Someone in the room is infected. Will it spread to everyone? Can a vaccination help eradicate the infection? How long will it take?
In this project we will look at several methods for investigating these and other questions related to infectious diseases. In the SIR model framework, S denotes a susceptible individual, someone who can become infected. I denotes an infected individual, and R denotes someone who is removed. Before becoming absorbed in morbid thoughts, let's clarify what removed means. It could mean that the individual has passed away, but in this experiment we will assume that a removed individual has attained immunity from the infection (chicken pox, influenza, measles, etc).
Here we will assume that no individuals die from the infection. To allow for that, we would need a slightly different model. We also assume that once recovered from the infection, an individual cannot become infected again, although we know that the immunity from infections such as chicken pox are not 100%. Again, a slightly more complicated model would be needed for this (begin simple, then expand).
Methods:
We begin our experiment with 3 members of the class being "infected". The rest of the class is susceptible. Each member of the class has a cup and three dice, red, blue, and white. We will move around the room having "encounters" where some members of the susceptible population may become infected. Some of the infected individuals may become removed. Here are the rules of play.
Movement:
Each member of the class must move around the room and must not stop in between encounters.
Rules for encounters:
When two members approach each other, there is an encounter. If you are susceptible, you will use the white dice; infected, the red dice; and removed, the blue dice. Each member will roll the appropriate dice and take action according to the following rules.
1. Susceptible : Susceptible - no changes.
2. Susceptible : Infected - if the sum of the rolls is even, the susceptible becomes infected.
3. Removed : Anyone - no changes.
4. Infected - if an infected rolls a 6, they become removed after the encounter.
Every minute we will stop and count the number of infected and the number of susceptible individuals. When the number of infected individuals becomes zero, the experiment stops. The counts will be done by a show of hands with eyes closed to minimize encounter bias.
Exercises:
1. Record this data and plot the results.
2. Describe the results. If we did the experiment again, what would you
suspect the results would be?
3. On average, what do you think the result would be (% who get infected, time
course of the disease, etc)
As you can tell from the experiment in class, it takes quite a while to run this experiment. It would be nearly impossible to run this experiment for a large population, with different probabilities, and including vaccination.
Instead, we turn to the computer. On the computer we can easily simulated 1000 or even 100,000 people, adding in the effect of vaccinations or quarantine (removing a percentage of either the susceptible or infected population). For this experiment, we will use Berkeley Madonna. (simulation file)
In the computer program, we will begin by simulating 60 students, similar to this class. Among them, 10 are infected. We will take into account one encounter at a time.
In the program, four random numbers are chosen. The first two are numbers between 1 and 60, the others between 0 and 1. These numbers determine whether the encounter will be between an infected individual and removed individual, a susceptible individual and an infected individual, or whichever. Each case is equally likely. If the encounter is between an infected individual and a susceptible individual, then the third random number is used to determine if the encounter results in an infection. If it is, the number of infected's is increased by one and the number of susceptibles is decreased by one. If the encounter involves any infected individual, the fourth number is used to determine if the infected individual becomes removed.
There are two parameters for the probability of infection and for the probability of removal. These parameters are in a slider box when the file is opened.
Exercises:
1. How does this experiment differ from the one we did with the class?
Do you expect similar or different results?
2. Adjust the parameters to the same as those used in the class experiment and
compare these results.
3. Click on <parameters> and <batch run> from the menu. Select
300 repetitions and have the program report the mean (average). This is
what the average behavior looks like. Is this what you predicted? Explain.
4. Suppose you have 10,000 people instead of 60. Experiment with different
infection scenarios by changing the initial conditions and report what you find.
A Berkeley Madonna file for a continuous dynamic model of this phenomenon is available here for comparison. Notice that the results match the average behavior of the discrete system.
1. How would you add vaccinations to this model?
2. What would happen if you return to the susceptible population after spending
time in the removed category? This could be a model of athlete's foot or
similar infection.
3. For diseases such as HIV, it has been suggested that reducing the number of
encounters can help to eliminate the disease. Using our model, can you
substantiate this?
4. An issue with diseases where the removed state implies a diseased individual,
what happens when the infected person is given medications that prolong their
life (lengthen the time they are in the infected class)?
5. We can also use differential equations to model infectious diseases.
This model is not appropriate for studying
eradication of infectious diseases. By looking at different parameter
values, explain why not.