In this assignment you will experience the system dynamics modeling process by exploring the                 propagation of an infectious disease through a population of susceptible individuals. Infectious diseases    present critical management and public policy challenges (consider SARS, Ebola, MERS, Avian               influenza, HIV, Covid,19, or the threat of bioterror attack). They also provide a useful setting to explore   feedback—how the states of a system (the levels or stocks) influence (or “feed back” to) theflows 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 can also help   to explain the growth of new products, the diffusion of innovations, the spread of financial panics, and      other forms of social contagion important in business settings. The assignment gives you practice with the concepts of dynamic modeling and the modeling process while enabling familiarity with the modeling      software we will use.

For the purposes of this assignment only, you have permission to get help from other students.  Feel free to work with others.  However, you must prepare your own assignment submission individually, and submit your individual work and write-up via Latte.

A.         Get the Software

We’ve mentioned that there are several excellent software packages available for dynamic modeling. For our work this semester we will use the Vensim PLE software.  You have previously received information on  how to download and install the Vensim PLE.

B.         Become Familiar with the Software and Build and Explore the Base Model

Start your work as soon as possible and ask the TA any questions you may have.

The class Latte site includes this assignment, a Vensim PLE tutorial, and a datafile. The datafile                <Sierra_Leone_Ebola_Cases.vdf> contains data for the Ebola epidemic in Sierra Leone as a Vensim Data File (.vdf). Put the Sierra_Leone_Ebola_Cases.vdf file in the directory where you want to save your         assignment model. The tutorial walks you through the development of the epidemic model step by step    and shows you how to use the data.

B1.      1 point.  Do the tutorial and build the model it describes.

    Be sure to document your model (both the diagram and the equations). This means the

following:

1.   Add sufficient commentary and explanation in the documentation field for every variable in your model 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 until later 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 the 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.

2.   Label causal links with the correct polarities.

3.   Denote causal loops with appropriate names and labels.

    Always provide the units of measurefor 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.  Dimensional   errors are usually symptoms of serious conceptual difficulties.

    Select valuesfor the constants (parameters) in your model.  As you finish the tutorial, you

will have chosen base case parameter values for the exogenous parameters we asked you to  select. Once you have a satisfactory overall pattern of behavior, don’t worry too much about the exact values, but use the base case settings as a basis for comparison in the questions that follow.

口       Name your model “<Last Name>-B1.mdl”, e.g. “Smith-B1.mdl” . Upload your model with your final submission (see part D below for naming conventions for your model files).

B2.       2 points.  Answer the following three questions.

口 a.    What happens when you initialize the stock of infected people at zero? Briefly explain how you    account for the behavior you observe with reference to the model’s structure (remember, structure generates behavior). How do the dynamics change if you assume that one or more members of     the population in question are already infected?

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

口 c.    How do the dynamics change as infectivity varies? Explain.

    Submit your answers to the above questions in a brief writeup: each answer may be a few

sentences and involve a handful of model runs. To back up your points, show only a few       graphs, tables, or any form of result summary based on model output. You need not show all model runs you have done—just select the minimum you need to answer the questions          concisely.

    Brevity is important, but you must also be complete.  Make sure your answers are model-

based explanations.  A good answer will refer to specific features of the model structure to explain why you observe a particular portion of behavior.  Your grade depends heavily on the quality of your explanations.

    Note: Place relevant selections of models and graphs next to your answers to a given

question. Placing graphs in an appendix makes it harder to follow your logic (and to grade your work appropriately).

B3.       1 point.  Critique your model.

口        The model you have developed so far is very simple. Briefly critique its formulation and list the major assumptions you view as unrealistic. Aim for one paragraph or so.

C.         Improve Your Model.

In developing your critique of the model, you should have identified a range of unrealistic  assumptions. In this section we will guide your exploration of what happens when one such assumption is relaxed.

C1.        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, so that people remain          infectious indefinitely. For most infectious diseases, including SARS, Ebola, smallpox, chicken pox, measles, and influenza, patients either recover or die in a relatively short period of time     (days to a few weeks).

We’ll help you reformulate your model to capture the notion that people eventually recover from Ebola and are thus no longer infectious. The revised model is known in epidemiology as the        “SIR” model (the “R” denotes “Removal” or “Recovery”). After following our instructions, your final model will look something like the following:

To modify your model, first create a new stock to represent the recovered population, placing it to the right of Population Infected with Ebola. Next create a new flow, called the Recovery Rate, that moves people from the Infectious to the Recovered states (see the tutorial for help in creating stock and flow variables). Set the initial number of Population Recovered from Ebola to zero.

    For simplicity, we will not distinguish death from recovery, so the recovery flow includes

those recovering and those dying.

Next add an auxiliary variable representing the average duration of infection (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 number of people who recover per day must depend on the number of people who are        currently infected and the average duration of their infectivity. There are several ways to model the recovery process, but the simplest (and most widely used) is:

Recovery Rate = Population Infected with Ebola/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, as shown on the previous model diagram.

Next, 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       Ebola to read:

Population Infected with Ebola = 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 Ebola accumulates the difference  between infection and recovery. Note that the infectious population is no longer equal to the cumulative number of cases.

Finally, estimate a value for Average Duration of Infection. Epidemiologists believe the           infectious stage of Ebola lasts about 1 to 3 weeks. Use your judgment to select a base case value. Check that your diagram now resembles that on the previous page.

    Make sure your revised model is documented and dimensionally consistent.

口        What kind of feedback loop have you added to your model? Make sure you have added an appropriate loop identifier and loop name to your diagram.

口        Include your model (the “<Last Name>-C1.mdl” model file you created in Vensim) with your final submission.

C2.       2 points.  Analyze your model to determine the effect of adding recovery.

口        Explore how your model now behaves by running simulation tests. How do the dynamics change as the duration of infectivity is increased or decreased? What is different now? Explain in a brief discussion with reference to the structure of your model. You need not include many runs or        tables, but do present a small selection of graphs or an overview of the data to back up your         points.

D.         Explore Your Improved Model.

D1.        1 point. Return to the questions you explored before. How does the epidemic change, once

recovery is added? Now, what is the effect when contact frequency and infectivity change? Vary key parameters over a 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?

D2.        1 point. In reality, people who contract Ebola are only contagious after symptoms appear, 2-21

days after being initially exposed to the virus.  This can be modeled by introducing a further stock of Exposed people, the “SEIR” model.  How might the dynamics of the SEIR model differ from   the SIR model you built in part C?  (You do not need to submit this model)