business dynamics figures and tables excerpted from business dynamics: systems thinking and modeling...
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Business Dynamics
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 fromBUSINESS DYNAMICS: SYSTEMS THINKING AND MODELING FOR A COMPLEX WORLDJohn 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 withBUSINESS DYNAMICS: SYSTEMS THINKING AND MODELING FOR A COMPLEX WORLDprovided 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.
0
Population/Carrying Capacity(dimensionless)
0 1
g*
0
Population/Carrying Capacity(dimensionless)
Stable EquilibriumUnstable
Equilibrium
Positive Feedback Dominant
•• (P/C)inf
= 0.5
NegativeFeedback Dominant
0 1
0.0
0.5
1.0
-4 -2 0 2 40
0.25
Population(Left Scale)
Net Growth Rate(Right Scale)
Time
PC
= 11 + exp[-g*(t - h)]
g* = 1, h = 0
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Figure 9-2 The growth of sunflowers and the best fit logistic model
0
100
200
300
0 14 28 42 56 70 84Days
Estimated Sunflower Height
Actual Sunflower Height
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Figure 9-3 Dynamics of epidemic diseaseSources: 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).
0
100
200
300
1/22 1/24 1/26 1/28 1/30 2/1 2/3
0
250
500
750
1000
0 5 10 15 20 25 30Weeks
Influenza epidemic at an English boardingschool, January 22-February 3, 1978.The data show the number of studentsConfined 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).
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
Figure 9-5 People remain infectious (and sick) for a limited time, then recover and develop immunity.
IR=(ciS)(I/N) N is total populationN=S+I
Business Dynamics
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.
0
500
1000
1500
2000
2500
0 4 8 12 16 20 24Days
InfectionRate
RecoveryRate
0
2500
5000
7500
10000
0 4 8 12 16 20 24Days
Susceptible
Infectious
Recovered
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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.
0
2500
5000
7500
10000
0 10 20 30 40 50 60Days
c = 6
c = 3
c = 2.5
c < 2
c = 2
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Figure 9-8 Dependence of the tipping point on the contact number and susceptible population
0
25
0 1Susceptible Fraction of Population (S/N)
(dimensionless)
No Epidemic(Stable; negative loops dominant)
Epidemic(Unstable; positive loop dominant)
cid(S/N) = 1
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Figure 9-9 Successive epidemic waves created by increasing contact rate
0
2500
5000
7500
10000
0 500 1000 1500 2000Days
Susceptible Population
0
50
100
150
0 500 1000 1500 2000Days
Infectious Population
0.0
0.5
1.0
1.5
0 500 1000 1500 2000Days
New Cases per InfectivePrior to Recovery
Tipping Point
PositiveLoop
Dominant
NegativeLoops
Dominant
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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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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.
Quarter-Year
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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).
0
1000
2000
3000
1981 1983 1984 1986 1988
Sales Rate
0
2000
4000
6000
8000
1981 1983 1984 1986 1988
Cumulative Sales( Installed Base)
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Figure 9-15 Fitting the logistic model of innovation diffusion
0.001
0.01
0.1
1
10
100
1000
1981 1983 1984 1986 1988
Data
Regression
Estimated Ratio of A/P(Adopters/Potential Adopters):ln(A/P) = -5.45 + 1.52(t - 1981); R 2 = .99
0
2000
4000
6000
8000
1981 1983 1984 1986 1988
Cumulative Sales( Installed Base)
Estimated Installed Base
0
1000
2000
3000
1981 1983 1984 1986 1988
Sales Rate
Estimated Sales Rate
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Figure 9-16 Fitting the logistic model to data for US cable TV subscribers
0
0.1
0.2
0.3
0 10 20 30 40 50 60Cable Subscribers
(million households)
Data
Best Linear Fit(g = 0.18 - 0.0024S; R 2 = .52)
0
25
50
75
100
1950 1960 1970 1980 1990 2000 2010
Estimated CableSubscribers
Cable Subscribers
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Figure 9-17 Predicted cable subscribers differ greatly depending on the growth model used.
0
50
100
150
1950 1960 1970 1980 1990 2000 2010 2020
Estimated CableSubscribers,
Gompertz Model
Cable Subscribers
Estimated CableSubscribers,
Logistic Model
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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
0
1000
2000
3000
1981 1983 1984 1986 1988
Sales Rate
Sales from Word of Mouth
Sales from Advertising
0
1000
2000
3000
1981 1983 1984 1986 1988
Sales Rate
Logistic Model
Bass Model
0
2000
4000
6000
8000
1981 1983 1984 1986 1988
Cumulative Sales( Installed Base)
Logistic Model
Bass Model
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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
0
25
50
75
100
0 1 2 3 4 5 6Years
PotentialCustomers
Installed Base
0
20
40
60
80
0 1 2 3 4 5 6Years
Sales Rate
Discard Rate
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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.
Business Dynamics
Figure 9-23 Behavior of the repeat purchase model
0
20
40
60
80
100
0 1 2 3 4 5 6Years
Adoption Rate(%/year)
Adopters(%)
PotentialAdopters
(%)
0
20
40
60
80
0 1 2 3 4 5 6Years
Sales Rate
InitialPurchases
Repeat Purchases