Classical SIR and Network Models

1. Classical SIR and Network Models 2012 TCM Conference January 27, 2012 Dan Teague NC School of Science and Mathematics

2. 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

3. The Susceptibles The Infectives. The Recovered

11. Fundamental Problem with the Classical DE Model

12. Just Like Predator-Prey

14. What does βmeasure? 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.

15. 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.

16. Network Model If every vertex connects to every other vertex, then we have the classical Mass Action model.

19. Network Models • Erdós-Renyí Random Network • Configuration Models • Small World Network • Power Law Network • Preferential Attachment Networks

21. Friendship Paradox: Your Friends Have More Friends Than You Have Nicholas Christakis http://www.youtube.com/watch?v=L-dPxGLesE4&feature=related

22. Network Dynamics Network Structure Dynamics on Networks Dynamics of Networks

23. 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).

24. Average Vertex Degree

25. When will there be a Giant Component?

26. 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.

31. How can we solve this equation for S?

32. Think Graphically

34. When is there a Giant Component?

35. Graph Implicitly

36. Analysis? Can we do anything analytic with this equation?

37. Think of all the Inverse Functions you know.

38. x is a solution to some “unsolveable” equation.

39. We need such a function.

40. Lambert-W function

43. MathCAD Lambert-W Function

44. When will there be a Giant Component?

45. Is this kind of talk useful? Dan Teague Image above from Annalisa Crannell, NC School of Science and Mathematics Franklin & Marshall Collegeteague@ncssm.edu