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Fig. 18.5. Bifurcation diagram showing that awareness-dependent rewiring gives rise to a large
oscillatory region. The rewiring rate is assumed to change proportional to the prevalence so that
w = w
0
I. Lines mark branches of stable steady states (solid) and saddle points (dashed/dotted).
The white line tentatively marks a branch of saddle limit cycles. Shaded regions indicate ranges
of I observed during long individual-based simulations in the neighborhood of the attractive limit
cycle (light gray) and of stable stationary solutions (dark gray). Computation of the Jacobian
eigenvalues reveal a subcritical Hopf bifurcation (A), two fold bifurcations (C,E) and a transcritical
bifurcation (F). One can suspect the presence of a fold bifurcation of cycles (B) and a homoclinic
bifurcation (D). Inset: time series on the limit cycle attractor at p = 0:0006. Other parameters:
w
0
= 0:06, r = 0:0002. Figure reprinted from Gross and Kevrekidis (2008).
to launch the epidemic into the growth phase and the disease goes extinct. Hence
only cycles of relatively small amplitude at high disease prevalence were observed
in the present model.
In many subsequent works no cyclic regime was found for two reasons: First, the
basin of attraction of the limit cycle is relatively small. Therefore in simulations
stochastic uctuations can cause the system to depart from the cycle unless the
number of simulated nodes is relatively high (e.g., N10
6
). Moreover, in some
model variants the SI-links are rewired to random targets and not just to susceptible
nodes. As the density of infected nodes increases, so does the quota of unsuccessful
rewiring, in which a susceptible node rewires an SI-link link from one infected node
to another. Therefore, the epidemic-suppressing eect of rewiring becomes weaker
with increasing density of infected. Rewiring can in this case no longer control
the epidemic at the high point of the cycle and hence oscillations are avoided at
the cost of higher prevalence. By contrast if there is an eect that increases the
eciency of rewiring with increasing density of infected then oscillatory behavior
is promoted. For instance in Gross and Kevrekidis (2008) it was assumed that the
awareness of the population is increased if the density of infected is high and thus
rewiring activity is increased. In this case the authors nd robust large-amplitude
oscillations over a considerable parameter range (see Fig. 18.5). Some evidence
suggests that such awareness-driven cycles occur in nature (Grassly et al., 2005).
