Tracing DAY-ZERO and Forecasting the Fade out of the COVID-19 Outbreak in Lombardy, Italy: A Compartmental Modelling and Numerical Optimization Approach.

Lucia Russo; Cleo Anastassopoulou; Athanassios Tsakris; Gennaro Nicola Bifulco; Emilio Fortunato Campana; Gerardo Toraldo; Constantinos Siettos

At this point we should note that the above optimization problem may in principle have more than one nearby optimal solutions, which may be attributed to the fact that the tuning of both DAY-ZERO and the transmission rate may in essence result in nearby values of the objective function. In order to quantify the above uncertainty in the optimization procedure, we created a grid of initial 205 guesses within the intervals in which the optimal estimates were sought: for the DAY-ZERO (t 0 ) we used a step of 2 day within the interval 27 December 2019 until the 5th of February 2020 i.e. ±20 days around the 16th of January, for β we used a step of 0.05 within the interval (0.3, 0.9) and for we used a step of 0.02 within the interval (0.01, 0.29). The numerical optimization pro-210 cedure was repeated 48 times for each combination of initial guesses. For our computations, we kept the best fitting outcome for each combination of initial guesses. Next we fitted the resulting cumulative probability distributions of the optimal values using several functions including the Normal, Log-normal, Weibull, Beta, Gamma, Burr, Exponential and Birnbaum-Saunders distribu-215 tions and kept the one resulting in the maximum Log-likelihood (see in the Supporting Information for more details). For the computed parameters of the corresponding best distributions, we also provide the corresponding 95% confidence intervals. Note, that the expected values of the resulting distributions do not correspond to optimal values, due to the approximation that is introduced 220 by the fitting procedure; however they correspond to expected/ most probable values around which an optimal solution is sought. Thus, for that purpose, we have used as initial guesses the mean values of the resulting distributions to the Levenberg-Marquard (implemented by the "lsqnonlin" function of matlab [23] ) to find the optimal solution around this "representative point" as well the 225 corresponding 95% confidence intervals. The step size for the finite differences 11 . CC-BY-NC-ND 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Note that there are three infected compartments, namely E, I, I c and two of them (E,I) that determine the outbreak. Thus, considering the corresponding equations given by Eq.(2),(3), (4) , and that at the very first days of the epidemic S ≈ N and D ≈ 0, the Jacobian of the system as evaluated at the disease-free 240 state reads: