Edited by: Zisis Kozlakidis, International Agency For Research On Cancer (IARC), France
Reviewed by: Gui-Quan Sun, North University of China, China; Sandro Rolesu, Istituto Zooprofilattico Sperimentale della Sardegna (IZS), Italy
This article was submitted to Infectious Diseases – Surveillance, Prevention and Treatment, a section of the journal Frontiers in Public Health
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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.
Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a novel coronavirus that has provoked the global pandemic of COVID-19. First reported in the city of Wuhan, China, its emergence quickly triggered a ‘lockdown’ within Wuhan and the surrounding cities (
Containment of the virus has proven challenging. Although some patients will require intensive care, others have unreported mild symptoms, with as many as 17.9% of infected individuals possibly being asymptomatic (
Here we consider a two-way balance which aims to (i) maximize the number of people able to work outside of lockdown, while (ii) ensuring that the number of people with COVID-19 requiring medical help at no point crosses a threshold beyond which hospitals are unable to cope. As an immediate termination of lockdown for all is likely to trigger a surge in infections, a graded easing of lockdown restrictions is likely required. The focus of our analysis is to understand the optimal pathway by which to release people as safely as possible back into a general and growing non-quarantined set of workers.
To understand how to restart the economy yet avoid the saturation of health services, we present decision-making as a problem in optimal control. To determine an optimal solution requires two definitions. The first is a system of process-based differential equations whose boundary conditions or other attributes can be varied by policy decisions. The second definition is an objective function metric, which depends on the balance and extent to which our two conditions are fulfilled. The aim is to solve the differential equations, and find decisions affecting their boundary conditions that are optimal and maximize the objective function. Our equation set is based on a standard Susceptible, Exposed, Infected, Recovered (SEIR) model framework (
SEIR-based equations are solved for non-quarantine and quarantine groups, connected by modeled release strategies from lockdown. That is, we allow different fractions of the quarantined group to move into the non-quarantined group, and at different times. For each potential strategy of movement between the two groups, an objective function is calculated—some metric describing the desirability of such a strategy. This is high when many people are removed from quarantine, as they are available to work—a desirable outcome. However, its value switches to a near-infinite negative should the health service threshold be crossed due to high infection numbers. Our model calculates the highest possible objective function (the optimal strategy) and returns the number of release dates, their time of occurrence, and the number of people at each time. For comparison, we perform parallel simulations, where we release all in quarantine to the non-quarantine pool, but allow the return to quarantine later if necessary, should infections risk exceeding the capacity of the health services. We herein refer to this as a lockdown “on-off release” strategy, and again find optimal timings and number of releases.
No mathematical model, especially for something as complicated as virus transmission and human behaviors, can make predictions accurate to within a small margin of error. However, models are especially useful in two circumstances, and that we exploit. First, although simulations may lack absolute precision, predictions will have some level of robustness. Such predictions give strong indications of expected responses to a range of different boundary conditions, i.e., alternative release scenarios. Numerical model flexibility and speed of operation enables “what if?” questions to be asked of alternative forms of graded lockdown release. Second, by repeated operation of a model, it is possible to scan across ranges of parameter values. After governments start to release people, changes to infection levels can be compared against ensembles of simulations with perturbed parameters. Data-model comparison allows selection of the most appropriate parameter value; an approach sometimes referred to as “adaptive learning.” The trajectory for that value becomes a more reliable forecast for the days and weeks beyond the available data. Evidence this approach is feasible is illustrated in data of infections in countries before and during a lockdown. Although there is substantial geographic variation, all curves have similar forms, amenable to parameterization. Indeed, politicians have frequently described a mathematical functional form, with the expression “flattening the curve,” used to explain why lockdown is essential to avoid overwhelming health services.
Our aim is to generate dynamical predictions and help inform the debate as to future lockdown release options. Each simulation can be readily understood in terms of policy decisions, and mathematically this implies careful parameterization. Our model is parameter sparse, yet sufficiently complex to capture a broad range of options. Critically, each parameter is related to understandable quantities characterizing infection levels or lockdown decisions. A central part of our analysis, in the absence of much knowledge of the SARS-CoV-2 virus, is to vary our fundamental parameters to determine their effect on the optimal strategy. This identification of sensitivity aids understanding and can identify research priorities crucial to enhancing our understanding and ability to manage the COVID-19 pandemic.
Our model considers two parallel SEIR (Susceptible, Exposed, Infected, Recovered) systems, one describing the spread of disease in the quarantined ‘lockdown’ population, and the other capturing the spread amongst those at work. The key difference between our two SEIR pools (those in lockdown and those who are not) is that transmission of the disease is assumed to be lower for those in a state of lockdown due to the self-isolation measures in place. This means that the susceptible population (
where
Our equations describe the movement of individuals through four stages, from being initially susceptible to the disease (
Schematic diagram depicting the movement of individuals through the SEIR network. The function
The lowercase Greek letters in Equations (1)–(8) represent our rate parameters. Firstly, β represents the transmission rate of the disease. Significant work early in the pandemic used available data to quantify the rate of SARS-CoV-2 transmission and a range of estimates have already been reported in the literature. Kucharski et al. (
μ represents the natural, background death rate of the population regardless of the impact of COVID-19, and can have important implications for the strength of herd-immunity effects on disease dynamics, as this is the only mechanism in our model through which the recovered population is reduced. The parameter α represents the rate of death directly attributed to SARS-CoV-2. While the mortality rate of SARS-CoV-2 has been demonstrated to vary substantially between age classes (
The parameter σ represents the incubation rate. The exposed population classes, (
In the present work, we used the population of the United Kingdom as an example to inform our initial proportion of the population in quarantine. Using Labor Force Survey data from 2018/19, the Institute for Fiscal Studies estimate that 7.1 million adults across the UK are in the set of key-worker guidelines set out by the UK government (
All model variables, parameters and the values used for these are presented in
Definitions of the variables and parameters used in the SEIR model, of Equations (1)–(4) for the non-quarantined population, and Equations (5)–(8) for quarantined groups.
Non-quarantined susceptibles | ( |
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Non-quarantined exposed | ( |
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Non-quarantined infected | ( |
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Non-quarantined recovered | ( |
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Total non-quarantined population | |||
Quarantined susceptibles | ( |
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Quarantined exposed | ( |
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Quarantined infected | ( |
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Quarantined recovered | ( |
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Total quarantined population | |||
β | Transmission rate | 0.6–2.35 | ( |
μ | Natural death rate | ||
σ | Incubation rate | 0.1961 | ( |
Reduced rate of transmission due to quarantine | 0.05 | ||
α | Disease-induced death rate | 0.00657 | ( |
γ | Recovery rate | 0.2222 | ( |
Total infection capacity | 4,000,000 | ( |
Firstly, the extent and longevity of immunity to SARS-CoV-2, and its effect on the dynamics of the pandemic, remain open, high priority research questions (
The parsimonious nature of our model was chosen to enhance the ease of interpretation of our results and, most importantly, to enable the model to be quickly adapted to non-UK populations. Different countries currently provide varying levels of epidemiological detail in their reporting of COVID-19 cases. By reducing the number of classes and parameters considered, our model is amenable to a wider range of countries and scenarios than the more specific model structures currently published (
While the model parameters are obtained from current research estimates (see
The primary challenge facing policy makers currently is in devising how to return the population to work most safely, ending the lockdown and its detrimental consequences on the economy. The objective is to release as many people from lockdown, as soon as possible, without overwhelming the health system with a subsequent resurgence of infections. This objective neatly fits the general framework of optimal control problems, a branch of mathematical study that seeks to maximize a certain objective functional through the use of available controls, while limited by constraints. In our model, the controls are the methods by which we release people from the quarantined classes, described by the function
We consider two distinct strategies for ending the lockdown; a “gradual release” strategy, whereby individuals are slowly, but permanently, released from quarantine in staggered waves until the entire population has been transferred from the quarantined class, and an “on-off release” strategy, whereby the lockdown is lifted for the entire population simultaneously, but can subsequently be reinstated when necessary (the mathematical formulation of these strategies is outlined below). In each case, we seek to ensure that any strategy employed does not cause the total number of infected individuals (
Many formal optimal control approaches employ the use of “adjoint equations” to minimize the Hamiltonian of the ODE system. While we also pursued this approach, it requires a continuous-time form for the control function
A gradual release strategy aims to end the lockdown of the public from quarantine through multiple staggered releases. Expressed mathematically, we seek to release
Therefore, the optimum choice of
To calculate these outputs, we used
Example of a gradual release from quarantine. Here, 20 million people are moved out of quarantine at
The “on-off” release strategy considers releasing the quarantined population all at once, with the aim to then return everyone to lockdown when required, should the number of infected exceed a threshold which threatens to overwhelm medical services. Formally, we seek
For each choice of
Example of an “on-off” release from quarantine. Here quarantine is ended at
The number of people to be released from quarantine,
Optimum gradual release strategies for a range of different values of
From
Additional to the graphical sensitivity analysis presented in
In essence, for each model parameter a value between 0 and 1 is calculated that describes how sensitive the optimum release strategy is to that parameter, with a value nearer unity being more sensitive. Total sensitivity indexes were calculated for β, σ, α, γ,
To determine the optimal timings for an “on-off” lockdown release strategy, both the times at which quarantine was ended,
Optimum on-off release strategies for a range of different values of
From
Just as with the gradual release strategy, total sensitivity indices were calculated via the same method for our optimum on-off strategy. In descending order, and for the on-off strategy, these now become: γ : 0.6371,
Here, we have investigated the optimal release of individuals from a state of lockdown. The primary conclusion of our work is that a gradual release strategy is preferable to an on-off release strategy. We conclude this from the finding that a population-wide instantaneous release would cause the number of infected individuals to rise dramatically, in a short period of time. Any decision to begin easing lockdown measures will require constant monitoring and a high-level of population testing to track the likely rise toward a second-peak of infections. We show that employing a gradual release strategy, where groups of the population are slowly released from quarantine sequentially, will slow the arrival of any subsequent infection peaks compared to an on-off strategy, where lockdown is ended for all individuals imminently and reinstated when subsequent infections begin to increase. In all considered instances (i.e., parameter variations), it will not be possible to end lockdown for the entire population for any longer than 2 weeks, as the number of infected individuals is then expected to quickly overwhelm the health service following such a release. By ensuring that the increase in the number of infected individuals is as slow as possible, this will enable health officials to monitor more accurately the evolving situation, and provide more time to respond to unexpected increases in the number of infected individuals. We note that our approach does not consider the ethical responsibilities that will also impact any policy decision. If enough hospital provision was available, many more people can return to employment, but we recognize this will result in increased risk of further mortalities. As many governments state however, a functioning economy is more able to provide health provision to those with life-threatening illnesses unrelated to COVID-19.
For a gradual release strategy, our simulations broadly suggest that a large section of the population should be released from lockdown initially, after the first peak of infections has fully passed. The rest of the population may then be released 3–4 months later following a likely second peak in infections. Again, in a general context, it is optimal to wait for 1–2 weeks after the end of an infection peak before releasing any of the population from lockdown. While it is desirable to return the population to work as early as possible, our optimal calculation states that this 1–2 weeks “wait” period is crucial in ensuring that the number of infected individuals is as low as possible when ending any lockdown measures, to reduce the growth of new cases. After this sufficient, cautious, wait period has ended, people should then be released from quarantine, with the knowledge that as many as 1 in 100 of them (under the worst-case scenario) may require critical care (
What we have not undertaken here is to investigate or advocate any particular forms of changed behaviors that might be needed by those released, although understanding them can allow parameters (such as transmission rates) to be adjusted in our framework. Additional measures proposed include: reopening local connections before connecting cities further apart (
Placing our analysis in the context of other studies, Mulheirn et al. (
The nearest analysis to ours found in the literature, based on both a SEIR framework and applied to COVID-19, is by German et al. (
Whilst we believe that our model framework does have predictive capability, we do raise a couple of caveats. We recommend exploring our findings within a variety of other model frameworks. Stochastic frameworks may be better suited to model the exact time periods when populations are first reintroduced, so as to better calculate the range of time frames until a second wave of infections in a probabilistic setting. Likewise compartmental infection models, such as those presented by Giordano et al. (
In preparing to monitor the situation upon easing lockdown measures, our sensitivity analysis highlights that the recovery rate of the disease, γ above, is the most critical parameter in understanding the magnitude of any subsequent peaks in infection. Our calculations can be trusted further if that value is well-understood. For example, if new hospitalized patients of COVID-19 appeared to be remaining symptomatic and infectious for longer than previously estimated, it is plausible to assume within the general community that the disease is therefore being transmitted faster than previously expected. This knowledge could trigger preparations for a potential need to reinstate lockdown measures. Hence further research efforts into the infectious period should also therefore be prioritized. In a similar vein, the parameter to which results are second-most sensitive is transmission rate, β, and so also worthy of precise research.
A potential benefit of the on-off release strategy is that it greatly increases the number of people subsequently moved to the recovered class, rapidly bolstering the acquisition of herd-immunity. This in theory would enable the full re-opening of the economy at an earlier date, however it makes the critical assumption that recovered individuals would remain immune to the disease. The nature of immunity to SARS-CoV-2 is an open question and efforts are being made to understand its strength and longevity, but currently the WHO advises that there is no evidence yet to suggest that recovered COVID-19 patients have ongoing immunity to a second infection (
In conclusion, using an optimal control methodology, we have shown that a gradual staggered release of individuals out of lockdown is recommended to ensure that health systems are not overwhelmed by a surge in infected individuals. It has been well-observed that older individuals are more likely to require critical care as a result of COVID-19 (
The ongoing threat of COVID-19 will require continual monitoring and study in the coming months. It is important to ensure that infections are kept to a minimum, and that the government and relevant services are given enough time to prepare for increases in infections. The findings of this study stress that gradual and cautious action must be taken when easing lockdown measures, to save resources, and lives, while adding to the evidence base of possible routes out of lockdown.
The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below:
TR, CH, and MB designed the model structure. TR performed the numerical simulations. DV and TB collated the parameter values from the literature. TB produced the figures. All authors reviewed and contributed to writing the manuscript.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The Supplementary Material for this article can be found online at: