Human Populations and the Spread of HIV

The goal of your work here is to develop a mathematical model which demonstrates the growth of the human population. Once you have done that, you will then modify the model to also reflect the spread of HIV. Clearly many people in the Biotechnology field are working on vaccines for HIV, so such a model would have significance for predicting its impact.

Part One – The Leslie Model

The foundation for this project is the Leslie Population model, developed in 1945 by the British mathematician P.H. Leslie. You will need to review this in the textbook (pages ).

The significant feature of this model is that it breaks a population down into age groups, or cohorts. This allows use of data particular to each group, regarding reproductive rates and mortality rates. Actuaries, for example, work with such data.

One you have finished the background reading, please set up a trial model in Maple. This will require little more than multiplication of 10x10 matrices and 10 x 1 columns.

Part Two – the Human Population

Here you will gather data on reproductive and mortality rates to build a working model of the human population. The model also requires an “initial condition” so you will have to characterize the current population, cohort by cohort.

You have a decision to make: do you want to model the global population ort the United States population?

Once you have gathered the data, modify your Maple worksheet accordingly. Then use the model to predict the population for 2010,2020,2030, 2040 and 2050.

Part Three – the HIV population

We now view the human population not as 10 age groups but as 10 groups subdivided into 2 subgroups, those infected and those not. We have three kinds of “births” to consider now: uninfected ordinary births, uninfected becoming infected, and infected having children. You will need to gather sufficient data to make the model work.

Part Four – a Vaccine

The goal here is simple: in Part Three, you modeled the spread of HIV. If a vaccine were available, what impact would it have?

Work to be handed in:

A paper describing

the basic Leslie Model
your understanding of the HIV positive subpopulation
how you built your final model including all assumptions that went into it
what impact you see a vaccine as having, with data from your model to support

your conclusions

all references

A diskette with all models on it, and a “readme” file indicating which is which.

The Acoustics of Whale Populations

The goal of this project is to use mathematical techniques to possibly study the sounds made by whale populations and the potential information provided by them. The mathematical methods are based on linear algebra concepts. The sounds have been gathered by various oceanographic research sites in MP3 files (see links below for specific sources).

Part One Fourier Analysis

First you must understand the mathematics of periodic functions, Fourier Analysis. Sounds such as those made by whales, voices or musical instruments are periodic functions. Fourier Analysis breaks such functions down into sums of sin and cosine functions. Such sums are called Fourier Series. It is your job to understand how this works. The following link will provide you with this.

Part Two – Matlab

Doing Fourier Analysis requires a great deal of computation, thus requiring software. Matlab is an ideal package for this work, for two reasons: it can read in WAV and MP3 files, and it can describe any signals within such files by Fourier Analysis. Your job in this part is to learn how to use Matlab.

Please follow the following link to go through a Matlab tutorial.

Once you have done this, try it out on some simple sounds files, which you may download from the following links: . In each case you have to decide what the frequency of the signal stored in the file is. Please provide these in your report and let me know if you have any problems.

Part Three – Whale Acoustics

Your job here is to develop some background in the fundamentals of whale acoustics. What sorts of information have they been known to communicate? Under what conditions? What whales have been studied? What value is there in understanding whale communications? What are some sites with technical information?

Part Four – Using Matlab for Whale Acoustics

Here you will try Matlab out on existing files. Do you see any repetition in the files?