Its similar to Rt but its much easier to calculate. To get Rt, you need to estimate how many people are infected on average by one person at any point in time. This is usually done by simulating the process in various ways.

To get the daily rate of growth on the other hand, all you need to do is look at the actual numbers. Assuming you had 110 cases today and 100 cases yesterday, the current rate of growth is 10%. Since the rate calculated this way will be a bit erratic, I use some averaging and exponential smoothing to get the general trend.

The code to generate these graphs is available here: https://colab.research.google.com/drive/1QDZgdGkOzs6dP74-iED... - it uses the UK coronavirus API and google mobility trends reports directly from their respective URLs, so it should be easy to make a copy of the notebook, review/modify it and re-run it.

Why is the ratio of positive cases to hospitalizations different? I'd say this is mostly due to testing capacities being very different between now and March. In March in the UK for example it was impossible to get a test unless things are so bad that you need to get hospitalized. This is very different from now where anyone with symptoms (or just saying they have symptoms) can get tested.

Based on the graph, another reason for the different ratio between cases and deaths seems to be the period July to mid August. This was a period where the growth rate of cases didn't match the growth rate of hospitalizations and deaths at all. This likely means that parts of the population that wasn't that succeptable to severe outcomes were getting infected (which makes sense given that bars were opened up again at the beginning of July - although that is speculation)

If one tries comparing the US data with the data from other developed countries, what can be seen is that only the third peak in the US has the "realistic" ratio of cases to deaths, that is, one which better corresponds to the ratio from the countries which were able to handle the outbreaks better, as in, being able to detect enough cases early enough, and for that appropriate test capability, adequate to the incidence, seems to be necessary. The approximate expected ratio, worldwide, seems currently (with obviously better testing infrastructure compared to Spring, and as long as the hospitals aren't too stressed) to be 1 case of death per 100 registered "cases", after a delay, and when using some averages across a moving window of data for both both cases and deaths, e.g. 7 day average.

In the first peak in the US it was as high as almost 1 death per 14 cases(!!) -- around 32K detected daily cases with 2.2K deaths were the peaks of the averages! In the second peak in the US it was 1 death per 60 cases. Both show that there weren't enough cases detected, compared to some other countries, and that more appropriate level of testing was achieved in the US only during the third wave. But if there are some systematic delays in reporting deaths during the third wave, it can still turn out that what we see at the moment will look worse, once the wave is over. Even that effect is known to exist in the reporting of one European country: Sweden, where the systematic reporting delay of deaths during the wave gives very wrong impression of the ratio as long as the wave is raising, compared to other countries: https://news.ycombinator.com/item?id=25122196 There are some signs that such an effect can be in play in the US as well.

Additionally, what one has to be aware of is that any statistics over a big population will at different points of time not always proportionally reflect the contribution of the events in different areas -- e.g. New York. That way, some big problems for the local population will appear "less bad" when averaged, which shouldn't be comforting.