R values, doubling times and halving times

R values can be used to describe whether an epidemic is increasing or decreasing, and how fast; but in most circumstances, it is better to use doubling times and halving times.

#### R values

The R value of an epidemic is the Reproduction Number, which is the average number of new cases that each existing case will spread to directly.

If the R value is 1, the epidemic will continue at a constant rate of new cases per day, until most of the population has been infected.

If R is greater than 1, then each case will lead to more than one new case, and the epidemic will increase. If R is less than 1, each existing case causes on average less than one new case, and the virus steadily peters out, eventually becoming locally extinct.

The R value of an epidemic cannot be measured directly, but it can be estimated from how fast the number of cases is increasing or decreasing, or from numbers of hospital admissions or deaths.

R values for the covid-19 pandemic have been estimated at around 3 when the virus was spreading before any restrictions were imposed (so each case lead to three new cases), and as low as 0.3 after a combination of radical measures were implemented. .

#### Doubling times

As well as describing epidemics via R values, we can describe an increasing epidemic by the doubling time. Rather than increasing by a constant amount daily, epidemics tend to increase in proportion to their current size (which is known as an exponential increase) - and so the time taken for the epidemic to double in size tends to stay constant (until measures are imposed). This is known as the doubling time.

Doubling times for the covid-19 epidemic were typically 2 - 3 days before restrictions were introduced.

#### Halving times

When epidemics are declining, they tend to do so in proportion to their size. This is known as an exponential decrease. It can be described by the halving time, which (no surprise) is the time for the number of new cases to reduce by half.

Halving times in the covid-19 pandemic have been as low as 5 days in several countries (e.g. China, South Korea, and New Zealand) [1]

#### Choice of R values or doubling times and halving times

Should we use R values or doubling times / halving times to describe epidemics?

R values have advantages that only a single measure of the epidemic is needed, and that they can be used in complicated mathematical procedures. But they cannot be measured directly, and they do not mean much to the average person, so their use can hamper the involvement of the general population in the control of the epidemic.

Doubling times and halving times are much more readily understood: most people can see that, for example, the number of cases has doubled in the last 3 days, having previously doubled in the preceding 3 days, and so it is likely to continue doubling every 3 days until something is done. Similarly, most people can see for themselves that, for, example, the number of new cases per day has halved every 7 days for the last two weeks, and so is likely to continue to halve every 7 days.

#### Conclusions

While R values are useful for experts, epidemics should be described in terms of doubling times and halving times. This will aid the involvement of the general population in the control of their epidemic.

#### References

 [1] Countries with the best covid-19 control: halving times of 5 days

First published: 31 May 2020
Last updated: 8 Jun 2020