# When will COVID-19 be Over? Introduction to Ro & Re

This introductory document will give you a basic understanding of how decisions will be influenced when deciding when the COVID-19 is over. This is one aspect and one of the most important aspects, among others how it will be decided when the virus replication is at a rate lower and its safe to open lockdowns.

### Remember George Box’s wise words about models:

“All models are wrong but some models are useful”.

### Or as he put it elsewhere, less dramatically but more specifically:

**“Models, of course, are never true, but fortunately it is only necessary that they be useful. For this, it is usually needful only that they not be grossly wrong”.**

**The basic reproduction number Ro**

*The reproductive value describes the average number of people an infected individual can expect to pass the coronavirus onto. It is, therefore, a measure of how transmissible, or contagious, a disease is.*

*The World Health Organization estimated at the start of March that the R0 for coronavirus stands somewhere between 2 and 2.5. By comparison, seasonal flu is estimated to be roughly 1.3 while measles has a reproductive value of between 12 and 18. *

The basic reproduction number is defined as the number of cases that are expected to occur on average in a homogeneous population as a result of infection by a single individual, when the population is susceptible at the start of an epidemic before widespread immunity starts to develop and before any attempt has been made at immunization. So if one person develops the infection and passes it on to two others, the R_{0} is 2.

If the average R_{0} in the population is greater than 1, the infection will spread exponentially. If R_{0} is less than 1, the infection will spread only slowly, and it will eventually die out. The higher the value of R_{0}, the faster an epidemic will progress.

R_{0} is estimated from data collected in the field and entered into mathematical models. The estimated value depends on the model used and the data that inform it.

**R _{0} is affected by:**

- the proportion of susceptible people at the start and the density of the population;
- the infectiousness of the organism;
- the rate of disappearance of cases by recovery or death, the first of which depends on the time for which an individual is infective;

The larger the population, the more people are susceptible, and the more infective the virus, the larger R_{0} will be for a given virus; the faster the rate of removal of infected individuals, by recovery or death, the smaller R_{0} will be.

The zero in “R zero” means that it is estimated when there is zero immunity in the population, even though not everyone will necessarily be susceptible to infection, although that is the usual assumption. In an epidemic with a completely new virus, the earlier the measurements are made the nearer the calculated value is likely to be to the true value of R_{0}, assuming high-quality data. For this reason, it is better to talk about the transmissibility of the virus at the time that it is measured, using a different symbol, R_{e}, the effective reproduction number.

**The effective reproduction number, R**_{e}

_{e}

The *effective *reproduction number, R_{e}, sometimes also called R_{t}, is the number of people in a population who can be infected by an individual at any specific time. It changes as the population becomes increasingly immunized, either by individual immunity following infection or by vaccination, and also as people die.

R_{e }is affected by the number of people with the infection and the number of susceptibles with whom infected people are in contact. People’s behaviour (e.g. social distancing) can also affect R_{e}.

The number of susceptibles falls as people die or become immunized by exposure. The sooner people recover or die, the smaller the value of R_{e} will be at any given time.

Unfortunately, the symbol R_{0 }is often used in publications when R_{e }is meant. This can be confusing.

### Which interventions can help reduce the R0?

There are lots of infection control measures experts can use to push this number down and “flatten the epidemiological curve”. A study in the Lancet earlier this month, for instance, estimated that travel restrictions in Wuhan caused R0 to drop from 2.35 to 1.05 after just one week.

According to a pre-print study from the London School of Hygiene and Tropical Medicine published at the start of April, the average number of people an individual comes into contact with each day dropped by 73 per cent since the UK’s lockdown began.

“This would be sufficient to reduce R0 from a value from 2.6 before the lockdown to 0.62 during the lockdown, indicating that physical distancing interventions are effective,” the study, which tracked over 1,300 adults and has not yet been peer reviewed, concluded. The government’s more recent estimates have since backed up this data.

### So if it is working, why are not out of lockdown yet?

The R0 is not the only figure the government is tracking. It is also making sure that the pandemic does not overwhelm the NHS, looking for a “sustained and constant” fall in death rates, ensuring there is enough personal protective equipment (PPE), and finally, being confident that any changes do not risk another peak.

Prime Minister Boris Johnson said the UK is close to achieving these aims – but the final test, preventing a second wave of coronavirus, comes back round to the R0 again.

On Monday, Professor Whitty said that it was the job of the government’s Scientific Advisory Group for Emergencies (Sage) to inform the government the effect on the R value of every decision taken on lifting the lockdown.

In Germany, where some lockdown measures were lifted once the R value made it to 0.7 – with some shops reopening, for example – R0 has crept back up to around 0.76, the head of the country’s Robert Koch Institute on said on Thursday.

As such, its government plans to keep many social distancing measures in place for a longer period than initially expected. It is likely that the UK, too, will have to walk a similar tightrope, lifting some restrictions while keeping a close eye on R0 and cases, and adapting its plan depending on what happens.

This could mean many more months before things go completely back to normal, or, as Scotland’s First Minister Nicola Sturgeon put it on Wednesday night: “People talk about lifting the lockdown, that is not going to be a flick of the switch moment – we’re going to have to be very careful, very slow, very gradual.

### Comparison of estimated values of R_{0} for a range of viruses, summarized from a variety of published sources.

The graph shows how variable estimated values of R_{0} can be from virus to virus. However, the numbers should be interpreted with caution, since the quality of data on which they were based will have varied from study to study, the data will have been taken at different times during the course of the spread of the virus (e.g. compare influenza in Spring and Autumn 1918), and the models used will have been different. It should also be clear from the graph, although the mortality data are not shown, that R_{0} is not related to the case fatality rate.

**Figure 4. **Estimated values of R_{0} in different viral infections, culled from a variety of published sources

**What is the R**_{0}** of SARS-CoV-2?**

_{0}

Estimates of the R_{0} of SARS-CoV-2 vary widely. Figure 5 shows nine estimates, all from studies in China or South Korea. The mean estimate of R_{0} is 2.63 (95% CI = 1.85,3.41). From a wider survey of 16 published estimates, the mean estimate is 2.65 (1.97, 3.09). In both cases, the range is 0.4-4.6.

**Figure 5. **Nine estimates of R_{0} in studies for which information was given about the time over which the measurements were made, from 1 December (day 1) to 6 March (day 97); the data are from studies in mainland China (red), Wuhan (black), Shenzhen (blue), and South Korea (green); the results show the large degree of variability in mean estimates, attributable to variations in the quality of the data and the models used; however, five of the results cluster around 2.6

This means that in order to protect us against future epidemics, herd immunity of around 62% will be needed, taking the mean estimate across all these studies. Taking the highest value (4.6), 78% immunization will be needed, and it would be wise to aim for at least that. When a test for neutralizing antibodies to SARS-CoV-2 and a vaccine are available, we shall be able to tell to what extent the population has been immunized by exposure and vaccination.

In the UK, one of the key findings that influenced the Government’s decision to start the lockdown was the model presented by the Imperial College COVID-19 response team. The model explores the effect of two strategies (a) suppression, by which interventions are instituted to bring R_{e} to below 1, and (b) mitigation, by which strategies are instituted to reduce the impact of the epidemic, but not interrupt viral transmission completely, thus reducing R_{e}, but not necessarily below 1. The model assumed an R_{0} of about 2.4 and predicted that, in the absence of any control measures, 81% of the UK population would become infected, with over half a million deaths. However, this number could be reduced to under 50,000 (and possibly under 6000 deaths) with a suite of suppression strategies in place. Little wonder, given these figures, that the UK government swiftly began the lockdown.

It is important to note that the value of R_{0} varies considerably in the models used in the COVID-19 pandemic. One systematic review reported that the mean of 29 reported values of R_{0} from 21 studies was estimated at 3.32 (2.81-3.82), with a range of 1.9 to 6.49; all the included studies were from China. In a statement on 23 January, 2020 about the outbreak of COVID-19 the World Health Organization (WHO) gave a preliminary R_{0} estimate of 1.4–2.5.

**Herd immunity**

Initial reports suggested that one of the UK Government’s strategies in tackling the pandemic was to allow the virus to spread within the community, in a controlled way, so that immunity, so-called herd immunity, could develop across the population. However, the health secretary, Matt Hancock, later said that this was not part of the UK response to the virus. In contrast, other countries, such as Sweden, have responded to the pandemic in ways that avoid full lockdown. The problem with leaving people to catch the infection spontaneously, leading to herd immunity, is that the death rate would increase as a result. For example, on 10 April, the number of confirmed cases in Sweden was 9685 with 870 deaths (9.0%), compared with Norway with 6219 confirmed cases and 108 deaths (1.7%) and Denmark with 5830 confirmed cases and 237 deaths (4.4%).

R_{0} predicts the extent of immunization that a population requires if herd immunity is to be achieved, the spread of the infection limited, and the population protected against future infection. To prevent sustained spread of the infection the proportion of the population that has to be immunized (P_{i}) has to be greater than 1 − 1/R_{0}. The relation between P_{i} and 1 – 1/R_{0} is shown in Figure 1.

For example, if R0 = 2, immunization needs to be achieved in 50% of the population. However, if R0 = 5 the proportion rises steeply, to 80%. Beyond that the rise is less steep; an increase in R0 to 10 increases the need for immunization to 90%. Measles has an R0 greater than 10, which is why immunization of a large proportion of the population is so important in preventing the disease.

Thus, if R0 is 10, a child with measles will infect 10 others if they are susceptible. When other children become immune to the infected child who encounters 10 children will not be able to infect them all; the number infected will depend on Re. When immunity is 90% or more the chances that the child will meet enough unimmunized children to pass on the disease falls to near zero, and the population is protected.

**Figure 1. **The relation between the basic reproduction number of a virus, R0, and the proportion of the population that needs to be immunized to achieve herd immunity; note the steep rise of the curve at values of R0 between 1 and 5; three examples are shown: R0 =2, proportion = 50%, R0 = 5, proportion = 80%; R0 = 10, proportion = 90%; the inset shows a linearization of the main graph, generated by plotting P against 1/R0

### References

1)

“When will it be over?”: An introduction to viral reproduction numbers, R0 and Re

2) What is the ‘R’ value and why is it so important for the easing of the coronavirus lockdown? In epidemiology, the basic reproduction value describes the average number of people an individual can expect to infect https://www.telegraph.co.uk/global-health/science-and-disease/what-r-value-mean-coronavirus-lockdown-uk/