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The term 1 d t t d represents the fraction of nodes

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Unformatted text preview: (t )d 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 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. 17...
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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.

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