Team Number: 080
School Name: Silver High School
Area of Science: Statistics and Probability
Project Title: Mathematical modeling of the AIDS virus

Problem Definition:
The purpose of this mathematical modeling project is to develop equations based on historical data, and to predict the spread of the AIDS virus in a statistically representative population. This project was selected because AIDS is adversely impacting populations throughout the world. A number of interesting viruses have been eradicated in recent history via improved medical technology and treatment; however no cure has yet been discovered for AIDS. The fact that there is no positive treatment for the cure of AIDS indicates that more in-depth research is required in both developed and undeveloped countries. Statistical analysis is readily applied in our program to evaluating the mathematical aspects of exposing a population to the AIDS virus and predicting the spread of AIDS. This program has a few goals: to model the spread of AIDS according to historical data, to use that model to predict the spread of AIDS in the future, to create different models which represent unexpected situations (a cure for AIDS, an increase in the spread of AIDS, etc.), and to compute the probability of error in the predictions.

Solution to problem:
Data collected on the spread of AIDS will be stored in a file used by the program. The correlation of this data will be found using least squares fitting techniques. The program will fit three equations to the historical data: a linear equation, a logarithmic equation, and a cubic equation. It will then compute the margin of error in each of the fitted equations to determine which of the three is the best fit. Once the best fit has been determined the selected equation will be used to make predictions beyond the known data (into the future). As well as the three equations mentioned above, a differential equation will also be tailored to fit the data. This equation will be changed slightly to represent different possible situations that could affect the spread of the AIDS virus.
Progress up to this point:
Currently, a program has been written which uses some of the techniques mentioned above. The current program fits linear, and logarithmic equations to data that is stored in a separate file. This program also computes the average margin of error for each of these equations. A second program has been written that makes use of a differential equation. This “differential equation” program makes predictions that have a similar correlation to the real world data, but are not yet accurate. This particular program will be greatly refined before it is put to use in the main program.

Expected Results:
After the completion of the programming, refining, and testing of this program it will be useful in predicting the spread of AIDS in various situations and places. This provides an envelope for examining the effects of the AIDS virus on a given population. However, the mathematical methods and a major part of the programming are applicable to many different situations. The mathematical models and computer program developed to analyze the impact of the AIDS virus in a selected population can also be used, by changing specific parameters, to predict the behavior of other diseases in other populations, as well as modeling other situations that do not involve diseases such as: the spread of a forest fire trough South America or the change of death rate in a small village due to a contaminated water supply.

Team Members: David Saxton, Adam Cummings, Cyrus Marcum
Teacher: Peggy Larisch

----(2004) “Statistics” http://dictionary.reference.com/search?q=Statistics
----(1996) “Forgotten Statistics: A Self-Teaching Refresher Course” Douglas Downing, Jeff Clark, and Jeffrey Clark
----(1990) “History of AIDS: Emergence and Origin of a Modern Pandemic”, Mirko D. Grmek
----(2004) “AIDS” http://en.wikipedia.org/wiki/AIDS
----(2004) “AIDS” http://www.free-definition.com/AIDS.html
----(2004) “WHO HIV/AIDS Programme” http://www.who.int/hiv/en/