Transcript BD Ch 9 Dynamics of Growth_ S Shape Growth

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
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BUSINESS DYNAMICS: SYSTEMS THINKING AND MODELING FOR A COMPLEX WORLD
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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.
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Figure 9-2 The growth of sunflowers and the best fit logistic model
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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).
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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.
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Structure of the SIR epidemic model
IR=(ciS)(I/N)
N=S+I
N is total population
Figure 9-5 People remain infectious (and sick) for a limited time, then recover and
develop immunity.
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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.
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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.
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Figure 9-8 Dependence of the tipping point on the contact number and
susceptible population
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Figure 9-9 Successive epidemic waves created by increasing contact rate
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Figure 9-10 Mad cow disease—the epidemic of BSE in the United Kingdom
Source: UK Ministry of Agriculture, Fisheries, and Food.
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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.
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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.
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Figure 9-13 Adoption of a new idea or product as an epidemic
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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).
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Figure 9-15 Fitting the logistic model of innovation diffusion
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Figure 9-16 Fitting the logistic model to data for US cable TV subscribers
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Figure 9-17 Predicted cable subscribers differ greatly depending on the
growth model used.
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Figure 9-18 The Bass diffusion model
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Figure 9-19 The Bass and logistic diffusion models compared to actual VAX sales
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Figure 9-20 Modeling product discard and replacement purchases
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Figure 9-21 Behavior of the Bass model with discards and repurchases
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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.
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100
Po tential
Ado pter s
(%)
80
Ado pter s
(%)
60
Ado ption Rate
(%/year )
40
20
0
0
1
2
3
Ye ar s
4
Figure 9-23 Behavior of the repeat purchase model
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5
6