Fig 1.
Flow diagram of the data-driven approach for the synthesis of optimal lockdown policies.
The initial step consists of a policy maker defining a performance measure based on sanitary and economic objectives, and a modeller selecting a consistent generic epidemiological model. Then, public healthcare/mobility data is used in conjunction with Approximate Bayesian Computation to calibrate the dynamical model and determine the degree of uncertainty in the model parameters. This assists the formulation of an optimal control problem where the original sanitary + economic performance measure is optimized constrained to the calibrated epidemiological model. An optimal lockdown policy is then computed via global optimization techniques, and the final output is an optimal lockdown policy. The optimised lockdown is then applied and its real-time effects can be sensed through public data, which fed back into the learning and optimization framework for re-computation and update.
Fig 2.
Graphical representation of the model, for each age group.
The green color represents a compartment that is observed independently for each age group, while blue represents a compartment whose sum across age groups is observed.
Fig 3.
Contact matrices at different locations in the UK for the age groups used in the present study (0–19, 20–39, 40–59, 60–79, 80+); these are obtained by aggregating and combining the contact matrices for 5-year bands provided by [18] and [19].
Fig 4.
Raw and elaborated mobility data in the UK.
In the raw mobility data, which is scaled with respect to a baseline value representing average mobility in the months prior to the pandemics, a strong weekly seasonality is present, which we mostly removed using a Savitzky–Golay filter. Moreover, we note that the mobility towards “residential” (which is not used in our analysis) locations is larger during the lockdown months than before, as people spend more time in their homes. Finally, we note the very large increase in people’s mobility towards parks around the end of the winter season. That contribution is however only one of the components in our aggregated mother mobility value, so the latter does not increase that abruptly.
Fig 5.
Flow diagram of the data-driven optimal control approach.
Starting from a generic-type SEIRD model, we learn optimal model parameters based on mobility/healthcare datasets and Approximate Bayesian Computation. The output is a posterior distribution of model parameters, which is used to generate calibrated SEIRD dynamics and a cost functional accounting both for sanitary and economic costs of a lockdkown. These two ingredients determine the formulation of an optimal control problem, which is solved by means of a global optimization algorithm. The final output of our approach is an optimal lockdown policy which can be recalibrated as new data is fed into the system.
Fig 6.
Comparison of predictions of our model with the real number of hospitalized people with COVID-19 and total daily deaths (green) on 23rd May for England.
The solid red line denotes the median prediction, filled spaces denote the 99% credible interval and the vertical dashed line denotes the observation horizon. The different columns represent number of people in hospital, IC (left), daily deceased (middle) and value of (right).
Fig 7.
Dependence on the prediction horizon Topt for determining and optimal control strategy for England.
Here, (ϵs, ϵw, ϵo) = (100, 100, 100) and the lockdown strategies were applied for Th = 90 days starting on the 24th of May.
Fig 8.
Different relative weights between the sanitary cost and economic cost of the lockdown measures with the lockdown strategies starting on the 24th of May and applied for Th = 120 days, for England.
Fig 9.
Different economic cost weights produce different opening strategies and predicted hospitalized people, with lockdown strategies starting on the 24th of May and applied for Th = 120 days, for England.
Fig 10.
Evolution of corresponding to different opening strategies starting on the 24th of May, learned using different economic cost weights for different ϵ values, with 99% credibility intervals; these plots are referred to England.
Fig 11.
Dynamically updated control strategy: We fit the model on data for England up to tobs = 11th of April and determine the optimal mobility strategy up to the next observation point tobs = 26th April.
Data until the latter is used to repeat the procedure, in order to update the optimal control strategy exploiting newly available information. Here (ϵschool, ϵwork, ϵother) = (150, 300, 3). In panel (a), we show the resulting optimal mobility values, with the corresponding values of and their credibility intervals shown in panel (b).
Fig 12.
Comparison of predictions of our model with the real number of hospitalized people with COVID-19 and total daily deaths (green) on 31st August in England and France.
The solid red line denotes the median prediction, filled spaces denote the 99% credible interval and the vertical dashed line denotes the observation horizon. The different rows represent different observation horizons, while the columns represent number of people in hospital (IC compartment, left column), daily deceased (middle column) and value of (right column).
Fig 13.
Optimal strategy for England with parameters fitted with data up to the 31st August.
The lockdown strategy is applied for 120 days starting on the 1st of September and uses a prediction horizon of Topt = 30 days.
Fig 14.
Optimal strategy for France with parameters fitted with data up to the 31st August.
The lockdown strategy is applied for 120 days starting on the 1st of September and uses a prediction horizon of Topt = 30 days.