Applying Abstract Algebra
and Graph Theory to
Model Flu Seasons
By Ben Hughes, schleyfox@gmail.com
Collaboration: Prateek Vasireddy and Casey Glass (High School students)
Olgamary Rivera-Marrero and Brandy Stigler (VT math graduate students)
April 30, 2005
DIMACS Conference on Linking Mathematics and Biology in High Schools
What is the flu?
• A respiratory illness
• The Influenza virus can range from mild to life
threatening
• 10%-20% of U.S. Residents get the flu
• 36,000 Americans die from complications of flu each
year.
Complications
Complications from the flu include pneumonia,
dehydration, and worsening of preexisting medical
conditions. The flu can be, and often is, fatal.
Graph: Flu Mortality rates
Mission
Abstract Algebra
• Predict which region in Virginia has a higher
possibility of an epidemic for the near-future
Graph Theory
• Determine the most central location to build a clinic
• Determine the minimal number of infectious disease specialists
to have on staff at the clinic
• Decide what types of prevention methods the specialist should use
• Find an optimal route for the R.N to travel & administer vaccinations
Predicting Epidemics
• Using data from Virginia Department of Health, a model was created
Virginia Department of Health Flu Activity Statistics
Northwest (NW) – 1
Northern (N) – 2
Southwest (SW)– 3
Central (C) – 4
Eastern (E) – 5
(http://www.vdh.state.va.us/).
NW
N
SW
C
E
1996
124
2
579
9
243
1997
116
22
310
1
68
1998
181
34
758
33
154
1999
407
132
1603
102
314
2000
337
110
1120
50
292
2001
362
141
903
161
396
2002
795
345
887
534
925
2003
3017
1791
4789
3599
5451
How we made the model
1. We took the flu activity data and normalized it by computing the highest value and
dividing everything by it and then multiplying by 100.
2. Then we created a basic model in DVD
(Discrete Visualizer of Dynamics) f  x  1
1
5
using the functions.
NW
N
SW
C
E
2001
0
0
1
0
0
2002
1
0
1
0
1
2003
0
0
1
0
1
f2  0
f3  1
f4  0
f5  1
3. Then we added zero-polynomials to the functions to account
for borders and other factors we believed might affect the
spread of the flu until we got the dependency graph on the
right.
Then we rendered the state space graph that is seen on the next slide.
Dependency graph
What the model showed
• Central and Northwest Virginia will have epidemics
on 2005
• Southwest Virginia borders both of those regions
• We chose Pittsylvania County for a clinic because of its
size (and we live there)
NW N SW C E
2001
2002
2003
2004
2005
Clinic and the Specialists
Using a shortest path and vertex coloring we determined both the most
central city (Chatham) and the minimum number of specialists needed (3).
Mathematical Modeling: Using Graphs and Matrices
Clinic and the Specialists
•One specialist will give speeches to the community
•Another will teach first aid classes to help make
people aware of what to do about the flu
•The other one will pass out posters like the one below.
RN's AKA Flu-Shooters
Because vaccination is an effective way of preventing the flu, RNs, or “flushooters,” must be sent to each town/city in the county.
By finding a Hamilton circuit
(a graph where one can pass
through each vertex exactly
once and return to the
starting location) we found one
of the most efficient path for an
RN to take.
http://www.utc.edu/Faculty/ChristopheMawata/petersen/lesson12b.htm
Conclusions
• We were able to make mathematical models for
our prediction and prevention of the flu in
Southwest Virginia
References
• ww.cdc.gov/flu/keyfacts
• Mathematical Modeling: Using Graphs & Matrices
Learning in motion software - http://www.learn.motion.com/products/modeling/index.html
• DVD - Discrete Visualizer of Dynamics
http://dvd.vbi.vt.edu/visualizer/new_dvd11.pl
• Graph Theory Lessons – Euler Circuit and Hamiltonian Circuit
http://www.utc.edu/Faculty/ChristopheMawata/petersen/lesson12b.htm