Social Media Mining: An Introduction

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CUUS2079-07 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 17:17


7.4 Epidemics 203

SI

β1

Figure 7.10. SI Model.

that when an infected individual meets a susceptible individual the disease
is being spread with probability 1 (this can be generalized to other values).
Figure7.10demonstrates the SI model and the transition between states that
happens in this model for individuals. The value over the arrow shows that
each susceptible individual meets at leastβIinfected individuals during
the next time step.
Given this situation, infected individuals will meetβNpeople on aver-
age. We know from this set that only the fractionS/Nwill be susceptible
and that the rest are infected already. So, each infected individual will
infectβNS/N=βSothers. SinceIindividuals are infected,βISwill be
infected in the next time step. This means that the number of susceptible
individuals will be reduced by this factor as well. So, to get different val-
ues ofSandIat different times, we can solve the following differential
equations:
dS
dt

=−βIS, (7.27)

dI
dt

=βIS. (7.28)

SinceS+I=Nat all times, we can eliminate one equation by replacing
SwithN−I:
dI
dt

=βI(N−I). (7.29)

The solution to this differential equation is called thelogistic growth
function,

I(t)=

NI 0 eβt
N+I 0 (eβt−1)

, (7.30)


whereI 0 is the number of individuals infected at time 0. In general, analyz-
ing epidemics in terms of the number of infected individuals has nominal
generalization power. To address this limitation, we can consider infected
fractions. We therefore substitutei 0 =IN^0 in the previous equation,

i(t)=

i 0 eβt
1 +i 0 (eβt−1)

. (7.31)

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