​Years ago, when I was director of computational science, the Chemical Database Service was undergoing a review. The presentations we made showed a good rate of growth in the number of users of the various components of the service. One of the professors on the review panel then asked why the user growth rate didn’t seem to be exponential - he said he expected exponential growth in the uptake of IT-based services.  Since the users of the CDS were UK research chemists, and the user numbers were already pretty high, it occurred to me to wonder what the maximum number of users should be.  How many research chemists are there in the UK and how many of them could reasonably be expected to use the CDS in their work?   What lay behind this question was the obvious thought that in a finite population one might expect exponential growth initially but not forever.  Even when user growth is driven by word of mouth, rather than targeted dissemination, this ought to be true. I probably made a hand waving argument along these lines at the CDS review, but I always thought that one day I would work out the dynamics of the transmission of information in a finite population of of individuals properly - it looked like a fairly easy little problem. But I must have forgotten about it until, years later, I was reading Martin Nowak’s book about how to mathematise aspects of biology, evolution etc, where an equivalent problem is discussed.

Something about coronavirus and lockdown must have brought this back to mind, and here is a brief note on it.  I was right; it is pretty easy.

Nowak also gives a very brief introduction to the discrete-time version of the model I came up with for CDS. This is the so-called “logistic map”; it has some amazing properties pointing the way to the field of deterministic chaos. Though not strictly relevant to the subject of this note, it is fun and I give some examples of results of the map.

Post 12 Transmission in a finite population.pdf