bdsm-ch9_modeling the s-shaped growth
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Business Dynamics and System Modelingy y g
Chapter 9: Modeling the S‐Shaped hGrowth
Pard TeekasapPard Teekasap
Southern New Hampshire University
OutlineOutline
1. Logistic Growth
2. Modeling Epidemics2. Modeling Epidemics
3. Innovation Diffusion
General Concept of S Shaped GrowthGeneral Concept of S‐Shaped Growth
• The S‐Shaped growth occurs if there is no feedback delay from the constrainty
• With the delay in negative feedback, the behavior will be S shaped growth withbehavior will be S‐shaped growth with overshoot and oscillation
• If the carrying capacity is consumed by the growing population, the behavior will begrowing population, the behavior will be overshoot and collapse
Logistic GrowthLogistic Growth
Net Birth Rate = g(P,C)P = g*(1‐P/C)P
= g*P‐g*P2/C g P g P /C
Pinf = C/2
• P = Population; C = Carrying Capacity
• g(PC) = fractional growth rateg(P,C) = fractional growth rate
• g* = maximum fractional growth
• Pinf = Population when the net growth rate is maximummaximum
Why Logistic Model is ImportantWhy Logistic Model is Important
• Many S‐Shaped growth processes can be approximated well by the logistic modelpp y g
• The logistic model can be solved analytically
h l i i d l b f d i• The logistic model can be transformed into a linear form so the parameters can be estimated based on OLS
Analytic Solution for Logistic EquationAnalytic Solution for Logistic EquationP
CPg
dtdP 1* ⎟
⎠⎞
⎜⎝⎛ −=
dtgP
CP
dP *1
=⎟⎠⎞
⎜⎝⎛ −
⎠⎝
dtgP
CP
dPC
*1
=⎟⎠⎞
⎜⎝⎛ −
⎠⎝
∫∫
dtgdPPCPPPC
CdP
C
*)(
11)(
=⎥⎦
⎤⎢⎣
⎡−
+=−
⎠⎝
∫∫∫
tgPPPCPtgPCP
ctgPCP
*)0()]0(ln[))0(ln(*)ln()ln(
*)ln()ln(−−+=−−
+=−−⎦⎣
tg
CtP
PCeP
PCP
)(
)0()0(
=
−=
−
tgePC
tP*1
)0(1
)(−
⎥⎦
⎤⎢⎣
⎡−+
=
Behavior of Logistic functionBehavior of Logistic function
Rat
e
g*
( C)The logistic model
0
al N
et G
row
th R
0 1
g(P,C)
Frac
tiona
Population/Carrying Capacity(dimensionless)
Positive Feedback Dominant
NegativeFeedback Dominant
1.0 0.25
ying
Cap
acity
nles
s)
Net B
irth Rate/(1
PC = 1
1 + exp[-g*(t - h)]
g* = 1, h = 0
wth
Rat
e
0.5
Population
Net Growth Rate(Right Scale)
Popu
latio
n/C
arry
(dim
ensi
o /Carrying C
apac1/tim
e)
0
Net
Gro
w
Stable EquilibriumUnstable
Equilibrium
•• (P/C)inf = 0.50 1 0.0-4 -2 0 2 4
0
Population(Left Scale)
Time
P ity
Population/Carrying Capacity(dimensionless)
Dynamics of DiseaseDynamics of Disease300
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
200
nfin
ed to
bed
Confined to bed for influenza at any time(the stock of symptomatic individuals). 100
Patie
nts
con
01/22 1/24 1/26 1/28 1/30 2/1 2/3
1000
Epidemic of plague, Bombay, India 1905-6. Data show the death rate (deaths/week).
500
750
ople
/wee
k)
250
500
Dea
ths
(peo
00 5 10 15 20 25 30
Weeks
SI ModelSI Model
SusceptiblePopulation
S
InfectiousPopulation
IInfectionRate
IRB R
S I
IR
+ +++
Depletion Contagion
-
Contact InfectivityRatec
InfectivityiTotal
PopulationpN
Equation for SI ModelEquation for SI Model
• I = INTEGRAL(IR,I0)
• S = INTEGRAL(‐IR,N‐I0)S INTEGRAL( IR,N I0)
• IR = (ciS)(I/N)
• S + I = N
• IR = ciI(1‐I/N)IR = ciI(1 I/N)
• Compare to logistic growth model
• Net Birth Rate = g*(1‐P/C)P
Assumptions for SI ModelAssumptions for SI Model
h d h d d• Births, deaths, and migration are ignored• Once people are infected, they remain p p , yinfectious indefinitely. Therefore, the model applies to chronic infections, not acute illnesspp ,
• The population is homogeneous: all members are assumed to interact at the same averageare assumed to interact at the same average rateTh i ti• There is no recovery, quarantine, or immunization
Is SI Model a second order model?Is SI Model a second‐order model?
• Absolutely not
• Even though the model has two stocks, theyEven though the model has two stocks, they are interdependent
S I N• S + I = N
SIR ModelSIR Model
SusceptiblePopulation
InfectiousPopulation
RecoveredPopulation
InfectionRate
IRB
DepletionR
Contagion
SB
Recovery
RecoveryRateRR
I R
+ +
Contact
++ - + -
ContactRate
c
InfectivityiTotal
PopulationN
Average Durationof Infectivity
dN d
Equation for SIR ModelEquation for SIR Model
• S = INTEGRAL(‐IR,N‐I0‐R0)
• I = INTEGRAL(IR‐RR,I0)I INTEGRAL(IR RR,I0)
• R = INTEGRAL(RR,R0)
• IR is the same as SI model
• RR = I/dRR = I/d
Is SIR model a second order?Is SIR model a second‐order?
• Yes
• Even though it has 3 stocks, only two areEven though it has 3 stocks, only two are independent
Simulation of SIR modelSimulation of SIR model2000
2500
Rat
es
Infection
Figure 9 6 Simulation of 1000
1500
nd R
ecov
ery
eopl
e/da
y)
Rate
Figure 9-6 Simulation of an epidemic in the SIR model. The total population is 10,000. The contact rate is 6 per person
500
1000In
fect
ion
an (pe
RecoveryRate
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
00 4 8 12 16 20 24
Days10000
ple)
initial infective population is 1, and all others are initially susceptible.
5000
7500
ulat
ion
(peo
p Susceptible Recovered
2500
5000
eptib
le P
opu
Infectious
00 4 8 12 16 20 24
Susc
e
Days
How can epidemic happen?How can epidemic happen?
• IR > RR
• ciS(I/N) > I/dciS(I/N) > I/d
• cid(S/N) > 1
• cid = contact number
• cid(S/N) = reproduction ratecid(S/N) = reproduction rate
Effect of contact rateEffect of contact ratec < 2
10000
ople
)
c = 2.5c = 2
7500
atio
n (p
e
5000
e Po
pula
c = 6
c = 3
2500
scep
tible
00 10 20 30 40 50 60
Sus
DDays
Contact Number VS Population Fraction
25
) cid(S/N) = 1
ber (
cid
nles
s)
Epidemic
( )
act N
umm
ensi
on Epidemic(Unstable; positive loop dominant)
Con
ta (dim
00 1
No Epidemic(Stable; negative loops dominant)
0 1Susceptible Fraction of Population (S/N)
(dimensionless)
Herd ImmunityHerd Immunity
• Situation that the contact number is small enough that the system is below the tipping g y pp gpoint
• With the herd immunity the arrival of infected• With the herd immunity, the arrival of infected individual does not produce an epidemic
• However, change in contact rate, infectivity, or duration of illness can push a system past theduration of illness can push a system past the tipping point
Epidemic WaveEpidemic Wave1.5
Positive
1.0
oduc
tion
Rat
ees
per
infe
ctiv
e)
Tipping Point
LoopDominant
Negative
0.0
0.5
0 500 1000 1500 2000
Rep
ro(n
ew c
ase
New Cases per InfectivePrior to Recovery
gLoops
Dominant
100
150
n (p
eopl
e) Infectious Population
Days
50
ectio
us P
opul
atio
00 500 1000 1500 2000
Infe
Days
7500
10000
n (p
eopl
e)
Susceptible Population
• Contact rate is continuously increased
2500
5000
eptib
le P
opul
atio
n • Contact rate is continuously increased• A single infected individual arrives every 50 days
00 500 1000 1500 2000
Susc
e
Days
Diffusion of new ideas and new product
Th diff i d d ti f id d• The diffusion and adoption of new ideas and new products follows S‐shaped growth
• What are the positive feedbacks?• What are the positive feedbacks?• What are the negative feedbacks?P l h d d h d i• People who adopted the product come into contact with those who haven’t, exposing them to it and infecting some of them with the desireto it and infecting some of them with the desire to buy
• However the system is limited by number ofHowever, the system is limited by number of population
New product adoption modelNew product adoption model
P t ti l
Adoption
PotentialAdopters
P
AdoptersA
AdoptionRateAR
B
Market
R
Word of
+ +++
MarketSaturation
Word ofMouth
-
ContactR t AdoptionRate
c
AdoptionFraction
iTotal
PopulationN
The case of DEC VAX 11/750The case of DEC VAX 11/7503000
Sales Rate
2000
s/Ye
ar
1000Uni
t
01981 1983 1984 1986 1988
8000Cumulative Sales
6000
Cumulative Sales(- Installed Base)
s
2000
4000
Uni
ts
01981 1983 1984 1986 1988
How to estimate the parameters?How to estimate the parameters?
eAN
AAN
A tg
0
0 0
−=
−
tAA
ANAN
0
0
ll ⎟⎞
⎜⎛
⎟⎞
⎜⎛ tg
ANAN 00
0lnln +⎟⎟⎠
⎜⎜⎝ −
=⎟⎠⎞
⎜⎝⎛
−
tgPA
PA
00lnln +⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎠⎞
⎜⎝⎛ g
PP 00⎟⎠
⎜⎝
⎟⎠
⎜⎝
Use linear regression to get the parameterUse linear regression to get the parameter
Fitting Logistic Model into Product Diffusion
100
1000
rs
Estimated Ratio of A/P(Adopters/Potential Adopters):
1
10
100
Pote
ntia
l Ado
pter
men
sion
less
)
Data
(Adopters/Potential Adopters):ln(A/P) = -5.45 + 1.52(t - 1981); R 2 = .99
0.001
0.01
0.1
1981 1983 1984 1986 1988
Ado
pter
s/P
(dim
Regression 6000
8000
Estimated Installed Base
2000
4000Cumulative Sales(- Installed Base)
Uni
ts01981 1983 1984 1986 1988
2000
3000
Sales Rater
Estimated Sales Rate
1000
S
Uni
ts/Y
ea
01981 1983 1984 1986 1988
However, it’s not practical forforecasting
• The entire sale history was used
• The final value of installed base is neededThe final value of installed base is needed
• How can we know what is the final value for a d ?new product?
Getting the fractional growth rateGetting the fractional growth rate
• From g(C,P) = g*(1‐P/C)
• Mapping the fractional growth with theMapping the fractional growth with the adopters
h l i */C h i i *• The slope is ‐g*/C. The intercept is g*
Logistic model for US cable subscribersLogistic model for US cable subscribers0.3
s)0.2
h R
ate
(1/y
ears
Data
0.1
ctio
nal G
row
th
Best Linear Fit(g = 0 18 0 0024S; R2 = 52)
00 10 20 30 40 50 60
Frac
Cable Subscribers(million households)
(g = 0.18 - 0.0024S; R2 = .52)
100
75Estimated Cable
Subscribers
ehol
ds
25
50
Mill
ion
Hou
se
01950 1960 1970 1980 1990 2000 2010
Cable Subscribers
However the prediction is uncertainHowever, the prediction is uncertain
l f l h• Actual fractional growth rate varies substantially around the best fit
• The best fit is changed when the historical period change, and also the forecast will p g ,change
• Logistic model presumes a linear decline inLogistic model presumes a linear decline in the fractional growth rate as population grows However no compelling theoreticalgrows. However, no compelling theoretical basis for linearity
Which forecast is correct?Which forecast is correct?
150
Estimated CableSubscribers
100
Subscribers,Gompertz Model
ehol
ds
50Estimated Cable
Subscriberson H
ouse
50 Subscribers,Logistic Model
Mill
io
01950 1960 1970 1980 1990 2000 2010 2020
Cable Subscribers
Logistic curve can fit data well, but you shouldn’t use it
Th bilit t fit th hi t i l d t d t• The ability to fit the historical data does not mean the forecast is correct
• “By the time sufficient observations have• “By the time sufficient observations have developed for reliable estimation, it’s too late to use the estimates for forecasting purposes ”use the estimates for forecasting purposes.
• A purpose of modeling is to design and test policies. The ability to fit the historical datapolicies. The ability to fit the historical data provides no information about if its response to policies will be correct
• The logistic model can’t generate anything but growth
So which model is right?So, which model is right?
h b l f d l l h l• The ability of a model to replicate historical data does not indicate that the model is useful
• Failure to replicate historical data does not mean a model should be dismissed
• The utility of a model requires the modeler to decide whether the structure and decisiondecide whether the structure and decision rules of the model correspond to the actual structure and decision rules used by the realstructure and decision rules used by the real people
Bass Diffusion ModelBass Diffusion Model
d l h bl• Logistic model has some startup problems • zero is equilibrium q• the positive feedback during the beginning of the growth process is weakthe growth process is weak
• There are several channels besides word of mouthmouth
• Bass solved the startup problem by assuming the potential adopters know the products through external information
Bass Diffusion ModelBass Diffusion Model
Adoption
PotentialAdopters
P
AdoptersA
RateARB R
MarketSaturation
Word ofMouth++
TotalPopulation
N
Saturation Mouth
Adoptionfrom
Adoptionfrom Word +
+
+
+
AdoptionFraction
N
Ad ertising
Advertising of Mouth
+++
-+
B
MarketSaturation Fraction
i
ContactR t
AdvertisingEffectiveness
aRate
c
Equation for Bass ModelEquation for Bass Model
• AR = Adoption from Advertising + Adoption from Word of Mouth
• Adoption from Advertising = aP
Ad i f W d f h i A/N• Adoption from Word of Mouth = ciPA/N
• AR = aP + ciPA/N/
Behavior of the Bass ModelBehavior of the Bass Model6000
8000
Logistic Model
4000
6000
Cumulative Sales(- Installed Base)
Uni
ts
Bass Model
0
2000
1981 1983 1984 1986 19882000
3000
Sales Ratear
Logistic Model
1000Uni
ts/Y
ea
Bass Model
01981 1983 1984 1986 1988 3000
1000
2000
Sales Rate
Uni
ts/Y
ear
Sales from Word of Mouth
01981 1983 1984 1986 1988
Sales from Advertising
The model with replacement purchaseThe model with replacement purchaseAverage Product Life
l
DiscardR
- l
BRate
+Replacement
Purchase
Potential Adopters
AdoptionRateAR
B RMarket Word of
AdoptersP
AdoptersA
ARTotal
PopulationN
MarketSaturation
Word ofMouth
Adoptionfrom
Adoptionfrom Word +
++
+
B N
AdvertisingEffectiveness
Advertising of Mouth+++
-+
MarketSaturation Adoption
FractioniEffectiveness
a ContactRate
c
i
Another model with replacement purchase
Sales Rate+ +Average
Consumptionper Adopter
Initial Salesper Adopter
++ InitialPurchase
Rate
RepeatPurchase
Rate
PotentialAdopters Adopters
+ +
AdoptionRateAR
B RMarket
S t tiWord of
AdoptersP A
TotalPopulation
N
Saturation Mouth
Adoptionfrom
Adoptionfrom Word +
++
+
M k tB
AdvertisingEffectiveness
Advertising of Mouth+++
-+
MarketSaturation Adoption
FractioniEffectiveness
a ContactRate
c
i
The difference between two modelsThe difference between two models
h fi d l h h d• The first model assumes that when adoptersdiscard the products, they will become the
t ti l d t h d t k d i ipotential adopters who need to make a decision again
• The second model assumes that the adopters still have the same decision and repurchase the
d i i h li i h lproduct again without listening to other people• The first model is for the products with long average life. When they need to buy it again, the decision environment change.
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