Figures and Tables excerpted from Business Dynamics: Systems Thinking and Modeling for a Complex World Chapter 9 Dynamics of Growth_ S Shape Growth John D. Sterman Massachusetts Institute of Technology Sloan School of Management Figures and Tables excerpted from BUSINESS DYNAMICS: SYSTEMS THINKING AND MODELING FOR A COMPLEX WORLD John D. Sterman Published by Irwin/McGraw-Hill, an imprint of the McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright  2001 by the McGraw-Hill Companies, Inc. All rights reserved. The contents, or parts thereof, may be reproduced in print form solely for classroom use with provided such reproductions bear copyright notice, but may not be reproduced in any other form or for any other purpose without the prior written consent the McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Business Dynamics

The logistic model Figure 9-1 Top: The fractional growth rate declines linearly as population grows. Middle: The phase plot is an inverted parabola, symmetric about (P/C) = 0.5 Bottom: Population follows an S-shaped curve with inflection point at (P/C) =0.5; the net growth rate follows a bell-shaped curve with a maximum value of 0.25C per time period. Business Dynamics

Figure 9-2 The growth of sunflowers and the best fit logistic model Business Dynamics

Dynamics of epidemic disease Influenza epidemic at an English boarding school, January 22-February 3, 1978. The data show the number of students Confined to bed for influenza at any time (the stock of symptomatic individuals). Epidemic of plague, Bombay, India 1905-6. Data show the death rate (deaths/week). Figure 9-3 Dynamics of epidemic disease Sources: Top: British Medical Journal, 4 March 1978, p. 587; Bottom: Kermack and McKendrick (1927, p. 714). For further discussion of both cases, see Murray (1993). Business Dynamics

Structure of a simple model of an epidemic Figure 9-4 Births, deaths, and migration are omitted so the total population is a constant, and people remain infectious indefinitely. Business Dynamics

Structure of the SIR epidemic model IR=(ciS)(I/N) N is total population N=S+I Figure 9-5 People remain infectious (and sick) for a limited time, then recover and develop immunity. Business Dynamics

Figure 9-6 Simulation of an epidemic in the SIR model Figure 9-6 Simulation of an epidemic in the SIR model. The total population is 10,000. The contact rate is 6 per person per day, infectivity is 0.25, and average duration of infectivity is 2 days. The initial infective population is 1, and all others are initially susceptible. Business Dynamics

Epidemic dynamics for different contact rates Figure 9-7 The contact rate is noted on each curve; all other parameters are as in Figure 9-6. Business Dynamics

Figure 9-8 Dependence of the tipping point on the contact number and susceptible population Business Dynamics

Figure 9-9 Successive epidemic waves created by increasing contact rate Business Dynamics

Figure 9-10 Mad cow disease—the epidemic of BSE in the United Kingdom Source: UK Ministry of Agriculture, Fisheries, and Food. Business Dynamics

Quarter-Year Figure 9-11 Incidence and mortality of AIDS in the US Source:: US Centers for Disease Control and Prevention. HIV/AIDS Surveillance Report, Midyear 1997 edition, vol. 9 (no. 1), figure 6 and caption, p. 19. Business Dynamics

Figure 9-12 Prevalence of AIDS in the United States Source:: US Centers for Disease Control and Prevention. HIV/AIDS Surveillance Report, 1996, vol. 8 (no. 2), p. 1. Business Dynamics

Figure 9-13 Adoption of a new idea or product as an epidemic Business Dynamics

Figure 9-14 Sales of the Digital Equipment Corporation VAX 11/750 in Europe Top: Sales rate (quarterly data at annual rates). Bottom: Cumulative sales (roughly equal to the installed base). Business Dynamics

Figure 9-15 Fitting the logistic model of innovation diffusion Business Dynamics

Figure 9-16 Fitting the logistic model to data for US cable TV subscribers Business Dynamics

Figure 9-17 Predicted cable subscribers differ greatly depending on the growth model used. Business Dynamics

Figure 9-18 The Bass diffusion model Business Dynamics

Figure 9-19 The Bass and logistic diffusion models compared to actual VAX sales Business Dynamics

Figure 9-20 Modeling product discard and replacement purchases Business Dynamics

Figure 9-21 Behavior of the Bass model with discards and repurchases Business Dynamics

Figure 9-22 Modeling repeat purchases Figure 9-22 Modeling repeat purchases. Total sales consist of initial and repeat purchases. Each potential adopter buys Initial Sales per Adopter units when they first adopt the product and continues to purchase at the rate of Average Consumption per Adopter thereafter. Business Dynamics

Figure 9-23 Behavior of the repeat purchase model Business Dynamics