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it is different from disease spreading in that the rumor dynamics is driven by direct
contacts between individuals of different classes. Prior research supports a valuable
basis for modeling and analyzing the information propagation process in online
social networks.
3.1.1 Epidemic Models
he study of epidemic models has attracted the attention of epidemiologists and
grown exponentially since the middle of the 20th Century, so that a tremendous
variety of models has now been formulated, mathematically analyzed, and applied
[2-6]. Epidemic models have been used in planning, implementing, and evaluating
various prevention, therapy, and control programs.
Here, we introduce a traditional model that describes disease spreading through
a population by contacts between infected and healthy individuals: the susceptible
infectedremoved (SIR) model. Other models and generalizations can be found in
References 3-7.
he theoretical approach to epidemic spreading is based on compartmental
models, that is, models in which the individuals in the population are divided into
a set of diferent groups [3, 4]. he SIR model describes diseases resulting in the
immunization or death of infected individuals, and assumes that each individual
can be in one of three possible states: susceptible (denoted by S ), infected ( I ), or
removed ( R ). Susceptible individuals are healthy persons who can catch the disease
if exposed to infected individuals. Once an individual catches the infection, he or
she moves into the infected (and infective) class, and then, after some time, into
the removed class. he model is based on two parameters: the transmission rate λ ,
and the recovery rate μ . he following model is based on the homogeneous mix-
ing hypothesis, meaning that individuals with whom a susceptible individual has
contact are chosen at random from the whole population.
ds t
dt
( )
= −
λ ρ
k
( ) ( ),
t s t
ρ
d t
dt
( )
= −
μρ
+
λ ρ
(3.1)
( )
t
k
( ) (
t s
t
),
dr t
dt
( )
= μρ
( )
t
where s ( t ), ρ ( t ), and r ( t ) are, respectively, the density ( i.e ., the fraction) of suscep-
tible, infected, and removed individuals at time t , and k is the number of contacts
per unit time that is supposed to be constant for the whole population.
More recently, starting with the works by Pastor-Satorras and Vespignani
[8], there has been a burst of activity on understanding the effects of complex
 
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