Optimal Control Policies to Address the Pandemic Health-Economy Dilemma
Rohit Salgotra, Thomas Seidelmann, Dominik Fischer, Sanaz Mostaghim, Amiram Moshaiov
11 Optimal Control Policies to Address the PandemicHealth-Economy Dilemma
Rohit Salgotra, Amiram Moshaiov
School of Mechanical EngineeringIby and Aladar Fleishman Faculty of Engineering
Tel Aviv University, IsraelEmail: [email protected], [email protected] Seidelmann, Dominik Fischer, Sanaz Mostaghim
Chair of Computational IntelligenceFaculty of Computer Science
Otto von Guericke University Magdeburg, GermanyEmail: [thomas.seidelmann,dfischer,sanaz.mostaghim]@ovgu.de
Abstract —Non-pharmaceutical interventions (NPIs) are effec-tive measures to contain a pandemic. Yet, such control measurescommonly have a negative effect on the economy. Here, wepropose a macro-level approach to support resolving this Health-Economy Dilemma (HED). First, an extension to the well-knownSEIR model is suggested which includes an economy model.Second, a bi-objective optimization problem is defined to studyoptimal control policies in view of the HED problem. Next, severalmulti-objective evolutionary algorithms are applied to performa study on the health-economy performance trade-offs that areinherent to the obtained optimal policies. Finally, the resultsfrom the applied algorithms are compared to select a preferredalgorithm for future studies. As expected, for the proposed modelsand strategies, a clear conflict between the health and economyperformances is found. Furthermore, the results suggest that theguided usage of NPIs is preferable as compared to refrainingfrom employing such strategies at all. This study contributes topandemic modeling and simulation by providing a novel conceptthat elaborates on integrating economic aspects while exploringthe optimal moment to enable NPIs.
Index Terms —SARS-CoV-2, COVID-19, pandemic model, eco-nomic model, control policies, multi-objective optimization.
I. I
NTRODUCTION
This study is motivated by the current SARS-CoV-2 pan-demic and the need to support policy-making on controlling thevirus spreading. In particular, this study is based on the under-standing that controlling the pandemic by Non-PharmaceuticalInterventions (NPIs), such as lockdowns, comes with an eco-nomic cost that creates an unavoidable dilemma to be resolvedby the Decision-Makers (DMs) [1]. This is hereby referred toas the Health-Economy Dilemma (HED) [2].A common approach to support a decision on NPI policiesis to use a compartment model such as the SEIR (Susceptible,Exposed, Infectious, Recovered) model [3]. This model hasbeen used to predict the number of infected individuals overtime for major epidemics [4]. However, most epidemiologi-cal models do not consider the economical consequences ofapplying NPIs.Epidemiological Economics (EE) focus on studying boththe economic and epidemiological aspects of managing theinfectious disease (e.g., [2, 5, 6, 7]). The study in [8] wasprobably the first to address the HED problem. It assumes that various rate parameters of the applied pandemic model can becontrolled. Viewing the problem as a bi-objective one, optimalrate-parameters were sought to influence both the economyand the health performances. This study follows some of theideas of [8]. Yet, the focus here is on different variables.Namely, rather than searching for optimal rate values, herethe search concerns the optimal time to onset the proposedcontrol actions. Moreover, in contrast to [8], which uses justone algorithm, here several algorithms are applied and theirsuitability for the problem is studied.Inspired by [9], we propose an extension of the SEIRmodel, which integrates the health and economy aspects.Augmented with the effects of NPIs strategies, the proposedmodel allows identifying optimal control policies for maximiz-ing the economic performance while minimizing infections.In other words, the extended SEIR model allows studyingthe involved health-economy performance trade-offs, whichsupports resolving the HED by the DMs.In addition to proposing the extended SEIR model, thispaper provides a study on optimal control policies (opti-mal NPIs). Particularly, a bi-objective optimization problemis defined to minimize extreme peaks of infection numbersand the economic damage simultaneously. This problem issolved using existing Multi-Objective Evolutionary Algorithms(MOEAs) and the extended SEIR model. The employed al-gorithms, which are taken from [10], search for the Pareto-optimal set of control policies and the associated Pareto-front which reveals their performance trade-offs. Finally, acomparison study is conducted to unveil the accuracy of theresults as obtained by the considered algorithms. This aims toidentify the preferred algorithm for future studies.The rest of this paper is organized as follows. The proposedextension to the SEIR model is described in Section II. SectionIII describes the considered bi-objective optimization problem.In Section IV, the experimental setup is presented. SectionV presents the results of the simulation and optimizationexperiments, which are discussed in Section VI. Finally, theconclusions of this study are provided in Section VII. a r X i v : . [ c s . N E ] F e b II. P
ANDEMIC AND E CONOMIC M ODELLING
This section presents the mathematical formulations of theproposed model. In Subsection II-A, the extended SEIR modelis introduced. Next, in Subsection II-B, the considered macro-economic model is integrated to the extended SEIR model.Finally, in Subsection II-C, the applied control policies aredescribed.
A. The Extended SEIR Model
We propose a new pandemic model to support dealingwith the economic consequences of having a portion of thepopulation quarantined during the pandemic. This portion ofthe population is dependent on the epidemiological statusof individuals, their clinical progression, and prospective in-tervention measures. Specifically, we extend the SEIR com-partment model in a manner inspired by [9]. In contrastto the original SEIR model, the proposed pandemic modelcontains seven compartments rather than four. The consideredpopulation is divided into sub-populations (compartments),including: susceptible ( S ), susceptible quarantined ( S q ), ex-posed ( E ), exposed quarantined ( E q ), infectious ( I ), infectiousquarantined ( I q ) and recovered ( R ). Figure 1 depicts the sevencompartments and their relations. The four compartments,which are highlighted by a gray background, take a part in theeconomic extension of the model. In other words, the unionof these sub-populations could be viewed as a compartment ofindividuals which influence the deterioration of the economy.Fig. 1: The extension of the SEIR modelIndividuals in ( S ) are those who can be infected by the virus.Individuals from ( S ) can move either to ( S q ), ( E ), or ( E q ). In( S q ) there are susceptible individuals who went into quarantineas a preventive measure. Individuals are moved to ( E ) if theyhave contracted the virus, but do not suspect the infection.On the other hand, ( E q ) includes exposed individuals thatare quarantined. An individual enters this compartment whenexposed and being suspected as infected, or when transferredinto a preventive quarantine for other reasons. Individuals ofthe exposed states inevitably enter the infected states. For both,( I ) and ( I q ) it can be said that the individuals are infectious,but only those in ( I ) can actually spread the disease, whereasthose in ( I q ) cannot. When showing symptoms while in ( I ),an individual might get diagnosed with the virus and thuswill enter ( I q ). If an individual was in ( E q ) before, then itis assumed that infection was already suspected. Hence, such individuals go straight to ( I q ) once the incubation period isover. Individuals from ( I ) and ( I q ) will eventually progressto ( R ) once recovered and gained immunity from the virus.However, we consider immunity to be non-permanent andhence individuals will go back to being susceptible, ( S ), aftera period of being in ( R ).The dynamics of the extended SEIR model are representedby a set of first-order differential equations, i.e., state equa-tions, as common to such compartment models. The equationsare listed in the following. S (cid:48) = − c r (1 − t p ) c dp IS − c r t p (1 − c dp ) IS − c r t p c dp IS − p qr S + p qer S q + i lr R (1) Sq (cid:48) = c r (1 − t p ) c dp IS + p qr S − p qer S q (2) E (cid:48) = c r t p (1 − c dp ) IS + p qer E q − p qr E − i r E (3) Eq (cid:48) = + c r t p c dp IS + p qr E − p qer E q − i r E q (4) I (cid:48) = + i r E − d r I − i rr I (5) Iq (cid:48) = + i r E q + d r I − i qrr I q (6) R (cid:48) = + i rr I + i qrr I q − i lr R (7)Ten parameters are used in the proposed pandemic model.Here, ( c r ) refers to contact-rate . It describes how often andhow closely individuals interact with each other. The contact-rate governs how fast the virus spreads, together with ( t p ),which is the transmission probability when an infected personinteracts with a susceptible one. Next is the contact detectionprobability ( c dp ). It is the likelihood by which an individualwill know that she was in contact with an infected person.An individual that is aware of the contact will either go intoquarantine, meaning ( S q ) if transmission did not actually occur,or into ( E q ) if it did. The individual moves to ( E ) if contactwas made without detection, but with transmission. With a rateof p qr ( preventive quarantine rate ), individuals who believe tobe susceptible will go into quarantine without any concretesuspicion of infection. Namely, from ( S ) to ( S q ), or from ( E )to ( E q ). On the other hand, they will also leave quarantineagain, with a rate of p qer ( preventive quarantine end rate ).Immune individuals in ( R ) might return to being susceptible( S ), according to i lr , the immunity loss rate . Individuals willmove from ( E ) and ( E q ) into ( I ) and ( I q ) according to the incubation rate ( i r ). Based on d r , the diagnosis rate , infectedindividuals are identified and moved from ( I ) to ( I q ). Finally,infected individuals recover either with the infected recoveryrate ( i rr ), or with the infected quarantined recovery rate ( i qrr ).In the equations, the term c r (1 − t p ) c dp IS corresponds toindividuals that had contact with the infected ones but didnot get the infection. They know they were in contact andthus move to ( S q ). The term c r t p (1 − c dp ) IS correspondsto individuals who contracted the virus without suspicionand move to ( E ). The term c r t p c dp IS concerns the infected individuals with a suspicion of contact who move to ( E q ).The term p qr S involves those that are susceptible and are sentto preventive quarantine ( S q ), without any concrete suspicion.The term p qr E includes individuals who move into preventivequarantine without concrete suspicion, but they are actuallyexposed and hence move to ( E q ). The term p qer S q describessusceptible individuals who leave the preventive quarantine,hence returning to ( S ). The term p qer E q represents the exposedindividuals which are assumed to be unaware of their infection.These individuals are shifted back to ( E ). The terms i r E and i r E q describe the movement of exposed individuals to thecorresponding infected compartments, ( I ) and ( I q ), followingthe incubation period of the virus. The term d r I representsinfectious individuals who are tested, correctly diagnosed, andfinally quarantined to ( I q ). The term i rr I and i qrr I q involveindividuals which recovered from the disease and gainedimmunity, thus moving to ( R ). Finally, i lr R are individualswho lost their immunity and go back to being susceptible ( S ).The total population N is assumed to be static and equals thesum of all the compartments: N = S + S q + E + E q + I + I q + R (8) B. The Economic Model
In a pandemic scenario, the risk of economic depressionis high. Applying actions such as lockdowns have a negativeimpact on the economic activities. Generally, such pandemiccontrol actions will deteriorate the economy and will lead toa loss of wealth [11]. Thus, the proposed model accounts forthe pandemic influence on the economy. Specifically, we addedan equation to represent the influence of the model compart-ments on the
GDP . It is to be viewed as a relative valuewith a neutral zero baseline, starting at the beginning of thepathogen’s diffusion. It will then develop positively based onan assumed innate growth of the
GDP , and negatively basedon the status of the pandemic compartments. The relationbetween economy and pandemic is assumed to be unilateral inthis model. This means that the economic model is influencedby the pandemic’s course, but not vice versa.The current economic model assumes that no forced quaran-tines are implemented. This means that all individuals in ( S ),( E ), and ( R ) act without any significant restrictions and there-fore will maintain their usual economic behavior. This appearsto be possible because all these individuals are symptom-free,not quarantined, and believe themselves to be uninfected, evenif that is not actually true for individuals in ( E ). On the otherhand, ( S q ), ( E q ), ( I ), and ( I q ) have a considerable impact onthe economy. This is because the concerned individuals areeither quarantined, which translates to inhibited spending andproductivity, or infected, thus their contribution to the economyis reduced noticeably. Next, the following new compartment( C e ) is introduced to reflect individuals that influence theeconomy. C e = p qi S q + e qi E q + i i I + i qi I q (9)The economy compartment influence the GDP according tothe following differential equation:
GDP (cid:48) = b g − p i ( C e ) (10) The above equation involves six parameters, which bringsthe total amount of parameters for the combined model to 16.The baseline growth is indicated by ( b g ) and represents theinnate growth of the GDP, which we assume to be positiveand linear. The other parameters concern the influence ofthe pandemic on the economy. The pandemic influence ( p i )is a scaling factor for the overall impact of the pandemicon the economy. This allows scaling the total impact of thepandemic on the economy. The last four parameters describehow strong the negative impact of ( S q ), ( E q ), ( I ), and ( I q )are on the GDP . We assume that economic behavior ischanging depending on the compartment, e.g., individuals in( S q ) might be more active as individuals in ( I q ). Hence thereis one parameter dedicated for each of these compartments: susceptible quarantined impact , ( s qi ), exposed quarantinedimpact , ( e qi ), infected impact , ( i i ), and infected quarantinedimpact , ( i qi ). C. Control Policies
In the context of our model, a control policy refers toan action, intervention, or event which may influence thepandemic and the economy, and will alter their predicted trend.In the current study, two major types of actions are consideredincluding: social distancing and lockdown.It is suggested here to implement the policies by simpleparametric adaptations. Each of the selected policies influencesa single parameter within the model over a certain time frame.The influenced model-parameters are ( c r ) for social distancing,and ( p qr ) for lockdown. It should be noted that for simplicity,in our model, individuals that are locked-down are consideredas in quarantine. Each of these model-parameters is effected bya time dependent influence curve. This curve is defined by thefollowing five variables. The amplitude defines how strong thepolicy will affect the parameter. At a zero amplitude, a policywould have no effect at all. Any other value will simply beadded to the corresponding model parameter, individually andindependently for each time step. The trigger time defines thetime stamp t when the policy is going to become active. Fromthat point in time, the policy influence will steadily rise beforereaching its peak. Once reaching its peak, the influence willfade away. The extent by which each of these phases is lastingis governed by additional three variables including buildup , peak , and fade . The total duration of the policy is equal to thesum of these three parameters. Describing the policy by such atime dependent influence curve allows modeling the expecteddelay in the public response to new regulations and the publicfatigue over time to follow the rules. In the current study allthe curve-variables are kept constant except for the triggeringtime.In the following simulation, B´ezier curves implement theinfluence of the policy model. Each such curve is multipliedby a specific scaling factor, ensuring that the peak of theB´ezier curve actually corresponds to the given amplitude ofthe policy. As such, the x -coordinate of the curve correspondsto the time relative to the starting point of the policy, whereasthe y -coordinate corresponds to the policy’s efficacy at that P o li cy S t r eng t h Social DIstancing
Policy Duration P o li cy S t r eng t h Lockdown
Fig. 2: The influence curves for the social distancing andlockdown policies.relative time. The control points of the B´ezier curve are givenas:
ControlP oint , ControlP oint buildup, ControlP oint buildup, ControlP oint buildup + peak, ControlP oint buildup + peak, ControlP oint buildup + peak + f ade, (11)The control points define the shape of the curve. Whenmultiplying the B´ezier curve by the scaling factor, the policywill reach its stated amplitude in exactly one point. The scalingfactor also accounts for negative amplitudes. With this scaledcurve and the trigger time the policies influence is fully definedover time. Figure 2 shows plots of the associated scaled B´eziercurves for the policies social distancing and lockdown with theparameters as listed in Table I.III. T HE O PTIMIZATION P ROBLEM
Next, we formulate a bi-objective optimization problem thatminimizes health and economic objectives: min f ( t ) , min f ( t ) , (12) where f ( t ) = max t ( E ( t, t ) + E q ( t, t ) + I ( t, t ) + I q ( t, t )) and f ( t ) = − min t GDP ( t, t ) (13)where f corresponds to the health objective and f cor-responds to the economic objective. The goal of f is tominimize the peak of concurrent amount of infected or exposedindividuals ( flatten the curve ). The objective of f is to avoidor minimize a recession of the economy. If f is optimal( f = 0 ), the GDP will never be lower than at the beginningof the observed period. Here, t refers to the specified timestamp, where t ∈ R : 0 ≤ t ≤ t max and t max indicatesthe last observed time stamp. In our experiments we define t max = 300 . The decision vector is denoted as t . It containsthe triggering times for social distancing and lockdown, whichare the optimization variables of the current study. According to the Pareto-optimality approach, both of theseobjectives are considered simultaneously without any a-prioriarticulation of the objective preferences. Given the conflictingobjectives, this results in a set of Pareto-optimal solutions andtheir associated front in the objective-space. The proposedsolution approach provides trade-off information, which isexpected to support a rational strategy selection.IV. E XPERIMENTAL S ETUP
In this section, the experimental setup is presented. All sim-ulations are performed using MATLAB R2019b on a MacBookPro with 2.2 GHz Intel Core i7 processor and 16 GB of 2400MHz DDR4 RAM. The employed parameters are shown inTable I. For the optimization study, four algorithms have beenused including: NSGA II (Non-dominated Sorting GeneticAlgorithm II) [12], NSGA III (Non-dominated Sorting GeneticAlgorithm III) [13], MOPSO (Multi-Objective Particle SwarmOptimization) [14], and MOEA/D (Multi-Objective Evolution-ary Algorithm based on Decomposition) [15]. Such MOEAsare designed to solve multi-objective problems by searching forthe Pareto-optimal solutions and their associated front. Unlessotherwise stated, these algorithms are implemented using theircodes in PlatEMO [10] with their default settings.For each algorithm and run, we used a population size of100 and 4,000 evaluations to find optimal solutions. For thisparameter setting, the trigger time’s upper bound is set to 100since we assume that there will be no appropriate solutionswhen on-setting the policies later. Due to the stochastic natureof the employed algorithms, 36 independent runs were per-formed for each algorithm to reach statistical conclusions. Tocompare the algorithm performances, we produced a referencefront and an associated reference solution set by combining allof the individual Pareto fronts from all runs and by the removalof all dominated solutions (see Section V).V. R
ESULTS
This section presents the results of the simulation andoptimization experiments: Based on the defined objectives,we optimize the control policy’s trigger times to find optimaltrade-off solutions between the health and economic objectives.We compare NSGA II, NSGA III, MOEA/D, and MOPSOto find possible strategies for activating the control policies(social distancing and lockdown).Figure 3 illustrates some vectors as obtained by the con-sidered algorithms after 600 evaluations. Also shown is acurve, which can be viewed as an estimation of the Pareto-front. In view of this curve, it can be observed that noneof the algorithms was able to reach a good representationof the entire front within 600 evaluations. Next, Figure 4shows, for each algorithm, the resulting front after 4,000evaluations. Observing the figure, it can be concluded thatall the algorithms reached at least some part of the referencefront after 4,000 evaluations. These results should be evaluatedwhen compared with the reference front, which is shown inFigure 5. The conclusions from both figures is that MOEA/Dfailed to find a good representation of the front, whereas theother three algorithms did. It is particularly noticeable that it
TABLE I: Parameters used in the experiments.
Parameter Value
Extended SEIR Model Parameters
Initial S E , I S q , E q , I q , R c r t p i r p qr c dp d r i rr i qrr i lr Economy Model Parameters
Initial
GDP b g p i p qi e qi i i Fixed Policy Parameters
Social Distancing Amplitude -5Social Distancing Buildup 5Social Distancing Peak 40Social Distancing Fade 400Lockdown Amplitude 1Lockdown Buildup 5Lockdown Peak 10Lockdown Fade 40
Optimization setup.
Population Size 100Evaluations 4,000Runs 36Decision Variables Upper Bound 100Decision Variables Lower Bound 0
Health Objective (f ) E c ono m y O b j e c t i v e ( f ) Optimization Results: Comparison at 600 Evaluations
Pareto frontNSGA IINSGA IIIMOEA/DMOPSO
Fig. 3: Comparison of four multi-objective optimization algo-rithms after 600 evaluations each.lacks diversity, with many solutions clustered around the lowerright part of the front. It should be noted that the points, whichwere found by MOEA/D, along the vertical line at f =0.42 aredominated by the lowest point in the reference front.In Figure 5, five points are highlighted, indicated by thelabels A, B, C, D, and E . These were selected at the edges ofeach of the apparent regions of the reference front. Each ofthese points is associated with a particular optimal strategy, asshown in Figure 6. The shape of the combined front in Figure5 strongly suggests a significant trade-off that is inherentto the presented HED problem. Generally, no strategy can Health Objective (f ) E c ono m y O b j e c t i v e ( f ) Optimization Results: Comparison at 4000 Evaluations
Pareto frontMOPSO
Pareto frontMOEA/D
Pareto frontNSGA III
Pareto frontNSGA II
Fig. 4: Comparison of four multi-objective optimization algo-rithms after 4,000 evaluations. A single, randomly chosen runis plotted for each algorithm.
Health Objective (f ) E c ono m y O b j e c t i v e ( f ) Optimization Results: Combined Pareto Front
A B C DE
Fig. 5: Combined Pareto front of four multi-objective opti-mization algorithms over 36 runs with 4,000 evaluations each.simultaneously satisfy the health and economic objectives toa large extend. Moreover, the obtained front is not convex.In fact, the shape of the front indicates that if the traditionalweighted-sum solution approach was used, the obtained frontwould have resulted in the two extreme points of the front,hence hiding most of the potential alternatives.Table II aims to further highlight the differences betweenthe performances of the employed algorithms. It shows thestatistics for three major performance indicators including:Hyper-Volume (HV), Inverted Generational Distance (IGD),and Spread (SD). The statistical significance was tested with aMann-Whitney U test and p-value for all preceding Kruskal-Wallis tests is . .TABLE II: Mean (standard deviation) Performance Indicatorsat 4,000 Evaluations. Algorithm HV IGD SDNSGA II † ‡ ( )MOEA/D 1.8E-01 (7.6E-03)† 1.9E-02 (1.2E-02)† 6.8E-01 (1.1E-01)†MOPSO 1.9E-01 (5.8E-03)† 5.1E-03 (1.4E-02)† 7.5E-01 (1.0E-01)†The NSGA II (HV and IGD) or NSGA III (SD) performs significantly (p-value ≤ . ) better (†) and equivalent (‡), respectively, in comparison with thecorresponding algorithm. Social Distancing Trigger Time (t) -20020406080100 Lo ck do w n T r i gge r T i m e ( t ) Optimization Results: Pareto Set
CD BE A
Fig. 6: Combined Pareto set corresponding to the solutionsdisplayed by the combined Pareto front.In case of HV, NSGA II performed significantly betterthan all other algorithms. When evaluating IGD, NSGA IIand NSGA III performed significantly better than the otheralgorithms, while showing no significant difference betweeneach other. Considering SD, NSGA III performed significantlybetter than all other algorithms. Overall, NSGA II and NSGAIII performed best, showing both good convergence and spreadof the solutions along the Pareto-front. MOPSO also foundgood solutions, whereas MOEA/D performed the worst out ofthe four algorithms.
300 600 1000 1300 1600 2000 2300 2600 3000 3300 3600 4000
Evaluations H y pe r v o l u m e Optimization Results: Hypervolumes Over Time
NSGA IINSGA IIIMOEA/DMOPSO
Fig. 7: The HV averages vs. evaluations.Figure 7 provides a comparison of the convergence per-formance of the four optimization algorithms. It shows theevolution of their HV with respect to increasing number ofevaluations. The shown curves corresponds to the combinedPareto front from all the runs of each algorithm. Evidently,NSGA II and NSGA III converged faster and reached highervalues, as compared with the other two algorithms. MOPSOseems to have convergence difficulties, which can also beobserved when viewing the results in Figure 3. The stagnatingconvergence of MOEA/D suggests that its performance wouldnot change significantly with additional evaluations.VI. S
TRATEGY E VALUATION
The marked strategies in Figure 6 are discussed in thefollowing. The discussion assumes that the ratio objective-space’s scales reflects trade-offs of interest to the DMs (i.e., shifting by 0.02 in the health performance is comparable insome way to shifting by 0.1 in the economy performance).Comparing strategies A and B, it can be observed that inboth strategies, lockdown is triggered in an early stage. Thesestrategies differ primarily by the triggering time of the otheraction, i.e., social distancing. The associated performance ofA is the best in terms of health but the worst in terms ofeconomy. When triggering social distancing earlier, as done instrategy B, the economy performance is improved at the costof deteriorating the health performance. A reversed situationexists when comparing strategies E and D. Both involvestriggering the social distancing at the earliest possible time.Strategy E provides the best economy performance but theworst health performance. When shifting from E to D, there isalmost no gain in the health performance as compared with theloss in the economy. Finally, its worth to compare strategy Cwith strategies B and D. When shifting from B to C, the healthcost appears small as compared with the economy gain. Whenshifting from D to C there is also a trade-off, which mightbe worthwhile to consider. It should be noted that there is nosingle optimal strategy of triggering both policies late, whichappears logical. The following provides a general descriptionabout the dynamics of the pandemic compartments and theGDP, as obtained by alternative strategies including a baselinestrategy of not acting at all.
A. Strategy I: No policies
The first strategy represents the baseline of what would havehappened if no policies were applied. As Figure 8 shows, thisapproach quickly develops herd immunity while sustainingeconomic health. However, the public-health system suffersin this no-strategy scenario, which causes many fatalities,and may result in overloading and possibly a collapse of thesystem. In the studied case, during the pandemic, a maximumof more than 54% ( f = 0 . ) of the population would beinfected and the GDP would decline by . . Time -0.500.511.522.5 S , S q , E , E q , I, I q , R , E gdp Pandemic Economic Policy Model : No Policies
SusceptibleSusceptible QuarantinedExposedExposed QuarantinedInfectedInfected QuarantinedRecoveredGDP
Fig. 8: No control strategy at all
B. Strategy II: Health-based policy
Next, we consider a case where policymakers prioritizehealth over economy. In such a case, the optimization suggestsan early lockdown, followed by late social distancing. Figure9 illustrates the dynamics of the process in the case of settingthe triggering of the lockdown to be at t = 8 . andfor the social distancing at t = 66 . . This correspondsto the highlighted solution A from Figures 5 and 6. As canbe observed form Figure 9, these points in time correspondroughly to the beginning of the pandemic’s first and secondwaves. It appears that the chosen strategy managed to reducethe impact of those waves. There is a maximum of 26.5%( f = 0 . ) infected individuals at a specific point intime. In this scenario, the economy declines significantly( f = 0 . ) and stays depressed for relatively long time.Figure 9 also shows that herd immunity develops much lateras compared with the first (baseline) strategy of no actionat all, while having around 50% less simultaneously infectedindividuals. Time -0.6-0.4-0.200.20.40.60.811.21.4 S , S q , E , E q , I, I q , R , E gdp Pandemic Economic Policy Model
SusceptibleSusceptible QuarantinedExposedExposed QuarantinedInfectedInfected QuarantinedRecoveredGDP
Wave 1 Wave 2
Fig. 9: Optimal strategy for minimizing the health objective.
C. Strategy III: Economy-based policy
This strategy focuses on minimizing economic losses whileconsidering health as a secondary objective. It is character-ized by a very early social distancing, triggering already at t = 0 . , and a very late lockdown, triggering at t = 100 .The solution presented here is also highlighted as point Ein Figures 5 and 6. While this strategy achieves the highestpossible objective value of f = 0 (the economy would notdecline), it neglects the health objective and allows 42.23% ofindividuals to be infected simultaneously.Still, this strategy offers better performances in both objec-tives, when compared with the baseline condition (no policies).Figure 10 shows that this strategy would delay herd immunitysignificantly. We also observe that our model’s constraint to useeach policy exactly once can be counterproductive. Withoutthis rule, there would have been herd immunity around timestep 100, but the second policy still needed to be used,triggering a second infection wave. Future studies shouldinvestigate this dynamic more closely. Time S , S q , E , E q , I, I q , R , E gdp Pandemic Economic Policy Model
SusceptibleSusceptible QuarantinedExposedExposed QuarantinedInfectedInfected QuarantinedRecoveredGDP
Wave 1 Wave 2
Fig. 10: Optimal strategy for maximizing the economy bene-fits.
D. Strategy IV: A trade-off policy
Finally, we present an optimal trade-off strategy betweenhealth and the economy.
Time -0.500.511.52 S , S q , E , E q , I, I q , R , E gdp Pandemic Economic Policy Model
SusceptibleSusceptible QuarantinedExposedExposed QuarantinedInfectedInfected QuarantinedRecoveredGDP
Wave 2Wave 1
Fig. 11: Optimal trade-off strategy.For the chosen strategy, both actions are in place compara-tively early. The social distancing first, at a time of t = 0 . ,followed by the lockdown on time step . . Figure 11shows a reasonably good health objective of f = 33 . maximal simultaneously infected individuals, while the econ-omy losses reached a point below the baseline at f = 0 . .Comparing this strategy to the two strategies of the abovesubsections, it is obvious that it cannot compete with theirstrong focus on a single objective, but concurrently it does notneglect any objective as much as both of them did.VII. S UMMARY AND C ONCLUSIONS
This study proposes an extended macro-level pandemicmodel that includes an economic perspective. The modelallows examining the impact of various policies on both the economy and the pandemic spread. Next, a bi-objectiveoptimization problem is presented with conflicting health andeconomy objectives. The proposed model is used to search forthe Pareto-optimal strategies, for the aforementioned problem.In this study, each strategy is composed of triggering timesfor social distancing and lockdown. To solve the presentedproblem, four multi-objective optimization algorithms are usedincluding: NSGA II, NSGA III, MOPSO, and MOEA/D.NSGA II and NSGA III provided significantly better resultsthan the two other algorithms. Furthermore, this study includesa discussion on various strategies and their trade-offs, asrevealed by the optimization. As expected, these strategiesprovide better performances as compared with no action atall. What stands out in the current study is that a secondwave might be inevitable once the effects of the appliedpolicies fade. Since that no pharmaceutical interventions isconsidered here, only herd immunity can end the pandemicin the proposed model.Future research may extend the presented model in manyways. For example, explicit compartments might be added forasymptomatic infections, including the number of case fatal-ities, the vaccination rate, or effects on mental health. Also,implementing sector-specific curves can extend the economicmodel, e.g., travel, agriculture, or services. Financial supportand long-term effects on the economy are further points to takeinto consideration. It is noted that the current study focuseson two types of control policies, whereas many more canbe suggested. Furthermore, the policies’ strength and durationshould be optimized as well. Apart from these, the consideredobjectives could be extended. For example by considering bothlong and short term impacts not just on the spread of thepandemic and the economy, but also on aspects such as mentalhealth and education.A
CKNOWLEDGEMENT
The authors would like to acknowledge the support of
The Volkswagen Foundation to carry out this research. Theauthors would also like to thank Prof. Bruria Adini andProf. Leonardo Leiderman from Tel Aviv University, Israel,and Prof. Ingrid Ott from Karlsruhe Institute of Technology,Germany, for fruitful discussions on the topic of this study.R
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