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θ (t ) =
.
�d �
14 Networks: Lecture 8 SIS Model–2 Let ν denote the transmission rate of infection and δ denote the recovery
rate of an infected individual.
We assume that the probability that a susceptible agent with degree d becomes infected in a period [t , t + �) is �νθ (t )d . Using a mean ﬁeld analysis, we can write the evolution of ρd (t ):
˙
ρd (t ) = (1 − ρd (t ))νθ (t )d − ρd (t )δ.
The term (1 − ρd (t ))νθ (t )d represents the fraction of nodes of degree d
that were susceptible and become infected and ρd (t )δ represents the
fraction that recover to become susceptible again.
Using this, we can characterize the steady state. Let θ (t ) → θ and ρd (t ) → ρd , and λ = ν/δ:
λθ d
ρd =
,
and therefore,
λθ d + 1
θ= P (d )λθ d 2 ∑ �d �(λθd + 1) .
d 15 Networks: Lecture 8 Nonzero Steady State Infection Rate
θ = 0 is always a solution: if nobody is infected, the system stays that way.
We next analyze when the steady state has a solution with θ > 0.
¯
Assume the degree distribution is regular, all nodes have degree d . Then:
¯
d λθ
,
θ= ¯
d λθ + 1
¯
implying a solution θ = 1 − λ1¯ , which is positive only if d > 1/λ = δ/ν.
d
If the number of meetings is suﬃciently large compared to the relative
recovery/infection rate, then the infection can be sustained.
It can be shown that for powerlaw degree distributions, there is always a
positive solution. 16 Networks: Lecture 8 Nonzero Steady State Infection Rate
In general, let H (θ ) be
H (θ ) = P (d )λθ d 2 ∑ �d �(λθd + 1) .
d We have H (0) = 0 and H (θ ) is increasing and strictly concave in θ .
Thus, for H to have a nonzero ﬁxed point, we must have H � (0) > 1.
�d 2 � Note that H � (0) = λ �d � .
Hence the condition for positive steady state infection is
λ > �d �
.
�d 2 � ¯
For regular graphs, the threshold is λ > 1/d , as before.
For power law distributions (with γ < 3), �d 2 � is divergent, hence the above
equation is satisﬁed for any positive λ.
Intuition: Individuals with highdegree nodes serve as conduits for
infection. Even very low infection rates can lead them to become
infected and infect many others.
17...
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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.
 Fall '09
 Acemoglu

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