A **logistic growth curve** is an S-shaped (sigmoidal) curve that
can be used to model functions that
increase gradually at first, more rapidly in the middle growth period, and
slowly at the end, leveling off at a maximum value after some period of
time. The initial part of the curve is exponential; the rate of growth
accelerates as it approaches the midpoint of the curve.
At the midpoint (*K*/2),
the growth rate begins to decelerate but continues to grow until it reaches an
asymptote, *K* which is called the "Carrying Capacity" for the environment.

This type of curve is frequently used to model biological growth
patterns where there is an initial exponential growth period followed by a
leveling off as more of the population is infected or as the food supply or
some other factor limits further growth. The form of the symmetric logistic
growth function is:

*y* = *K* / (1 + exp(*a* + *b***x*))

where *K*, *a*, and *b* are parameters that shape and scale the
function. The value of *b* is negative.

The following NLREG program fits a logistic curve to the number of new cases of
AIDS reported in the United States during the period 1981 through 1992. The
computed function fits the data remarkably well showing that the AIDS
infection rate followed a classic logistic curve and leveled off at about
47,500 new cases per year (in the United States). The **domain** option
on the **plot** statement causes NLREG to extrapolate the plot of the
function through 1995. Note, because of recent advances in drugs that
greatly slow the progression of AIDS, the morbidity curve does not follow
its natural form through recent years.

Here is the NLREG program for fitting the logistic growth curve:

Title "New Cases of AIDS in The United States";
Variable Year; // Year for new cases
Variable Newcases; // Total new cases during year
Variable Childcases; // New cases for children under 13
Parameter K=50000; // Initial estimate of max cases
Parameter a=5;
Parameter b=-.8;
constant baseyear=1980;
Function newcases = K / (1+exp(a+b*(year-baseyear)));
Plot Domain=1980,1995, ylabel="New Cases";
Data;
[ data values go here ]

Here is the plot generated by NLREG of the function fitted to the data: