## Basic Reproduction Number (R)

**Estimated R Number for Key Datasets**

The graph above, which updates live, shows a best fit overlay for the UK Basic Reproduction Number (R) number using swab test positive data, the ONS infection survey, and the COVID Symptom Study, overlaid with UK government data published 8th may 2020 - 26th March 2021. The methodology used to create these R number estimates is shown below.

**R Number from UK Government**

The graph above shows the UK SARS-CoV-2 Basic Reproduction Number (R) range published by the UK government. The UK government stopped publishing this data for the UK (*light blue above*) on 26th March 2021, opting instead to publish separate R numbers for England (*dark blue above*), Wales, Scotland, and Northern Ireland.

**Calculated R Range Fitted to UK Government Data**

Using symptom study app data, and plugging it into __‘the Simple Model’__, we can derive a rough estimate of the current UK R value.

For this we need to know serial interval for SARS-CoV-2. The serial interval (tau) is defined as: “*the period of time between a primary case patient (infector) with symptom onset, and a secondary case patient (infectee) with symptom onset* ”; it is represented in basic reproduction number calculations with the Greek letter ‘Tau’.

There are a number of different values for the serial interval published in the scientific literature, ranging between 4.0 – 7.5 days. The large range in this value makes it difficult to accurately calculate the basic reproduction number, using the ‘simple method’. To help pick a representative serial interval, if we calculate the basic reproduction number for a number of different ‘tau’ values, and fit them to published government data, we get the graph below.

**Calculated R Number with UK Government Data**

From this graph, we see that a serial interval in the range 2.6 – 4.4 gives a sensible fit to published government data.

Nishiura *et al.*, in a paper published in the ‘International Journal of Infectious Diseases’, calculate the serial interval for SARS-CoV-2 to be 4.0 – 4.6. This gives us a more narrow range to choose a serial interval from to help calculate R from published symptom study app data.

Another useful piece of information is: on 27th November 2020, the R value published by the symptom study app team, in the daily report sent to the UK government was 0.8. On the 26th November 2020 their published R value was 0.9. The implication is that at this time, their published R value was around 0.85.

If we use a serial value of 4.6, which is the upper value calculated by Nishiura *et al.*, then we get an R value for the 27th November 2020 of 0.85, which fits well with the R value published by symptom study app team. The graph below shows the basic reproduction number (R) over time, calculated from published symptom study data, using a serial interval of 4.6 days.

Further to this, as time passed, and more data was generated, on the 9th December 2020, the symptom study team published R to be 0.9, and 0.8 on the 8th December. If published R is 0.9, then R is actually in the range 0.9499 – 0.8500, and when published R is 0.8, then R is actually in the range 0.8499 – 0.7500.

On the 13th December 2020, the R number predicted by the team changed to 1.0, putting it in the range 1.0499 – 0.9500.

We can use these observed changes in the R number published in the reports sent to the UK government by the symptom stud app team, to help figure a serial interval (tau) that produces an R number consistent with those published by the symptom study team.

Fitting our calculated R number with the reported change in R number by the symptom study team in November (26th – 27th) and December (8th – 9th), gives a serial interval (tau) in the range - (see graph above). Using the median of this (), gives the graph below. The published symptom study data used to calculate the R number here, represents the situation on the ground four days prior to the date that the data is published. Thus the data in these graphs is time shifted by four days to allow for this.

**Infection Survey R Fitted to UK Government Data**

We can do the same with Infection Survey data published by the ONS. Using a serial interval (tau) value of , we get the daily R number shown in the graph above.

**Swab Test Positive R Fitted to UK Government Data**

Applying the same methodology as above, we can use swab test data to estimate R number. Using a serial interval (tau) of 6.5 gives the R values shown in blue below. This is best visualised using 5 day averaged data, which is shown in red below.