How and When to End the COVID-19 Lockdown: An Optimization Approach

Abstract

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 optimization methods, combined with our adapted SEIR model, determine how to maximize those working while preventing the health service from being overwhelmed. The optimal strategy is broadly found to be to release approximately half the population 2-4 weeks from the end of an initial infection peak, then wait another 3-4 months to allow for a second peak before releasing everyone else. We also modeled 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.

Keywords: COVID-19; SARS-CoV-2; epidemiology; mathematical model; optimization; quarantine.

Figure 1

Schematic diagram depicting the movement of individuals through the SEIR network. The function u(t) 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.

Figure 2

Example of a gradual release from quarantine. Here, 20 million people are moved out of quarantine at t = 80 days, followed by the remaining population at t = 200 days. Variables E_Q and E are not plotted. Model parameters are those of Table 1. (A) Quarantined population. (B) Non-quarantined population.

Figure 3

Example of an “on-off” release from quarantine. Here quarantine is ended at t = 50 days, and then reinstated at t = 80 days. Quarantine then ends again at t = 150 days. Variables E_Q and E are not plotted. Parameter values are those of Table 1. (A) Quarantined population. (B) Non-quarantined population.

Figure 4

Optimum gradual release strategies for a range of different values of I_thresh (infection threshold), β (transmission rate) and c (lockdown effectiveness), as marked. Plots (A–C) show the total quarantined population, displaying when releases from quarantine are made by the instantaneous decreases. Plots (D–F) depict the associated total infected population (I + I_Q) associated with each optimum release strategy.

Figure 5

Optimum on-off release strategies for a range of different values of I_thresh (infection threshold), β (transmission rate) and c (lockdown effectiveness), as marked. Plots (A–C) show the total quarantined population, displaying when releases and re-entry to quarantine are made. Plots (D–F) depict the associated total infected population (I + I_Q) associated with each optimum release strategy.