R

If I have 2 children, they each have 2, those 2 have 2 and so on, how many generations of Grants will there be before the Grant population reaches one million? I believe the answer is 20; 2 to the power 20 = 1,048,576. So that's about 600 years.

If I have 3 children, they each have 3, those 3 have 3 and so on, how many generations of Grants will there be before the Grant population reaches one million? Now it's only 13 generations; 400 years.

So I've been slacking, and my grandkids need to sharpen up if we are to get there any time soon.

Obviously this assumes we are all still around to enjoy our millionth family member. So maybe a bit flawed.

But if ten people re-tweet one of Donald Trump's tweets, and so on, how long would it take for everyone in the USA to have read the tweet? There are just under 400 million people in the US, so just 9 iterations of the power of ten are needed; that could be just 9 days! Add one more day and the  whole world has seen it! Such is the power of social media that attracts the attention of demagogues and advertising executives.

Now, if 2 people read my blog and only one of them recommends it to someone else and so on, how long will it take for there to be no readers? Given that there is no such thing as half a person, very quickly. However, if I publish a second post and the original readers (so enthralled by the original post) each recommend it to one different person, then the readership expands at...some rate or other (my A Level Maths is pretty rusty now). In my case, quality is paramount; in Trump's case, quantity is everything.

The reproduction rate (R) matters in the case of COVID-19 and other viruses. The PM told us last Sunday that R was estimated at that time to be between 0.5 and 0.9. This in itself is not very helpful, given that 0.5 (my pessimistic blog R above) would eradicate the virus in the UK in 16 months, statistically, whereas a rate of 0.9 would take over 8 years.

Note the word "estimate". How do you know that one person with the virus will infect, on average, say 0.75 people? In short, "know" is the wrong word. You would only "know" if you tested every person in the country every day. However, mathematicians will use sample data to give estimates for the whole population so contact tracing, which has apparently been very successful in some Asian countries, can give good quality data. If you identify 1,000 people who have the virus and trace all of their recent contacts and test those, you will in fact have a good idea of the reproduction rate.

There are, of course, many problems with this. For us in the UK, the lack of effective contact tracing is undoubtedly problematic; perhaps the imminent use of a contact tracing app will help, perhaps it won't. I'll certainly give it a try. One of the most significant problems may be that R may be higher in highly dense city populations, e.g. New York, than in isolated rural communities. So the government has talked about the possibility of different rates of relieving the lockdown in, say, Cornwall and in London.

But at the moment there are pressures for easing the lockdown protocols for such as schools and businesses - for different reasons. Evidence is presented to politicians regarding the effects of this easing but no-one will really know what will happen until it happens. I would be more comfortable about schools resuming if there were widespread tests for both teachers and pupils. As the head of the World Health organisation said back in March, 'Our key message is: test, test, test'. I'm not sure we listened soon enough.

If we are doing 100,000 tests a day on average, how long before the whole population of the UK has been tested? Nearly two years. And the test results will be out of date after a day, so the only answer would be test everyone every day. Is that practicable? Why not? In any case, start with schoolkids.