DCS/CSCI 2350: Social & Economic Networks How does a disease - - PDF document

3/26/18 1

DCS/CSCI 2350: Social & Economic Networks

How does a disease propagate in a network? Chapter 21 of EK

Mohammad T . Irfan

Flu outbreak (2018)

u January 11, 2018

3/26/18 2

Zika outbreak (2015—2016)

u March 2016

Zika outbreak (2015—2016)

u November 21, 2016

3/26/18 3

Example (February 2015)

u Measles outbreak (CA, December’14—Feb’15)

Image: Fox 40

Example (December 2014)

u Ebola epidemic

3/26/18 4

Example (January 2014) Important factors of epidemics

u Pathogen

u How contagious is it? u How long is the infectious period? u How severe is it?

u Contact network

u “Contact” depends on pathogen: flu vs. STD u Examples

u Human diseases– travel pattern u Animal diseases (e.g., 2001 F&M disease in the UK) u Plant diseases– spatial footprint

3/26/18 5

Diffusion vs. epidemics

u Similar mechanism of spread u No decision making in epidemics u Epidemics: probabilistic model

u A person having flu will infect another person in

his contact network with some probability

Modeling epidemics

u Branching model

u Network is a tree

u SIR model

u One cannot be infected multiple times u General network structure (directed graph)

u SIS model

u One can be infected multiple times

3/26/18 6

Branching model (p, k)

u Contact network is a tree with k branches

from each internal node

u An infected node infects others in contact

with a probability p

u If probability p is high

This node is infected first

3/26/18 7

u If probability p is low

Will it become an epidemic?

u Basic reproductive number, R0 u R0 = Expected # of new cases of the disease

caused by a single person

u R0 = p k u Dichotomy

u R0 < 1 => disease will die out for sure u R0 > 1 => disease will persist with positive prob.

u Knife-edge

u R0 = 1: critical value

3/26/18 8

Insights from the branching model

u R0 = p k u How to prevent an epidemic?

u Reduce the value of p – sanitary practice u Reduce the value of k – quarantine

SIR Model

u General directed graph as contact network u 3 possible stages for each node

u Susceptible (S): Not yet infected, but susceptible u Infectious (I): Infected and may infect others

within tI period (or steps)

u Removed (R): Cured (will never be susceptible or

infectious) u Each step:

u An infectious node infects its neighbors with

probability p

3/26/18 9

• S: No shade
• I: Shaded+

dark border

• R: Shaded+

thin border

• tI = 1
• p = 0.5

Basic reproductive number

u Dichotomy does not hold for SIR model on

general graph

u R0 = expected number of new infections

caused by a node

u R0 can be > 1, but the disease may still die

• ut (next: an example)

3/26/18 10

Example: R0 > 1 doesn't cause epidemic in SIR

Model parameters: p = 2/3, tI = 1 R0 = 4/3 (why?) Probability that a layer will be uninfected = ?

Percolation: static view of SIR

u Coin flips are done in advance u Paths originating from the initially infected

nodes denote future infections

3/26/18 11

Other models: SIS

u SIS model

u S: Susceptible u I: Infectious

u A node can become infected multiple times u Dichotomy result exists (not covered here)

NetLogo experiment on SIR

u Models Library à Networks à Virus on a

network

u Edit the "go" button and uncheck "Forever" u Edit the max to 100% for the following:

u virus-spread-chance (p) u recovery-chance [proxy for infectious period tI]

3/26/18 12

NetLogo (continued)

u Then set the slider to 50% for:

u virus-spread-chance (p) u recovery-chance [proxy for infectious period tI]

u Set the slider to 100% for:

u gain-resistance-chance

u Experiment by varying the virus-spread-

chance (p) and average-node-degree (k)