By the end of March 2020, over 100 countries around the world had in place either a full or partial lockdown in an attempt to slow the spread of Covid-19. The ambition has been to ‘flatten the curve’. The ‘curve’ is the number of infections over time, and ‘flattening’ it spreads those cases over a longer period, avoiding a situation where the number of simultaneous cases exceeds the capacity of our hospitals.

In many countries, these lockdowns have proved to be partially successful, and several governments have now begun to relax their initial lockdowns. However, any plan for doing so must involve a *workable testing strategy*: who will be tested for Covid-19, and how often? As others have noted, the economic benefit of a faster recovery would be measured in trillions of dollars, which would easily justify spending billions on testing. However, it is not yet clear what a feasible and effective testing strategy might look like.

## The purpose of a testing strategy

The reproduction number *R* in an epidemic is the expected number of cases directly generated by one infected person. The *basic *reproduction number, *R _{0}*, is the initial value of

*R*. An epidemic can only be controlled if the value of

*R*is brought below one. When that happens, each infected person infects less than one new person, and the epidemic dies out.

The aim of any testing strategy for Covid-19 is to bring the *effective *reproduction number, *R’*, below one, by testing people for infection and isolating those who are found to be infected. As a result of doing this, *R’ *will fall below *R _{0}*, because a fraction of the population will be tested each day and those found to be infectious will be isolated. Clearly, doing this becomes essential if lockdowns are to be relaxed, since, up until now, lockdowns have been the only really reliable way of bringing

*R’*down.

## An unworkable strategy: Universal random testing

One proposed testing strategy is ‘universal random testing’. In this strategy, testing would be ‘universal’ because tests will be carried out on the entire population, and ‘random’, because a fraction of the population will be randomly selected each day for testing.

Professor Paul Romer, who won the Nobel Prize in economics in 2018, has made a valuable contribution by proposing such a strategy (Romer 2020). Drawing on a simple model, he argues that randomly testing 7% of the population per day for evidence of infection, and then isolating those who test positive, would be sufficient to bring *R’* down to 0.75 and thus to halt the epidemic. With a population of 300 million in the US, testing on this scale would require about 20 million tests a day. In the UK, with a 60 million population, it would require about 8.5 million tests a day.

However, we argue that Romer’s conclusion rests on a mistaken calculation, and that it also ignores the difficulties caused by asymptomatic cases. We show that, if half of all cases are asymptomatic, then more than 20% of the population would need to be tested every day to reduce *R’* to the desired value of 0.75. We derive this result using a corrected method for calculating the impact of an infectious person on others, when testing and isolation is taking place. Our method allows for the fact that, although self-isolation will help to deal with the problem, those who are asymptomatic will go on infecting additional people. It is this factor which makes particularly difficult to devise a testing strategy.

Clearly, carrying out tests every day on at least 20% of all the people who are not in isolation is not a workable strategy. This is because it would require everybody in the population to be tested about every five days. This is just not feasible. Furthermore, we believe that even attempting to implement such a testing strategy would be a waste of valuable testing resources. It will be possible to do better than this.

The general results from our calculations are shown in Figure 1. This figure plots *R’* as a function of the share of the population tested. This figure assumes that anyone who tests positive is isolated immediately that day. It also assumes that all those who are infected are asymptomatic for five days (and are subject to random testing during that time). After five days, a proportion ‘alpha’ (*α*) of those infected people display symptoms and self-isolate even if they have not been tested. In Iceland this number was found to be roughly 50%, but we will consider a range of possibilities. In Figure 1, the upper schedule, where *α* = 0%, assumes that nobody displays symptoms after five days and so nobody self-isolates. The lower schedule, where *α* = 100%, assumes that everyone displays symptoms after day five and self-isolates.

When *α* = 50%, half of those infected display symptoms after five days (which appears to have been the case in Iceland) and then self-isolate. It is apparent from Figure 1 that the proportion of the population tested must be as high as 21% to ensure that *R’* = 0.75. This number is very different from that put forward by Romer, who claimed that such a result could be achieved if as few as 7% of the population were tested each day.

Figure 1 also suggests that we should draw the following pessimistic conclusion: for random universal testing to be workable – testing, say, less than 10% of the population per day – policymakers would need to be confident of several factors. First, that nearly all infected patients are symptomatic and self-isolate (so that outcomes are close to those shown by the bottom line in Figure 1). Second, that the tests are sufficiently effective (i.e. that they capture at least 70% of the infected cases, which is what we have assumed in drawing this figure). Third, that testing is conducted quickly and that people are isolated on the day that the test is carried out (which is what we have assumed here). And fourthly, that whole households are isolated where any member is infected (since otherwise self-isolation would not be effective).

**Figure 1** Over 20% of the population needs to be tested to stop the infection if no-one self isolates when they show symptoms

We do not feel that all these conditions can be met, given our current state of knowledge about the virus – particularly the first condition. Thus, we think it is right to be very cautious about universal random testing as a workable testing strategy.

## A workable strategy: Stratified periodic testing

We argue instead that ‘stratified periodic testing’ would be a workable testing strategy. According to this strategy, testing at rates of well above 20% would be carried out on individuals within carefully selected groups and we claim that doing this could eliminate the spread of Covid-19 in those groups by bringing their effective reproduction numbers, *R’*, below one.

Testing would be ‘stratified’ because it is performed only on particular groups, based on their likelihood of infecting others. This likelihood could be deduced from their occupation, geography, and other factors. And testing would be ‘periodic’ because the same people would be tested at regular intervals. (Periodic testing of 20% of a group means that, on days one to five, a different fifth of the group is tested each day; on day six, the first fifth is tested again, and so on.)

Testing health workers would be the most obvious and urgent part of such a strategy. But other groups whose members are likely to infect others should also be tested very frequently. These include those who make contact with large numbers of people (such as bus drivers or supermarket employees), those who are exposed to people who are particularly susceptible to the infection (e.g. prison warders and care workers), and those who cannot work from home and have been given permission to work despite the lockdown (e.g. those involved in construction and manufacturing).

In our paper (Cleevely et al. 2020), we provide a simple formula that could be used by those who implement our testing strategy. Use of this formula would enable officials to calculate the minimum testing rate required for any group, in order to prevent the spread of the epidemic in that group, and thus to bring it under control within that group. Such a formula could also be used to calculate the minimum testing rate required in particular parts of the country, and so allow policymakers to respond to different patterns of infectivity in different places. These features would be important because some parts of the population have a much lower infection reproduction rate than others, and therefore need to be tested much less regularly, if at all.

Adopting the targeted approach which we propose would clearly reduce the spread of Covid-19 within the groups which were singled out for frequent testing. But, more than this, it would also enable the general lockdown to be eased more rapidly. Lower-risk groups might be allowed to return to work earlier than would otherwise have been possible, simply because the problem of spread in high-risk groups would have been brought under control.

Figure 2 shows the periodic testing rates that would be required to obtain a range of different values for *R’* in different groups, for various values of *α* (i.e. the proportion of those who are infectious and self-isolate after five days).

**Figure 2** With periodic testing the proportion of the population checked each day can vary depending on how likely it is that infected people self-isolate

The calculations underlying this figure, in contrast with those underlying Figure 1, make the more realistic assumption that those testing positive only self-isolate the day after the test is carried out. It is clear from this figure that the testing rate is very sensitive to *α*, the proportion of the infected population who self-isolate after five days. For testing rates that are below 20% per day (i.e. less frequent than once every five days, the period in which a person is assumed to be asymptomatic) we can clearly see the huge risk which comes from a low value of α (either because a large share of the population who are infectious remain asymptomatic, or because those who are symptomatic nevertheless refuse to self-isolate). The effective reproduction rate, *R’*, is suddenly much larger, at any testing rate, the smaller is the value of *α*. To get the value of *R’* down to, say, 0.75 would consequently require much more frequent testing of those belonging to the group in question.

## Benefits of stratified periodic testing

Stratified periodic testing has three critical benefits. To begin with, the tests do not have to be perfect. The British government has suggested that “no test is better than an unreliable test”. But this is not right. Like Paul Romer, we have assumed in our calculations a false negative of 0.3 (i.e. 30% of those who are infectious test negative). Yet, it is still possible to bring *R’* below one within the group with unreliable tests like these.

Secondly, periodic testing (the testing of the same person, periodically) is far more efficient than random testing (a random choice of who is tested in each period). Intuitively, random testing of say 20% of a group per day wastes resources. On any day, some of those tested will have been tested the previous day, whilst others who have not been tested for a long time will still not be tested. By contrast, periodic testing of 20% of a group means that each person is tested every five days. In our paper, we show that this increases the efficiency of any given number of tests by approximately 35%, at no extra cost.

Finally, it is feasible to run these infection tests (i.e. do you have the infection?) alongside antibody tests (i.e. *did *you have the infection?) to reduce the spread of Covid-19 even further.

A practical combined strategy could proceed as follows.

1. Immediately and frequently perform infection tests on groups and areas with a high reproduction number and immediately isolate those found to be positive. Track and test the contacts of those who test positive, as these would now have a higher probability of also testing positive.

2. Self-isolate anyone developing symptoms for a minimum of seven days. These people would not need to be tested unless medically necessary. Trace and test their contacts.

3. If there are enough tests, then test people at the *end *of their isolation period, to show that they are clear of the virus before they are allowed to come out of isolation.

4. Provide widespread home-kit antibody testing to enable everyone to see if they have had the virus. These would be one-off tests that would not need to be repeated.

5. Introduce a system to track people with immunity who would be allowed to circulate freely if they had i) a positive antibody test, or ii) a positive active infection test more than a specified number of days ago, or iii) a negative active infection test after their symptoms resolved.

This strategy would also need to involve some additional random testing of groups in the population, and some random testing of samples taken from the whole population, for *informational* purposes. Such informational testing would involve testing small samples of those involved, not for purposes of isolation, but instead in order to guide operation of the strategy outlined above. Such informational testing would serve two purposes.

First, some random testing, taking small samples from the particular groups in which the basic reproduction number, *R _{0}*, is already known to be high, will make it possible to track the situation in these groups. These are the groups which need very frequent testing of everyone in the group, for the reasons which we have been discussing in this paper. The amount of testing required for the group will depend on the value of

*R*for that group.

_{0}Second, testing of small samples of wider groups in the whole population will also be needed, in order to identify new groups where the basic reproduction number *R _{0}* is high. This may happen for idiosyncratic reasons that are hard to anticipate. That is why such testing of widespread groups is needed. Once identified, such groups will then also need very frequent testing of everyone in the group. The amount of testing required for any newly identified group will – of course – also depend on the value of

*R*which has been identified for that group.

_{0}The accuracy of this testing for informational reasons will be determined by the sample size, rather than population size. The samples required for these informational purposes will be *very *small relative to the size of the whole population.

The strategy outlined above would make possible an efficient use of testing resources. And it would achieve two important things at once. Stratified periodic infection testing would slow the spread of Covid-19 in key groups, and antibody testing – for the entire population – would separate out the immune population and allow them to return to work.

## Conclusion

In conclusion, we have demonstrated that ‘universal random testing’ is not likely to be an effective tool for simultaneously reducing the spread of Covid-19 and getting the economy back to work. With that strategy for infection testing, we have found that more than 21% of the population would need to be tested every day if the aim was to reduce the Covid-19 reproduction rate, *R’*, to 0.75. That is much more than the 7% testing rate which Paul Romer has claimed would be sufficient.

Clearly, carrying out tests every day on at least 21% of the whole population, other than those in isolation, is not a workable strategy. It would imply that everybody – other than those in isolation be tested at least every five days. Even if – as our calculations suggest – such a strategy would be effective, it is just not feasible, at least at present. Attempting such a testing strategy would therefore be a waste of currently scarce and valuable testing resources.

Instead, we argue that ‘stratified periodic testing’ is a realistic strategy for simultaneously reducing infection spread and getting back to work. Such a strategy for infection testing would have three important benefits: (i) it would not require tests to be perfect, (ii) it would make use of the fact that periodic testing is far more efficient than random testing, and (iii) it would allow antibody testing to be run alongside stratified periodic infection testing in a powerful way (as and when the former kind of test becomes widely available).

## References

Cleevely, M, D Susskind, D Vines, L Vines and S Wills (2020), “A Workable Strategy for Covid-19 Testing: Stratified Periodic Testing rather than Universal Random Testing”, Covid Economics 8: 44-70.

Romer, P (2020), “How to re-start the economy after COVID-19”, Online Lecture for the Bendheim Center for Finance at Princeton University, 3 April.