key: cord-0868627-mgbvutrk
authors: Humphrey, L.; Thommes, E. W.; Fields, R.; Hakim, N.; Chit, A.; Cojocaru, M. G.
title: A path out of COVID-19 quarantine: an analysis of policy scenarios
date: 2020-04-29
journal: nan
DOI: 10.1101/2020.04.23.20077503
sha: 29de94efc5287cdc02b774615b20fbfe5fb816c8
doc_id: 868627
cord_uid: mgbvutrk

In this work we present an analysis of the two major strategies currently implemented around the world in the fight against COVID-19: Social distancing & shelter-in-place measures to protect the susceptible, and testing & contact-tracing to identify, isolate and treat the infected. The majority of countries have principally relied on the former; we consider the examples of Italy, Canada and the United States. By fitting a disease transmission model to daily case report data, we infer that in each of the three countries, the current level of national shutdown is equivalent to about half the population being under quarantine. We demonstrate that in the absence of other measures, scaling back social distancing in such a way as to prevent a second wave will take prohibitively long. In contrast, South Korea, a country that has managed to control and suppress its outbreak principally through mass testing and contact tracing, and has only instated a partial shutdown. For all four countries, we estimate the level of testing which would be required to allow a complete exit from shutdown and a full lifting of social distancing measures, without a resurgence of COVID-19. We find that a "brute-force" approach of untargeted universal testing requires an average testing rate of once every 36 to 48 hours for every individual, depending on the country. If testing is combined with contact tracing, and/or if tests are able to identify latent infection, then an average rate of once every 4 to 5 days is sufficient.

Korea and Italy, have clearly passed a (first) peak. With 23 few exceptions, countries have taken drastic social distancing 24 measures (e.g. (7-10)) in an effort to suppress the disease, or 25 at least prevent it from overwhelming a country's critical care 26 capacity. However, these measures have had a massive effect 27 on daily lives and economies throughout the world, and are 28 clearly unsustainable. It is thus critical to devise strategies 29 which will allow the phasing out of social distancing measures 30 while preventing a resurgence of outbreaks ((10),(11)). 31 In this work, we fit a disease transmission model to daily on social distancing via universal shelter-in-place measures. 37 We demonstrate that the rate at which these measures can be 38 eased while preventing future outbreaks is prohibitively slow. 39 We compare these scenarios to that of South Korea, a coun-40 try which has successfully controlled COVID-19 principally 41 through highly aggressive testing and contact tracing, while 42 maintaining a comparatively normal operating of society. We 43 show that an analogous strategy provides a feasible path back 44 to normalcy in Italy, Canada, the US, and by extension other 45 countries. 46 The structure of the paper is as follows: In Section 2 we 47 introduce our model main ideas, notation, and assumptions. 48 We follow in Section 3 with the presentation of results for Italy, 49 Canada, the US and South Korea, wherein we infer the net 50 effect to date of social distancing (in the three former) and 51

We demonstrate how current quarantine measures can be lifted after the current pandemic wave by large-scale, frequenttesting and contact tracing on the remaining susceptible populations. We present an analysis of the two major strategies currently implemented around the world against COVID-19: Social distancing & shelter-in-place measures to protect the susceptible, and testing & contact-tracing to identify, isolate and treat the infected. We find that a "brute-force" approach of untargeted universal testing requires an average testing rate of once every 36 to 48 hours for every individual, depending on the country. If testing is combined with contact tracing, and/or if tests are able to identify latent infection, then an average rate of once every 4 to 5 days is sufficient.

testing plus contact tracing (in the latter). We then evaluate scenarios for countries to phase out social distancing while with s = S N , e = E N , i = I N , r = R N , s + e + i + r = 1 and where β is the rate of effective contacts, 1/σ = T lat is the mean latent period (which may differ from the incubation period), and 1/γ = T inf is the mean duration of infectiousness, with both times having exponential distributions. We also have the auxiliary equation for the cumulative number of cases,

The daily incidence of cases on day i is then

In turn, a SIR model is similar to (1) 

In the SIR model, the growth factor ρ is given by

One then can express R0 in terms of ρ:

In a similar manner, in the SEIR model we have (13):

[6] 90 and by solving for β from Equation 6 we get:

[7]

In the limit as σ → ∞, the SEIR model reduces to the 93 SIR model, and accordingly, as can readily be shown by 94 L'Hospital's rule:

For COVID-19, as for other pandemics (e.g. SARS, MERS, 97 the 1918 Spanish flu), we can assume the entire population 98 to be initially susceptible. Therefore, in the early stages of 99 an outbreak, R ef f ≈ R0. We will also assume that infection 100 with COVID-19 confers subsequent immunity, which does not 101 wane significantly over the time horizon considered here. Thus, 102 whether they die or recover, an infected person is considered 103 removed from the pool of susceptibles. In the absence of a 104 vaccine or other control measures, 105 is the cumulative number of people infected at time t. 106 From Equation 2, assuming β to be given, we see that R ef f 107 can be decreased in two ways: by decreasing s at a rate higher 108 than that due to infection alone; or by increasing γ. The 109 former can be considered an abstraction of social distancing 110 measures, since these effectively take a part of the population 111 "out of circulation" as far as disease transmission is concerned. 112 The latter can be achieved by identifying and isolating infected 113 individuals early, thus cutting short T inf .

To explicitly depict the role of control measures, we adapt 115 the SEIR model to a pandemic setting by adding a Quarantined 116 (Q) and an isoLated (L) compartment. As before, we include 117 the auxiliary equation for C, the cumulative number of infected. 118 The resulting SQEIRL model, a variant of a model introduced 119 by Brauer (14) in the context of SARS, is described by:

where as in the standard SEIR model, β is the mean rate 128 of effective contacts, 1/σ = T lat is the mean latent period, 129 and 1/γ = T inf is the mean infectious period. Addition-130 ally, 1/α = Tquar is the mean time for susceptibles to be 131 quarantined-where "quarantined" constitutes an abstraction 132 of social distancing measures-and 1/ = Treturn is the mean 133 time for people to leave quarantine. Finally, 1/κ1 = T isol,lat 134 and 1/κ = T isol,inf are the mean times for the latent and 135 

150 and the rate of effective contacts is

We can also express the effective reproduction number in 153 terms of ρ S(Q)EIRL : 

Estimates of the serial interval of COVID-19 range from 3.95 184 to 6.6 days(16-19). We adopt a value of Tser = 5 days. The 185 latent period of the disease is not well constrained, but it can 186 be shown (Appendix A) that for a given value of Tser and ρ, the 187 maximum value of R0 is obtained when T lat = T inf = Tser/2. 188 We assume this "worst-case" scenario and let T lat = T inf = 189 2.5 days.

Italy. The initial linear phase corresponds to exponential growth, which subsequently turns over into sub-exponential growth. The exponential growth factor ρ and the corresponding doubling time are estimated via a regression fit to the initial phase. A value for R0 is calculated using Equation 9, and assuming σ = γ = (2.5d) −1 . For each country, we fit the S(Q)EIRL model solutions for daily incidence {inc model,1 , ..., inc model,n } to daily case reports. The model output is multiplied by a factor kf , where k is an estimate of the fraction of symptomatic cases reported, obtained using delay-adjusted case fatality rates (24) , and f is the fraction of cases which are symptomatic, estimated to be 0.5 from from a widespread testing campaign conducted in Iceland (25). We compute − log L, the normal negative log likelihood of the time series of observed daily incidences, {inc obs,1 , ..., inc obs,n }, given the model output, as a function of model parameters

where i0 is the initial number of infected and the qi are the frac- Linear and semi-log plots of daily incidence data of confirmed cases in Italy, together with maximum-likelihood model fit ("with measures, confirmed") and 95% CI envelope. Also shown is the inferred true number of infected, taking into account under-reporting and asymptomatic cases ("with measures, all"). Shown for comparison are the number of confirmed cases ("no measures, confirmed") and all cases ("no measures, all") expected to have occurred in the absence of countermeasures. Lower two panels: Cumulative incidence (top) and inferred weekby-week part of the susceptible population effectively quarantined (bottom).

It should again be emphasized that "percentage quaran-217 tined" represents an abstraction of the net effect of social dis-218 tancing measures, rather than a quantity to be taken literally. 219 Nevertheless, it provides a relatively interpretable measure 220 of the week-by-week impact of social distancing/shelter-in-221 place measures on the normal functioning of society. Across 222 the three countries, 40-70% of the susceptible population is 223 inferred to be effectively quarantined by the third week of 224 April. . 6 . U.S. daily COVID-19 data and model fit for incidence and cumulative incidence, and the inferred part of the population effectively quarantined; see caption of Figure 3 for details C. Long-term prospects with social distancing alone. As an 226 example, we evolve Italy forward in time. The currently-227 inferred level of effective quarantine, 70%, is kept in place 228 until 1 June 2020, by which time daily incidence is predicted 229 to have dropped to a level of several hundred cases per day. 230 From that point forward, we make monthly adjustments to 231 the effective quarantine level in order to maintain R ef f ≈ 1. 232 Results are shown in Figure 7 . This allows reducing the 233 quarantined population to about 60%. However, thereafter the 234 quarantined can only be released extremely slowly if R ef f is to 235 remain at 1. This is because, at a rate of several hundred cases 236 per day, only a very small fraction of the pool of susceptibles 237 is removed each day. Under this scenario, we predict that the 238 quarantined fraction of susceptibles in Italy would still need 239 to be about 55% even by the end of 2021. 

Korea. As we have seen in the previous section, South Korea ex-242 perienced a very similar early exponential growth in cases, and 243 hence has a similar inferred R0, as the other three countries. 244 However, its mitigation and control measures have differed 245 substantively. The country employed a rapid scale-up of test-246 ing, concurrent with contact tracing and isolating of infected 247 individuals. There are social distancing measures imposed, 248 but no shelter-in-place, which is the main difference in this 249 region. The population has not only agreed, but volunteered 250 to participate in a continual surveillance of contacts in order 251 to identify potential spread early.

In reproducing South Korea's daily case counts with the 253 S(Q)EIRL model, we accordingly allow for some quarantining 254 inate the rate at which exposed/infected people are removed. We present next pandemic forecasts under different test-281 ing and contact tracing rates, in the four countries under 282 consideration. In Figure 10 below, we clearly see that for shorter than the thresholds given in 284 Table 2 , a second pandemic wave is averted in each respective 285 country. This threshold is the least stringent in Italy, where 286 about 10% of the population (accounting for asymptomatic 287 Modeling studies unanimously show that, barring a pro-310 portion of asymptomatic cases so large that the majority of 311 people have already been infected, a second wave of disease 312 is inevitable if distancing measures are fully halted. We show 313 that a "slow-burn" approach of gradually relaxing measures 314 while avoiding subsequent outbreaks requires an infeasibly long 315 time; even by the end of 2021, such measures would still need 316 to be in force at a level equivalent to quarantining roughly 317 half of the population.

The example of South Korea has successfully demonstrated 319 (so far at least) an alternate hybrid strategy, in which a mas-320 sive nationwide program of testing and contact tracing allows 321 society to avoid a complete shutdown, albeit with extensive 322 protective measures in place. In this work, we have attempted 323 to quantify the level of testing which would be needed to 324 allow a country to make a near-complete return to a normal 325 functioning of its society. Among the countries considered 326 here, we estimate that a strategy of universal testing alone 327 would require an average testing frequency ranging from about 328 once every 48 hours (Italy) to once a day (the U.S.) for every 329 person. If testing is combined with contact-tracing, and/or 330 if testing is able to already detect infections in their latent 331 phase, then a testing rate ranging from once every 5 days 332 (Italy) to once every 3 days (the U.S.) would be sufficient. 333 These estimates assume a test with sensitivity at or near 100% 334 and immediate isolation once a subject tests positive. Though 335 reaching these targets would necessitate an undeniably enor-336 mous logistic effort, the prospect of an indefinitely-prolonged 337 global shutdown is more daunting still. Furthermore, South 338 Korea has demonstrated (and is continuing to do so) that such 339 an exercise is far from impossible. Assuming social distancing 340 at about a third the level of Italy, we infer an effective mean 341 isolation time well below 5 days from 1 April 2020 onward 342 in South Korea. Development and universal distribution of a 343 self-testing solution combined with further advances in mobile 344 device-based contact tracing would bring a testing-based exit 345 strategy from global quarantine even closer. Nevertheless, it 346 is vital not to rush into a relaxing of distancing measures. 347 First, testing capacity and strong sustainable supply chains 348 for delivery of tests and results must be built. Second, as 349 illustrated in Figures 10 to 13 , if a change in control strategy 350 causes R ef f to exceed 1, how quickly a second wave builds 351 depends on the number of cases at that time. South Korea 352 D R A F T has few cases, so (slightly) exceeding R ef f = 1 would result in a gradual climb of cases, whereas doing so in Canada, which 354 in the near term is predicted to have a significantly higher 

In this case, we conduct a similar computation as in (13) , 410 but considering the 4 dimensional system of equations for 411 s, e, i, l leads us to the Jacobian of the S(Q)EIRL:

If computed at the disease free equilibrium (s, 0, 0, 0) we further 414 obtain:

Again we note that the linearized equations for s and l are 417 decoupled from the equations of e and i, thus, to get informa-418 tion on the growth rate of the infected compartment, let us 419 try to solve the linearized reduced system in (e, i) based on 420 the reduced Jacobian:

Its characteristic equation is:

The eigenvalues of this matrix can be computed to be

ρ1,2 = −(σ + κ1 + γ + κ) ± ((σ + κ1) − (γ + κ)) 2 + 4σβs 2 [13] 429 We first note that ((σ + κ1) − (γ + κ)) 2 + 4σβs > 0, given 430 all parameters are positive. This implies that ρ1 = ρ2 ∈ R 431 and clearly ρ2 < 0. We check whether ρ1 > 0 by looking at 432 ((σ + κ1) − (γ + κ)) 2 + 4λβs > σ + κ1 with no exponential growth in infected) which will be outlined 449 in detail in the next section.

Continuing as in (13), we express β as a function of 451s , σ, γ, κ, κ1 from (13) and we get:

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All authors contributed to the paper equally, from inception, model implementation to data fitting and forecast.