Читать книгу Maths on the Back of an Envelope - Rob Eastaway - Страница 14

There are some situations when a small change in the value assigned to one of the ‘inputs’ has an effect that grows dramatically as time elapses.

Take chickenpox, for example. It’s an unpleasant disease but rarely a dangerous one so long as you get it when you are young. Most children catch chickenpox at some point unless they have been vaccinated against it, because it is highly infectious. A child infected with chickenpox might typically pass it on to 10 other children during the contagious phase, and those newly infected children might themselves infect 10 more children, meaning there are now 100 cases. If those hundred infected children pass it on to 10 children each, within weeks the original child has infected 1,000 others.

In their early stages, infections spread ‘exponentially’. There is some sophisticated maths that is used to model this, but to illustrate the point let’s pretend that in its early stages, chickenpox just spreads in discrete batches of 10 infections passed on at the end of each week. In other words:

N = 10^T,

where N is the number of people infected and T is the number of infection periods (weeks) so far.

After one week: N = 10¹ = 10.

After two weeks: N = 10² = 100.

After three weeks: N = 10³ = 1,000,

and so on.

What if we increase the rate of infection by 20% to N = 12, so that now each child infects 12 others instead of 10? (Such an increase might happen if children are in bigger classes in school or have more playdates, for example.)

After one week, the number of children infected is 12 rather than 10, just a 20% increase. However, after three weeks, N = 12³ = 1,728, which is heading towards double what it was for N = 10 at this stage. And this margin continues to grow as time goes on.