This preview shows page 1. Sign up to view the full content.
Unformatted text preview: (t )d
θ (t ) =
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 + 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 + 1
implying a solution θ = 1 − λ1¯ , which is positive only if d > 1/λ = δ/ν.
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 power-law 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 high-degree nodes serve as conduits for
infection. Even very low infection rates can lead them to become
infected and infect many others.
View Full Document
This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.
- Fall '09