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.