1. System Dynamics
S-Shape Growth
Shahram Shadrokh

2. 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

3. Net Birth Rate = g(P, C) P = g* (1 - P/C) P
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

4. MODELlNG S-SHAPED GROWTH
The logistic model
Logistic Growth
The first term g*P is a standard first-order linear positive feedback process; the second term, - g*P2/C, is nonlinear in the population and represents the ever stronger negative feedback caused by the approach of the population to its carrying capacity.
In the logistic model the net birth rate is an inverted parabola which passes through zero at the points P = 0 and P = C.
The maximum net birth rate occurs when Pinf = C/2
The maximum net growth rate occurs precisely halfway to the carrying capacity.
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.

5. 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*.

6. 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.

7. 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 =
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.

8. DYNAMICS OF DISEASE: MODELlNG EPIDEMICS
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
Epidemic of plague, Bombay, India Data show the death rate (deaths/week).

9. 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.

10. 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.

11. 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

12. 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

13. 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

14. 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.

15. 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.

16. IR > RR => ciS(I/N) > I/d or cid(S/N)>1,
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.