Social Media Mining: An Introduction

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


208 Information Diffusion in Social Media

recovered will lose immunity after a certain period of time and will become
susceptible again. A new parameter has been added to the modelλthat
defines the probability of losing immunity for a recovered individual. The
set of differential equations that describe this model is
dS
dt

=λR−βIS, (7.45)

dI
dt

=βIS−γI, (7.46)

dR
dt

=γI−λR. (7.47)

Like the SIR model, this model has no closed-form solution, so numer-
ical integration can be used. Figure7.17demonstrates a simulation of the
SIRS model with given parameters of choice. As observed, the simulation
outcome is similar to the SIR model simulation (see Figure7.13). The major
difference is that in the SIRS, the number of susceptible and recovered indi-
viduals changes non-monotonically over time. For example, in SIRS, the
number of susceptible individuals decreases over time, but after reaching
the minimum count, starts increasing again. On the contrary, in the SIR,
both susceptible individuals and recovered individuals change monotoni-
cally, with the number of susceptible individuals decreasing over time and
that of recovered individuals increasing over time. In both SIR and SIRS,
the infected population changes non-monotonically.

7.4.6 Intervention

A pressing question in any pandemic or epidemic outbreak is how to stop
the process. In this section, we discuss epidemic intervention based on a
recent discovery by Nicholas A.Christakis [2004]. In any epidemic out-
break, infected individuals infect susceptible individuals. Although in this
chapter we discussed random infection in the real world, what actually
takes place is quite different. Infected individuals have a limited number
of contacts and can only infect them if said contacts are susceptible. A
well-connected infected individual is more dangerous to the epidemic out-
break than someone who has no contacts. In other words, the epidemic
takes place in a network. Unfortunately, it is often difficult to trace these
contacts and outline the contact network. If this was possible, the best way
to intervene with the epidemic outbreak would be to vaccinate the highly
connected nodes and stop the epidemic. This would result in what is known
asherd immunityand would stop the epidemic outbreak. Herd immunity
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