Biology Reference
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back to the susceptible state with rate m , modeling the
possibility of reinfection.
A cornerstone of epidemic processes is the presence of
the so-called epidemic threshold [19] . In a fully homoge-
neous population the behavior of the SIR model is
controlled by the reproductive number R 0 ¼ b / m ,where
b¼l<
of dynamic self-organization and the lack of characteristic
scales
35] .In
particular, the various statistical distributions character-
izing social networks are generally heavy-tailed, skewed,
and varying over several orders of magnitude. This is a very
peculiar feature typical of many natural and artificial
complex networks, characterized by virtually infinite
degree fluctuations, where the degree k of a given node
represents its number of connections to other nodes. In
contrast to regular lattices and homogeneous graphs char-
acterized by nodes having a typical degree k close to the
average
typical hallmarks of complex systems [31
e
e
is the per-capita spreading rate, which takes into
account the average number of contacts k of each individual.
The reproductive number simply identifies the average
number of secondary cases generated by a primary case in
an entirely susceptible population and defines the epidemic
threshold such that only if R 0
k
>
1 ( b ε m ) can epidemics
reach an endemic state and spread into a closed population.
The SIS and SIR models are indeed characterized by
a threshold defining a transition between two very different
regimes. These regimes are determined by the values of the
disease parameters, and characterized by the global
parameter i N which identifies the density of infected indi-
viduals (nodes in a network) in the infinite time limit. In the
limit of an infinitely large population, this density is zero
below the threshold and assumes a finite value above the
threshold. In this perspective we can consider the epidemic
threshold as the tipping point of the system. Below the
critical point the system relaxes into a frozen state with null
dynamics
, such networks are structured in a hierarchy
of nodes, with a few nodes having very large connectivity
<
k
>
e
the hubs
while the vast majority of nodes have smaller
degrees (see Chapter 9). This feature usually finds its
signature in a heavy-tailed degree distribution, often
approximated by a power-law behavior of the form P(k)
e
f
x - g , which implies a non-negligible probability of finding
vertices with very large degree [32
35] .
The presence of large-scale fluctuations virtually acting
at all scales of the network connectivity pattern calls for
a mathematical analysis where the variables characterizing
each node of the network are explicitly entering the
description of the system. Unfortunately, the general solu-
tion, handling the master equation of the system, is hardly if
ever achievable even for very simple dynamical processes.
For this reason, a viable theoretical approach considers the
use of techniques such as mean-field and deterministic
continuum approximations, which usually provide the
understanding of the basic phenomenology and phase
diagram of the process under study. In both cases the
heterogeneous nature of the network connectivity pattern is
introduced by aggregating variables according to a degree
block formalismwhich assumes that all nodes with the same
degree
e
the healthy phase. Above this point, a dynam-
ical state characterized by a macroscopic number of infected
individuals sets in, defining an infected phase ( Figure 27.3 ).
The above results are generally obtained by using the
so-called homogenous assumption in which all individuals
in the populations are considered statistically equivalent
and no structure (spatiotemporal, connectivity pattern, etc.)
is included in the system description.
One of the most important features affecting dynamical
processes in real-world networks, however, is the presence
e
38] . This
assumption allows the grouping of nodes in degree classes,
yielding a convenient representation of the system. For
instance, if for each node i we associate a corresponding
state s i characterizing its dynamical state, a convenient
representation of the system is provided by the quantities S k
that indicates the number of nodes of degree k in the
dynamical state s ¼
k
are
statistically equivalent
[36
e
s and the corresponding degree block
density of nodes of degree k in the state s
S k
V k
where V k is the number of nodes of degree k. Finally the
global averages on the network are then given by the
expressions
s k ¼
X
r s ¼
P
ð
k
Þ
s k
k
where r s is the probability that any given node is in the
state s. This formalism is extremely convenient in networks
FIGURE 27.3 Phase diagram. Above R 0 ¼ 1 the total number of
infected individuals becomes a finite fraction of the population size.
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