Social Media Mining: An Introduction

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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)