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Modeling, state estimation, and optimal control for the US COVID-19 outbreak
Calvin Tsay, Fernando Lejarza, Mark A. Stadtherr, Michael Baldea
TL;DR
COVID-19 policy makers need to limit infections while managing the socioeconomic costs of distancing, testing, and quarantining. The paper builds an optimization-based modeling, estimation, and control framework for the US outbreak, finding that early action, especially quarantining confirmed infections, is most effective and that alternating strict and relaxed distancing can flatten the curve. Its conclusions are bounded by assumptions about population flows, future rates, and detection bias.
Problem
COVID-19 requires effective containment through social distancing, testing, and quarantining, but these measures carry significant socioeconomic costs.
Method
The paper combines epidemiological modeling, estimation of parameters and hidden states, and globally optimized control of social distancing, quarantining, and testing inputs.
Results
Quarantining confirmed infected subjects has the greatest impact, while early interventions and alternating strict and relaxed social distancing can help flatten the infection curve.
Takeaways & Limitations
Testing is most important immediately before relaxed social-distancing periods, and periodically updated policies can account for new data and hidden-state uncertainty.
Takeaways & Limitations
The model excludes population influx and outflux, assumes constant recovery and death rates over the next 100 days, and omits surveillance or detection bias.
Abstract
from arXiv · showhide
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.
Introduction
The rapid spread of COVID-19 created an urgent need for policies that limit infections while accounting for socioeconomic costs. Epidemiological models and optimal-control methods can help determine social-distancing and testing policies, but parameter estimation remains challenging early in an outbreak.
- COVID-19 spread rapidly worldwide, prompting lockdowns, quarantining, and social-distancing measures to contain infections.
- Policy makers must choose among mobility restrictions, resource investments, and improved screening to influence epidemic spread.
- Optimal-control studies use epidemiological models to select policies that minimize infections or deaths under resource and implementation constraints.
- The paper develops a complete optimization-based workflow for estimating model parameters and inputs and controlling social distancing and testing in the US outbreak.
- Open-source implementations of the estimation and optimal-control problems are provided freely.
Mathematical Modeling
The paper extends SEIR-style compartmental modeling with hidden asymptomatic or unconfirmed infections and deaths, while representing social distancing, quarantining, and testing as time-varying inputs.
- The modified SEAIR model adds asymptomatic or unconfirmed infections and a death compartment to a conventional SEIR structure.The a(t) state captures infected subjects absent from confirmed counts, while p(t) tracks deaths.
- The model contains six states: susceptible, exposed, asymptomatic or unconfirmed, confirmed infected, recovered, and perished populations.
- αa(t), αi(t), and κ(t) represent exposure from asymptomatic subjects, exposure from confirmed infected subjects, and confirmation through screening or testing.
- The model uses constant virus-related parameters over the considered horizon, including a zero assumed rate for loss of immunity.
- The SEAIR formulation includes three time-varying inputs and parameters whose values are partly estimated because hidden states are unmeasured.
Parameter estimation results
The authors estimate epidemic parameters and time-varying interventions across the US, Italy, Spain, and Germany, using historical data and model simulations. The fitted trajectories broadly reflect the chronology of containment measures, while some mortality trajectories suggest that a constant death rate is an approximation.
- The early US outbreak data were insufficient to fit parameters reflecting prevention measures already in place, so Italy, Spain, and Germany were analyzed for comparison.
- Historical trajectories and model predictions were compared for infected, recovered, and perished populations using fitted time-varying inputs.
- The simulated mortality trajectories for Italy and Spain deviate slightly from historical data, especially around day 80, suggesting that μ may vary over time.
- The estimated containment and testing trajectories broadly follow the observed chronology, with social distancing appearing earlier in Europe than in the US.
- Relaxing social distancing, quarantining, and testing can produce an alarming number of infections and deaths, particularly in the US.
Dynamic optimization results
The optimization results support alternating control policies that flatten infection growth, with quarantining of confirmed infections consistently more influential than social distancing. Early intervention and moving-horizon updates improve control when hidden states or future trajectories are uncertain.
- Optimization setup: The optimization uses deterministic mean predictions and constrains peak infections while accounting for socioeconomic cost.Model uncertainty is represented with two-standard-deviation bands, but optimization itself uses mean predicted values.
- Future control: A peak of 612,493 infected subjects is the minimum feasible value when optimization begins at day 85, making a 700,000-person cap very challenging.Meeting the cap requires immediately lowering both exposure inputs for approximately 25 days.
- Future control: Quarantining confirmed infections remains at its lower bound across optimized policies, indicating a larger impact than exposure reduction from asymptomatic or unconfirmed cases.The analysis identifies quarantining as more important than social distancing for reducing the infected population.
- Moving-horizon control: The 100-day optimization horizon can produce concerning infection growth near its endpoint, motivating periodic policy revision.This limitation is addressed by the moving-horizon strategy.
- Constraint sensitivity: Relaxing the lower bound on social distancing requires more frequent distancing periods, whereas relaxing the quarantine bound requires social distancing nearly all the time.In both cases, the optimal quarantine input remains at its lower bound.
- Future control: Alternating strict and relaxed social-distancing periods can flatten infection growth while allowing periods of social mobility.The resulting policies resemble bang-bang control, with inputs switching between lower and upper bounds.
Discussion
The study concludes that quarantining confirmed infected subjects has the greatest mitigation impact, while social distancing helps flatten infection peaks. Effective policies depend on early action, hidden-state estimates, and periodically updated control, but the model has important scope limitations.
- Quarantining infected subjects is the most important considered mitigation strategy and should be maximized throughout.
- Early interventions have much larger effects than later, more drastic interventions.
- Testing is most useful immediately before relaxed social distancing to reduce unconfirmed infections during increased mobility.
- Moving-horizon control and state estimation incorporate new data to reduce inaccuracies in epidemic models and hidden-case counts.
- On-off policies are characteristic of linear-objective, input-affine nonlinear control problems and alternate strict limitations with relative freedom of movement.
- The model excludes population flows, assumes constant recovery and death rates, and omits treatment, vaccine, mutation, and surveillance-bias effects.
Methods
The methods combine nonlinear least-squares parameter estimation with dynamic optimization of social-distancing, quarantining, and testing inputs. A moving-horizon formulation updates policies using revised state information while balancing infection constraints against societal cost.
- Nonlinear least-squares regression estimates model parameters by minimizing mean squared error between predicted and measured states.
- The estimation problem removes asymptomatic recovery contribution ρa(t) because the observed recovered data represent confirmed recoveries attributed to βi(t).
- The estimation includes time-varying αa(t), αi(t), and κ(t), time-invariant β and µ, and the initial exposed-state value e0.
- Time-varying inputs are constrained to five-day piecewise-constant intervals to limit over-fitting and reflect delayed policy effects.
- The optimal-control problem minimizes socioeconomic cost subject to a specified upper bound on peak infections.
- Testing cost is weighted with λκ = 0.1, and moving-horizon policies are recomputed every 25 days over 100-day optimization windows.
- The control problem is discretized by orthogonal collocation and solved with BARON to a 0.1% optimality gap.
Supplementary Information
The supplementary materials document parameter estimates, covariance information, cost-versus-peak analyses, and moving-horizon control results. They also show daily hidden-state estimation alongside policies updated at regular intervals.
- Table S1 reports mean values of the time-invariant model parameters, while Table S2 reports their covariance matrices.
- Cost analyses evaluate socioeconomic cost as peak-infection targets vary, with costs reported relative to a no-action scenario.
- Moving-horizon control updates policies in 25-day intervals using current measured or estimated populations as initial conditions.
- Supplementary figures show optimal policies found at days 110, 135, and 160, including predicted and true populations with containment and testing profiles.
- Hidden states e(t) and a(t) are estimated daily while control policies are updated every 25 days.