classical sir and network models 2012 tcm conference january 27, 2012 dan teague nc school of...
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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 [email protected]
Is this kind of talk useful?