Social Media Mining: An Introduction

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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