How many tests do we need in the US?

As the US contemplates a return to business as usual on May 1st, officials everywhere face the daunting question of how to navigate a safe exit from lockdown. With no restrictions in place, health services would soon be overwhelmed. Yet the social and economic costs of widespread lockdowns have been ruinous so far and continue to grow. Clearly, some degree of compromise is needed.

Experts agree that frequent testing of the population will be essential. But how many tests is enough? Harvard’s Jha, Tsai and Jacobson have estimated a figure of 500,000 tests per day which is significantly higher than the 150,000/day being currently carried out. Here we go over this question more carefully and find that the answer depends crucially on three factors:

1. the post-lockdown rate of disease spread, captured by R

2. the fraction of new cases that actually get identified for contact tracing, and

3. the efficacy of contact tracing

The effective reproduction number R can be kept down by continued, voluntary and regulated, social distancing and contact tracing can be turbo-charged by using privacy protecting smartphone app based technology being rapidly developed in the United States.

So how many tests are needed on May 1st?

Assuming perfect and fast contact tracing starting May 1st, we find that the number can vary from 300,000-1,500,000 tests per day as we move from R=2 to R=3. At this level of testing the epidemic will continue to recede and ``digital herd immunity'' will be reached in roughly a month. In other words the actual R will be driven below 1.

Our estimates assume that each individual needs to be tested only once and that each test is 100% accurate. We also neglect the additional tests that arise due to ``influenza-like illness'' other than COVID-19. Finally, we assume that contact networks do not overlap, in the sense that a given infection cannot be traced to more than one source. A mathematical derivation for specialists is included in the pdf below but first we elaborate on this bottom line.

We start, like Jha, Tsai and Jacobson from a reasonable estimate of around 60,000 new US cases on May 1st. In line with current pessimistic estimates we assume that only 50% of these cases will be symptomatic. So at best, we have 30,000 symptomatic cases to act upon. Existing estimates for the required testing rate, with contact tracing in place, assume that tracing around N=10 contacts of these symptomatic infections, say 300,000 infections, is enough to stop further spread. In fact, the challenge on May 1 is that R will jump from its present value R<1 to a value R>1. The adequacy of testing capacity will depend on what this new value is:

• If the new R, absent testing, is less than 2, then testing symptomatic people and their contacts alone will bring the effective reproduction number to R/2 < 1 (assuming perfect, instant contact tracing). In this case, N_T = 300,000 tests per day is a reasonable estimate, and the number of new cases will continue declining as the effective R is now <1.

• If the new R, absent testing, is greater than 2, then it is not sufficient to test forwards (i.e. only the contacts encountered during the pre-symptomatic infectious period). One needs to test prior contacts in search of source asymptomatic transmitters, which on May 1st will be also be around 30,000. Such “backwards testing” is much more costly in terms of contacts traced; instead of testing N=10 contacts per case, we need to test N x (R+2) > 40 contacts per case, in the worst case. This cost is illustrated in figure below, and was not considered in previous analyses. Taking this effect into account, for a typical pre-lockdown reproduction ratio R=3 our analysis predicts N_T = 1,500,000.

A few comments on the estimate

1. Our estimates of N_T = 300,000-1,500,000 tests per day are large numbers and with less than perfect contact tracing and other frictions they would need to be somewhat larger. The numbers will be towards the lower end if prudent levels of continued social distancing keep the untested and untraced ``naive'' R below 2, contact tracing is highly efficient and all 50% of the infections which involve significant symptoms are handled by the testing and tracing system.

2. It seems to us that highly efficient contact tracing more or less requires using smartphone apps. This ideas has been gaining ground lately, culminating in the joint Apple-Google commitment to build contact tracing capabilities into their phones. The basic idea is that when enough people sign up to an app-based contact-tracing scheme, information about new cases spreads faster than the virus itself, and epidemics can no longer spread. This is “digital herd immunity”, a notion that we introduced in a recent paper. The point is that if everyone were running a contact-tracing app by the time May 1st rolls around, we could prevent future epidemics by testing only people with symptoms, and their immediate contacts.

3. While our numbers are large, they are much smaller than the most ambitious numbers, such as those from economist Paul Romer. In his highly secure ``Romer equilibrium” every single person in the US would get tested roughly once per fortnight. This would keep the disease in firm check. But this would require testing around 30 million people per day. Compared with today’s rate of around 150,000 tests per day, this goal seems completely out of reach anytime soon. By contrast digital technology can be scaled up extremely rapidly.

4. At the other extreme, research out of Switzerland shows that if we just want to monitor how an epidemic is spreading in a relatively homogeneous country, we need to be running around 15,000 tests per day – this extends standard polling methods to measure gradients and not just levels. Scaling this up to the US, it looks like today’s testing rate of 150,000 per day is good enough to track the epidemic accurately provided it is devoted solely to this aim. But this would do nothing to prevent it from spreading.

5. Finally, a third strategy, which we might call the “Swedish” approach, would be to use testing to protect health-care workers and vulnerable members of society, say those over 70 and with co-morbid conditions. For the latter group this would mean regular testing of those who come into close contact with them. Very roughly this might gain us a factor 10-20 over the Romer number and thus require 1.5-3 million tests per day. It would leave the rest of the population unmonitored and let the epidemic take its course at a rate set by social distancing. Perhaps in as little as 2-3 weeks we will have a pretty good idea of how this can work from the Swedish experience---for now this is not on the table.