Covid-19 Hospitalisation and Death Rates per Day (explanations below)
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What does this chart show?
The red curve shows the number of daily deaths where Covid-19 was mentioned on the death certificate.
This number is about 20% higher than the "headline" daily number, which is the number of deaths of people who had a positive test in the preceding 28 days and is quoted widely, e.g. by the BBC. Relying on death certificates is more accurate, but data come in 2-3 weeks later.
The light-blue curve on the chart shows the rate of admissions to hospital, i.e. the number of admissions in the UK each day (brown curve) divided by the number of positive tests (black curve) nine days earlier.
Nine days is the time between the maximum of the hospitalisations (12 Jan 2021) and the maximum of positive tests (3 Jan 2021) at the height of the "second wave".
The dark-blue curve on the chart shows the rate of deaths in hospital, i.e. the number of deaths (red curve) divided by the number admitted to hospital seven days earlier (brown curve). It shows how likely are you to die once you are admitted to hospital. (Again, seven days is the lag between the maxima of the curves in January.)
Seven days is the time between the maximum of the deaths (19 Jan 2021) and the maximum of hospitalisations (12 Jan 2021). It was eight days in April 2020. "second wave".
The purple curve on the chart shows the rate of deaths per positive test, i.e. deaths (red curve) divided by the number of positive tests 16 days earlier (black curve). It shows how likely are you to die once you have tested positive.
All underlying data for the black, brown and red curves are downloaded daily from the UK Government's website.
Averages:
For each day's government numbers, the average of the data for the seven days around that day is taken, which is then averaged again over the surrounding two weeks. This serves to smooth out fluctuations and gives a clearer picture of trends.
Discussion:
Hospital admissions
In the early days of the pandemic, testing was not widely available and up to early July 2020 the government counted only tests that were performed in hospital for patients and staff. Thus the rate of hospitalisations per test started out very near 1 (i.e. 100%). It dropped to about 30-40% with more testing capacity coming online in early April and more NHS staff testing negative, and finally stabilised in August once testing became widely available.

From about mid-August 2020 until end of February 2021, roughly 10% of positive cases ended up in hospital, with only one dip below 7% in October, possibly driven by students returning to school and universities and falling ill, but less seriously than their elder and with no need for hospital.

Beginning in March 2021, there is a significant drop in the level of hospitalisations, down to just over 2% in early July 2021. This is the clear effect of vaccinations: they eased the severity of Covid-19 as well as changing the age profile of patients, with the younger and more robust making up the bulk of the patients.

Deaths
In the early days of the pandemic, as is the case for hospitalisations, poor availability of tests lead to staggering statistics: in March 2020, more people died than had been tested positive sixteen days earlier. Back then, few people got tested other than those already seriously ill in hospital and many care-home residents died without ever being tested or admitted to hospital. Also, tests were often performed much later during the illness or even just before death, which invalidates our assumptions, which are based on the January 2021 situation, of a nine-day delay between test and admission and a further seven days between admission and death.

The numbers become more reliable and more instructive by Summer 2020: death rates fell rapidly to well under 5% of positive tests and touched 1% in September. They then rose slowly again to a plateau at 2.5 - 3.5% between November and February. After that, the rate fell steadily and dramatically as vaccinations of the elderly and vulnerable, which had started in December, became effective. The latest published data (mid-August) show a 0.2% fatality rate, i.e. one person died for each 500 positive tests.

Surprising is the strong upwards bump to around 33% in deaths per hospital admission between October and March. It appears that at this stage mostly very seriously ill people were sent to hospital.