Applying Abstract Applying Abstract Algebra and Graph Algebra and Graph

Theory toTheory toModel Flu SeasonsModel Flu Seasons

April 30, 2005DIMACS Conference on Linking Mathematics and Biology in

High Schools

By Ben Hughes, schleyfox@gmail.comCollaboration: Prateek Vasireddy and Casey Glass (High

School students)Olgamary Rivera-Marrero and Brandy Stigler (VT math

graduate students)

• 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.

What is the flu?What is the flu?

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

ComplicationsComplications

MissionMission

• Predict which region in Virginia has a higher possibility of an epidemic for the near-future

• 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

Abstract Algebra

Graph Theory

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ECSWNNW

Virginia Department of Health Flu Activity Statistics

• Using data from Virginia Department of Health, a model was created

Predicting Epidemics

Northwest (NW) – 1 Northern (N) – 2 Southwest (SW)– 3 Central (C) – 4Eastern (E) – 5 (http://www.vdh.state.va.us/).

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.

1 5

2

3

4

5

1

0

1

0

1

f x

f

f

f

f

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.

How we made the modelHow we made the model

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ECSWNNW

2. Then we created a basic model in DVD(Discrete Visualizer of Dynamics) using the functions.

Dependency graph

• 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)

What the model showedWhat the model showed

2001

2002

2003

2004

2005

NW N SW C E

Clinic and the SpecialistsClinic and the Specialists

Using a shortest path and vertex coloring we determined both the mostcentral city (Chatham) and the minimum number of specialists needed (3).

Mathematical Modeling: Using Graphs and Matrices

•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.

Clinic and the SpecialistsClinic and the Specialists

Because vaccination is an effective way of preventing the flu, RNs, or “flu-shooters,” 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.

RN's AKA Flu-ShootersRN's AKA Flu-Shooters

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

ReferencesReferences

• 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 Dynamicshttp://dvd.vbi.vt.edu/visualizer/new_dvd11.pl

• Graph Theory Lessons – Euler Circuit and Hamiltonian Circuithttp://www.utc.edu/Faculty/ChristopheMawata/petersen/lesson12b.htm