The Epidemic Model 1 A set of lessons called "Plagues and People," designed by John Heinbokel, scientist, and Jeff Potash, historian, both at The Center for System Dynamics at the Vermont Commons School, develop the argument that epidemics have changed the course of history. Both medical practitioners and those whose services are needed at such critical times are interested in the dynamics of epidemics. This is a classic scenario in the study of the dynamic of a very important system. Problem 1a: The Basic Model In any epidemic there have to be at least two categories of people: those who are infected with a certain disease, and those who are not infected, but are susceptible to the disease. Yes, there could be others, like those who are immune, but we will not consider that group at this time. There are factors to consider in the transfer of a disease from an infected person to an uninfected person. Some of these factors will be introduced shortly. J1. In your journal, sketch the graph of what you think the Susceptible Population will look like over the course of the epidemic (say 20 days). Also, sketch the graph of what you think the Infected Population will look like. 1. Susceptible 2. Infected high Draw your graphs in your journal. low 0 20 days We will start very simply for this first model. Assume there are 20,000 people (in a remote location) who are healthy, but susceptible to a certain disease, and that there are 10 people who are already infected with the disease. No new people are traveling into or out of our remote location, and the infected people are not dying so the total population remains constant during the entire simulation. The healthy people can become infected (flow to the infected stock) when they are contacted by an infected person. The number of people becoming infected each day depends upon two components: the total number of contacts the infected population makes per day, and the probability that the disease will be transferred on any given contact. The flow should be defined as the product of these two components. 1 This lesson was written by Diana M. Fisher. Thanks to George Richardson for guidance in the development of this model scenario. Thanks to John Heinbokel and Jeff Potash for their excellent "Plagues and People" models. 7.3 The Epidemic Model Student Lessons Page 7-17
The total number of contacts the infected population makes depends upon the number of people who are infected, and the number of contacts each infected person makes each day. Let's assume that each of the infected people makes one contact each day, and that the contact results in a transfer (100% chance) of the disease to that healthy person. Try modeling this simple situation for 20 days (Set DT=0.125, Runge-Kutta 4 Integration Method. Be sure your stocks do not have the non-negative box checked.) If your model is set up correctly, you should find that the healthy people all become infected quickly and the model causes the healthy population to become negative (which is impossible, of course). It might seem that the best fix is to check the non-negative box in the healthy population stock. However, even though this technique is convenient, it allows modelers to build less robust models that work, because the non-negative box saves them. You should build a model that behaves correctly without relying on nonnegative stocks. Here s one suggestion: The chance of creating a newly infected person has to decrease as the number of healthy people decrease because there is less likelihood that an infected person will meet an uninfected person on the one contact they have per day. So create a converter called probability of meeting a susceptible person. This probability will depend upon the number of healthy people there are. You may remember from your math class that simple probability of an event happening can be calculated by setting up a fraction. The numerator of the fraction is the number of desired outcomes, i.e., the number of healthy people there are to meet on any given day. The denominator is the total number of outcomes possible, i.e., the total population. The total population in our simulation is a constant value. That value should include both the number of healthy and number of infected people. Create a new converter called total population and define it as a constant, the numeric value for all of the people in our remote location. Draw a causal connection from the total population converter to the probability of meeting a susceptible person. Now we can define the fraction for the probability of meeting a susceptible person. One more slight adjustment to the denominator is necessary. It is not reasonable to meet oneself, so the denominator (total population) should be reduced by 1, to take this into account. The total number of contacts the infected population makes will no longer directly affect the flow, since not all of those contacts will be with a susceptible person. (Remove the connection from total infected contacts per day to the becoming infected flow.) Instead, a new converter, representing the number of contacts the infected population has with the susceptible population, needs to be constructed. It depends upon both the total number of contacts the infected population makes and the probability that the contact will be with a susceptible person. Should these two factors be multiplied or added? Those people who are becoming infected (flow) will depend upon the number of contacts infected people have with susceptible and the probability that the disease will be transferred upon such a contact. Before we continue, let's make a change that reflects a better modeling practice. Redefine the susceptible population as total population minus infected population, 7.3 The Epidemic Model Student Lessons Page 7-18
instead of using the value 20,000. 2 (Note: This new definition can be made without drawing any additional connections. Every stock dialog box allows full access to any component in the model.) Design the diagram. Insert the values. Be sure to place units after each value or formula. Create a graph that includes the number of susceptible people, number of infected people, and the number of people becoming infected every day (new infections). Set the vertical scale for susceptible and infected people from zero to 22,000. Run the simulation. Print the diagram, graph and equations (with units) on one sheet of paper. How do you know this model segment is working properly? Use the curves on the graph to help you explain. Problem 1b S-shaped growth of infected people should make sense. In this section you will modify a few converters. In real epidemics, the rate of transfer of the disease must include at least two factors. The first is the average number of times an infected person comes into contact with another person. If infected people go out and meet a lot of people every day, healthy people are more likely to become sick than if infected people stay at home and see no one. For our model, let s assume that each infected person makes, on average, two contacts per day, instead of one. Next, each disease has a probability of transfer of the disease, on any given contact. If a disease is airborne the probability is significantly higher than if the disease required physical contact with the infected person. Assume that the probability of transfer of the disease is 50% on any given contact of an infected person with a healthy person, instead of 100%. 2 Computer programmers often call this process "good housekeeping" as it allows modifications made to a program to automatically update formulas dependent upon those changes. 7.3 The Epidemic Model Student Lessons Page 7-19
Change your model to include this new information. What do you think will be different in the graph? Why? Run the simulation. Hopefully you were not surprised by the result. Problem 2a: Adding Recovered People Assume that people who recover from the illness are to be considered in the simulation. (There are no recovered people at the start of the simulation.) The recovered people will be immune to the illness. Once the illness is contracted, it lasts for 10 days, on average, before a person recovers. Change your model accordingly. Before you run the simulation determine what you think should happen to the size of the infected population. Write what you think will be different below. Set the simulation time to 80 days. Set up a table and a graph (each containing susceptible, infected, and recovered populations). Run the simulation. Print the diagram, graph, and the equations (with units) on two sheets of paper. Use the graph and/or the table to answer the following questions. 1. When does the number of infected people reach its highest point? How many people were infected when the epidemic was at its height? 7.3 The Epidemic Model Student Lessons Page 7-20
2. Explain why the system produces the shapes of the susceptible, infected, and recovered curves. Susceptible: Infected: Recovered: 3. How many people became ill with the disease at some time during the first 14 days? 7.3 The Epidemic Model Student Lessons Page 7-21
How did you figure this out? Problem 2b Now we will assume that a recovered person does not remain immune to the disease. This person will lose immunity after 30 days, on average. On the grids below sketch what you think the pattern of behavior for each population will follow over an 80-day time span. Use a dashed curve for your hypotheses. Susceptible 22000 11000 Infected Recovered 22000 22000 11000 11000 20 40 60 80 days 20 40 60 80 20 40 60 80 days days Run the simulation for 80 days. Print the diagram, graph and equations (with units). Identify one feedback in the system. Draw the loop diagram for the feedback and explain how it works. Identify whether it is reinforcing or counteracting and why. 3 3 The loop from Susceptible to Infected to Recovered and back to Susceptible is NOT a feedback loop. It is a circular flow of "material" (in this case, people). 7.3 The Epidemic Model Student Lessons Page 7-22
Problem 3: Not Showing Immediate Symptoms of Illness Remove the component allowing the recovered people to become susceptible again. In this scenario the recovered people will be immune after contracting the disease. A new epidemic, with much the same characteristics as in the previous scenario, has the additional characteristic that those who become infected do not show symptoms for 5 days, on average. (Note: The 10 people who are infected to begin with are infected, but not showing symptoms. There are no infected people who are showing symptoms at the beginning of the simulation.) Once the infected people show symptoms, they still require an average of 10 days to recover, so people are infected for an average of 15 days now: 5 days not showing symptoms and 10 days showing symptoms. You will want to add a converter, possibly named illness duration without symptoms. (Note: We will assume both groups of infected people can spread the disease.) Have the graph record the Susceptible, Infected Not Showing Symptoms, Infected Showing Symptoms, and Recovered populations. Run the simulation for 80 days. Print the diagram, equations (with units), graph (including all stocks), and a table (including all stock and flow values, and probability of meeting susceptible). The table may be set to display values only every 2 days so it will fit on one sheet of paper. 7.3 The Epidemic Model Student Lessons Page 7-23
How does the new factor, infected not showing symptoms, affect the overall dynamic of the epidemic? In your journal, based on the structure of the model, answer both of the following questions (J1 and J2): J2. In an epidemic situation, what does a vaccine (immunization) do to the dynamic of the epidemic? (Be specific. Which part of the structure is most affected by a vaccine?) What would a quarantine do? (Again, be specific. Remember, use your model structure as part of your answer.) J3. When there is a vaccine (immunization) shortage, why give priority for rationed vaccine to the elderly, the very young, and health care workers? (Extra credit for including a news article on epidemics - state date and source.) 7.3 The Epidemic Model Student Lessons Page 7-24