Classical SIR and Network Models

2012 TCM ConferenceJanuary 27, 2012

Dan TeagueNC School of Science and

Mathematics

Slides borrowed from the MAA Invited Lecture, Mathematical Approaches to Infectious Disease Prediction and Control by Lauren Ancel Meyers, University

of Texas, during Mathfest, 2011

The Susceptibles

The Infectives.

The Recovered

dSS I

dt

dIS I I

dt

dRI

dt

0 when 0

0 or

dISI I

dt

S I S

1S

S

0 00

( ) lnS

I S S I SS

1dI SI I

dS SI S

dSS I

dt

dIS I I

dt

dRI

dt

Fundamental Problem with the Classical DE Model

Just Like Predator-Prey

dSaS bWS

dtdW

cW dSWdt

What does β measure?

In , β is a product of probabilities.

SI counts the number of possible S-I interactions.

Not all of them happen. β accounts for this.

Not all that happen lead to an Infective. β accounts for this as well.

dSSI

dt

The Mass Action Assumption

If, according to β (SI), my quota of possible infectionable interactions is 5, I can infect 5 individuals today.

Tomorrow, I get a new group of 5.

In the real world, if I have 5 close friends I can infect, I don’t get a new set of 5 tomorrow.

Network Model

If every vertexconnects toevery other vertex,then we have the classical Mass Action model.

Network Models

•Erdós-Renyí Random Network

•Configuration Models

•Small World Network

•Power Law Network

•Preferential Attachment Networks

Friendship Paradox: Your Friends Have More Friends Than You

Have

Nicholas Christakishttp://www.youtube.com/watch?v=L-dPxGLesE4&feature=related

Network Dynamics

Network Structure

Dynamics on Networks

Dynamics of Networks

What does the Mathematics of

Networks look like?The Giant Component in a Random

Erdós-Renyí Graph

Suppose we have a graph with V vertices in which the edges are created at random. Each possible edge is created with probability p.

This graph is denoted G(V, p).

Average Vertex Degree

221.83

12k

When will there be a Giant Component?

When will the GC exist and how large will it be?

Without a Giant Component, any outbreak will be small.

A network must contain a Giant Component for and epidemic to become established.

How can we solve this equation for S?

Think Graphically

When is there a Giant Component?

Graph Implicitly

Analysis?

Can we do anything analytic with this equation?

1 k SS e

Think of all the Inverse Functions you know.

1If sin 0.3 , what is ?

If ln 0.3 , what is ?

If 0.3 , what is ?

x x

x x

x x

x is a solution to some “unsolveable” equation.

1

n

If sin 0.3 , what is ?

If ln 0.3 , what is ?

is the solution to sin 0.3.

is the solut

If 0.

ion to 0.3.

is the solution

3 , what is ?

to 0.3.

x

n

x x

x e

x x

x x

x x

x x

We need such a function.

Lambert-W function

MathCAD Lambert-W Function

When will there be a Giant Component?

Dan Teague Image above from Annalisa Crannell, NC School of Science and Mathematics Franklin & Marshall Collegeteague@ncssm.edu

Is this kind of talk useful?