System Dynamics for Business Policy
In this assignment you will use the Vensim software to explore how an infectious disease propagates through a population of susceptible individuals. Infectious diseases are critical management and public policy challenges (consider SARS, Avian influenza, HIV, and the threat of bioterror attack) and provide a useful setting to explore feedback€”how the states of a system (the levels or stocks) influence (or feed back to) the flows that alter those states. The assignment also introduces you to the structure and behavior of fundamental feedback systems. These systems are the building blocks from which more complex systems are composed. In particular, the feedback structure governing the spread of contagious disease is also important in the growth of new products, the diffusion of innovations, the spread of rumor and riot, and other contexts of social contagion. The assignment gives you practice with the concepts of dynamic modeling and with the modeling process, along with familiarity with the Vensim software.
1 Getting Familiar with the Software and Building the Base Model
The class Moodle site includes a VensimPLE tutorial, called <VensimPLE_Tutorial_S2008>, developedfor the SD class at MIT. Make sure you also download the file <SARSDATA.vdf>, which has the Taiwanese SARS data already embedded in it. Put the SARSDATA file in the Vensim directory. The tutorial will walk you through the development of the epidemic model step by step.
1.1 Complete the tutorial and build the model it describes
? Be sure to document your model. Documenting your model means adding enough commentary and explanation so that the meaning of each variable, the reasoning behind each formulation, and the data sources for each parameter value are clear to your audience. Do not wait till you are done to fill in the documentation. Document each equation as you create it. Doing so saves time in the long run. Forcing yourself to describe, in writing, the rationale for each formulation and sources for each parameter tests your understanding: if you can’t write a concise, clear description of a formulation you probably don’t understand it yourself. Documenting as you go avoids costly delays in the discovery of errors and the need for rework.
? Always provide the units of measure for every variable, and check that every equation is dimensionally consistent. Vensim can test your model for dimensional consistency (see the tutorial). Models that fail the dimensional consistency test are meaningless. Often dimensional errors are symptoms of more serious conceptual difficulties.

1.2 Answer the following questions (2 points)
€¢ A. What happens when you initialize the stock of infected people at zero? Briefly explain why. How do the dynamics change if you assume that one or more members of the population in question are already infected?

€¢ B. How do the dynamics change as the contact frequency increases? Does changing the contact frequency influence the total number of people who get SARS? Explain why or why not with reference to the structure of the model.

€¢ C. How do the dynamics change as infectivity varies? Explain.

€¢ D. How does the time horizon (the length of the simulation) change the apparent shape of the curves? Why?

? To answer the question on time horizon, you need to examine your simulation over different periods of time. You can run the model once with a long time horizon (e.g., 1000 days) and then change the time range viewed in graphs and tables. Select Time Axis from the Control Panel dialog box to change the time frame displayed in graphs and tables (see The Control Panel in the tutorial or User’s Guide).

? Vensim allows you to load multiple runs (or datasets) and then view the output of different runs in the same graphs and tables. See tutorial and the section on The Control Panel in the User’s Guide for details on selecting datasets.

€¢ E. Hand in your model (diagram and equation listing) and answers to the above questions. Show only a few graphs or tables of model output, the minimum you need to answer the questions concisely.
1.3 Critique your model (2 point)
€¢ The model you have developed so far is very simple. Briefly critique its formulation and list the major assumptions you view as unrealistic.
2 Improving Your Model.
All models are wrong (the map is not the territory). In your critique of the model you may have identified a wide range of unrealistic assumptions. In this section you will relax one such assumption. Make one change at a time and test it thoroughly, rather than making multiple changes at once. With multiple changes it is impossible to determine which altered assumption is responsible for any changes in behavior.
2.1 Reformulate your model (2 points)
€¢ So far we’ve assumed people remain infectious indefinitely. In epidemiology this is known as the SI (for Susceptible-Infectious) model. The SI model is appropriate for chronic infections that the body cannot clear and for which there is no cure. For most infectious diseases, however, including SARS, smallpox, chicken pox, measles, and influenza, patients either recover (or die).

Reformulate your model to include the removal of infectious people from the stock of infectious individuals. The revised model is known in epidemiology as the SIR (Susceptible-Infectious-Removed) model.

To modify your model, first create a new stock, the Recovered population, placing it to the right of the population infected with SARS. Next create a new flow, called the Recovery Rate, that moves people from the Infectious to the Recovered states (see the tutorial for creating stock and flow variables). Note: For simplicity, we will not distinguish mortality from recovery, so the recovery flow includes those recovering and those dying.

Next add a variable representing the average duration of infectivity (that is, how long people remain infectious, on average).

Now you need to capture the feedback structure determining the rate at which people recover. The recovery rate must depend on the number of people who are currently infected and the average duration of their infectivity. There are several formulations for recovery, but the simplest (and most widely used) is:

Recovery Rate = Population Infected with SARS/Average Duration of Infectivity

To implement this formulation you need to add causal links connecting the infected stock and the average duration of infectivity to the recovery flow.

Next, you must change the equation for the infected population, so that the recovery flow is now included as an outflow that drains the infected stock. Alter the equation for the Population Infected with SARS to read:
Population Infected with SARS= INTEG(Infection Rate €“ Recovery Rate)
The equation states that the infectious population increases as people become infectious and decreases as people recover; the population infected with SARS accumulates the difference between infection and recovery.
Note that the infectious population is no longer the same as the cumulative number of cases. Your diagram should now look something like:

Finally, you must estimate a value for the average duration of infectivity. Epidemiologists believe the infectious stage of SARS lasts about 1 to 3 weeks (7 €“ 21 days). Use your judgment to select a base case value.
Make sure your revised model is documented and dimensionally consistent.
2.2 Analyze your model to determine the effect of adding recovery (3 points)
€¢ A. What kind of feedback loop have you added to your model?

€¢ B. How do the dynamics change as the duration of infectivity is increased or decreased? As contact frequency and infectivity change? Vary the key parameters over a wide range to be sure you understand the possible dynamics of the system. Specifically, consider how the timing and severity of the epidemic change with parameters. How do the dynamics differ from those of the SI model in which there is no recovery? Explain with reference to the structure of your model.

€¢ C. Your original model assumed the total population was about 350, roughly the same as the total number of cumulative cases reported. This is, of course, incorrect: in 2003 the population of Taiwan was about 22 million. Change total population to 22 million and run your model again. How large is the epidemic now? Can you match the data for cumulative cases now, while keeping the parameters within reasonable ranges? Why/why not? What happened in the real system to alter the course of the epidemic compared to what your model suggests?
2.3 Policy Implications (1 point)
€¢ Briefly discuss the policy implications of your analysis. If you were given a fixed budget to prevent the spread of SARS, how would you spend it? How would these interventions differ from those you might use on a disease with very different characteristics, such as HIV (which has lower infectivity, but much longer duration)?