Modeling, state estimation, and optimal control for the US COVID-19 outbreak

Abstract

The novel coronavirus SARS-CoV-2 and resulting COVID-19 disease have had an unprecedented spread and continue to cause an increasing number of fatalities worldwide. While vaccines are still under development, social distancing, extensive testing, and quarantining of confirmed infected subjects remain the most effective measures to contain the pandemic. These measures carry a significant socioeconomic cost. In this work, we introduce a novel optimization-based decision-making framework for managing the COVID-19 outbreak in the US. This includes modeling the dynamics of affected populations, estimating the model parameters and hidden states from data, and an optimal control strategy for sequencing social distancing and testing events such that the number of infections is minimized. The analysis of our extensive computational efforts reveals that social distancing and quarantining are most effective when implemented early, with quarantining of confirmed infected subjects having a much higher impact. Further, we find that "on-off" policies alternating between strict social distancing and relaxing such restrictions can be effective at "flattening" the curve while likely minimizing social and economic cost.

Figure 1

Digraph representation of SEAIR model; model inputs and parameters.

Figure 2

Historical data and model fit for infected (i(t)), recovered (r(t)), and perished (p(t)) subjects (in thousands). Solid lines represent the mean of 500 Monte Carlo simulations, shaded areas represent two standard deviations from the mean, and circle markers are historical data. The right-most column shows the fitted trajectories of the time-varying inputs.

Figure 3

Simulation of future infected (i), new confirmed cases $(\kappa a)$, and perished (p) subjects (in thousands) for current control policies. Shaded grey area indicates historical data, color solid lines represent the mean of 500 Monte Carlo simulations, and color shaded area represents two standard deviations form the mean.

Figure 4

Optimal control policy to limit peak infections to 700,000 (left) or 1,400,000 (right) in the next 100 days. Top: population numbers, with two standard deviations shaded. Bottom: containment and testing profiles. The shaded grey area indicates past days, which were simulated using historical inputs (not optimized).

Figure 5

Optimal containment and testing strategies to limit peak infections to 1,000,000 in the next 100 days for different constraints on ${\alpha }_a$ and ${\alpha }_i$. Top: ${\alpha }_a(t)$. Bottom: $\kappa (t)$. The shaded grey area indicates the solution found using the normal bounds, replicated from Fig. 4 (right).

Figure 6

Optimal control policy to limit peak infections to 700,000 (left) or 70,000 (right) in the past 35 days and next 100 days. Top: population numbers, with two standard deviations shaded. Bottom: containment and testing profiles. The shaded grey area indicates past days, for which the true historical inputs and outputs are shown as dashed lines and the optimized are shown as solid lines.

Figure 7

Optimal moving horizon control policy (right) to limit peak infections to 1,400,000, with e and a underestimated by a factor of three at $t=74$, and comparison to the same situation without a moving horizon strategy (left). Top: predicted (dash-dotted) and true (solid) population numbers. Bottom: containment and testing profiles. The shaded grey area indicates past days, which were simulated using historical inputs (not optimized). The policies are updated every 25 days with daily state estimation.