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 205

probability of contacting others. Similarly, a parameterγin the SIR model

defines how infected people recover, or the recovering probability of an

infected individual in a time periodt.

In terms of differential equations, the SIR model is

dS

dt

=−βIS, (7.32)

dI

dt

=βIS−γI, (7.33)

dR

dt

=γI. (7.34)

Equation7.32is identical to that of the SI model (Equation7.27). Equa-

tion7.33is different from Equation7.28of the SI model by the addition of

the termγI, which defines the number of infected individuals who recov-

ered. These are removed from the infected set and are added to the recovered

ones in Equation7.34. Dividing Equation7.32by Equation7.34, we get

dS

dR

=−

β

γ

S, (7.35)

and by assuming the number of recovered at time 0 is zero (R 0 =0),

log

S 0

S

=

β

γ

R. (7.36)

S 0 =Se

βγR

(7.37)

S=S 0 e−

βγR

(7.38)

SinceI+S+R=N, we replaceIin Equation7.34,

dR

dt

=γ(N−S−R). (7.39)

Now combining Equations7.38and7.39,

dR

dt

=γ(N−S 0 e−

βγR

−R). (7.40)

If we solve this equation forR, then we can determineSfrom7.38and

IfromI=N−R−S. The solution forRcan be computed by solving

the following integration:

t=

1

γ

∫R

0

dx

N−S 0 e−

βγx

−x

. (7.41)