# Statistical Modeling of The AIDS Virus

### An excursion into the modeling of infectious diseases

Executive Summary

Introduction

Project Proposal

Analytical Methodology

Results

Conclusions

Future Work

References

Appendix 1
Understanding The Model

Appendix 2
Code, and Computations

Appendix 3
History of the AIDS Virus

### Conclusions

##### Mathematical Models

The SI model was chosen to be used in this program because it is used to model an infectious disease without any recovery rate. However, the original SI model was not completely applicable to the problem faced by this program due to the fact that the original equation only took into account a static population (one that does not change its total population over time) and it is obvious that real-world populations never remain static for any length of time. Due to the fact that the SI model lacked the complexity to model population change or population response to a dangerous disease it was modified extensively by Team 80 to suit the purposes of this program. These modifications (discussed further in Appendix 4) caused it to be more applicable to the situation, as it allowed the program to use a changing contact rate, calculate a population's response to a disease, and also take into account age, adjusting the contact rate based on that as well [1].

##### Computer Program

The computer program that was the final product of the work of Team 80 more than adequately runs the modified SI model. Along with running the model it also implements the use of random numbers to make its operations more similar to reality. The program also simulates a population's response to a disease that is perceived as a threat. Using these features this statistically based program operates in a very realistic way. The program in this study produces results that are similar to observed AIDS virus activity in a selected country (United States). The population selected for this study was not observed to die off; however AIDS is having a detrimental effect on the population by causing many people to have shortened lives and a declining birth rate in the general population.

##### Results

The computer program developed by Team 80 was able to produce results that were very similar to real-world data. The program did this through use of an extensively modified version of the mathematical equation known as the SI model. The SI model, when unmodified, takes the number of infected (I) and susceptible (S) people in a population, as well as the contact rate and infectivity of the disease, and uses these number to generate the number of new infected for the next year[1]. This program also takes into account the birth and death rates of the population being modeled, the age division, and also the chance that a person is immune to the disease (either through a cure/immunization or a genetic immunity to the disease). The program also uses randomly generated numbers to determine the chance that a person will die, be born, or be infected, as well as for many other of this program's functions. This allows for varied, yet statistically accurate results each time this program is run. Another way the program achieves accurate results is by allowing for a population's possible response to an epidemic. This assumes that once a critical percentage of the population becomes infected, it will trigger more education and preparation against the disease, thus lowering the contact rate.

The results of the repeated tests of the program indicate that AIDS/HIV will have a detrimental effect on any population it is introduced to, causing much loss of life. The results showed that while the disease is highly unlikely to eliminate the entire population, those who are susceptible to it are affected negatively. The unsusceptible population was mostly unaffected, and grew relatively smoothly throughout the tests. The results also show what the effect of a cure/immunization will have on the spread of the virus, based on how effectively the cure is distributed and at what point it is developed. These results show that the later a cure is developed, the less influence it will have on the disease, showing the importance of developing an early cure. The infected population is directly dependent upon the susceptible population so the development of an immunization would be extremely beneficial. An immunization stops the problem at its source; as a result; it is significantly more beneficial to use an immunization than a cure. This project shows the importance of taking preparations against the AIDS epidemic in a timely and efficient manner in order to prevent the greater loss of human life.

##### Word Processor Program

Several programs were used in the formation of the program written by this team. Dev C++ provided an adequate compiler and bug checker for the C++ programming language. Microsoft Word and Open Office were used for word processing and document formatting. Microsoft paint was used multiple times for the preparation of graphics used in the report. Notepad was used for reading and editing input and output files in this program. Microsoft Excel created the charts shown in this report and was used to test the equations used in the program. Scientific Notebook was used to type out the formulas used in a graphically presentable way. Finally three separate G-mail accounts provided by www.google.com were used by the members of this group for correspondence with each other and data storage.

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