Social Media Mining: An Introduction

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

204 Information Diffusion in Social Media

60000

New Cases of AIDS in The United States

40000

New Cases 20000

1980 1983 1986 1989

Year

1992 1995

0

100

80

60

S(t)I(t)

40

20

(^00102030)
t
Population
40 50
(a) SI Model Simulation (b) HIV/AIDS Infected Population Growth
Figure 7.11. SI model simulation compared to the HIV/AIDS growth in the United
States.
Note that in the limit, the SI model infects all the susceptible population
because there is no recovery in the model. Figure7.11(a) depicts the logistic
growth function (infected individuals) and susceptible individuals forN=
100,I 0 =1, andβ= 0 .003. Figure7.11(b) depicts the infected population
for HIV/AIDS for the past 20 years. As observed, the infected population
can be approximated well with the logistic growth function and follows the
SI model. Note that in the HIV/AIDS graph, not everyone is getting infected.
This is because not everyone in the United States is in the susceptible
population, so not everyone will get infected in the end. Moreover, there
are other factors that are far more complex than the details of the SI model
that determine how people get infected with HIV/AIDS.
7.4.3 SIR Model
The SIR model, first introduced byKermack and McKendrick [1932], adds
more detail to the standard SI model. In the SIR model, in addition to theI
andSstates, a recovery stateRis present. Figure7.12depicts the model.
In the SIR model, hosts get infected, remain infected for a while, and then
recover. Once hosts recover (or are removed), they can no longer get infected
and are no longer susceptible. The process by which susceptible individuals
get infected is similar to the SI model, where a parameterβdefines the
SIR
β1 γ
Figure 7.12. SIR Model.