Modelling Spread of Disease
This is a simulation activity designed to help students understand both the application of modelling in the context of disease spread and the potential complexity of doing so. In this activity we will set up a human simulation of the spread of a fictitous disease and model is spread through a number of iterations. The simulation will allow us to vary some things so that we can examine the effects. It may be an occasion when you want to get together with other DP classes that will also benefit from the exercise!
The following needs to be read and understood by the person who is going to run the simulation. That will most likely be a classroom teacher, but students may want to come back to this page to remind themselves of what they did.
Space - You will need an open space big enough for the number of people you have to walk around freely. For a group fof 30 people you might need a space of about 8m by 5m that you mark out somehow as a rectangle. You will see the difference this size makes at some point during the simulations.
Identifiers - You will need a set of numbered cards so that you can give one to each of your students so they can be identified without using their names. A pack of playing cards can do this just as well to save time.
Photographs and videos - If you can designate a photographer, it is great to keep photographic records of the event to feed off as this idea develops.
Scorekeeper - If you can, it is extremely useful to have a designated score keeper who will count the numbers of people infected at different stages and do some live plotting of the results, using a tool like Desmos for example.
Whats the disease? - Well this simulation is designed to model the notion of a generic 'disease'. Doing this allows us to circumvent issues that might pertain to specific diseases and also allows us to be sympathetic. That said, it is entirely possible to pick a specific context for such a simulation and change the parameters to fit it. For example, it could be interesting/relevant to look at sexually transmitted diseases. So that this can be widely used, we will speak about a generic 'disease'.
Introduction - The Island
At the start, we understand that we are all citizens of the Island (this of course is easily adapted to suit your location) and that we will go about our daily business that will take us around the island, but not off the island. The boundaries are marked by the rectangular boundaries you have set up.
Everyone is given a numbered card or identifier (as explained above) that they must keep to themselves.
Simulation 1 - Unrestrained spread
- Everyone wanders around the island until you say 'stop'. When stopped you pick a number (card) at random and announce that this person has just been diagnosed with a contagious disease.
- In this case, the person may identify themselves as the origin of the disease and 'the original case - stage 1'.
- This person should choose 2 people to infect with this disease and then sit down - they are allowed to move freely. The two new people who have been infected at this stage are infected at stage 2. They should remember that they were infected at stage 2.
- Each of these two people now need to find 2 uninfected people to infect at 'stage 3' and then sit down. The 4 people who have just been infected remember that they were infected at 'stage 3' and then sit down.
- The simulation continues like this until there is no one left to infect.
Summary - The number of people infected at each stage should read like a geometric sequence with a first term of 1 and a common ratio of 2. The point should be made that this translates to ' the number of new cases at each stage' but that the total number of cases will be the sum of the number of terms of that sequence. Thus, we have one model for number of new cases per day which leads to another model for total number of cases.
Questions and limits - At this point it is useful to consider how realistic we think the model might be. How likely is that that every person infected will find exactly 2 people to infect and so on.
Variations - Obviously, this exponential spread goes pretty quickly through small group so the idea of varying it to a bigger number may not be too practical. Clearly decimal ratios can't be used in this kind of simuation, but might be more realistic when actually modelling data.
Simulation 2 - Limited movement
A quick note - For this simulation, the size of the space makes a huge difference. Too big and the disease wont spread at all, too small and the limited movement will have not effect. For that reason it is useful to look at the space as people are moving around and adjust or accordingly. It might be equally useful to run it a few times with different size spaces. It is quite important though not to tell people what is going to happen in advance.
It is worth noting that, despite the context, this can be a lot of fun as people strain to reach each other. A bit of a laugh helps to make this activity quite memorable and so I anticiapte and allow for this.
- The second simulation is set up in the same way as the first but the key difference is that once the first person is diagnosed, no one is allowed to move. It might be useful to define this by saying that everyone's feet must stay in exactly the same position.
- The first person is diagnosed (by a randomly selected number/card). They are then asked to infect 2 people by touching them if they can (remember that no one is allowed to move their feet). Then they sit down as 'Stage 1'
- These two people (if there were two) then try to infect 2 people by touching them (no one moves their feet) and sit down at stage 2.
- This continues until either a) everyone is infected or b) the remaining infected people are not in reach of anyone else.
Summary - So this model should show a slightly slower spread of the disease because of the limited movement. Not everyone will be able to reach 2 peope to infect them. Some people might escape uninfected. Both of the models should be slightly different. Collect the numbers so that they can be plotted and we can see the difference.
Questions and limits - What realism has this model introduced? How is this idea manifest in some real situations? What difference does the size of the space make here and what does this mean in reality. (eg population density)
Variations - As suggested already - this simulation can be explored by varying the size of the space to simulate different population densities in reality. If using a bigger space, then spread will be slower and maybe even stopped by the space between people. In this context a higher rate might be worth exploring. For example - big space but infection rate of 3 or 4.
Simulation 3 - Planning ahead
This one is particularly good fun. Hopefully by this stage you are happy with the size of the space you have chosen as, again, it is quite important. Read the below before starting.
- This 3rd simulation is exactly the same except for the key difference that you have told the population in advance that this is going to happen and asked them to position themselves accordingly. As people mingle you have asked them to think about where they want to be when the disease starts to spread. Of course, no one knows who is going to get the disease first.
- For this reason it is quite important that the space is not big enough for everyone to find their own space that keeps everyone at arms length.
- What follows here is a really interesting exercise in human behaviour as people to choose to try and isolate themselves or club together in groups and take their chances.
- The simulation is then run the same way.
Summary - This is a little less predictable. Invariably, a number of individuals will have tried to isolate themselves, whilst others will be in communes in the corners, taking their chances with the company they choose and protecting themselves for the rest. As such, if the disease happens to go to one person who has isolated themsleves then it doesn't go very far and you might run it again with another number. In anycase, the likelyhood of some people escaping is high and certainly we would expect a slower spread.
Questions and limits - The questions here are really about what metaphor for human behaviour each decision that was taken is.
Variations - Clearly size is everthing here. Later on the page I have written about the introduction of 'behaviours' which could work for variation here.
Simulation 4 - Incubation
In all of the simulations so far, we have known from the beginning who the first person infected was. In this one, that will be different in attempt to simulate the notion of an incubation period in which a person might be infected but not know. This one needs to be run with a good bit of care to make sure everyone understands the rules as it is a little bit different from the others.
- To start with, the islanders mingle as usual until you say 'stop'. At that point you pick the number/card of the person who will first be infected, BUT make it clear that this person must keep their best poker face on so that no one knows it is them. It is key, that, whenever a person is infected, they keep this quiet.
- So now the crowd has an infected person but only they know who they are. They are stage 1.
- You ask the crowd to mingle and then when you say stop, they are asked to pair up. At this point they are to have a noisy conversation with eachother and shake hands 'hello how are you today?' In all but one case the answer will be fine. The infected person will tell their partner that they have the disease and have, unfortunately, just passed it on to them. This person was affected at stage 2 and must remember that.
- There are now 2 people infected, but only they know. The crowds mingle again until you say stop, at which point they air up with sime one new and have the same noisy conversation again. At this point our two infected people will pass the disease on to 2 more people (who will remember stage 3) and we will have a total of 4, but again, only they will know and they dont know about each other either.
- Repeat - This conversation is stage 4 and here we might get 4 more, but we might not, because 2 of the 4 infected people might meet each other in which case their is no further infection. Up to 4 new people will be infected here at stage 4 and must remember that
- This cycle then repeats through a few more stages - depending on the jumber of people you have. with 30-40 people you might run 8 or 9 stages. It doesn't matter if you do a couple more.
- End by asking if there is anyone left who is uninfected.
Summary - At the end I suggest making a human graph of the new cases per stage. Ask the first person to identify themselves and then come and sit on a line as stage 1. Then ask who was infected by that person at stage 2. Invite that person to come and sit next to them. Then, in turn, ask the people who were infected at different stages to come and sit in. single file line that corresponds to the stage. Help by making sure people sit at an even spacing so that they are making a bar chart of the frequency of new cases. Its great if you can get a good photograph of this. The graph tells a story. You can observe how it begins exponentially but the tails off completely and stops altogether as the number of new people to infect reduced to zero. At this point, hopefully, your graph plotter can plot the sums so we can see how the actual graph appears to be a logistic function.
Questions and limits - As above, we reflect on how well this simulates human behaviour and how the simulation might be developed.
Variations - In this case, the key question is about whether or not we allow people to shake hands with the same people more than once. This should be handled carefully since if they pair up early, the disease wont spread. This could of course be a metaphor for human behaviour too. Some rules about this and again, introduction of behaviours could change things.
A ToK moment
A global view
So this activity can be run at varying levels of complexity and that depends entirely on the time you have, the number of students you have and what you want to get out of it. The simulation works really well with 30 + people or more. The results get more interesting the more people you can manage. You need a big space and a little energy reserve to make it happen.