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node, [SS]; and the density of II-Links per Node, [II]. The density of susceptible
nodes, S, and the density of SI-Links, [SI], are then given by the conservation
relations S + I = 1 and [SS] + [SI] + [II] =hki. An advantage of this normalization
is that we can write all subsequent equations as if we were dealing with a number
of individual nodes and links instead of densities.
Let us start by writing a balance equation for the density of infected nodes.
Infection events occur at the rate p[SI] increasing the number of infected nodes by
one; Recovery events occur at a rate rI and reduce the number of infected nodes
by one. This leads to
d
dt
I = p[SI]rI :
(18.1)
The equation contains the (presently unknown) variable [SI] and therefore does
not yet constitute a closed model. One way to close the model were a mean eld
approximation, in which the density of SI-Links is approximated by [SI]hkiSI.
However, in the present case this procedure is not feasible: Rewiring does not alter
the number of infected and hence does not show up in Eq. (18.1). Thus the mean-
eld approximation is not able to capture the eect of rewiring. Instead, we will
treat [SI], [SS], and [II] as dynamical variables and capture their dynamics by
additional balance equations. This approach is often called moment expansion as
the link densities can be thought of as the rst moments of the network.
For the sake of conciseness it is advantageous to write balance equations only for
the densities of SS- and II-links and obtain the density of SI-links by the conservation
relation stated above. First the II-links: A recovery event can destroys II-links if
the recovering node was part of such links. The expected number of II-links in
which a given infected node is involved is 2[II]=I. (Here, the two appears since a
single II-link connects to two infected nodes.) Taking the rate of recovery events
into account, the total rate at which II-links are destroyed is simply 2r[II].
To derive the rate at which II-links are created is only slightly more involved.
In an infection event the infection spreads across a link, converting the respective
link into an II-link. Therefore every infection event will create at least one II-
link. However, additional II-links may be created if the newly infected node has
other infected neighbors in addition to the infecting node. In this case the newly
infected node was previously the susceptible node in one or more ISI-triplets. In
the following we denote the density of triplets with a given sequence of states A,
B, C as [ABC]. Using this notation we can write the number of II-links that are
created in an infection event as 1 + [ISI]=[SI]. In this expression the `1' represents
the link over which the infection spreads while the second term counts the number
of ISI-triplets that run through this link. Given this relation we can write the total
rate at which II-links are created as p[SI](1 + [ISI]=[SI]) = p([SI] + [ISI]).
Now the SS-links: Following a similar reasoning as above we nd that infection
destroys SS-links at the rate p[SSI]. Likewise SS-links are created by recovery at
the rate r[SI]. In addition SS-links can also be created by rewiring of SI-links. Since
