Biology-Biotechnology Majors

 

You have a choice of  2 areas to work in. Please pick one.

 

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  -  due Monday, November 17

 

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 on the web as well as in the notes linked here. Please see me when you have questions or when you think you have a good grasp of it.

 

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.

 

Specific activities that you need to work on:

            1) Maple worksheet

                        a) set up a 10x10 array, call it A

                                    makeup some numbers for birth rates and put them in the top row

                                    pick some fractions between 0 and 1 – put them on the subdiagonal for survival rates

                        b) have a variable k which is an integer

                        c) set up a 10x1 array X0    (stands for the starting population)

                                    put 1 into all the entries

                        d) have Maple compute  Ak X0     -   this is the population after k periods of time

                        e) have Maple compute the eigenvalues of A and make note of what the largest positive one is

                                    "What is an eigenvalue and how on earth do I find it?".

                                                i) open the linalg library using the with command

                                                ii) apply the eigenvals function to A                eigenvals(A);

                                    we will worry about what an eigenvalue is later…

 

            2) Actuarial information

                        find estimates of the actual birth rates per cohort for the US population  ( or any other

                                    population you wish to study!)

                        find estimates for the survival coefficients – in actuarial literature these may be called

                                    mortality rates

 

Part Two – the Human Population

 

In this part of the project, the original Maple sheet developed in Project One should be duplicated and given to all members of the group.

This project should produce two versions:

            a) one for the overall population (your group may have done some of the work in Project #1)

                        if you have not done research to acquire real parameters for your Leslie matrix, now is the time!

                        use the worksheet to predict the population distribution for 2010,2020,2030,2040 and 2050

                        compute the eigenvalues of your matrix and make note of the largest one

 

            b) one for a population with a subset which have the HIV virus.  This means twice as many cohorts. For each age group,

                        there will be two groups, uninfected and infected. Thus X will be 20 by 1 and A 20 by 20. You will want to set

                        up A on paper first (and perhaps talk it over with me)  Then you need to implement it in Maple and proceed to

                        the population distribution for 2010,2020,2030,2040 and 2050.

 

            c) compare results. Discuss in your group if you feel the results to be realistic. See if there are any reference which predict

                        what future levels will be

 

Please turn in a paper with the following components:

 

  • Cover page with members, group # etc
  • Introduction
  • A Review of how the Leslie Model works

this should be a mix of  English and mathematics

  • Your models and their results (  a) and  b) above )
  • Your thought on  c)
  • Conclusion
  • References

 

            Note: please edit and trim your Maple sheet so only the relevant material is left….

                       

 

Part Three – a Vaccine

 

The purpose of this project is to review and summarize your work done from the first two projects.  What you turn in should be a mix of writing and mathematics, with more of the former.

 

Part One

 

Summarize the basic Leslie population model, explaining  how  it works so that if a student in this course read it in the future, he or she would come away with an understanding of it.  Your explanation should occur at both the entry level and the matrix level

 

Part Two

 

Summarize your model from Project II in the same manner as Part One.  This model broke the population into HIV-positive and negative

components.

 

Part Three

 

Suppose you worked for a biotech firm that had a potentially effective HIV-vaccine.  In this part of the project:

 

discuss how you would use and modify your model to forecast it's impact.

what sociological and economic factors would keep the vaccine from being as effective as it might be?

 

 

Part Four

 

Summarize what things you have learned as individuals and as a group about project work during this course, especially as it might be applicable to future work in IQPs and MQPs.  Areas to include are

 

      • time management (meetings, schedules)
      • communication skills
      • importance of writing
      • using technology
      • dividing and coordinating work
      • producing a finished product

 

Conclusion

 

Summarize what you learned and accomplished working on the Leslie Population model

 

 

 

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?