Infectious Disease Spread
MELT Modeling
Someone in the room is infected. Will it spread to everyone? How fast will it spread? We
will consider how to model this situation (the spread of disease in a closed environment)
in a couple of different ways.
First, list any assumptions you think we need to make. Be prepared to explain.
One model framework that is popular and useful for this situation is the SIR model. This
includes three populations – S (susceptible1), I (infected2), and R (removed3). What
assumptions do we need to make in order to sort such a division of the population?
A Hands-on Experiment
First, let’s consider an experiment that models the situation. We will begin our
experiment with three 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, and some of the infected individuals may become
removed. Here are the rules:
Movement: Each participant must move around the room and must not stop between
encounters.
Rules for encounters: When two participants approach each other, there is an encounter.
If you are susceptible, you will use the white die; infected, the red die; and removed, the
blue die. Each participant will roll the appropriate die and take action according to the
following rules:
i.e., and individual who is uninfected and susceptible to infection
this meaning is obvious, isn’t it?
3 before you get too morbid, “removed” could mean that an individual has passed
away, but it could also mean that the individual has recovered from the infection
and developed an immunity.
1
2
1. Susceptible: Susceptible – No changes
2. Susceptible: Infected – If the sum of the rolls is even, the susceptible participant
becomes infected, otherwise, no change
3. Removed: Anyone – No changes
4. Infected: Infected – No changes
5. 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. We will conduct these counts by a show of hands with eyes
closed to minimize encounter bias. When the number of infected individuals becomes
zero, the experiment stops.
Extensions and Creating a Model
1. How could we alter the simulation to incorporate different infection rates? How could
we incorporate vaccinations?
2. What would happen if participants returned to the susceptible population after
spending time in the removed category?
3. How could be extend our simulation to include a deceased (D) category? This is an
SIRD model.
4. Create a mathematical model for the situation.
Infectious Disease Spread
MELT Modeling
Teacher Notes
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It’s useful to start this activity with a discussion about the spread of disease and what
assumptions are necessary in our model. Depending on the level, you could do the
activity and then discuss the assumptions, or discuss assumptions and try to guide it
toward the necessary assumptions for the activity. Assumptions are very important in
modeling this situation, and it can highlight for students that assumptions aren’t really
a bad thing.
The SIR model makes the assumption that no one dies from the infection, that, once
removed, an individual does not become infected again (even though immunity from
certain infections is not 100%). These situations can be modeled, but they are
somewhat more complicated – not beyond the scope of what we can do, but it’s often
best to start simple, then expand our model.
The experiment can be altered with different numbers of infected to begin, and
changing the criteria for people to move from one group to another – that is, changing
the probability that someone gets infected or is removed. You could also limit the
number of encounters an individual can have while infected – say, after n encounters,
they become removed regardless. This could simulate recovery after a set time frame
(you could also vary n for each individual to vary how many interactions different
participants might have before recovering).
It’s helpful to record the data in Excel, and then graph and discuss what you see. With
a large enough group, dividing it into two (or more) environments might show
different progressions of the disease.
Note that the activity is a simulation, which differs somewhat from a model. After the
simulation, we will consider how to model the situation. It’s tempting to simply try to
model the data that we collected, but then we’re modeling the simulation, not the
situation itself.
An Excel document for an SIRD model can be provided. It allows one to adjust the
various rates and observe the impact.