in Technology in Society 23(2):131-146 (2001)
Death and the
Human Environment:
The United
States in the 20th Century
Jesse H. Ausubel
Perrin S. Meyer
Iddo K.Wernick
Program for the Human Environment
The Rockefeller University
New York, NY, 10021
Abstract:
Causes of death varied systematically in the United States during the
20th century as the human environment came under control. Infections
became less deadly, while heart disease grew dominant, followed by
cancer. Logistic models of growth and multi-species competition in which
the causes of death are the competitors describe precisely the
evolutionary success of the killers. We shows the dossiers of typhoid,
diphtheria, cholera, tuberculosis, pneumonia/influenza, heart disease,
cancer, and AIDS. Improvements in water and air supply and other aspects
of the environment provided cardinal defenses against infection. We
project cancer will overtake heart disease as the leading cause of death
about 2015, and infections may gradually regain their deadly edge.
AN INTRODUCTION TO DEADLY COMPETITION
Our subject is the history of death. Researchers have analyzed the
time dynamics of numerous populations-nations, companies, products,
technologies--competing to fill a niche or provide a given service.
Here we review killers, causes of death, as competitors for human
bodies. We undertake the analysis to understand better the role of the
environment in the evolution of patterns of mortality. Some of the
story will prove familiar to public health experts. The story begins in
the environment of water, soil, and air, but it leads elsewhere.
Our method is to apply two models developed in ecology to study
growth and decline of interacting populations. These models, built
around the logistic equation, offer a compact way of organizing numerous
data and also enable prediction. The first model represents simple
S-shaped growth or decline.[1] The
second model represents multiple, overlapping and interacting processes
growing or declining in S-shaped paths.[2] Marchetti first suggested the
application of logistic models to causes of death in 1982.[3]
The first, simple logistic model assumes that a population grows
exponentially until an upper limit inherent in the system is approached,
at which point the growth rate slows and the population eventually
saturates, producing a characteristic S-shaped curve. A classic example
is the rapid climb and then plateau of the number of people infected in
an epidemic. Conversely, a population such as the uninfected sleds
downward in a similar logistic curve. Three variables characterize the
logistic model: the duration of the process (Dt), defined as the time
required for the population to grow from 10 percent to 90 percent of its
extent; the midpoint of the growth process, which fixes it in time and
marks the peak rate of change; and the saturation or limiting size of
the population. For each of the causes of death that we examine, we
analyze this S-shaped “market penetration” (or withdrawal) and quantify
the variables.
Biostatisticians have long recognized competing risks, and so our
second model represents multi-species competition. Here causes of death
compete with and, if fitter in an inclusively Darwinian sense,
substitute for one another. Each cause grows, saturates, and declines,
and in the process reduces or creates space for other causes within the
overall niche. The growth and decline phases follow the S-shaped paths
of the logistic law.
The domain of our analysis is the United States in the 20th century.
We start systematically in the year 1900, because that is when
reasonably reliable and complete U.S. time series on causes of death
begin. Additionally, 1900 is a commencement because the relative
importance of causes of death was rapidly and systematically changing.
In earlier periods causes of death may have been in rough equilibrium,
fluctuating but not systematically changing. In such periods, the
logistic model would not apply. The National Center for Health
Statistics and its predecessors collect the data analyzed, which are
also published in volumes issued by the U.S. Bureau of the Census.[4]
The data present several problems. One is that the categories of
causes of death are old, and some are crude. The categories bear some
uncertainty. Alternative categories and clusters, such as genetic
illnesses, might be defined for which data could be assembled. Areas of
incomplete data, such as neonatal mortality, and omissions, such as
fetal deaths, could be addressed. To complicate the analysis, some
categories have been changed by the U.S. government statisticians since
1900, incorporating, for example, better knowledge of forms of
cancer.
Other problems are that the causes of death may be unrecorded or
recorded incorrectly. For a decreasing fraction of causes of death, no
“modern” cause is assigned. We assume that the unassigned or “other”
deaths, which were numerous until about 1930, do not bias the analysis
of the remainder. That is, they would roughly pro-rate to the assigned
causes. Similarly, we assume no systematic error in early records.
Furthermore, causes are sometimes multiple, though the death
certificate requires that ultimately one basic cause be listed.[5] This rule may hide environmental
causes. For example, infectious and parasitic diseases thrive in
populations suffering drought and malnutrition. The selection rule
dictates that only the infectious or parasitic disease be listed as the
basic cause. For some communities or populations the bias could be
significant, though not, we believe, for our macroscopic look at the
20th century United States.
The analysis treats all Americans as one population. Additional
analyses could be carried out for subpopulations of various kinds and by
age group.[6] Comparable analyses
could be prepared for populations elsewhere in the world at various
levels of economic development.[7]
With these cautions, history still emerges.
As a reference point, first observe the top 15 causes of death in
America in 1900 (Table 1). These accounted for about 70 percent
of the registered deaths. The remainder would include both a sprinkling
of many other causes and some deaths that should have been assigned to
the leading causes. Although heart disease already is the largest
single cause of death in 1900, the infectious diseases dominate the
standings.
Death took 1.3 million in the United States in 1900. In 1997 about
2.3 million succumbed. While the population of Americans more than
tripled, deaths in America increased only 1.7 times because the death
rate halved (Figure 1). As we shall see, early in the century
the hunter microbes had better success.
Table 1. U.S. death rate per 100,000 population for
leading causes, 1900. For source of
data, see Note 4.
|
|
Cause
|
Rate
|
Mode of Transmission
|
|
1.
|
Major
Cardiovascular Disease
|
345
|
[N.A.]
|
|
2.
|
Influenza,
Pneumonia
|
202
|
Inhalation,
Intimate
Contact
|
|
3.
|
Tuberculosis
|
194
|
Inhalation,
Intimate
Contact
|
|
4.
|
Gastritis,
Colitus,
Enteritis,
and Duodenitis
|
142
|
Contaminated
Water
and
Food
|
|
5.
|
All
Accidents
|
72
|
[Behavioral]
|
|
6.
|
Malignant
Neoplasms
|
64
|
[N.A.]
|
|
7.
|
Diphtheria
|
40
|
Inhalation
|
|
8.
|
Typhoid
and Paratyphoid
Fever
|
31
|
Contaminated
Water
|
|
9.
|
Measles
|
13
|
Inhalation,
Intimate Contact
|
|
10.
|
Cirrhosis
|
12
|
[Behavioral]
|
|
11.
|
Whooping
Cough
|
12
|
Inhalation,
Intimate Contact
|
|
12.
|
Syphilis
and Its Sequelae
|
12
|
Sexual
Contact
|
|
13.
|
Diabetes
Mellitus
|
11
|
[N.A.]
|
|
14.
|
Suicide
|
10
|
[Behavioral]
|
|
15.
|
Scarlet
Fever and Streptococcal Sore Throat
|
9
|
Inhalation,
Intimate Contact
|
DOSSIERS OF EIGHT KILLERS
Let us now review the histories of eight causes of death: typhoid,
diphtheria, the gastrointestinal family, tuberculosis, pneumonia plus
influenza, cardiovascular, cancer, and AIDS.
For each of these, we will see first how it competes against the sum
of all other causes of death. In each figure we show the raw data, that
is, the fraction of total deaths attributable to the killer, with a
logistic curve fitted to the data. In an inset, we show the identical
data in a transform that renders the S-shaped logistic curve linear.[8] It also normalizes the process of
growth or decline to one (or to 100 percent). Thus, in the linear
transform the fraction of deaths each cause garners, which is plotted on
a semi-logarithmic scale, becomes the percent of its own peak level
(taken as one hundred percent). The linear transform eases the
comparison among cases and the identification of the duration and
midpoint of the processes, but also compresses fluctuations.
Typhoid (Figure 2) is a systemic bacterial infection caused
primarily by Salmonella
typhi.[9] Mary Mallon, the
cook (and asymptomatic carrier) popularly known as Typhoid Mary, was a
major factor in empowering the New York City Department of Health at the
turn of the century. Typhoid was still a significant killer in 1900,
though spotty records show it peaked in the 1870s. In the 1890s, Walter
Reed, William T. Sedgewick, and others determined the etiology of
typhoid fever and confirmed its relation to sewage-polluted water. It
took about 40 years to protect against typhoid, with 1914 the year of
inflection or peak rate of decline.
Diphtheria (Figure 2) is an acute infectious disease caused by
diphtheria toxin of the Corynebacterium diphtheriae. In
Massachusetts, where the records extend back further than for the United
States as a whole, diphtheria flared to 196 per 100,000 in 1876, or
about 10 percent of all deaths. Like typhoid, diphtheria took 40 years
to defense, centered in 1911. By the time the diphtheria vaccine was
introduced in the early 1930s, 90 percent of its murderous career
transition was complete.
Next comes the category of diseases of the gut (Figure 2).
Deaths here are mostly attributed to acute dehydrating diarrhea,
especially in children, but also to other bacterial infections such as
botulism and various kinds of food poisoning. The most notorious
culprit was the Vibrio
cholerae. In 1833, while essayist Ralph Waldo Emerson was working
on his book Nature, expounding
the basic benevolence of the universe, a cholera pandemic killed 5 to 15
percent of the population in many American localities where the normal
annual death rate from all causes was 2 or 3 percent.
In 1854 in London a physician and health investigator, John Snow,
seized the idea of plotting the locations of cholera deaths on a map of
the city. Most deaths occurred in St. James Parish, clustered about the
Broad Street water pump. Snow discovered that cholera victims who lived
outside the Parish also drew water from the pump. Although consumption
of the infected water had already peaked, Snow's famous removal of the
pump handle properly fixed in the public mind the means of cholera
transmission.[10]
In the United States, the collapse of cholera and its relations took
about 60 years, centered on 1913. As with typhoid and diphtheria,
sanitary engineering and public health measures addressed most of the
problem before modern medicine intervened with antibiotics in the
1940s.
In the late 1960s, deaths from gastrointestinal disease again fell
sharply. The fall may indicate the widespread adoption of intravenous
and oral rehydration therapies and perhaps new antibiotics. It may also
reflect a change in record-keeping.
Tuberculosis (Figure 2) refers largely to the infectious
disease of the lungs caused by Mycobacterium tuberculosis. In
the 1860s and 1870s in Massachusetts, TB peaked at 375 deaths per
100,000, or about 15 percent of all deaths. Henry David Thoreau, author
of Walden: or, Life in the
Woods, died of bronchitis and tuberculosis at the age of 45 in
1862. TB took about 53 years to jail, centered in 1931. Again, the
pharmacopoeia entered the battle rather late. The multi-drug therapies
became effective only in the 1950s.
Pneumonia and influenza are combined in Figure 3. They may
comprise the least satisfactory category, mixing viral and bacterial
aggressors. Figure 3 includes Influenza A, the frequently
mutating RNA virus believed to have induced the Great Pandemic of
1918-1919 following World War I, when flu seized about a third of all
corpses in the United States. Pneumonia and influenza were on the loose
until the 1930s. Then, in 17 years centered on 1940 the lethality of
pneumonia and influenza tumbled to a plateau where "flu" has
remained irrepressibly for a half century.
Now we shift from pathogens to a couple of other major killers.
Major cardiovascular diseases, including heart disease, hypertension,
cerebrovascular diseases, atherosclerosis, and associated renal diseases
display their triumphal climb and incipient decline in Figure 3.
In 1960, about 55 percent of all fatal attacks were against the heart
and its allies, culminating a 60-year climb. Having lost 14 points of
market share in the past 40 years, cardiovascular disease looks
vulnerable. Other paths descend quickly, once they bend downward. We
predict an 80-year drop to about 20 percent of American dead.
Cardiovascular disease is ripe for treatment through behavioral change
and medicine.
A century of unremitting gains for malignant neoplasms appears neatly
in Figure 3. According to Ames et al., the culprits are
ultimately the DNA-damaging oxidants.[11] One might argue caution in
lumping together lung, stomach, breast, prostate, and other cancers.
Lung and the other cancers associated with smoking account for much of
the rising slope. However, the cancers whose occurrence has remained
constant are also winning share if other causes of death diminish. In
the 1990s the death rate from malignancies flattened, but the few years
do not yet suffice to make a trend. According to the model, cancer's
rise should last 160 years and at peak account for 40 percent of
American deaths.
The spoils of AIDS, a meteoric viral entrant, are charted in
Figure 3. The span of data for AIDS is short, and the data
plotted here may not be reliable. Pneumonia and other causes of death
may mask AIDS' toll. Still, this analysis suggests AIDS reached its
peak market of about 2 percent of deaths in the year 1995. Uniquely,
the AIDS trajectory suggests medicine sharply blocked a deadly career,
stopping it about 60% of the way toward its project fulfillment.
Now look at the eight causes of death as if it were open hunting
season for all (Figure 4). Shares of the hunt changed
dramatically, and fewer hunters can still shoot to kill with
regularity. We can speculate why.
BY WATER, BY AIR
First, consider what we label the aquatic kills: a combination of
typhoid and the gastrointestinal family. They cohere visually and phase
down by a factor of ten over 33 years centered on 1919 (Figure
5).
Until well into the 19th century, towndwellers drew their water from
local ponds, streams, cisterns, and wells.[12] They disposed of
the wastewater from cleaning, cooking, and washing by throwing it on the
ground, into a gutter, or a cesspool lined with broken stones. Human
wastes went to privy vaults, shallow holes lined with brick or stone,
close to home, sometimes in the cellar. In 1829 residents of New York
City deposited about 100 tons of excrement each day in the city soil.
Scavengers collected the “night soil” in carts and dumped it nearby,
often in streams and rivers.
Between 1850 and 1900 the share of the American population living in
towns grew from about 15 to about 40 percent. The number of cities over
50,000 grew from 10 to more than 50. Increasing urban density made
waste collection systems less adequate. Overflowing privies and
cesspools filled alleys and yards with stagnant water and fecal wastes.
The growing availability of piped-in water created further stress. More
water was needed for fighting fires, for new industries that required
pure and constant water supply, and for flushing streets. To the extent
they existed, underground sewers were designed more for storm water than
wastes. One could not design a more supportive environment for typhoid,
cholera, and other water-borne killers.
By 1900 towns were building systems to treat their water and sewage.
Financing and constructing the needed infrastructure took several
decades. By 1940 the combination of water filtration, chlorination, and
sewage treatment stopped most of the aquatic killers.
Refrigeration in homes, shops, trucks, and railroad boxcars took care
of much of the rest. The chlorofluorocarbons (CFCs) condemned today for
thinning the ozone layer were introduced in the early 1930s as a safer
and more effective substitute for ammonia in refrigerators. The ammonia
devices tended to explode. If thousands of Americans still died of
gastrointestinal diseases or were blown away by ammonia, we might
hesitate to ban CFCs.
Let us move now from the water to the air (Figure 6).
“Aerial” groups all deaths from influenza and pneumonia, TB, diphtheria,
measles, whooping cough, and scarlet fever and other streptococcal
diseases. Broadly speaking these travel by air. To a considerable
extent they are diseases of crowding and unfavorable living and working
conditions.
Collectively, the aerial diseases were about three times as deadly to
Americans as their aquatic brethren in 1900. Their breakdown began more
than a decade later and required almost 40 years.
The decline could be decomposed into several sources. Certainly
large credit goes to improvements in the built environment: replacement
of tenements and sweatshops with more spacious and better ventilated
homes and workplaces. Huddled masses breathed free. Much credit goes
to electricity and cleaner energy systems at the level of the end
user.
Reduced exposure to infection may be an unrecognized benefit of
shifting from mass transit to personal vehicles. Credit obviously is
also due to nutrition, public health measures, and medical
treatments.
The aerial killers have kept their market share stable since the
mid-1950s. Their persistence associates with poverty; crowded
environments such as schoolrooms and prisons; and the intractability of
viral diseases. Mass defense is more difficult. Even the poorest
Bostonians or Angelenos receive safe drinking water; for the air, there
is no equivalent to chlorination.
Many aerial attacks occurred in winter, when indoor crowding is
greatest. Many aquatic kills were during summer, when the organic
fermenters were speediest. Diarrhea was called the summer complaint.
In Chicago between 1867 and 1925 a phase shift occurred in the peak
incidence of mortality from the summer to the winter months.[13] In America and
other temperate zone industrialized countries, the annual mortality
curve has flattened during this century as the human environment has
come under control. In these countries, most of the faces of death are
no longer seasonal.
BY WAR, BY CHANCE?
Let us address briefly the question of where war and accidents fit.
In our context we care about war because disputed control of natural
resources such as oil and water can cause war. Furthermore, war leaves
a legacy of degraded environment and poverty where pathogens find prey.
We saw the extraordinary spike of the flu pandemic of 1918-1919.
War functions as a short-lived and sometimes intense epidemic. In
this century, the most intense war in the developed countries may have
been in France between 1914-1918, when about one-quarter of all deaths
were associated with arms.[14]
The peak of 20th century war deaths in the United States occurred
between 1941-1945 when about 7 percent of all deaths were in military
service, slightly exceeding pneumonia and influenza in those years.
Accidents, which include traffic, falls, drowning, and fire follow a
dual logic. Observe the shares of auto and all other accidents in the
total kills in the United States during this century (Figure 7).
Like most diseases, fatal non-auto accidents have dropped, in this case
rather linearly from about 6 percent to about 2 percent of all
fatalities. Smiths and miners faced more dangers than office workers.
The fall also reflects lessening loss of life from environmental hazards
such as floods, storms, and heat waves.
Auto accidents do not appear accidental at all but under perfect
social control. On the roads, we appear to tolerate a certain range of
risk and regulate accordingly, an example of so-called risk
homeostasis.[15]
The share of killing by auto has fluctuated around 2 percent since about
1930, carefully maintained by numerous changes in vehicles, traffic
management, driving habits, driver education, and penalties.
DEADLY ORDER
Let us return to the main story. Infectious diseases scourged the
19th century. In Massachusetts in 1872, one of the worst plague years,
five infectious diseases, tuberculosis, diphtheria, typhoid, measles,
and smallpox, alone accounted for 27 percent of all deaths. Infectious
diseases thrived in the environment of the industrial revolution's new
towns and cities, which grew without modern sanitation.
Infectious diseases, of course, are not peculiarly diseases of
industrialization. In England during the intermittent plagues between
1348-1374 half or more of all mortality may have been attributable to
the Black Death.[16]
The invasion of smallpox into Central Mexico at the time of the Spanish
conquest depopulated central Mexico.[17] Gonorrhea depopulated the Pacific
island of Yap.[18]
At the time of its founding in 1901, our institution, the Rockefeller
Institute for Medical Research as it was then called, appropriately
focused on the infectious diseases. Prosperity, improvements in
environmental quality, and science diminished the fatal power of the
infectious diseases by an order of magnitude in the United States in the
first three to four decades of this century. Modern medicine has kept
the lid on.[19]
If infections were the killers of reckless 19th century urbanization,
cardiovascular diseases were the killers of 20th century modernization.
While avoiding the subway in your auto may have reduced the chance of
influenza, it increased the risk of heart disease. Traditionally
populations fatten when they change to a “modern” lifestyle. When
Samoans migrate to Hawaii and San Francisco or live a relatively
affluent life in American Samoa, they gain between 10 and 30 kg.[20]
The environment of cardiovascular death is not the Broad Street pump
but offices, restaurants, and cars. So, heart disease and stroke
appropriately roared to the lead in the 1920s.
Since the 1950s, however, cardiovascular disease has steadily lost
ground to a more indefatigable terminator, cancer. In our calculation,
cancer passed infection for the #2 spot in 1945. Americans appear to
have felt the change. In that year Alfred P. Sloan and Charles
Kettering channeled some of the fortune they had amassed in building the
General Motors Corporation to found the Sloan-Kettering Cancer Research
Center.
Though cancer trailed cardiovascular in 1997 by 41 to 23 percent,
cancer should take over as the nation's #1 killer by 2015, if long-run
dynamics continue as usual (Figure 8). The main reasons are not
environmental. Doll and Peto estimate that only about 5 percent of U.S.
cancer deaths are attributable to environmental pollution and
geophysical factors such as background radiation and sunlight.[21]
The major proximate causes of current forms of cancer, particularly
tobacco smoke and dietary imbalances, can be reduced. But if Ames and
others are right that cancer is a degenerative disease of aging, no
miracle drugs should be expected, and one form of cancer will succeed
another, assuring it a long stay at the top of the most wanted list. In
the competition among the three major families of death, cardiovascular
will have held first place for almost 100 years, from 1920 to 2015.
Will a new competitor enter the hunt? As various voices have warned,
the most likely suspect is an old one, infectious disease.[22] Growth of
antibiotic resistance may signal re-emergence. Also, humanity may be
creating new environments, for example, in hospitals, where infection
will again flourish. Massive population fluxes over great distances
test immune systems with new exposures. Human immune systems may
themselves weaken, as children grow in sterile apartments rather than
barnyards.[23]
Probably most important, a very large number of elderly offer weak
defense against infections, as age-adjusted studies could confirm and
quantify. So, we tentatively but logically and consistently project a
second wave of infectious disease. In Figure 9 we aggregate all
major infectious killers, both bacterial and viral. The category thus
includes not only the aquatics and aerials discussed earlier, but also
septicemia, syphilis, and AIDS.[24] A grand and orderly succession
emerges.
SUMMARY
Historical examination of causes of death shows that lethality may
evolve in consistent and predictable ways as the human environment comes
under control. In the United States during the 20th century infections
became less deadly, while heart disease grew dominant, followed by
cancer. Logistic models of growth and multi-species competition in
which the causes of death are the competitors describe precisely the
evolutionary success of the killers, as seen in the dossiers of typhoid,
diphtheria, the gastrointestinal family, pneumonia/influenza,
cardiovascular disease, and cancer. Improvements in water supply and
other aspects of the environment provided the cardinal defenses against
infection. Environmental strategies appear less powerful for deferring
the likely future causes of death. Cancer will overtake heart disease
as the leading U.S. killer around the year 2015 and infections will
gradually regain their fatal edge. If the orderly history of death
continues.
FIGURES

Figure
1. Crude Death Rate: U.S.
1900-1997. Sources of data: Note 4.

Figure 2a.
Typhoid and Paratyphoid Fever as a Fraction of All Deaths: U.S.
1900-1952. The larger panel shows the
raw data and a logistic curve fitted to the data. The inset panel shows the same data and a transform that renders
the S-shaped curve linear and normalizes the process to 1. "F" refers to the fraction of the
process completed. Here the time it
takes the process to go from 10 percent to 90 percent of its extent is 39
years, and the midpoint is the year 1914.
Source of data: Note 4.
Figure 2b.
Diphtheria as a Fraction of All Deaths: U.S. 1900-1956. Source of data: Note 4.
Figure 2c.
Gastritis, Duodenitis, Enteritis, and Colitis as a Fraction of All
Deaths: U.S. 1900-1970. Source of data: Note 4.
Figure 2d. Tuberculosis, All Forms, as a Fraction of
All Deaths: U.S. 1900-1997. Sources of data: Note 4.

Figure 3a.
Pneumonia and Influenza as a Fraction of All Deaths: U.S. 1900-1997.
Note the extraordinary pandemic of 1918-1919. Sources of data: Note 4. Figure
3b. Major Cardiovascular Diseases
as a Fraction of All Deaths: U.S. 1900-1997.
In the inset, the curve is decomposed into upward and downward logistics
which sum to the actual data values.
The midpoint of the 60-year rise of cardiovascular disease was the year
1939, while the year 1983 marked the midpoint of its 80-year decline. Sources of data: Note 4. Figure 3c. Malignant Neoplasms as a Fraction of All
Deaths: U.S. 1900-1997. Sources of data: Note 4. Figure 3d. AIDS as a Fraction of All Deaths: U.S.
1981-1997. Sources of data: Note 4.

Figure 4. Comparative Trajectories of Eight Killers:
U.S. 1900-1997. The scale is
logarithmic, with fraction of all deaths shown on the left scale with the
equivalent percentages marked on the right scale. Sources of data: Note 4.

Figure 5. Deaths from Aquatically Transmitted Diseases
as a Fraction of All Deaths: U.S. 1900-1967.
Superimposed is the percentage of homes with water and sewage service
(right scale). Source of data: Note 4.

Figure 6. Deaths from Aerially Transmitted Diseases as
a Fraction of All Deaths: U.S. 1900-1997. Sources of data: Note 4.

Figure 7. Motor Vehicle and All Other Accidents as a
Fraction of All Deaths: U.S. 1900-1997.
Sources of data: Note 4.

Figure 8.
Major Cardiovascular Diseases and Malignant Neoplasms as a Fraction of
All U.S. Deaths: 1900-1997. The
logistic model predicts (dashed lines) Neoplastic will overtake Cardiovascular
as the number one killer in 2015.
Sources of data: Note 4.

Figure 9.
Major Causes of Death Analyzed with a Multi-species Model of Logistic
Competition. The fractional shares are
plotted on a logarithmic scale which makes linear the S-shaped rise and fall of
market shares.
NOTES