Fig 1.
(a)–(c) Marginal densities on (β, τ, α) for noisy SIR data, obtained from the neural scheme (blue) and the MALA sampler (pink). The ground truth (dotted line) was calculated using a simple grid search on (β, τ) ∈ [0, 1] × [1, 30] with 10.000 grid points. (d) Predicted average time series for the S (lightgreen), I (red), and R compartments. Shown are the predictions (solid line) and standard deviation (shaded area) generated by drawing 10.000 samples from the predicted joint distribution of (β, τ) using the neural scheme and solving the noiseless ODE model Eq (6) each. Also shown are the true data for each compartment (dots). The neural network was trained for 100 epochs from 300 different initial conditions, and 50 MALA chains were run until stationarity was reached, with a thinning factor of 5. Here, stationarity is defined via a Gelman-Rubin statistic of below 1.2, see S2 Fig in the S1 Appendix. Both the neural and MCMC samplers are parallelised.
Table 1.
Hellinger distances (Eq (9)) between the estimated and true marginals.
Fig 2.
Schematic illustration of the SEIRD+ model, as originally presented in [13].
Each parameter λi indicates the transition rate between the respective compartments. S, E, and I are the susceptible, exposed, and infected agents. Upon contact with an infected agent, each may be contacted by the contact tracing agency (CT) and ordered to quarantine (QS, QE, QI compartments). λQ models the rate of compliance with the contact tracing agency’s instructions. SY, H, and C are the symptomatic, sick, and critically sick agents. Agents from these as well as the I and QI compartments can recover and transition to the R compartment, where they are assumed to stay, at least for the period under consideration (< 9 months). Finally, critically ill patients may die of the disease (D), though this compartment is not included in the loss function. We assume the exposure rate λE varies as public health measures change. The parameter λQ is further assumed to be a function of λCT and CT, and is thus not learned; see S1 Appendix. Figure adapted from [13].
Fig 3.
Evolution of COVID-19 in Berlin for the period from February 16 to October 27 2020.
Shown are the ABM data [31] for the symptomatic, hospitalised, and critical compartments (orange, red, purple), the sum of all three (light brown, solid line), as well as the official infection figures (light brown, dots) [32]. The red period is the calibration period, with the shades representing varying levels of government restrictions and correspondingly different exposure levels λE: from mid-March, businesses and factories started closing; in late March, the German government imposed broad contact restrictions; in early May, schools and kindergartens started reopening across the country, followed by further loosening of restrictions in mid-June, before the start of the summer holidays. The blue period is the projection data on which we evaluate the prediction. The ABM data only contains a single Q compartment and no CT compartment. It also does not produce a D compartment, for the reasons given in the text. See S1 Appendix for details.
Fig 4.
Marginals on the parameters Λ = (λS, …, λCT).
Shown are the neural marginals (blue, left side) and MCMC estimates (pink, right side), which in both cases were smoothed using a Gaussian kernel. Also shown are the means (green dots) and modes (yellow dots) of the marginals. We employ a three-layer neural network with 20 neurons per layer and sigmoid activation functions on all but the last layer, where we again use the absolute value function.
Fig 5.
Comparison of the neural calibration results (left) and the MCMC calibration results (right) for the symptomatic, hospitalized, and critical compartments.
Red lines are the true data, green lines the prediction using the estimated mean of the joint density, calculated by drawing 1000 samples from the joint distribution. The green shaded areas represent one standard deviation. The blue shaded area is the test period for which projections are generated. Calibration results for the remaining compartments are shown in S3 Fig in the S1 Appendix.
Table 2.
Calibration and projection error of the neural and MALA schemes on the different compartments.
Fig 6.
Results on a reduced training dataset, consisting only of the symptomatic, hospitalized, and critical compartments.