what-when-how
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network topology on the rate and patterns of disease spread. hey [8] ana-
lyzed real data from computer virus infections and defined a dynamical model
for the spreading of infections on scale-free networks that could help in the
understanding of other spreading phenomena on communication and social
networks. Yamir Moreno et al. [9] exploited the mean-field-like rate equations
describing the system and studied the susceptible-infected-removed epide-
miological model on assortative networks, providing numerical evidence of
the absence of epidemic thresholds. J. G´omez-Gardenes et al. [10] studied an
immunization strategy that could be used for designing and deploying a digital
immune system.
3.1.2 Rumor Models
Based on the epidemic model, Daley and Kendal [11] proposed the basic rumor
model (also called the DK model). he basic DK rumor model is deined as fol-
lows. Each of the
N
elements of the network can be in one of three possible states.
Following the original terminology [12], these three classes correspond to
ignorant
(denoted by
I
),
spreader
(
S
), and
stifler.
(
R
) nodes. Ignorants are those individu-
als who have not heard the rumor and hence are susceptible to being informed.
Spreaders comprise active individuals who are spreading the rumor. Finally, stiflers
are those who have heard the rumor but are no longer spreading it.
In the homogeneous mixing hypothesis, the DK model can be described in
terms of the densities of ignorants, spreaders, and stiflers, that is,
i
(
t
),
s
(
t
), and
r
(
t
),
respectively, as a function of time:
di t
dt
( )
= −
λ
ki t s t
( ) ( ),
ds t
dt
( )
=
λ
ki t s t
( ) ( )
−
α
t
ks
(
)[ ( )
s t
+
r t
( )],
(3.2)
dr t
dt
( )
=
α
+
ks t
( )[ ( )
s t
r t
( )]
where
k
is the number of contacts per unit time that is supposed to be constant
for the whole population; and when an ignorant meets a spreader, it turns itself into
a new spreader at rate
λ
; spreaders become stiflers with probability
α
if they are in
contact with another spreader or a stifler. he decay of spreading may be due to a
process of “forgetting” or because spreaders learn that the rumor has lost its “news
value.”
Recently, several investigators [13-16] have explored this model on top of com-
plex network topology. he heterogeneity of the connectivity distribution inherent
to scale-free networks makes it necessary to take into account that nodes could not
only be in three different states but they also belong to different connectivity classes
