Biology Reference
In-Depth Information
HMF level we have a vanishing threshold, implying that for
power-law networks characterized by a divergent second
moment of the degree distribution the disease will always
spread except when g
¼
1
[38]
. For finite networks this
fraction is
1 but still close to this value. It is then
extremely relevant to research activity concerning the
developing of dynamical ad-hoc strategies for network
protection: targeted immunization strategies and targeted
prophylaxis that evolve with time might be particularly
effective in the control of epidemics on heterogeneous
patterns, compared to massive uniform vaccinations or
stationary interventions
[51
<
54]
.
For instance, nodes with high degree are the ones more
likely to spread the disease. Immunizing them through
a targeted scheme is the most efficient way to protect the
network from the disease. Analytical evaluation confirmed
by numerical simulations shows that in heterogeneous
networks targeted schemes allow complete protection from
the disease at the cost of immunizing
e
<
10% of the nodes
[38,51]
.
Although targeted immunization strategies are
extremely powerful they rely on a complete knowledge of
the network structure. Unfortunately, in most real cases this
is partial and limited. To overcome this problem several
methods based on local exploration mechanisms have been
proposed
[51,53,55
FIGURE 27.4
Epidemic progression in a scale-free network. Node
size represents the degree and the color varies between yellow and red to
indicate the time of infection. Blue nodes are never infected and remain
susceptible.
differences between the results for SIR and SIS models
[21]
. While the SIR model results are in good agreement
with the HMF theory also in quenched networks, the SIS
model results can be completely different from those pre-
dicted by the HMF. Exact solutions have recently shown
that quenched networks characterized by a maximum
degree that diverges with the system size, always have
a zero threshold
[21,49]
. In other words, the threshold of
a SIS model is zero also for networks of finite second
moment if the largest degree is a growing function of the
network size, contrary to the indications of the HMF theory.
The reasons behind the differences between the SIR and
SIS models in this context are not really clear and are still
a matter of debate, but are probably related to the absence/
existence of a steady state of infected individuals
[21]
.
Heterogeneous networks are extremely permeable
epidemics attacks
[38,50]
and at the same time uniform
immunizations are clearly not effective in this type of
network. They give the same importance to very connected
nodes and to nodes with very few connections, which
instead have completely different roles in the spreading of
the epidemic. The introduction of fraction g of nodes
immune to the disease, immunized, is equivalent to a simple
rescaling of the per capita spreading rate that becomes
57]
.
In one of the most ingenious strategies a fraction of
nodes is selected at random and each one of them is asked
to point to one of its neighbors. Each of these neighbors is
then immunized. This strategy is based on the heteroge-
neity of the systems. In these types of networks following
links at random gives a higher probability of reaching high
degree nodes that have many links pointing to them
[56]
.
This property is often cited as 'the friendship paradox',
which states that your friends have more friends than you
do
[35,38]
.
Many others variation involving chained versions of the
previous strategy
[57]
, shortest-path of different sizes
[58]
,
and other propagation properties have been proposed
[59]
.
It is important to stress that all of them are based on the
heterogeneous features of the network structure.
e
COMPLEX NETWORKS AND THE LARGE-
SCALE SPREADING OF INFECTIOUS
DISEASES
The network mindset is necessary not only in describing the
connectivity pattern of single individuals in a population.
A simple example is provided by the large-scale description
of epidemic spreading. The spread of the plague epidemic
in the 14th century (the 'Black Death')
[60]
was mainly
a spatial diffusion phenomenon. Historical studies have
established that disease propagation followed a simple
2
b
m
¼
<
k
>
1
k
2
<
>
ð
1
g
Þ
The change in the parameter is just proportional to the
fraction of nodes not immunized. For a SIR model at the
