Bi-Logistic Growth

This paper first appeared in the journal Technological Forecasting and Social Change, published by Elsevier Science Inc., New York.

Introduction

Many processes in biology and other fields exhibit S-shaped growth. Often the curves are well modeled by the simple logistic growth function, first introduced by Verhulst in 1845. Although the logistic curve has often been criticized for being applied to systems where it is not appropriate, it has proved useful in modeling a wide range of phenomena. Kingsland [1] provides a thorough history of the applications of the simple logistic curve in population ecology, its successes and failures. Marchetti and colleagues at IIASA [2,3], as well as many others [4], have shown thousands of examples, mainly in socio-technical systems. Recently, Young [5] surveyed and compared growth curves used for technological forecasting, including the logistic function. Almost all the analyses and successes apply to the case of a single growth process operating in isolation. Here, I extend the analysis of logistic functions to cases where dual processes operate.

The carrying capacity of a human system is often limited by the current level of technology, which is subject to change. More generally, species can sometimes alter and expand their niche. If the carrying capacity of a system changes during a period of logistic growth, a second period of logistic growth with a different carrying capacity can superimpose on the first growth pulse. For example, cars first replaced the population of horses but then took on a further growth trajectory of their own. We call such a system with two logistic growth pulses, growing at the same time or sequentially, “Bi-logistic.” As I will show, the Bi-logistic is useful in modeling many systems that contain complex growth processes not well modeled by the simple logistic.

The plan of this paper is as follows. First, a model based on the sum of two simple logistic growth pulses is presented in order to analyze systems that exhibit Bi-logistic growth. A nonlinear least-squares algorithm is described that allows values for the model parameters to be estimated from time-series growth data. Then, model sensitivity and robustness are discussed in relation to error structure in the data. Finally, a taxonomy and examples of systems that exhibit Bi-logistic growth are discussed.

Logistic Growth

The logistic law of growth assumes that systems grow exponentially until an upper limit or “carrying capacity” inherent in the system is approached, at which point the growth rate slows and eventually saturates, producing the characteristic S-shape curve [6]. In the simple exponential growth model, the growth rate of a population, N(t), is proportional to the population

.(1)

As a consequence, there are no limits to growth; as t® ¥N(t)® ¥. In the familiar analytic form, a is a growth rate parameter and b is a location parameter that shifts the curve horizontally but does not alter its shape:

(2)

The logistic model adds to the exponential model (1) a feedback term that slows the growth rate of the system as the “carrying capacity” or saturation parameter k is reached

(3)

For values of N(t) << k, equation (3) closely resembles exponential growth. As the population N(t) approaches k, the feedback term causes the rate of growth to slow to zero, giving rise to the familiar symmetrical S-shaped curve. The logistic law of growth arises as a solution to equation (3)

(4)

where a is a rate parameter; b is a location parameter (it shifts the function in time but does not affect the function’s shape); and k is the asymptotic value that bounds the function and therefore specifies the level at which the growth process saturates [7]. Symmetry implies that the logistic function has a point of inflection at k/2. It is convenient to define tm as the midpoint of the growth process: N(tm) = k/2. The location parameter b can be replaced by tm by defining b = -tma. It is also convenient to define a parameter Dt as the length of the time interval required for the growth process to grow from 10 to 90 percent of the saturation level k. The length of this interval (derived through simple algebra) is Dt = (ln81)/a.

An equivalent form of the standard 3-parameter logistic model (4) with parameters convenient for the analysis of historical time-series data can be defined as

.(5)

Figure 1. Growth of a sunflower fitted with a single logistic curve. The inset shows the logistic curve and the data linearized with the Fisher-Pry transform. The lower panel shows the residuals in percent deviation from the fitted curve. Source of data: [8].

Figure 1 shows the growth of a sunflower [8] and the corresponding logistic curve. The residuals (in percent deviation) are plotted in t he panel beneath the logistic curve.

The logistic growth curve can be linearized by a change of variable (first discussed by Fisher and Pry [9]), by defining

(6)

and substituting into equation (4)

.(7)

Plotting equation 7 with a logarithmic y-axis produces a straight line, and Dand tm can be easily read off this plot if the corresponding percents of saturation are marked. Plotting the logistic linearly also facilitates the rapid comparison to other logistic growth processes because all the curves are normalized by k. The inset of Figure 1 shows the sunflower data and the corresponding logistic curve plotted linearly. If k is known, the parameters Dand tm can be determined by using a linear regression technique to fit a straight line through the transformed data.

As discussed, the logistic growth model has been successfully applied to a wide range of biological and socio-technical systems. To explain why the logistic is so pervasive, Montroll [10] postulates “laws” of social dynamics modeled after Newton’s laws of particle dynamics. The first law of social dynamics states that “in the absence of any social, economic, or ecological force, the rate of change of the logarithm of a population, N(t), of an ‘organism’ is constant”,

.(8)

This is equivalent to Newton’s first law, which states that a particle in motion in the absence of any external forces will travel in a straight line with constant velocity. Equation (8) is also equivalent to exponential growth.

Montroll’s second law of social dynamics states that equation (8) is violated when a social, economic, or ecological force is applied. One of the simplest “forces” that could replace the right-hand side of equation (8) is a linear force proportional to the population:

(9)

which represents a deterrence to population growth. If g is replaced by a/k, where k is the carrying capacity, equation (8) becomes

(10)

which is equivalent to the logistic model (3). Thus, logistic growth can be viewed as a canonical form of growth for a system that is subject to forces that slow unconstrained growth. If multiple forces operate, a system can undergo more than one logistic growth pulse, as will be discussed shortly.

Bi-Logistic Growth

The standard 3-parameter form of the logistic growth model describes one period or “pulse” of growth as the system proceeds from rapid exponential growth to slow growth as the carrying capacity k is approached. Multiple growth pulses characterize many systems. In the case of a system with two well-defined serial logistic growth pulses, it is possible to split the time-series data set in two and model each set with a separate 3-parameter logistic function. This method is limited because it is often unclear exactly where to split the data set. Cases appear rare where one process ends entirely before the second begins. Problems arise in assigning values from the “overlap” period to the first or second pulse.

A superior alternative is to analyze systems that exhibit Bi-logistic growth by using the time-series data to estimate the parameters of a model comprised of the sum of two 3-parameter logistic growth pulses. The Bi-logistic growth model is then

.(11)

Selection of a method to estimate the parameters depends on the assumed distribution of the measurement errors in the data. A standard procedure is to assume that the measurement errors are independently and normally distributed with constant standard deviation. The best-fit parameters can then be found by minimizing the sum of the squares of the residuals. The residuals are defined as the difference between the time series data set (ti,yi) with m data points and the Bi-logistic model N(t)

Residuals = i = 1…m(12)

The parameter estimates can then be found by using a nonlinear regression technique to minimize the sum of the squares of the residuals

Minimize(13)

Figure 2. Example of a Bi-Logistic growth curve generated with 3% relative Gaussian error. The inset shows component growth curves.

The measurement errors of many historical data sets are unknown. Thus the common assumption that the errors are independently and normally distributed is often invalid. A least-squares method of regression can still be used to estimate parameters for these data sets, but the estimates are no longer guaranteed to be unbiased. When the measurement errors of a time series data set are unknown, as in the examples presented later in this paper, an ordinary least-squares regression technique, which gives equal weight to all of the data points, may be preferable.

In the following analyses, the Levenberg-Marquardt (L-M) method [11] of nonlinear least-squares regression is used to estimate the 6 parameters of the Bi-logistic function (11). The L-M algorithm requires provisional estimates to initiate its search for the parameters. That is, some a priori or external knowledge of the system is needed to derive estimates reasonably and efficiently. Usually, simple visual examination of the plotted raw data suffices. The L-M implementation used allows any number of the 6 parameters to be held at a constant value (if, say, the carrying capacity of a system is known). This feature also facilitates the derivation of initial parameters, because the regression routine has better convergence properties when fewer parameters are estimated.

Figure 2 shows a generated time-series data set fit with a Bi-logistic curve. The data set is the sum of two identical logistic growth pulses with the midpoints (tm) separated by 40 years. The first pulse has reached 90% of saturation (k1) before the second pulse begins, and thus two overlapping S-shaped curves are visible. Once the fit is obtained, a simple deconvolution can be defined as follows

(14)

where y1i and y2i are the component growth variables, which are plotted in the inset of figure 2.

Figure 3. Example of a Bi-Logistic growth curve generated with 3% relative Gaussian error. The inset shows the component growth curves linearized with the Fisher-Pry transform. The lower panel shows the residuals from the fit in percent deviation from the fitted curve.

The two data sets (ti, y1i) and (ti, y2i), can also be plotted as a linear function of time by utilizing the Fisher-Pry transform, as shown in the inset of Figure 3, with the circles designating (ti, y1i) and the squares designating (ti, y2i). When the second logistic pulse is below 1% of saturation (k2), the first component data set (ti, y1i>) is essentially identical to the raw data, (ti, yi), and it is plotted with solid circles. After this, the data are plotted with open circles to indicate that the data have been transformed. The second component growth data set (ti, y2i) is plotted with hollow squares to indicated that these data have also been transformed. The linear form of the Bi-logistic facilitates morphological analysis and comparison to other Bi-logistic processes.

Figure 4. Average height of American Boys with a Bi-Logistic growth curve. Note that the Bi-logistic curve is offset by 30 inches in order to account for early growth (ages 0 to 3). Source of data: [12].

A well-known growth process involving two growth spurts is shown in Figure 4, the average height of boys ages 3 to 19, in this case, American [12]. Two S-shaped growth pulses are clearly visible. The first growth pulse shown is centered at 5 years and has a characteristic growth time, Dt, of 10 years. The second growth pulse, called the “prepubertal acceleration” or the “adolescent spurt” is shorter and is centered at 13 years old. This growth pulse saturates at 68.7 inches, the average height of American men. The inset shows the Fisher-Pry linear transform of the two growth pulses.

The residuals are useful in determining how well the Bi-logistic model fits the data. If a system is well modeled by the Bi-logistic function, then the residuals will contain only noise, and the residuals will be randomly distributed around zero. The residuals can also tell a lot about the error structure. The lower part of Figures 3 and 4 show the residuals of the fit on the two time-series data. The residuals are shown as the percent deviation from the estimated value

residuals in percent deviation =.(15)

Many time series data sets from systems that are studied with logistics contain error that is relative to the growth level, which can change by orders of magnitude in the duration of the process. Accordingly, it is useful to analyze the residuals in percent deviation. While the ordinary least-squares technique used for analysis assumes constant error variance, it might be advantageous to use a regression method that assumes constant relative error, thus weighing the early growth data more heavily than the later data. However, early growth data are often unreliable, as processes may also not be well recorded or established. Thus there is a trade off between assuming relative error and constant error. The effect can be seen in the comparatively high levels of error present in the early data on the residual plot of the generated time-series data (figure 3), which was fit assuming constant error. More research is needed to determine the error structure of historical data-sets and on regression techniques that yield the best parameter estimates. Monte-Carlo techniques could be used to generate sample data sets with different error structures, and the subsequent analysis would be useful in determining confidence intervals for the Bi-logistic model parameters. Residual analysis could also identify “slices” of data that are especially noise-free and might be more heavily weighted when fitting.

Taxonomy of the Bi-logistic

A continuous spectrum of curves can be generated from the Bi-logistic model. However, it is useful to distinguish four basic patterns of Bi-logistic growth in order to develop a taxonomy of curves that can be used as a reference when analyzing systems where the shapes of the two underlying logistic trends are not already known. As mentioned, the regression routine used requires initial estimates, and a taxonomy is useful in this regard.

Figure 5. Taxonomy of the Bi-logistic growth model.

Figure 5 shows four hypothetical curves and their linearized versions.

Curve A of Figure 5 shows a Bi-logistic curve with two almost non-overlapping logistic growth pulses, dubbed the “sequential logistic”. The second pulse does not start growing until the first pulse has reached about 99% of saturation k1. This shape Bi-logistic characterizes a system which pauses between growth phases.

The B curve shows a Bi-logistic where the second pulse starts growing when the first pulse has reached about 50% of saturation. This “superposed” Bi-logistic growth model characterizes systems that contain two processes of a similar nature growing concurrently except for a displacement in the midpoints of the curves.

Curve C shows a growth process where a first pulse of logistic growth is joined by a second faster pulse, dubbed the “converging” logistic model, as the two pulses culminate about the same time. Often an advance in technology will allow both the carrying capacity and the growth rate of a system to increase, causing the second pulse to rise from the first with both a faster characteristic Dt, and higher carrying capacity,k.

Curve D shows a “diverging” Bi-logistic curve where two logistic growth processes begin at the same time but grow with different rates and carrying capacities defined from the start. It is noteworthy that curves C and D are S-shaped but asymmetric. They do not “look logistic.”

Examples and Results

A wide variety of historical time-series data sets were analyzed with the Bi-logistic model. The data sets exhibited here show the four types of Bi-logistic growth described above. The data sets chosen all show growth processes that have neared saturation in order to permit analysis of the residuals for the entire growth process. The data sets were also fitted with a single logistic growth pulse to check the improvement in fit by the Bi-logistic.

Figure 6. Growth of U.S. universities with a Bi-Logistic growth curve. Source of data: [13].

A sequential Bi-logistic is shown in figure 6, the growth of U.S. universities since 1700, as tabulated from the founding dates provided in Webster’s New Collegiate Dictionary [13]. The first pulse saturates at a k of 500 universities with the point of inflection and fastest rate of growth, tm, occurring in 1885. This is when many states inaugurated their public university systems. The second, smaller logistic pulse starts in 1950 when the first pulse has reached about 95% of saturation and has a much quicker characteristic growth time, Dt, of about 15 years. This pulse represents largely the creation of additional daughter campuses of state universities, a smaller niche to fill than the founding of universities for the U.S. as a whole.

Figure 7. Growth of U.S. universities with a single logistic growth curve. Source of data: [13].

To indicate the superiority of the Bi-logistic, consider Figure 7, which shows the same university data fitted with a single logistic curve. Optical inspection of the fitted curve as well as analysis of the residuals show that the Bi-logistic model fits the data much better than a single logistic curve. The residuals of the early data (1600-1800) have small absolute error but because the growth level is low the percent error is very high (~100%). As mentioned, this deviation is caused in part by the non-weighted regression algorithm used. The early growth of systems (below 10% of final saturation) is also suspect because feedback mechanisms that are assumed for logistic growth might not be in place yet, and thus the growth is probably not firmly logistic until a growth level of about 10% of the final saturation value has been reached.

Figure 8. U.S. nuclear weapons tests with a Bi-logistic growth curve. Source of data: [14].

Figure 8 shows the cumulative number of U.S. nuclear weapons tests [14] with a superposed Bi-logistic curve. The Bi-logistic provides an excellent fit, as shown by the residuals. The fastest rate of growth of the first pulse occurred in 1963, following the Cuban missile crisis. While the first logistic pulse was largely the race to develop bombs with higher yields, the second pulse, centered in 1983 and nearing saturation now, is probably due to research on reliability and specific weapons designed for tactical use. The Bi-logistic model predicts that we are at 90% of saturation of the latest pulse. Processes often expire around 90%, though sometimes processes overshoot. The residuals show the extraordinary, deviant increase in U.S. tests after the scare of the 1957 sputnik launch.

Figure 9. U.S. installed electric generating capacity with a Bi-logistic growth curve. Source of data: [15, 16].

Figure 9 shows the U.S. installed electric generating capacity [15,16] with a converging Bi-logistic curve. The first logistic pulse saturates at about 43 GW and is centered in 1926. A second shorter but much higher pulse begins in about 1940 and is at about 90% of saturation now. Ausubel and Marchetti [17] provide a detailed analysis of the underlying mechanisms affecting the electrification of the U.S. The first pulse of growth is associated with pure substitution, for example, the replacement of water mills and gaslight by more efficient and convenient electric devices. The second and much larger growth pulse is due to the increase in demand of electricity for energy functions that could not be easily fulfilled before electrification, ranging from TV’s to space cooling. The pair of pulses have more or less saturated now. A third electric pulse might be starting with the rapid increase in demand for information handling and new concepts in electric transport.

Figure 10. Cumulative number of published works, Jesse H. Ausubel, with a Bi-logistic growth curve. Source of data: personal communication.

Figure 10 shows the cumulative number of publications written or edited by my colleague, Jesse H. Ausubel, fitted with a diverging Bi-logistic curve. The first smaller, steeper pulse consists of committee reports and other collective documents associated with his work as a staff officer and study director. The second longer and higher pulse consists of research papers of which Ausubel is an author. His dual professional career is made neatly apparent by the Bi-logistic.

Issues and Conclusion

To analyze time series data sets where the simple logistic curve provides a poor fit, many other growth models have been examined [18] , such as the Gompertz function. These data sets might contain multiple growth pulses that would be better modeled by the Bi-logistic. Some of the other models introduce higher-order parameters where the physical interpretation is less clear than in the case of the Bi-logistic. More research is needed in order to determine if the Bi-logistic model performs as well as more complex non-symmetrical growth functions. Complex systems can also undergo or consist of more than two pulses of growth, and research is needed into the feasibility of extending the Bi-logistic model into an arbitrary sum of simple logistics. Of course, with enough parameters anything can be fit.

The logistic growth function has proven useful in modeling a wide variety of phenomena in the growth of systems. However, complex systems rarely follow a single S-shaped trajectory. The Bi-logistic function is effective in modeling systems that contain two logistic growth pulses. The Bi-logistic is attractive because it is a parsimonious model to which we can still attach clear physical interpretations.

Acknowledgments: I am grateful to Jesse Ausubel, John Helm, Robert Herman, Arnulf Grübler, Cesare Marchetti, Nebojsa Nakicenovic, and Andy Solow for advice and assistance.

References

  1. Kingsland, S., The Refractory Model: The Logistic Curve and the History of Population Ecology, The Quarterly Review of Biology 57, 29-52 (1982).
  2. Marchetti, C., Branching out into the Universe, in Diffusion of Technologies and Social Behavior, N. Nakicenovic and A. Grübler, eds., Springer-Verlag, New York, NY, 1991.
  3. Grübler, A., The Rise and Fall of Infrastructures, Springer-Verlag, New York, NY, 1990.
  4. Oliver, R. M., Saturation Models: A Brief Survey and Critique, Journal of Forecasting (Special Issue on Predicting Saturation and Logistic Growth) 7, 15-255 (1988).
  5. Young, P., Technological Growth Curves: A Competition of Forecasting Models, Technological Forecasting and Social Change 44, 375-389 (1993).
  6. Stone, R., Sigmoids, Bulletin in Applied Statistics 7, 59-119 (1980).
  7. Nakicenovic, N., U.S. Transport Infrastructures, in Cities and Their Vital Systems, J. Ausubel and R. Herman, eds., National Academy Press, Washington, D.C., 1988.
  8. Reed, H.S. and Holland, R. H., The Growth of an Annual Plant HelianthusProceedings of the National Academy of Sciences (USA), 5, 135-144 (1919).
  9. Fisher, J.C., and Pry, R. H., A Simple Substitution Model of Technological Change, Technological Forecasting and Social Change 3, 75-88 (1971).
  10. Montroll, E. W., Social Dynamics and the Quantifying of Social Forces, Proceedings of the National Academy of Sciences (USA) 75(10), 4633-37 (1978).
  11. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical Recipes in C: The Art of Scientific Computing 2nd ed., Cambridge University Press, New York, NY, 1992.
  12. Krogman, W. M., Child Growth, University of Michigan Press, Ann Arbor, MI, 1972.
  13. Woolf, H. B., ed., Webster’s New Collegiate Dictionary, Merriam-Webster, Springfield, MA, 1979.
  14. Stockholm International Peace Research Institute Yearbook 1992, Oxford University Press, New York, 1992.
  15. U.S. Bureau of the Census, Historical Statistics of the United States, Washington, D.C., 1975.
  16. U.S. Bureau of the Census, Statistical Abstract of the United States, Washington, D.C., (Various Years).
  17. Ausubel, J. and Marchetti, C., Elektron, Forthcoming in Technological Trajectories and the Human Environment, J. Ausubel and D. Langford, eds., National Academy Press, Washington, D.C.
  18. Posch, M., Grübler, A., and Nakicenovic, N., Methods of Estimating S-Shaped Growth Functions, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1987.

Carrying Capacity: A Model with Logistically Varying Limits

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Death and the Human Environment: The United States in the 20th Century

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.

 CauseRateMode of Transmission 
1.Major Cardiovascular Disease345[N.A.]
2.Influenza, Pneumonia202Inhalation,Intimate Contact
3.Tuberculosis194Inhalation,Intimate Contact
4.Gastritis, Colitus,Enteritis, and Duodenitis142Contaminated Waterand Food
5.All Accidents72[Behavioral]
6.Malignant Neoplasms64[N.A.]
7.Diphtheria40Inhalation
8.Typhoid and ParatyphoidFever31Contaminated Water
9.Measles13Inhalation, Intimate Contact
10.Cirrhosis12[Behavioral]
11.Whooping Cough12Inhalation, Intimate Contact
12.Syphilis and Its Sequelae12Sexual Contact
13.Diabetes Mellitus11[N.A.]
14.Suicide10[Behavioral]
15.Scarlet Fever and Streptococcal Sore Throat9Inhalation, 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

[1] On the basic model see: Kingsland SE. Modeling Nature: Episodes in the History of Population Ecology. Chicago: University of Chicago Press, 1985. Meyer PS. Bi-logistic growth. Technological Forecasting and Social Change 1994;47:89-102.

[2] On the model of multi-species competition see Meyer PS, Yung JW, Ausubel JH. A Primer on logistic growth and substitution: the mathematics of the Loglet Lab software. Technological Forecasting and Social Change 1999;61(3):247-271.

[3] Marchetti C. Killer stories: a system exploration in mortal disease. PP-82-007. Laxenburg, Austria: International Institute for Applied Systems Analysis, 1982. For a general review of applications see: Nakicenovic N, Gruebler A, eds. Diffusion of Technologies and Social Behavior. New York: Springer-Verlag, 1991.

[4] U.S. Bureau of the Census, Historical Statistics of the United States: Colonial Times to 1970, Bicentennial Editions, Parts 1 & 2. Washington DC: U.S. Bureau of the Census: 1975. U.S. Bureau of the Census, Statistical Abstract of the United States: 1999 (119th edition). Washington DC: 1999, and earlier editions in this annual series.

[5] Deaths worldwide are assigned a “basic cause” through the use of the “Rules for the Selection of Basic Cause” stated in the Ninth Revision of the International Classification of Diseases. Geneva: World Health Organization. These selection rules are applied when more than one cause of death appears on the death certificate, a fairly common occurrence. From an environmental perspective, the rules are significantly biased toward a medical view. In analyzing causes of death in developing countries and poor communities, the rules can be particularly. For general discussion of such matters see Kastenbaum R, Kastenbaum B. Encyclopedia of Death. New York: Avon, 1993.

[6] For discussion of the relation of causes of death to the age structure of populations see Hutchinson GE. An Introduction to Population Ecology. New Haven: Yale University Press, 1978, 41-89. See also Zopf PE Jr. Mortality Patterns and Trends in the United States. Westport CT: Greenwood, 1992.

[7] Bozzo SR, Robinson CV, Hamilton LD. The use of a mortality-ratio matrix as a health index.” BNL Report No. 30747. Upton NY: Brookhaven National Laboratory, 1981.

[8] For explanation of the linear transform, see Fisher JC, Pry RH. A simple substitution model of technological change. Technological Forecasting and Social Change 1971;3:75-88.

[9] For reviews of all the bacterial infections discussed in this paper see: Evans AS, Brachman PS, eds., Bacterial Infections of Humans: Epidemiology and Control. New York: Plenum, ed. 2, 1991. For discussion of viral as well as bacterial threats see: Lederberg J, Shope RE, Oaks SC Jr., eds., Emerging Infections: Microbial Threats to Health in the United States. Washington DC: National Academy Press, 1992. See also Kenneth F. Kiple, ed., The Cambridge World History of Disease. Cambridge UK: Cambridge Univ. Press, 1993.

[10] For precise exposition of Snow’s role, see Tufte ER. Visual Explanations: Images and Quantities, Evidence and Narrative. Cheshire CT: Graphics Press, 1997:27-37.

[11] Ames BN, Gold LS. Chemical Carcinogens: Too Many Rodent Carcinogens. Proceedings of the National Academy of Sciences of the U.S.A. 1987;87:7772-7776.

[12] Tarr JA. The Search for the Ultimate Sink: Urban Pollution in Historical Perspective. Akron OH: University of Akron Press, 1996.

[13] Weihe WH. Climate, health and disease. Proceedings of the World Climate Conference. Geneva: World Meteorological Organization, 1979.

[14] Mitchell BR. European Historical Statistics 1750-1975. New York: Facts on File, 1980:ed. 2.

[15] Adams JGU., Risk homeostasis and the purpose of safety regulation. Ergonomics 1988;31:407-428.

[16] Russell JC. British Medieval Population. Albuquerque NM: Univ. of New Mexico, 1948.

[17] del Castillo BD. The Discovery and Conquest of Mexico, 1517-1521. New York: Grove, 1956.

[18] Hunt EE Jr. In Health and the Human Condition: Perspectives on Medical Anthropology. Logan MH, Hunt EE,eds. North Scituate, MA: Duxbury, 1978.

[19] For perspectives on the relative roles of public health and medical measures see Dubos R. Mirage of Health: Utopias, Progress, and Biological Change. New York: Harper, 1959. McKeown T, Record RG, Turner RD. An interpretation of the decline of mortality in England and Wales during the twentieth century,” Population Studies 1975;29:391-422. McKinlay JB, McKinlay SM. The questionable contribution of medical measures to the decline of mortality in the United States in the twentieth century.” Milbank Quarterly on Health and Society Summer 1977:405-428.¥r¥r

[20] Pawson IG, Janes, C. Massive obesity in a migrant Samoan population. American Journal of Public Health 1981;71:508-513.

[21] Doll R, Peto R. The Causes of Cancer. New York: Oxford University Press, 1981.

[22] Lederberg J, Shope RE, Oaks SC Jr., eds. Emerging Infections: Microbial Threats to Health in the United States. Washington DC: National Academy, 1992. Ewald PW. Evolution of Infectious Disease. New York: Oxford, 1994.

[23] Holgate ST, The epidemics of allergy and asthma. Nature 1999;402supp:B2-B4.

[24] The most significant present (1997) causes of death subsumed under “all causes” and not represented separately in Figure 9 are chronic obstructive pulmonary diseases (4.7%), accidents (3.9%), diabetes mellitus (2.6%), suicide (1.3%), chronic liver disease and cirrhosis (1.0%), and homicide (0.8%). The dynamic in the figure remains the same when these causes are included in the analysis. In our logic, airborne and other allergens, which cause some of the pulmonary deaths, might also be grouped with infections, although the invading agents are not bacteria or viruses.