Logistic growth in the human world

Today I presented my research for upper secondary school teachers as part of a day on ‘Mathematics in the human world’ (‘Den mänskliga världens matematik’). This is probably one of the most important messages to get out from applied mathematics. Mathematics isn’t just for physicists, engineers and computer scientists. It is needed everywhere in society, not least in understanding society itself. I am impressed that the Swedish Royal Society (KVA), who organized the day, had the foresight to choose this theme. I hope we gave some inspiration to the 50 or so teachers who gave up there time to came along to listen.

Given the vast subject of ‘humanity and mathematics’, it was interesting that three of the four scientific talks (my own, Tom Britton’s and Kimmo Eriksson’s) all included a detailed description of the logistic growth equation. In modeling three completely different contexts---disease spread, adoption of mobile phone usage and applause after an academic talk---this innocuous little equation plays a central role.

The logistic equation can be derived from simple assumptions about social behavior. Assume you have heard a rumour and you tell 3 randomly chosen people about it during 24 hours. If there are N people and X of them have heard the rumour, then the probability that each of these random people have not already heard the rumour is 1-X/N. On average, you will ‘infect’ 3(1-X/N) people with the rumour. Now, if all the X people who have heard the rumour behave in the same way as you do then the average hourly increase in people knowing the rumour will be

dX = (3/24)X(1-X/N)

which is the logistic growth equation. The growth dX is small when either X is small (no-one has heard the rumour) or X is nearly equal to N (everyone has heard the rumour). The rumour spreads fastest when X=N/2 and half the population know about the rumour. This leads to the sigmoidal growth curve shown here.

We published a paper earlier in the year looking at clapping in a small audience.  First, we showed that both the onset and cessation of applause followed a logistic growth curve. Then we tried to address a problem that Kimmo raised during his talk today: “lots of mechanisms produce logistic growth curves, how do we know that it is ‘social contagion’ in any particular case?” We looked at clapping events, and compared the fit of a social contagion model with various models where individuals chose a randomly distributed number of claps to do irrespective of the behavior of others. Social contagion models were the most important factor in predicting the individual clap patterns seen in the data.

This paper got a lot of media attention, mainly because of our prediction that long applauses can occur not just because a presentation is good, but also because of a failure to co-ordinate stopping. Richard Mann gave a well-balanced interview on BBC about this. Richard also did a fun analysis, again using the logistic curve, of the media contacts he received after publication.

Of course, neither my presentation today nor the others at the humanity mathematics day were limited to logistic growth. I talked about Schelling’s model of segregation and collective motion, Kimmo about popularity of names and cultural evolution, and Tom about disease networks. Klas Markström, who helped plan the day together with Ingvar Isfeldt at KVA, described the mathematics and paradoxes of voting and democracy. There is a diverse mathematics to humanity.