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```									System Dynamics
S-Shape Growth

MODELlNG S-SHAPED GROWTH

The nonlinear population model developed in chapter 8 is quite general.
The population in the model can be any quantity that grows in a fixed
environment, for example, the number of adapters of an innovation, the
number of people infected by a disease, the fraction of any group adhering to
an idea or purchasing a product, and so on.
If the population is driven by positive feedback when it is small relative to its
limits, then the resulting behavior will be S-shaped growth, provided there are
no significant delays in the negative feedbacks that constrain the population.
If there are delays in the response of the population to the approaching
carrying capacity, the behavior will be S-shaped growth with overshoot and
oscillation; if the carrying capacity is consumed by the growing population, the
behavior will be overshoot and collapse
MODELlNG S-SHAPED GROWTH

Logistic Growth
An important special case of S-shaped growth is known as logistic growth
The logistic growth model posits that the net fractional population growth rate
is a (downward sloping) linear function of the population. That is:
Net Birth Rate = g(P, C) P = g* (1 - P/C) P
g(P, C) is the fractional growth rate,
g* is the maximum fractional growth (the fractional growth rate when the
population is very small)
The logistic model conforms to the requirements for S-shaped growth: the
fractional net growth rate is positive for P < C, zero when P = C, and negative
for P > C.
Rearranging the above equation gives:

Net Birth Rate = g*(1 - P/C)P = g*P - g*P2/C
Fractional Net Growth Rate
g*
MODELlNG S-SHAPED GROWTH
0
The logistic model
0                                   1

Logistic Growth
Population/Carrying Capacity
(dimensionless)
The first term g*P is a standard first-order linear positive feedback process; the
1.0                                    0.25

Net Birth Rate/Carrying Capacity
2/C, is nonlinear in the =population and represents the ever
second term, - g*P Negative                  P        1

Population/Carrying Capacity
Positive                                C 1 + exp[-g (t - h)]                                            *

Feedback        Feedback
stronger negative feedback caused by the 1, approach of the population to its
Dominant        Dominant                 g = h=0                                                *

(dimensionless)
carrying capacity.

(1/time)
0.5

In the logistic model the net birth rate is an inverted parabola which passes
Net Growth Rate
Net Growth Rate

(Right Scale)
through zero at the points P = 0 and P = C.
Population
0   •
0
(P/C) inf = 0.5       •
1                                                     0.0
(Left Scale)
0
Stable
Equilibrium                                            -4        -2             0    2   4
maximum net birth rate occurs when Pinf = C/2
TheUnstable
Equilibrium
Time

The maximum net growth rate occurs precisely halfway to the carrying capacity.

Population/Carrying Capacity
(dimensionless)

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.
MODELlNG S-SHAPED GROWTH

Logistic Growth
We can analytically solve the logistic growth:

where h is the time at which the population reaches half its carrying capacity;
setting P(h) = 0.5C and solving for h yields h = In[(C/P(0)) - 1]/g*.
MODELlNG S-SHAPED GROWTH

Other Common Growth Models
there are many other models of S-shaped growth.
The Richards curve is one commonly used model (Richards 1959). In
Richards' model the fractional growth rate of the population is nonlinear in the
population:

When m = 2, the Richards model reduces to the logistic.
The solution of the Richards model is:

A special case of the Richards model is the Gompertz curve, given by the
Richards model in the limit when m = 1.
MODELlNG S-SHAPED GROWTH

Other Common Growth Models
So the Gompertz curve is given by
In the Gompertz model, the maximum growth rate occurs at P/C = 0.368.
Another commonly used growth model is based on the Weibull distribution:

However, there is no guarantee that the data will conform to the assumptions
of any of the analytic growth models. Fortunately, with computer simulation,
you are not restricted to use the logistic, Gompertz, Richards, Weibull, or' any
other analytic model.
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

300
Influenza epidemic at an English boarding

Patients confined to bed
school, January 22-February 3, 1978.
The data show the number of students                                   200

Confined to bed for influenza at any time
100

0
1/22    1/24   1/26   1/28    1/30    2/1        2/3

1000

Epidemic of plague, Bombay,
Deaths (people/week)

India 1905-6. Data show the                                             750

death rate (deaths/week).
500

250

0
0     5      10    15        20         25         30
Weeks
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

A Simple Model of Infectious Disease
The total population of the community or region represented in the model is
divided into two categories: those susceptible to the disease, S, and those who
are infectious, I (for this reason the model is known as the SI model).
As people are infected they move from the susceptible category to the
infectious category.
The SI model invokes a number of simplifying assumptions.
- First, births, deaths, and migration are ignored.
- Second, once people are infected, they remain infectious
indefinitely, that is, the model applies to chronic infections, not acute
illness such as influenza or plague.
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

A Simple Model of Infectious Disease

The infectious population I is increased by the infection rate IR while the
susceptible population S is decreased by it:
I= INTEGRAL(IR, l0)
S = INTEGRAL( - IR, N – l0)
People in the community interact at a certain rate (the Contact Rate, c,
measured in people contacted per person per time period, or 1/time period).

Thus the susceptible population generate Sc encounters per time period.
Some of these encounters are with infectious people. If infectious people
interact at the same rate as susceptible people (they are not quarantined or
confined to bed), then the probability that any randomly selected encounter is
an encounter with an infectious individual is I/N.

Not every encounter with an infectious person results in infection.
The infectivity, i, of the disease is the probability that a person becomes.
infected after contact with an infectious person.
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

A Simple Model of Infectious Disease

IR = (ciS )(I/N)

The dynamics can be determined by noting that without births, deaths, or
migration, the total population is fixed: S+I=N

Though the system contains two stocks, it is actually a first-order system
because one of the stocks is completely determined by the other. Substituting
N - I for S in yields: IR = (c)(i) I (1 – I / N)

This equation is similar to the net birth rate in the logistic model.
The carrying capacity is the total population, N.

In the SI model, once an infectious individual arrives in the community, every
susceptible person eventually becomes infected, with the infection rate
following a bell-shaped curve and the total infected population following the
classic S-shaped pattern of the logistic curve
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

Modeling Acute Infection: The SIR Model

While the SI model captures the basic process of infection, it contains many
simplifying and restrictive assumptions.

The most restrictive and unrealistic feature of the logistic model as applied to
epidemics is the assumption that the disease is chronic, with affected
individuals remaining infectious indefinitely.

While the assumption of chronic infection is reasonable for some diseases
(e.g., herpes simplex, chicken pox), many infectious diseases produce a
period of acute infectiousness and illness, followed either by recovery and the
development of immunity or by death.

Here the model contains three stocks:
The Susceptible population, S, the Infectious population, I, and the Recovered
population, R, known as the SIR model
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

Modeling Acute Infection: The SIR Model

Those contracting the disease become infectious for a certain period of time
but then recover and develop permanent immunity.

The greater the number of infectious individuals, the greater the recovery rate
and the smaller the number of infectious people remaining.

I= INTEGRAL(IR - RR, l0)
S = INTEGRAL( - IR, N- l0- R0)
R = INTEGRAL(RR, R0)

In the SIR model, the average duration of infectivity, d, is assumed to be
constant and the recovery process is assumed to follow a first-order negative
feedback process:
RR = I/d
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

Modeling Acute Infection: The SIR Model
Model Behavior: The Tipping Point

Unlike the models considered thus far, the system is now second-order (there
are three stocks, but since they sum to a constant, only two are independent).

First, unlike the SI model, it is now possible for the disease to die out without
causing an epidemic.

If the infection rate is less than the recovery rate, the infectious population will
fall. As it falls, so too will the infection rate. The infectious population can
therefore fall to zero before everyone contracts the disease.

For any given population of susceptible, there is some critical combination of
contact frequency, infectivity, and disease duration just great enough for the
positive loop to dominate the negative loops. That threshold is known as the
tipping point.
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

Modeling Acute Infection: The SIR Model
Model Behavior: The Tipping Point

Below the tipping point, the system is stable: if the disease is introduced into
the community, there may be a few new cases, but on average, people will
recover faster than new cases are generated. Negative feedback dominates
and the population is resistant to an epidemic.

Past the tipping point, the positive loop dominates. The system is unstable and
once a disease arrives, it can spread like wildfire. that is, by positive feedback
limited only by the depletion of the susceptible population.

Unlike the chronic infection model in which everyone eventually contracts the
disease, in the SIR model the epidemic ends before the susceptible population
falls to zero.
DYNAMICS OF DISEASE: MODELlNG EPIDEMICS

Modeling Acute Infection: The SIR Model
Model Behavior: The Tipping Point

The exact tipping point in the SIR model can easily be calculated. For an
epidemic to occur, the infection rate must exceed the recovery rate:

IR > RR => ciS(I/N) > I/d or cid(S/N)>1,
cid, is known as the contact number.

The duration of the infectious period for diseases such as measles and
chicken pox is very short, a matter of days, but these diseases have high
contact numbers because they are easily spread through casual contact.

In contrast, the contact rate and infectivity of HIV are much lower.
Nevertheless, the contact number for HIV is high among those who engage in
risky behaviors because the duration of infection is so long.

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