Lecture 8 - Diffusion through Networks

Lecture 8 - Diffusion through Networks - 6.207/14.15:...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
6.207/14.15: Networks Lecture 8: Diffusion through Networks Daron Acemoglu and Asu Ozdaglar MIT October 7, 2009 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Networks: Lecture 8 Outline Spread of epidemics in networks Models of diffusion without network structure Bass model Models of diffusion that explicitly incorporate network structure Diffusion with immune nodes SIR model (susceptible, infected, removed) SIS model (susceptible, infected, susceptible) Reading: Jackson, Chapter 7, Sections 7.1,7.2. EK, Chapter 21. 2
Background image of page 2
Networks: Lecture 8 Introduction The study of epidemic disease has always been a topic where biological issues mix with the social ones. The patterns by which epidemics spread through a society is determined not just by the properties of the pathogen carrying it (including its contagiousness, the length of its infectious period, and severity), but also by the network structure within the population. Opportunities for a disease to spread from one person to another is given by the contact network , indicating who has contact with whom on a regular basis. We are interested in the following questions: Under what conditions will an initial outbreak spread to a nontrivial portion of the population? What percentage of the population will eventually become infected? What is the effect of immunization policies? The problem is relevant not only to disease transmission, but also to diffusion through a network of information, opinions, and adoption of new 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Networks: Lecture 8 Bass Model–1 An early model of diffusion is the Bass model. Although it does not capture any explicit social network structure, it still incorporates imitation. The model is built on two parameters: p captures the rate at which agents spontaneously get infections (in response to outside stimuli); and q captures the rate at which agents get infected through others (secondary infections). In the context of adoption of technologies, p can be interpreted as the rate of innovation and q as the rate of imitation due to social effects. Consider a discrete-time model and let F ( t ) be the fraction of agents infected at time t . The Bass model is described by the difference equation: F ( t ) = F ( t - 1 ) + p ( 1 - F ( t - 1 )) + q ( 1 - F ( t - 1 )) F ( t - 1 ) . The term p ( 1 - F ( t - 1 )) is the infection rate times the fraction of uninfected agents. The term q ( 1 - F ( t - 1 )) F ( t - 1 ) is the contagion rate times the frequency of encounters between healthy and infected agents. 4
Background image of page 4
Networks: Lecture 8 Bass Model–2 A continuous time version of this model is described by d F ( t ) dt = ( p + qF ( t ))( 1 - F ( t )) , with F ( 0 ) = 0. This is a nonlinear differential equation, but admits a closed form solution F ( t ) = 1 - e - ( p + q ) t 1 + q p e - ( p + q ) t . Note that the levels of p and q scale time, the ratio of q to p determines the overall shape of the curve.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/17/2011 for the course ECONOMICS 14.14 taught by Professor Daronacemoglu during the Spring '11 term at MIT.

Page1 / 17

Lecture 8 - Diffusion through Networks - 6.207/14.15:...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online