key: cord-335679-dpssd1ha
authors: Rawson, T.; Brewer, T.; Veltcheva, D.; Huntingford, C.; Bonsall, M. B.
title: How and when to end the COVID-19 lockdown: an optimisation approach
date: 2020-05-02
journal: nan
DOI: 10.1101/2020.04.29.20084517
sha: 
doc_id: 335679
cord_uid: dpssd1ha

Countries around the world are in a state of lockdown to help limit the spread of SARS-CoV-2. However, as the number of new daily confirmed cases begins to decrease, governments must decide how to release their populations from quarantine as efficiently as possible without overwhelming their health services. We applied an optimal control framework to an adapted Susceptible-Exposure-Infection-Recovery (SEIR) model framework to investigate the efficacy of two potential lockdown release strategies, focusing on the UK population as a test case. To limit recurrent spread, we find that ending quarantine for the entire population simultaneously is a high-risk strategy, and that a gradual re-integration approach would be more reliable. Furthermore, to increase the number of people that can be first released, lockdown should not be ended until the number of new daily confirmed cases reaches a sufficiently low threshold. We model a gradual release strategy by allowing different fractions of those in lockdown to re-enter the working non-quarantined population. Mathematical optimisation methods, combined with our adapted SEIR model, determine how to maximise those working while preventing the health service from being overwhelmed. The optimal strategy is broadly found to be to release approximately half the population two-to-four weeks from the end of an initial infection peak, then wait another three-to-four months to allow for a second peak before releasing everyone else. We also modelled an ''on-off'' strategy, of releasing everyone, but re-establishing lockdown if infections become too high. We conclude that the worst-case scenario of a gradual release is more manageable than the worst-case scenario of an on-off strategy, and caution against lockdown-release strategies based on a threshold-dependent on-off mechanism. The two quantities most critical in determining the optimal solution are transmission rate and the recovery rate, where the latter is defined as the fraction of infected people in any given day that then become classed as recovered. We suggest that the accurate identification of these values is of particular importance to the ongoing monitoring of the pandemic.

. Schematic diagram depicting the movement of individuals through the SEIR network. The function u describes the action of the strategy employed to end lockdown, as people are released from the quarantined group. The arrows linking the two groups operate in both directions, to allow for any "on-off" strategy where people are returned to quarantine. 106 The lowercase Greek letters in equations (1) -(8) represent our rate parameters. Firstly, β represents the transmission rate . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 2, 2020. . https://doi.org/10.1101/2020.04.29.20084517 doi: medRxiv preprint informing policy. For this reason our sensitivity analyses (below) also consider transmission rates up to twice as high as these 114 values. Note that we consider the population of both I and I Q to impact the spread of disease, as the quarantined group are still 115 assumed to occasionally mix with the population (for instance, when leaving their homes to shop for essential items). The 116 parameter c is a scalar between 0 and 1 that captures how effective the self-isolation (i.e. lockdown) measures enforced are in 117 reducing the the rate of SARS-CoV-2 transmission. 118 119 µ represents the natural, background death rate of the population regardless of the impact of COVID-19, and can have 120 important implications for the strength of herd-immunity effects on disease dynamics, as this is the only mechanism in our 121 model through which the recovered population is reduced. The parameter α represents the rate of death directly attributed to 122 SARS-CoV-2. While the mortality rate of SARS-CoV-2 has been demonstrated to vary substantially between age classes 10-12 , 123 in its current form our model does not incorporate age-structure and we therefore adopt an age-invariant mortality rate.

The parameter σ represents the incubation rate. The exposed population classes, E/E Q , capture the effect of the lag be-126 tween people becoming infected (and incubating the disease for several days) and becoming infectious. Understanding the size 127 of this effect is of great importance when assessing strategies in which a second lockdown may be enforced because efforts to 128 monitor the subsequent spread of infection must consider the upcoming, but lagged, threat posed by the exposed class. Lastly, γ 129 represents the recovery rate and describes how long individuals remain infectious. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 2, 2020. The parsimonious nature of our model was chosen to enhance the ease of interpretation of our results and, most impor-148 tantly, to enable the model to be quickly adapted to non-UK populations. Different countries currently provide varying levels of 149 epidemiological detail in their reporting of COVID-19 cases. By reducing the number of classes and parameters considered, our 150 model is amenable to a wider range of countries and scenarios than the more specific model structures currently published 9, 17 .

The result of this modelling choice is that our system captures the broad-scale dynamics of the disease resulting from different 152 lockdown exit-strategies rather than making accurate predictions of the number of infected individuals, which will require 153 continuous, data-driven adaptations applied to our framework.

The primary challenge facing policy makers currently is in devising how to return the population to work most safely, ending 156 the lockdown and its detrimental consequences on the economy. The objective is to release as many people from lockdown, as release which, even if gradual, will still be managed with distinct groups of people leaving at different times. Our primary 179 results presented in the following section are instead derived from an iterative process in which multiple different release times 180 and portions of the population are trialled across various ranges, with the optimal choice being that which maximises our 181 objective function. All code used to perform these optimal control approaches was performed in Matlab, and is available at: 182 https://osf.io/hrt2k/.

Definition Initial Conditions and definition of N and N Q Reference Source

Non-quarantined exposed. A gradual release strategy aims to end the lockdown of the the public from quarantine through multiple staggered releases.

Expressed mathematically, we seek to release M 1 people at time T 1 , while ensuring that I + I Q < I thresh at all times. We . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 2, 2020. . https://doi.org/10.1101/2020.04.29.20084517 doi: medRxiv preprint Therefore, the optimum choice of M 1 and T 1 are those which maximise C 1 . In short, this approach calculates how to release as 192 many individuals as possible, as early as possible, without breaking the infection carrying capacity. After this optimum solution 193 is found, a second release of M 2 people at time T 2 can be similarly calculated after the first release, if people still remain in 194 quarantine.

To calculate these outputs, we used ode45, a fourth-order Runge-Kutta solver in Matlab, to solve the system of equa-197 tions (1)-(8) using the initial conditions in Table 1 for t from 0 to T 1 . At this point we subtracted M 1 individuals proportionally 198 from S Q , E Q , I Q and R Q and added these to S, E, I and R. The system was then solved again from these new points for t from T 1 199 to 400 days. To allow understanding of the effect of different values of some of the parameters presented in Table 1 , we operate Table  1 . . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 2, 2020. Table 1 .

Figures 2 and 3 are example simulations, to illustrate general model behavior, but are not optimal solutions. We now consider outcome considered is the objective function, C, for our optimum strategy. Defined formally, the total sensitivity index for

, where Y is the model outcome monitored, and X i is the parameter considered. 

To determine the optimal timings for an "on-off" lockdown release strategy, both the times at which quarantine was ended, T off i , 258 and the times at which it was reinstated, T on i , were iterated on a mesh of 500 evenly spaced points across a timespan of 0 to 400.

Once one optimum release pair was found, the process was repeated up to two further times to identify subsequent optimum . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 2, 2020. . Here we have investigated the optimal release of individuals from a state of lockdown. The primary conclusion of our work 282 is that a gradual release strategy is far preferable to an on-off release strategy. We conclude this from the finding that a . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 2, 2020. . https://doi.org/10.1101/2020.04.29.20084517 doi: medRxiv preprint of time. Any decision to begin easing lockdown measures will require constant monitoring and a high-level of population 285 testing to track the likely rise towards a second-peak of infections. We show that employing a gradual release strategy, where 286 groups of the population are slowly released from quarantine sequentially, will slow the arrival of any subsequent infection 287 peaks compared to an on-off strategy, where lockdown is ended for all individuals imminently and reinstated when subsequent 288 infections begin to increase. In all considered instances (i.e. parameter variations), it will not be possible to end lockdown 289 for the entire population for any longer than two weeks, as the number of infected individuals is then expected to quickly 290 overwhelm the health service following such a release. By ensuring that the increase in the number of infected individuals is as 291 slow as possible, this will enable health officials to monitor more accurately the evolving situation, and provide more time to 292 respond to unexpected increases in the number of infected individuals. We note that our approach does not consider the ethical 293 responsibilities that will also impact any policy decision. If enough hospital provision was available, many more people can 294 return to employment, but we recognise this will result in increased risk of further mortalities. As many governments state 295 however, a functioning economy is more able to provide health provision to those with non-COVID19 life-threatening illness.

For a gradual release strategy, our simulations broadly suggest that a large section of the population should be released 298 from lockdown initially, after the first peak of infections has fully passed. The rest of the population may then be released three 299 to four months later following a likely second peak in infections. Again, in a general context, it is optimal to wait for one-to-two 300 weeks after the end of an infection peak before releasing any of the population from lockdown. While it is desirable to return 301 the population to work as early as possible, our optimal calculation states that this one-to-two week "wait" period is crucial in 302 ensuring that the number of infected individuals is as low as possible when ending any lockdown measures, to reduce the growth 303 of new cases. After this sufficient, cautious, wait period has ended, people should then be released from quarantine, with the 304 knowledge that as many as 1 in 100 of them (under the worst-case scenario) may require critical care 10 in the coming months. 305 It is expected that a second peak in infections may be observed one to two months after this release date, and that the remaining 306 population in quarantine should remain so until, once again, several weeks of low newly infected cases daily have been observed.

In conclusion, using an optimal control methodology, we have shown that a gradual staggered release of individuals out 362 of lockdown is recommended to ensure that health systems are not overwhelmed by a surge in infected individuals. It has been 363 well observed that older individuals are more likely to require critical care as a result of COVID-19 10 . Although our analysis 364 does not as yet differentiate by age who should be in any partial lockdown releases, this does indicate that, potentially, the 365 younger population could be the first to be released from lockdown. This would further ease any subsequent strain on the 366 health system, and potentially further bolster a herd-immunity effect. Similarly, our analysis does not model the capability of 367 businesses and individuals who have the infrastructure and availability to continue to work remotely.

The ongoing threat of COVID-19 will require continual monitoring and study in the coming months. It is important to 370 ensure that infections are kept to a minimum, and that the government and relevant services are given enough time to prepare for 371 increases in infections. The findings of this study stress that gradual and cautious action must be taken when easing lockdown 372 measures, to save resources, and lives, while adding to the evidence base of possible routes out of lockdown. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 2, 2020. . https://doi.org/10.1101/2020.04.29.20084517 doi: medRxiv preprint

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