Quantifying the effect of quarantine control in Covid-19 infectious spread using machine learning

Raj Dandekar; George Barbastathis

The copyright holder for this preprint . https: //doi.org/10.1101 //doi.org/10. /2020 thereafter, the neural network learns how to approximate it based on the local data from each region under study. Initial conditions The starting point t = 0 for each simulation was the day at which 500 infected cases were detected, i.e. I 0 = 500. The number of susceptible individuals was assumed to be equal to the respective regional populations, i.e. S(t = 0) = 11 million, 60 million, 52 million and 327 million for Wuhan, Italy, South Korea and USA respectively. Also, in all simulations, the number of recovered individuals was initialized to a small number R(t = 0) ≈ 10. Parameter estimation The time resolved data for the infected, I data and recovered, R data for each locale considered is obtained from the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University. The neural network-augmented SIR ODE system was trained by minimizing the mean square error loss function L NN (W, β, γ) = log(I(t)) − log(I data (t)) 2 + log(R(t)) − log(R data (t)) 2 (4.16) that includes the neural network's weights W . Minimization was carried out through local adjoint sensitivity analysis (Cao et al. 2003; Rackauckas et al. 2019 ) following a similar procedure outlined in Rackauckas et al. (2020) and implemented using the ADAM optimizer (Kingma & Ba 2014) for 300 ∼ 500 iterations. To avoid over-fitting, the training is stopped when the loss function, L is seen to stagnate and the first derivative I ′ (t), R ′ (t) is seen to match that of the data. (Lyons 2020) , leads to smaller recovery rate and hence a smaller fraction of the population being transferred from the infected compartment to the recovered compartment in the model described in (4.12) -(4.15). Thus, simultaneous training of W, β, γ as described above leads to our model over-estimating the infected case count. As a result, for the USA, we first find the optimal γ by minimizing (4.16), and then use this value of γ for minimizing the loss function, L(W, β) = log(I(t)) − log(I data (t)) 2 . Such an independent optimization procedure for estimating γ may lead to small errors in the estimation of R(t). Such errors are seen to be negligible in this case (figure 8a), thus validating this approach for the USA. The estimates of β, γ, Q(t) obtained using this procedure are shown in table 1. Table also shows the intervention efficiency defined as the number of days elapsed between detection of 500 th case and the first time when the effective reproduction number reached R t < 1 in the chosen locale.