Greenwood press

58
LOGISTIC FUNCTIONS
University admission rates for students
with different SAT verbal scores. The
acceptance rate of high school students
into a certain college based on their
SAT verbal score.

---

If you cover up the right side of the curve (SAT verbal scores greater than 550),
the remaining curve looks like an exponential curve. 
Consider the spread of rumors. Suppose that every hour a person who hears
a rumor passes it on to four other people. During the early life of the rumor the
equation that represents the spread of the rumor at each hour would be N = 4
t
,
where N is the number of people hearing the rumor at t hours. The exponential
growth equation would require 65,536 new listeners at the eighth hour. But what
if the rumor starts with a student in a 1,000-student high school overhearing the
principal saying, “We are going to dismiss school early today”? If every student
passing on the rumor could find someone who had not heard it, then the rumor
would pass through the entire student body before five hours were up. However,
after four hours, people spreading the rumor will be telling it to students who
already know. This means that the rate at which new listeners receive the rumor
has to decrease as the day goes on. People who learn about the rumor later in the
day are not likely to find anybody who hasn’t heard it. A logistic equation that 
models the spread of this rumor is 
N =
1
1
1000
+0.25
t
, where N is the number of 
students in the high school who have heard the rumor, and t is the number of
hours since the rumor started. This model would predict that half the student
body would have heard the rumor by the fifth hour.
Studies of diseases indicate that the early stages of an epidemic appear to
show an exponential growth in infected cases, but after a while the number of
people infected by the disease does not increase very rapidly. Like the spread of
rumors, diseases cannot be easily spread to new victims after much of the popu-
lation has encountered it. Logistic models describe the number of people infected
by a new disease if the entire population is susceptible to it, if the duration of the
disease is long so that no cures occur during the time period under study, if all
infected individuals are contagious and circulate freely among the population,
and if each contact with an uninfected person results in transmission of the dis-
ease. These seem like restrictions that would make it unlikely that logistic mod-
els would be good for studying epidemics, but the federal government’s Centers
for Disease Control and Prevention (CDC) make effective use of logistic models
for projections of the yearly spread of influenza through urban populations. CDC
statisticians adapt the model in a variety of ways for other types of diseases.
Logistic models are useful for tracking the spread of new technologies
throughout the country. The proportion of schools in the United States that have
Internet connections increased exponentially during the first half of the decade
(1991–2000), then leveled off at the end, with 95 percent of the schools having
Internet connections in 1999. A logistic function describes this pattern quite well.
Logistic curves describe the spread of other technologies such as the proportion
of families owning cell phones, the proportion of homes with computers, and the
number of miles of railroad track in the country from 1850 through 1950. The
logistic growth function carries a warning for companies that introduce new tech-
nologies: enjoy exponential growth in early sales, because it cannot last. When the
market is saturated with the technology, new sales are very difficult to make.