Sample Write Up for a Problem

The graph below shows the number of AIDS cases reported in Jamaica 1982-1999.

Develop both a power function and an exponential model to fit the data on reported AIDS cases in Jamaica 1982-1999.

Comment on the goodness of your fits. Which model fits the data better?

Key Questions: Can we fit functional models of the forms y = Kx^r and y = A(1 + r)^x respectively to the AIDS data provided in the graph above?

Preconceptions: The data suggests that for the time period in question, 1982-1999, the number of AIDS cases in Jamaica was increasing at an increasing rate. So perhaps a power function model or an exponential function model might fit the data reasonably well. The shape of the graph suggests a power function with exponent r > 1. It is difficult to predict what the annual (compounded) percent growth rate might be in an exponential model, but it will probably be between 20% and 50%. I think a power function will prove to be the better fit.

Discussion: Our models will have very limited predictive value. The situation in the US suggests that educational, research, and other preventative methods might slow the growth in the number of new AIDS cases. Also, our models will have little value in explaining the causes of the spread of AIDS. We are simply trying to fit a functional model to some data for a specific period of time.

Organization & Analysis of Data and Model Fitting: We will employ an Excell spreadsheet to perfom our analysis. The spreadsheet is designed so we can modify the parameters r and K in the power function model and r and A in the exponential model.

In formulating our models we introduced the following variables:

t = the number of years since 1983

C(t) = the number of AIDS cases reported in Jamaica t years after 1983.

The power function model: C(t) = 1.2t^2.32

The exponential function model: C(t) = 1.5(1.46)^t

Link to Excel spreadsheet for the data analysis and model development.

Summary and Critique: The models’ parameters were determined by a “guess and test” strategy. Visual comparison was used to produce curves approximating data points, and numerical comparison was used in comparing error sums. Although neither function provides what we would call an excellent fit, the power function provides the “better fit” as one can see from the graphs on the Excel spreadsheet. The sum of errors for our power function model is 790 while the sum of errors for the exponential model is 1341. So, by the numerical criterion we have been using in this course, for our choice of models, the power function provides the better fit. Of course, as noted above, neither model has any particular value in either predicting the future or in explaining the past. It is likely that others will suggest models with some significantly different parameters.