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18.3. An Adaptive SIS Model
In order to study the interplay of epidemic spreading and topological evolution
we consider a network of N nodes and L links. Each of the nodes represents
an individual while each of the links represents a contact over which the disease
can potentially spread. In epidemiological models discrete states are generally as-
signed to the individuals that signify the stage of the disease. In the simplest
case there are only two of these states called S, for susceptible, and I, for infected.
Thus the network eectively consists of boolean nodes, and the links can be de-
noted as SS-links, SI-links, and II-links according to the states of the nodes that
they connect.
Susceptible individuals can become infected if they are in contact with an in-
fected individual. The transmission of the disease along a given SI-link is assumed
to occur at a rate p. Once an individual has been infected she has a chance to
recover, which happens at a rate r and immediately returns the individual to the
susceptible state. Together these processes of infection and recovery constitute the
dynamical rules for a so-called SIS-model, a standard model of epidemiology. In the
model proposed in Gross et al. (2006) another process is included: If a susceptible
individual is connected to an infected individual she may want to break the link and
instead establish a new link to another susceptible individual. On a given SI-link
this rewiring occurs at a rate rate w.
Rewiring, rather than cutting links, captures that the number of social contacts
cannot be reduced arbitrarily; If you knew that the owner of your neighborhood
bakery has some infectious disease, you might decide to buy your bread somewhere
else while stopping to eat entirely is usually not the preferred option.
In the adaptive SIS-model the rewiring process has been introduced `optimisti-
cally': Only susceptible nodes rewire, and they manage unerringly to rewire to a
node that is also susceptible. Under these conditions rewiring always reduces the
number of links that are accessible for epidemic spreading and therefore the preva-
lence of the disease, i.e., the density of infected, is always reduced by this form of
rewiring behavior.
Despite the positive primary eect of rewiring, it has also some adverse con-
sequences, that may limit the benet from rewiring if the epidemic is already es-
tablished in the population. Figure 18.2 shows results from simulations that have
previously reported in Gross et al. (2006). The results were obtained by explicit
individual-based simulation of a network with N = 10
5
nodes and L = 10
6
links. In
this case the average degree of a node ishki= 2L=N = 20, as every link connects
to two nodes. The gure shows in the bottom row the degree distributions of sus-
ceptible and infected nodes and, in the top row, the mean degree of the neighbors
hk
nn
iof a node with given degree. Let us rst consider the left and center column
which correspond to (1) frozen network topology w = 0 and (2) frozen epidemic
dynamics p = r = 0.
