistence threshold for epidemics. In contrast to the case without rewiring the
two thresholds are discontinuous (rst order) transitions. Between them a region of
bistability is located, in which both the healthy and the endemic state are stable.
Thus, a hysteresis loop is formed. First order transitions, bistability, and hysteresis
are generic features of the model that can be observed at all nite rewiring rates.
Increasing the rewiring rate further hardly reduces the size of the epidemic in the
endemic state, however both thresholds are shifted toward higher infection rates. At
higher rewiring rates the the nature of the persistence threshold changes. First, a
subcritical Hopf bifurcation emerges. In this bifurcation the stability of a stationary
solution is lost by the interaction with an unstable limit cycle. Although the fold
