Epidemic the card game

Play the game

Take two packs of playing cards and separate out the same 31 cards from each pack. Perhaps choose aces to eights minus the ace of spades. Now deal one card into a column and then take that pack, turn it face up and spread it out.

The rule for creating a new column is to go through all the cards in the previous column, remove the corresponding card (white background in the animation) from the second pack, shuffle then deal two cards (yellow background in the animation) from that pack. If any of those two cards are still in the cards that were spread out then move them into the new column. Then re-form the second pack by returning the three cards and repeat as required. Now repeat this exercise until you can no longer form a new column.

The columns correspond to the generations of infected and their height to the number in that generation. The remaining cards from the first pack that do not form a column correspond to herd immunity - the 7 cards in the bottom left at the end of the animation.

If you have a third pack of cards then you, or any observer, can always draw a card from that and see whether the corresponding card has herd immunity at the end of the game.

The reproduction rate

The reproduction rate can be calculated by dividing the number of cards in a column by the number of cards in the previous column. The initial reproduction rate, R nought, for the game is two.

Exponential growth

You may note that the board in the animation has markings for increasing numbers of cards. 1, 2, 4, 8 and 16 which corresponds to exponential growth and the sum of which happens to be 31 - the number of cards in the packs. However the example game in the animation doesn’t grow exponentially which raises the question of what odds are there that it will. Calculating this number comes down to how many cards there are to choose at each draw. The first card is always a certainty. Any subsequent cards are drawn from 30 cards then 29 cards which means the second and third cards are always certainties. Then the odds start to drop.

Thus exponential growth is not impossible and it is left as an exercise for the reader to reproduce...

In the animation the first 11 cards match exponential growth, the odds of which are an achievable approximate 1 in 5.

Probability distributions

From the third column onward it is possible to construct a distribution for the probability that the column contains no cards to the probability that it contains twice the number of cards as the previous column where both extremes are not impossible. For a large number of cards this distribution is approximated by the logit-normal distribution for which the extremes are impossible and the expected number of cards in the second column is a fraction less than two.

How realistic is the game?

By using a pack of 1000 cards and playing the game 100 times then averaging the result then, unless you are unlucky, the numbers match those of the standard SIR model of Kermack-McKendrick from 1927. This model is illustrated in the textbooks by matching an influenza outbreak in an English boarding school.

However as every card is equally likely to be drawn the population is said to be well mixed and that doesn’t scale.

Flattening the curve

The underlying network that the cards form is known as a complete graph where each node is connected to each other node. For actual populations this is unrealistic and we must turn to social networks to model the correct structure. From a study of voles in Northumberland, England it was possible to extract a network of 1610 nodes. If this were a complete graph then it would be said to have a degree of 1609. However the network of voles varied in degree from 1 to 37 which means that applying the game means dealing from a pack of, at most, 37 cards.

The game was started from each node 10 times and the results averaged resulting in a flattened curve compared to a complete graph of the same number of nodes. The social network results took a maximum of 105 generations leaving 892 with herd immunity whereas the complete graph took 25 generations leaving 323 with herd immunity. Obviously herd immunity is not actual immunity and the voles, especially if they start being more social, are at greater risk of a subsequent wave of infections.

Further reading

If you’d prefer to see the formal Mathematics then there is a pre-print.

There is also a video recording of myself playing the game.