Classical SIR and Network Models 2012 TCM Conference January 27, 2012 Dan Teague NC 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

Fundamental Problem with the Classical DE Model

Just Like Predator-Prey

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.

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 vertex connects to every 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 Christakis

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

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?

Think of all the Inverse Functions you know.

x is a solution to some “unsolveable” equation.

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 College Is this kind of talk useful?