Robust and optimal predictive control of the COVID-19 outbreak

Johannes Köhler; Lukas Schwenkel; Anne Koch; Julian Berberich; Patricia Pauli; Frank Allgöwer

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Robust and optimal predictive control of the COVID-19 outbreak

Johannes Köhler, Lukas Schwenkel, Anne Koch, Julian Berberich, Patricia Pauli, Frank Allgöwer

arXiv:2005.03580v2May 7, 2020q-bio.PEeess.SYmath.OCphysics.soc-ph

TL;DR

The paper asks how to reduce COVID-19 fatalities while avoiding excessive social costs under uncertainty about outbreak dynamics and measurements. It develops open-loop, MPC feedback, and robust MPC strategies for Germany, finding that adaptive feedback is needed for reliable control and that optimal policies outperform simpler baselines.

Problem
COVID-19 control must reduce fatalities while limiting the economic and psychological costs of social distancing, despite uncertainty in disease dynamics and measurements.
Method
The study adapts SIDARTHE to German data and develops optimal open-loop, MPC feedback, and robust MPC-based social-distancing strategies.
Results
Optimal policies have significant advantages over simpler baselines, while adaptive MPC feedback addresses uncertainty and robust MPC handles model, measurement, and input inaccuracies.
Takeaways & Limitations
Reliable outbreak control requires continually measuring, monitoring, estimating, and adapting social-distancing policies through feedback.
Takeaways & Limitations
The analysis is based on a simple model fitted with limited data and uses information available before the paper’s initial submission in May 2020.

Abstract

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We investigate adaptive strategies to robustly and optimally control the COVID-19 pandemic via social distancing measures based on the example of Germany. Our goal is to minimize the number of fatalities over the course of two years without inducing excessive social costs. We consider a tailored model of the German COVID-19 outbreak with different parameter sets to design and validate our approach. Our analysis reveals that an open-loop optimal control policy can significantly decrease the number of fatalities when compared to simpler policies under the assumption of exact model knowledge. In a more realistic scenario with uncertain data and model mismatch, a feedback strategy that updates the policy weekly using model predictive control (MPC) leads to a reliable performance, even when applied to a validation model with deviant parameters. On top of that, we propose a robust MPC-based feedback policy using interval arithmetic that adapts the social distancing measures cautiously and safely, thus leading to a minimum number of fatalities even if measurements are inaccurate and the infection rates cannot be precisely specified by social distancing. Our theoretical findings support various recent studies by showing that 1) adaptive feedback strategies are required to reliably contain the COVID-19 outbreak, 2) well-designed policies can significantly reduce the number of fatalities compared to simpler ones while keeping the amount of social distancing measures on the same level, and 3) imposing stronger social distancing measures early on is more effective and cheaper in the long run than opening up too soon and restoring stricter measures at a later time.

1. Introduction

The paper frames COVID-19 control as a multi-objective problem: reducing fatalities while limiting the economic and psychological costs of social distancing. It develops optimal, feedback, and robust MPC strategies for Germany under model and measurement uncertainty.

Social distancing can contain COVID-19 spread but creates substantial economic and psychological costs, making outbreak control a multi-objective decision problem.
The study proposes optimal open-loop and feedback strategies over a two-year horizon to minimize fatalities without excessive social distancing.
The model extends prior work with a health-care-capacity-dependent mortality rate and parameters fitted to German outbreak data.
Optimal inputs can reduce fatalities relative to simpler baseline policies while keeping distancing costs low.
Precomputed optimal inputs become dangerous under model uncertainty, motivating adaptive feedback through model predictive control.
Robust MPC accounts for model inaccuracies, uncertain state measurements, and inexact inputs to control the outbreak cautiously and safely.

2. Modeling of the COVID-19 epidemic

The epidemic model adapts SIDARTHE to Germany by fitting parameters to newer German data and increasing mortality when critical illness exceeds health-care capacity. It represents the outbreak with eight epidemiological states.

The model is adapted from SIDARTHE and uses newer data to estimate parameters for the German COVID-19 outbreak.
Mortality increases when the number of critically ill patients exceeds German health-care capacity.
SIDARTHE represents eight states: Susceptible, Infected, Diagnosed, Ailing, Recognized, Threatened, Healed, and Extinct.

A AILING symptomatic

The paper adapts the SIDARTHE epidemic model to Germany by fitting constrained parameters to German case, death, recovery, and ICU data, while modeling health-care-capacity effects on mortality. The model distinguishes infection, detection, symptoms, life-threatening illness, recovery, and death states for control-policy analysis.

Mortality modeling: The model makes mortality and recovery rates for threatened patients depend on the number of threatened individuals, approximating health-care-capacity effects.This modification is intended to represent increased mortality when hospitals are overwhelmed.
Model structure: The model distinguishes detected and undetected cases, symptomatic and asymptomatic individuals, and a threatened state for life-threatening symptoms.Its eight states include Susceptible, Infected, Diagnosed, Ailing, Recognized, Threatened, Healed, and Extinct.
Parameter estimation: German parameters are estimated by constrained least-squares fitting of normalized confirmed cases, deaths, recoveries, and ICU data.The data are filtered, normalized by Ntotal = 8.3 · 10^7, and supplemented with prior assumptions to avoid over-fitting.
Parameter estimation: The fitting procedure imposes assumptions including negligible asymptomatic-case detection, a fixed initial confirmed-case count, approximately constant testing rate, and policy-dependent infection rates.The social-distancing policies are ordered from no countermeasures to full lockdown and modify selected infection rates.
Parameter validation: The fitted parameter set activates several realism constraints, including α = γ, while the model remains sensitive to parameter changes.Different estimated parameter ranges can produce substantially different parameter values, motivating robust control analysis.

3. Open-loop optimal control of the COVID-19 outbreak

The section compares baseline distancing and testing policies, then examines control goals and consistent-lockdown scenarios for the German outbreak. The simulations show that eliminating the virus requires prolonged strict measures, while ending an incomplete lockdown produces a second wave.

Testing policy: Testing is held at the current policy: available tests are used daily for symptomatic people, with ǫ(t) = 0.The paper notes that improved capacity or asymptomatic-contact tracing is not included in the present analysis.
Control goal: The model considers herd immunity as a control goal but assumes a vaccine will become available in approximately two years.The fitted model gives a herd-immunity timescale of more than six years without exceeding health-care capacity.
Optimal-control comparison: Optimal control techniques are introduced to improve the baseline policies while assessing the significance of those improvements.The control input u represents distancing policies or related measures and is changed at most weekly.
Baseline policies: 305 days are required for the fitted model to meet the virus-eradication threshold under continued lockdown.Eradication is defined as fewer than 0.5/Ntotal active contagious cases.
3.1.1. Consistent lockdown: A consistent lockdown that is not maintained until eradication produces a second outbreak wave after measures are suspended.Across lockdowns ending immediately, after 50 days, or after 150 days, longer lockdowns delay the peak but leave its amplitude almost unchanged.
3.1.1. Consistent lockdown: At least another 70.4% of the German population becomes infected in the second wave under the fitted post-lockdown state.The reported equilibrium value is Seq(x0, α3, γ3) = 0.9956, which exceeds the herd-immunity threshold condition.

3.2. Optimal control strategy

The paper formulates a two-year optimal-control problem that minimizes accumulated fatalities while keeping social-policy costs below a baseline. Under idealized model and measurement assumptions, optimized policies reduce fatalities, avoid ICU-capacity violations, and favor gradual relaxation over repeated tightening and loosening.

Optimization setup: The control problem minimizes accumulated fatalities over 100 weeks while using fewer accumulated social-policy resources than the baseline.The baseline remains a feasible solution, and policy inputs change weekly.
Results: 26% fatalities remain under the optimal policy versus the cautious baseline, despite slightly more infections and fatalities during the initial period.After roughly 200 days, infections and threatened individuals are lower under the optimal strategy.
Results: 39% fatalities remain under the optimal policy versus the more relaxed baseline, which exceeds ICU capacity.The optimal strategy avoids the baseline’s second wave by loosening restrictions slowly and steadily.
Discussion: The optimal policy smoothly increases the infection rate without exceeding ICU capacity and without increasing social cost over the full horizon.Compared with a consistent full lockdown, optimized policies permit an average doubling of the infection rate without significantly increasing fatalities.
Discussion: A stronger initial lockdown followed by gradual loosening performs better than repeatedly tightening and relaxing distancing measures.The paper also identifies finite-horizon “turnpike” behavior, in which restrictions are released aggressively near the horizon’s end.
Limitations: The open-loop results are highly sensitive to infection-rate changes, making accurate control of infection rates through governmental policies difficult.This motivates a robust feedback strategy that accounts for model and state uncertainty.

4. Optimal feedback control of the COVID-19 outbreak

Because open-loop optimization relies on accurate models, exact states, and precisely imposed infection rates, the paper updates the policy weekly using measured states. This MPC feedback remains effective under parameter mismatch and can reduce fatalities or social-distancing effort relative to open-loop control.

Motivation: Open-loop optimal control may perform poorly under model mismatch because it relies on accurate model identification, exact states, and precisely imposed infection rates.The model is identified from sparse data and prior estimates, so mismatch is expected in practice.
MPC feedback: MPC repeatedly solves the finite-horizon optimization from the current measured state, applies the policy for one week, and then updates it.The feedback mechanism incorporates online measurements at every weekly step.
Adaptive cost bound: The feedback policy adapts its social-policy cost bound upward when predicted ICU demand reaches 90% of capacity and downward when it remains below 10%.The adjustment increases containment effort when predicted demand is high and relaxes it when demand is persistently low.
Validation: Validation models alter the assumed stationary ratio of confirmed cases while retaining the identification procedure, testing robustness to parameter uncertainty.Two parameter sets use intervals [0.3, 0.6] and [0.3, 0.4] instead of the nominal [0.3, 0.45].
Results: MPC feedback effectively controls the outbreak with inaccurate models and can dramatically reduce fatalities or the required social distancing compared with open-loop control.Under the lower-reproduction validation model, MPC achieves almost identical performance while using significantly lower social and economic cost.

5. Robust and optimal feedback control of the COVID-19 outbreak

The robust MPC strategy accounts for biased state measurements and infection-rate uncertainty by predicting state intervals and minimizing worst-case fatalities. Applied weekly to validation models, it reduces fatalities while using no more restrictive overall measures than nominal or open-loop policies.

Uncertainty modeling: Biased measurements are handled by computing intervals guaranteed to contain the true state before robust prediction.The interval bounds use known measurement-bias limits, including especially high uncertainty for undetected infections.
Robust MPC formulation: The robust formulation propagates interval predictions under uncertain infection rates and minimizes the worst-case number of fatalities.The analysis focuses on ±5% uncertainty in the infection rate α and uses interval arithmetic for robust MPC.
Robust MPC formulation: The feedback algorithm updates the policy weekly, applying the optimized control for Ts = 7 days before recomputing it.Each update starts from the current interval state estimate and predicts over the remaining horizon.
Numerical results: For parameter set B, robust and nominal MPC require similar overall resources, while robust MPC reduces fatalities by 33% and both MPC policies outperform open-loop policies.The robust formulation initially uses unnecessarily high control effort, but applied resources differ by less than Δu from nominal MPC.

6. Conclusions and Discussion

The paper concludes that adaptive feedback is needed to handle model mismatch and measurement bias reliably, while optimized policies reduce fatalities without increasing social-distancing costs. Robust feedback favors strong initial measures followed by gradual loosening, although other outbreak influences remain outside the analysis.

Main findings: Neither virus eradication nor herd immunity without a vaccine is considered viable for managing the COVID-19 outbreak.
Main findings: Optimization can significantly reduce fatalities without increasing the costs of lowering infection rates through social-distancing policies.
Main findings: Because the model and measurements are imperfect, reliable outbreak control requires continuously updated adaptive feedback rather than a nominal fixed policy.The paper argues that monitoring and estimating current case numbers is necessary for adapting the policy.
Robustness: Robust MPC using interval arithmetic accounts for model mismatch and uncertainty, avoiding intermediate infection increases that could require another lockdown.The robust strategy thereby significantly reduces fatalities relative to feedback without robust uncertainty handling.
Policy implications: The robust controller recommends strict initial measures followed by gradual increases in the infection rate, with slow loosening beneficial in the long run.
Scope: Testing capacity, infection tracking, and identifying measures that achieve target infection rates are important influences not included in the paper.

Appendix A. Optimal control formulation using terminal constraints

The appendix replaces the modified finite-horizon cost with terminal constraints on contagious compartments and fatalities. These constraints ensure the final state improves on the baseline while containing the outbreak.

Terminal constraints: Terminal constraints are introduced to avoid finite-horizon artifacts such as ending with many infected people.They constrain the contagious population IDART = (I, D, A, R, T) at the end of the horizon.
Terminal constraints: The formulation replaces the modified cost F with the number of fatalities E and adds terminal constraints to the optimal-control problem.
Terminal constraints: The terminal conditions require a final state better than the baseline and ensure that the outbreak can be contained.The conditions apply element-wise to the contagious-state variables.

Appendix B. Alternative average constraint formulation

A stricter transient cost constraint keeps accumulated policy costs below the baseline at every time rather than only over the 100-week horizon. Even then, optimized control reduces fatalities, highlighting the value of early measures.

Transient constraint: The alternative formulation constrains accumulated policy cost to remain below the baseline cost at every time.This is stricter than restricting total cost only over the horizon N = 100 weeks.
Results: Under the stricter setting, fatalities are reduced by 33% and 37% for the two considered baselines.
Results: Early measures are crucial because policies that differ substantially beforehand become essentially equivalent during the 110–150-day period when ICU capacity is exceeded.

Appendix C. Interval predictions

Appendix C derives interval-prediction dynamics for uncertain epidemic parameters and uses projections to improve potentially conservative bounds. The appendix also presents a figure comparing optimal control with baseline policies under cautious and aggressive social-policy baselines.

Interval dynamics: Interval predictions are derived for uncertain parameters bounded within specified intervals, using positivity and a scalar property to obtain differential equations.The resulting formulation yields 2 · 8 ordinary differential equations for the interval bounds.
Illustrative comparison: Figure B.11 compares optimal control and baseline policies against ICU capacity for cautious and aggressive baselines under a transient social-policy constraint.Optimal control is shown in blue solid lines, baseline policy in red dashed lines, and ICU capacity as a black dotted line.
Bound refinement: The interval dynamics may produce conservative overapproximations, motivating projections that exploit the constraint ∑_{i=1}^8 x_i = 1 to improve bounds for S.The projections are applied after deriving the interval dynamics.
Implementation choice: The appendix notes that one bound could be set directly instead of simulating the corresponding interval equations, but this may not ensure the desired property.The alternative is presented as a possible implementation choice rather than the main derivation.