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System Dynamics S-Shape Growth Morteza Bazrafshan 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 The logistic model 0 0 1 Logistic Growth Population/Carrying Capacity (dimensionless) Population/Carrying Capacity (dimensionless) Net Growth Rate Net Birth Rate/Carrying Capacity (1/time) The first term g*P is a standard first-order linear positive feedback process; the 1.0 0.25 P = 1 2 second term, - g*P Negative /C, is nonlinear in theC population and represents the ever 1 + exp[-g (t - h)] Positive Feedback Feedback g = 1, approach h=0 stronger caused by the of the population to its Dominantnegative feedback Dominant carrying capacity. * * 0.5 In the logistic model the net birth rate is an inverted parabola which Net Growthpasses Rate (Right Scale) through zero at the points P = 0 and P = C. 0 •0 (P/C)inf = 0.5 • 1 Stable Equilibrium Population (Left Scale) 0.0 0 -4 -2 0 Time 2 4 Unstable TheEquilibrium maximum net birth rate occurs when Pinf = C/2 The maximum net growth rate occurs precisely halfway to the carrying capacity. Population/Carrying Capacity (dimensionless) Top left: The fractional growth rate declines linearly as population grows. Middle left: The phase plot is an inverted parabola, symmetric about (P/C) = 0.5 Right: 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 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 Patients confined to bed 300 200 100 0 1/22 1/24 1/26 1/28 1/30 2/1 2/3 Epidemic of plague, Bombay, India 1905-6. Data show the death rate (deaths/week). Deaths (people/week) 1000 750 500 250 0 0 5 10 15 Weeks 20 25 30 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., 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. INNOVATION DIFFUSION AS INFECTION: MODELlNG NEW IDEAS AND NEW PRODUCTS The Bass Diffusion Model One of the flaws in the logistic model of innovation diffusion is the startup problem. In the logistic, zero is an equilibrium: the logistic model cannot explain the genesis of the initial adopters. There are several channels of awareness that can stimulate early adoption of new innovations besides word of mouth and related feedback effects that depend on the size of the adopter population. These include advertising, media reports, and direct sales efforts. Frank Bass (1969) developed a model for the diffusion of innovations that overcomes the startup problem. The Bass diffusion model has become one of the most popular models for new product growth and is widely used in marketing, strategy, management of technology, and other fields. Bass solved the startup problem by assuming that potential adopters become aware of the innovation through external information sources whose magnitude and persuasiveness are roughly constant over time. INNOVATION DIFFUSION AS INFECTION: MODELlNG NEW IDEAS AND NEW PRODUCTS The Bass Diffusion Model The positive feedback is usually interpreted as word of mouth (social exposure and imitation) and the external sources of awareness and adoption are usually interpreted as the effect of advertising INNOVATION DIFFUSION AS INFECTION: MODELlNG NEW IDEAS AND NEW PRODUCTS The Bass Diffusion Model The external influences cause a constant fraction of the potential adopter population to adopt each time period. Hence the adoption rate, AR, is: AR = Adoption from Advertising + Adoption from Word of Mouth Adoption from Advertising = aP Adoption from Word of Mouth = ciPA/N where the parameter a, advertising effectiveness, is the fractional adoption rate from advertising (1/time period). The two sources of adoption are assumed to be independent. Collecting term the model can be expressed more compactly as: AR = aP + ciPA/N INNOVATION DIFFUSION AS INFECTION: MODELlNG NEW IDEAS AND NEW PRODUCTS 8000 3000 Logistic Model 3000 6000 Bass Model 2000 4000 2000 Units/Year Units Units/Year Logistic Model Bass Model Sales from 2000 1000 1000 Sales Rate 0 1981 0 0 1981 1981 Cumulative Sales (- Installed Base) Sales Rate Word of Mouth 1983 1983 1983 Sales from Advertising 1984 1986 1984 1984 1986 1986 1988 19881988 The Bass and logistic diffusion models compared to actual VAX sales