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CUUS2079-07 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 17:17
7.4 Epidemics 20710080604020(^001020)
S(t)
I(t)
30
t
Population
40 50
Figure 7.15. SIS Model Simulated withS 0 =99,I 0 =1,β= 0 .01, andγ= 0 .1.
By replacingSwithN−Iin Equation7.43,wearriveat
dI
dt
=βI(N−I)−γI=I(βN−γ)−βI^2. (7.44)
WhenβN≤γ, the first term will be negative or zero at most; hence,
the whole term becomes negative. Therefore, in the limit, the valueI(t)
will decrease exponentially to zero. However, whenβN>γ, we will have
a logistic growth function as in the SI model. Having said this, as the
simulation of the SIS model shows in Figure7.15, the model will never
infect everyone. It will reach a steady state, where both susceptibles and
infecteds reach an equilibrium (see the epidemics exercises).
7.4.5 SIRS Model
The final model analyzed in this section is the SIRS model. Just as the
SIS model extends the SI, the SIRS model extends the SIR, as shown in
Figure7.16. In this model, the assumption is that individuals who have
SIR
β1 γ
λ
Figure 7.16. SIRS Model.