Also, I've got a maths degree,
Oh good.
So there is some hope that you will understand this if I provide the numbers.
The UK has been at max testing capacity for about two weeks now. This means that even if everyone is positive then the positive cases are limited by the number of tests. The highest daily number of test results (not the same as tests, and presumably lagged by 48 hours or so) has been 8911.
The NHS announced an increase in testing capacity on 11 March. www.england.nhs.uk/2020/03/nhs-to-ramp-up-coronavirus-testing-labs/ In the following week the number of tests was 2.65x that of the week to 11 March.
Therefore without any increase in the number of infections you would expect an 'exponential' rise in the number of reported infections, IF the testing is done at random.
Clearly the testing is not random, but as of 28th February, only 0.2% of all tests had been positive, so in comparison to some other countries we were doing a lot of testing of people with possible symptoms (or even no symptoms at all) that turned out to not be covid-19+.
We know that symptoms take on average 5 days from infection to onset, and that people are likely infectious before symptoms show, or without being aware that they were positive.
So as of late-February when around 0.5% of daily tests were coming back positive, then it's likely that much less then 0.5% of the population was infected at the relative point (since it's reasonable to assume that the testing was much better than random). This is why all those people who said they thought they had covid-19 in November or whatever are certainly wrong. It's highly infectious and vast numbers of people would have caught it.
Anyway as we got into early March, around 3% of results were coming back positive. At that point we were I think still tracing travellers and their contacts.
By the 11th March announcement of testing expansion, 7% of daily results were positive and we had uncontrolled local transmission. But, still, 93% of those tested did NOT have covid-19, which is a pretty big proportion considering that we were only testing a small proportion of those wanting tests.
Where we are now is that 36% of the latest test cohort were positive, up from 33% the previous day. This is the highest daily % yet, but as I observed in my previous post, as the number of people with severe symptoms indicative of possible covid-19 increases then unless testing increases (it hasn't), then a higher % of those tested will be positive with no underlying increase in infections.
In other words, in late February people were being tested simply because they had contact with an infected person or had been to a bad area, with NO symptoms, and now, to put it bluntly, they are being tested because they are dead, and we need to know the cause.
I.e. in late-February less (probably much less) than 0.5% of the population was infected with covid-19, and we knew this because we had a test that was likely considerably better than random in selecting its subjects.
We know that 36% of yesterday's tests, which is therefore the upper bound for the infected population, but in reality it should be much lower than that, since testing resources are concentrated in hotspots.
For example, London is horrible infected with close to 1000 confirmed cases /million in many boroughs (Southwark, Lambeth), four times higher than Surrey or Hertfordshire, whereas Hull has just 20 confimed cases per million. Of course this might reflect testing to an extent.
It might be that movement restrictions (e.g., students stay at university, Telegraph journalists stay in London and don't go their second home) mean that some areas avoid largescale infection, when combined with the shutting down of restaurants, etc.
So it does make sense to stop people travelling long distances for whatever reason, but it also follows that infection rates are clearly very high in London so it is both lawful and sensible if you are trying to avoid infection to drive somewhere away from crowds rather than exercise outside your flat.
Anyway as far as exponential growth goes, the data are insufficient to draw conclusions given sampling bias, and especially when you consider that it's possible that certain areas are already saturated while others are at early stages in growth. These latter places may experience similar growth to the UK as a whole - you have a handful of cases in a particular area, and they spread to others via supermarkets, etc. until a large % of the population is infected. This will be true if the virus is as infectious as feared, but at the very least current strategies delay that process such that by the time it's a major problem around say Hull, others parts of the country have more capacity.