Epidemiological tag
There have been recent reports of schools banning the game of tag - bah humbug. This is unfortunate because, as well as getting children fit, tag can be used as part of a Mathematics lesson.
The fun rules
Very simply:
- When first tagged, tag two others and tell them (in admittedly bad English) that they are
It - If ever tagged again run around but don't tag any more children
- Choose one player to be the initial
It - The game ends when there is no one left who is
It
The game represents an SIR model:
- S for susceptible - never, or not yet,
It - I for infected - currently
It - R for recovered and immune to re-infection - no longer
It
At the end of the game the recovered players have acquired immunity, the remaining susceptible were never infected due to the effect of herd immunity. If the game is re-started then the some of the remaining susceptible can get infected showing that herd immunity is not a state of being. The game can be altered to model prior immunity by secretly telling some children that they can't accept a tag.
Lessons learned:
- Basic epidemic modelling
- That Maths can deal with randomness
- Abstraction. i.e. The game is equivalent to Epidemic, the card game
The game is an implementation of the minimal SIR model which averages to the ordinary differential equations of the standard SIR model. That model produces a classical infection curve due to competition between logistic growth and exponential decay. The logistic function (a linear transform of the hyperbolic tangent) forms an S-shaped curve which initially looks like an exponential curve and may be a reason why so many people incorrectly state that epidemics grow exponentially. Although a better reason may be that it is easier to draw a straight line on a log-plot than do the actual Maths.
A boring alternative
Very simplistically:
- Line the children up facing you
- From the leftmost to you ensure that they have two other children at arm's length to their left
- Tell the children to tag the children to their left when tagged
- Tell the leftmost child that they are
It - Watch the epidemic proceed like dominoes toppling
Lessons learned:
- That exponential growth is contrived
- That it ends when either it runs out of resources
- Or if the rules are changed mid-game
- Or else it continues to the largest number that you can think of plus one
Other examples of exponential growth are:
- Compound interest, which dates back to the Babylonians at least
- Grains of wheat on a chessboard, according to folklore from ancient India and Persia
- Pyramid schemes
Typically, nuclear chain reactions are described as growing exponentially, however the fallout includes unused isotopes showing that they are not. The (debunked decades previously by Mandlebrot) assumption that stock prices grow exponentially contributed to a market crash and the bail out of LTCM.