key: cord-1003656-hf8flijd
authors: Berger, Thomas
title: Feedback control of the COVID-19 pandemic with guaranteed non-exceeding ICU capacity
date: 2021-12-25
journal: Syst Control Lett
DOI: 10.1016/j.sysconle.2021.105111
sha: 186c4f64a157e94a52f73a52dbaf1f859b0f0815
doc_id: 1003656
cord_uid: hf8flijd

In this paper we investigate feedback control techniques for the COVID-19 pandemic which are able to guarantee that the capacity of available intensive care unit beds is not exceeded. The control signal models the social distancing policies enacted by local policy makers. We propose a control design based on the bang–bang funnel controller which is robust with respect to uncertainties in the parameters of the epidemiological model and only requires measurements of the number of individuals who require medical attention. Simulations illustrate the efficiency of the proposed controller.

One of the most dicult problems in the containment of the COVID-19 pandemic (or any other epidemic) is to generate reliable recommendations for action for policy makers in the presence of uncertain data and model parameters. In the present work, a robust control approach is taken, which is able to generate reliable signals for social distancing measures (or against them) in the pres- It is well documented [11, 15] that social distancing measures help to reduce infection rates and have an eect on the containment of the spread of SARS-CoV-2. On the other hand, social distancing has negative eects on both the economy and the mental and emotional health of the population. Therefore, policy makers face the hard decision of when to enact social distancing and when to relax the measures. To obtain recommendations for action, a reliable decision making mechanism based on a robust feedback control design would be desirable. In contrast to model-based techniques such as MPC (model-predictive control) used in [8, 12, 16] for instance, in the present paper we use a control methodology which does not require a specic model and is robust with respect to uncertainties in the system parameters. Because of the latter, it is not necessary to precisely identify all parameters such as the infection, recovery or death rates. Therefore, the approach may allow for an easier scalability, i.e., it may be applicable to dierent countries or regions Email address: thomas.berger@math.upb.de (Thomas Berger) or cities, without the need to precisely (re-)identify all the parameters for this region.

The above described question of when to enact social distancing measures or even a lockdown is a typical control theoretic question. Modelling this question utilizing a control input which takes only a nite number of different values, as suggested e.g. in [12] , a suitable feedback controller is able to generate the required signals. In the present work we restrict ourselves to a binary control input, i.e., with values in {0, 1}, in order to rst show the feasibility of the control design for this simple case. To this end, we combine a widely used model for the description of the COVID-19 pandemic from [17] with a control component proposed in [16] . The latter adds additional dynamics to the model which account for the eects of social distancing policies represented by the value of the control input and the response of the population to them (paying heed to possible delays).

The control objective is to keep the number of infected with moderate to severe symptoms (which typically require hospitalization) below a threshold dened by the number of available ICU (intensive care unit) beds. Another approach discussing control strategies which are able to bound the hospitalized population, by using control barrier functions, can be found in [1] . In the present paper, to achieve the control objective we exploit the bang-bang funnel controller developed in [14] , which is able to guarantee error margins in tracking problems and the control input switches between only two values. Funnel control proved an appropriate tool in several applications such as temperature control of chemical reactor models [10] , termination of brillation processes [3] , control of industrial servosystems [9] and underactuated multibody systems [4, 5] , voltage and current control of electrical circuits [7] , DClink power ow control [18] and adaptive cruise control [6] .

We stress that the focus of the present paper is not on modelling aspects, for which we rely on the available literature. The essential contribution is to show the fun-damental functionality of the proposed control law and to prove that it achieves that the available ICU capacity is not exceeded and that it is robust with respect to uncertainties in the parameters of the epidemiological model. 

with the gain function

In (1) the total population of a considered region is split into the following compartments: susceptible individuals S(t), infected but asymptomatic individuals I A (t), infected and symptomatic individuals I S (t), recovered individuals R(t), deceased individuals D(t) (due to the disease).

It is easily seen that the derivative of the sum of the above quantities is zero, d dt S(t)+I A (t)+I S (t)+R(t)+D(t) = 0 for all t ≥ 0, thus it stays constant over time and we may dene the initial population (assuming D(0) = 0) by

The other parameters used in (1) are summarized in Ta 

The last equation in (1) models the dynamics of social distancing policies and contains additional parameters to be explained in due course. The simplest way to model the population response would be to replace the transmission coecient β x , where x stands for either A or S, by β x (1 − u(t)) as in [1] , i.e., the control directly inuences the infection rates. However, the population response is typically not instantaneous, but people change their behavior with a certain delay this is accounted for by the last equation in (1) . As introduced in [16] , the function ψ can be seen as a time-varying population response which decreases the transmission coecients β A and β S in the case that social distancing measures are in place. Possible delays in the response are modelled by the parameters γ 0 , γ 1 . Note that we have ψ(t) ∈ [0, 1] for all t ≥ 0, where ψ(t) = 1 stands for the case of no distancing at all and ψ(t) = 0 would mean that any contacts between people are suppressed. Of course, the latter case is unachievable in practice, which is accounted for in the model. The control input u is assumed to take only binary values, i.e., u(t) ∈ {0, 1} for all t ≥ 0. This control signal models the policy enacted by the government, where u(t) = 0 means that no isolation measures are in place, and u(t) = 1 means that policy makers have determined social distancing.

The dynamics with which these measures inuence the response of the population are modelled by the last equation in (1) . Here, the value of the parameterψ ∈ (0, 1), see Table 1 , is additionally inuenced by a gain function K ψ dened in (2); the value of K ψ (t) decreases as the proportion of asymptomatic infected individuals increases. Assuming that initially ψ(0) = 1 (no isolation) we may infer

For further details on the model (1) we refer to [16, 17] . 

We now turn to the specication of a class of systems 

Before stating the assumptions (A1)(A3), we introduce the following denitions for a set of parameters from Σ:

With these quantities the assumptions are

The assumptions (A1)(A3) are mainly of technical nature and required in the proof of the main theorem. Nevertheless, we like to note that (A1)(A3) are typically satised in real epidemiological scenarios, see also Section 5.

The initial values S 0 , R 0 , I 0 S in (A2) for S(0) = S 0 , R(0) = R 0 and I S (0) = I 0 S are assumed to be positive to avoid technicalities and keep the proofs simple. In fact, these assumptions are true for the COVID-19 pandemic in practically every country in the world and nearly every region by the date this article is written (and taken as t = 0). Furthermore, in (A2) it is assumed that the initial value

restrictive assumption since the number of asymptomatic infected is typically much larger than the number of symptomatic infected individuals and usually the case at the beginning of a (local) outbreak. Finally, assumption (A3) denes a lower bound for the available ICU capacity φ + and is required to determine a non-empty range of control design parameters, see Section 3. It is easy to see that the assumptions are not contradictory and hence Σ is non-empty.

In order to obtain a feedback control law which is able to guarantee the transient behavior in (4) (and hence to achieve the control objective), we exploit the idea of bangbang funnel control from [14] . Bang-bang means that the control input switches between only two dierent values, hence this technique is suitable for our purposes. We stress that system (1) does not belong to the class of systems investigated in [14] and hence feasibility of bang-bang funnel control needs to be investigated separately. However, due to the special structure of (1) a simpler control law is possible here. To be precise, we consider the controller

In [14] , φ + and φ − are time-varying functions which determine a performance funnel for a certain signal to evolve in and ε + , ε − are safety distances required to guarantee that the signal evolves within this funnel. In the present paper, the aim is to achieve

thus, in view of the control objective formulated in Subsection 2.3, we may choose

We emphasize that the feedback control strategy (6) only requires the measurement of the number I S (t) of symptomatic infected individuals at time t, which, as mentioned in Section 2, can be measured accurately.

In the following we identify conditions on the safety distances ε + , ε − so that the application of the controller (6) to a system (1) with a tuple of parameters from Σ is feasible. In essence, these assumptions mean that by maintaining a strict lockdown, i.e., u(t) = 1 for t ≥ 0, it is possible to guarantee (4). If this is not possible, then no switching strategy can be successful.

The pairs of feasible control parameters depend on the choice of system parameters in general. For any Z ∈ Σ we dene the control parameter set

under the assumptions, where we use the constants dened in (5) in terms of Z,

Assumptions (A4) and (A5) are quite conservative since they are designed for worst-case scenarios, and hence they may impose hard restrictions in practice. However, the controller (6) may even be feasible if these assumptions are not satised and appropriate values for ε + and ε − can be identied by simulations.

It can be seen that, while (A4) simply gives a lower bound for ε − , assumption (A5) requires ε + to be within a range, where both the lower and upper bound depend on ε − . Therefore, the question arises whether (A4) and (A5) can be simultaneously satised. The following result gives an armative answer to this question, which is based on the assumptions (A1)(A3) on the system parameters Z. Proof. We show that there exist ε − , ε + > 0 which satisfy (A4) and (A5). This is true if, and only if, there exists M2

Then we may dene ε + := φ + − ε and choose ε − < ε, which satisfy (ε − , ε + ) ∈ C Z . The existence of ε as required above follows immediately from the observation that q(·) is continuous on the interval M2

. This shows that the set C Z is non-empty.

Finally, from assumptions (A4) and (A5) it is clear that

Since the values of the input u have the interpretation of enacted social distancing policies, it is typically not desired that fast switching between the two binary values occurs. To this end, in the remainder of this section we derive a lower bound for the interval of time between successive switches, the so-called dwell time. If u(t 0 ) = 1 with u(t 0 −) = 0, then I S (t 0 ) = φ + − ε + and as long as

Lower bounds for the dwell times are given in the following result.

Proposition 3.2. Let Z ∈ Σ, (ε − , ε + ) ∈ C Z and assume that (S, I A , I S , R, D, ψ) : R ≥0 → R 6 is a global solution of (1) under the control (6) . Then we have the following implications for all 0 ≤ t 0 ≤ t 1 :

Proof. First consider the case that I S (t 0 ) = φ + − ε + and

from which the rst implication follows. Note that φ + −ε + ε − > 1 and hence the lower bound is positive. Next consider the case that I S (t 0 ) = ε − and I S (t 1 ) = φ + − ε + . From (1), (3), Lemma A.1 and the fact that S(t) ≤ N − D(t) we may infer thaṫ

from which the second implication follows.

Note that the lower bound for the dwell time in u(t) = 0 in Proposition 3.2 depends on I A (t 0 ), and if this value is very large, then the lower bound may become negative, thus not giving any result. Furthermore, both lower bounds essentially depend on the parameters φ + , ε + , ε − , which can be adjusted in order to shape the minimum dwell time. In particular, for φ + → ∞ any desired minimum dwell time can be achieved (since I A (t 0 ) is bounded by N ).

In the following rst main result of the present paper we prove that, for any tuple of system parameters Z ∈ Σ and any pair of controller parameters (ε − , ε + ) ∈ C Z , the application of the bang-bang control law (6) Theorem 4.1. Let Z ∈ Σ and consider the associated system (1) . Further let (ε − , ε + ) ∈ C Z be a pair of controller parameters, φ + be as in (7) and assume that 6 such that (4) holds and u dened by (6) has nitely many jumps in each compact set.

Proof. Throughout the proof we use the constants dened in (5) without further notice. We divide the proof into several steps, where we rst show the existence of a local solution (Step 1). In Step 2 we show that u has only nitely many jumps in each compact set, which enables us to prove that the solution is actually global (Step 3). It then remains to show (4) in Step 4.

Step 1 : We show the existence of a local solution. Step 2 : We show that u has only nitely many jumps in each interval [a, b) ⊆ [0, ω). This is an immediate consequence of the rst implication in Proposition 3.2.

Step 3 : We show that ω = ∞. Since (S, I A , I S , R, D, ψ)

is bounded as S(t) + I A (t) + I S (t) + R(t) + D(t) = N and ψ(t) ≤ 1 for all t ∈ [0, ω), the case ω < ∞ is only possible when the jumps in u accumulate for t → ω, but this is excluded by Step 2.

Step 4 : It remains to show (4) . For brevity, set ε := φ + − ε + . Let t 1 := inf { t ≥ 0 | I S (t) = φ + } and, seeking a contradiction, assume that t 1 < ∞. Set

We show that I S (t 1 ) < φ + , which contradicts the assumptions and thus proves the claim.

Note thatİ S is almost everywhere dierentiable since u is piecewise constant and hence ψ is almost everywhere dierentiable. Furthermore, by (6) we have u(t) = 1 for all t ∈ [t 0 , t 1 ] and henceψ(t) = γ 1 K ψ (t)ψ − ψ(t) < 0.

Then we may calculatë 

then q(·) is monotonically decreasing on [t 0 , t 1 ], thus q(t) ≤ q(t 0 ) = ε < φ + . This nishes the proof.

We like to note that in Theorem 4.1 it is assumed that the initial value I S (0) lies within a certain interval. If the upper bound I S (0) ≤ φ + − ε + does not hold, then even a strict lockdown with u(t) = 1 for t ≥ 0 may not be sucient to guarantee (4). 

Assumption (A6) is of pure technical nature and typically satised in real epidemiological scenarios. Since the feasibility of (6) follows from Theorem 4.1, it is sucient to show that (ε − , ε + ) ∈ C Z is contained in any set CZ forZ in a neighborhood of Z ∈ Σ rob . Theorem 4.2. For all interior points Z of Σ rob and all (ε − , ε + ) ∈ C Z there exists δ > 0 such that B δ (Z) ⊆ Σ rob and

Proof. Let Z ∈ Σ rob and (ε − , ε + ) ∈ C Z be xed. Since Z is an interior point of Σ rob there exists δ 1 > 0 such that B δ1 (Z) ⊆ Σ rob . First, consider the function q : (8), where the appearing constants are dened in terms of Z. We show that q is strictly monotonically increasing. To this end, observe that q ′ (ε) = q1(ε) For anyZ ∈ Σ rob , let qZ be the corresponding function as in (8) as considered above. Dene the functions

then we nd that, for allZ ∈ B δ1 (Z),

) .

, and since f 3 , f 4 are continuous andZ → f 1 (Z, ε + ) is continuous as well on B δ1 (Z), there exists δ ∈ (0, δ 1 ) such that for allZ ∈ B δ (Z) we have ε − ∈ (0, f 1 (Z, ε + )), ε + ∈ (f 2 (Z), f 3 (Z)), which nishes the proof. Nevertheless, if a dierent epidemiological model is chosen, replacing (1), then Theorem 4.1 and its proof can be accordingly adjusted (even though a new analysis would be necessary) so that the controller (6) is still feasible for a suitable choice of parameters (ε − , ε + ) and its robustness properties are retained.

In this section we illustrate our ndings by a simulation of the epidemiological model (1) i.e., we have fty symptomatic or asymptomatic infected and we assume that 10% of the population already developed immunity due to a prior disease and hence belong to the class of recovered individuals. This number may be reasonable for the beginning of the COVID-19 pandemic (at least in some regions in Germany) as e.g. the COVID- 19 Case-Cluster-Study [19] (Heinsberg study) suggests.

We further assume that the disease spreads in Example City according to the parameters that have been used in [16] and which are summarized in Table 2 . Note that we chose a smaller value for p than in [16] , since the number of patients who require intensive care is typically much lower than the number of symptomatic infected in the model from [16] . In contrast to this, Fig. 2 shows the same scenario but with social distancing measures enacted according to the control (6) . For this simulation we have chosen ε + = 2.5ξ · n ICU , which is able to guarantee (4). Furthermore, we have chosen two dierent values for ε − , namely ε − 1 = 2ξ · n ICU and ε − 2 = 5ξ · n ICU . We stress that the chosen parameters satisfy assumptions (A1)(A3), where in particular the lower bound for φ + in (A3) is φ + > max{ M2 M1 , M 3 } ≈ 23.9 and hence satised. Furthermore, both pairs of controller parameters (ε − 1 , ε + ) and (ε − 2 , ε + ) satisfy (A4)(A5). Finally, (A6) is satised as well, so that the controller (6) is also robust.

Similar to various studies before, the simulations depicted in Fig. 2 show that social distancing measures are capable of reducing the total number of infected individuals and, as a consequence, the total number of disease induced deaths. The feedback controller (6) is able to guarantee (4) as shown in Fig. 2 (a) . It can be seen that, show that larger values of ε − lead to a faster switching with shorter periods between the switches, but, on the other hand, the pandemic is defeated at an earlier time point (i.e., the time T > 0 for which u(t) = 0 for all t ≥ T can be made smaller the larger ε − is chosen). These are two conicting objectives (few switches vs. shorter pandemic) and the policy makers have to decide which should be favored; the controller design parameters may be adjusted accordingly.

Another observation reveals that the total number of deaths, i.e., D max = lim t→∞ D(t) depends on the choice of ε − . Since minimizing D max seems a reasonable goal we have performed a set of simulations so that this is achieved, which led to the above value of ε − = ε − 1 . As shown in Fig. 2 (c) the number of deaths is indeed higher for ε − 2 and, as shown in Fig. 2 (b) , the total number of infected Finally, by way of comparison, we like to note that effective control methods for the COVID-19 pandemic based on model-predictive control (MPC) have been developed in [12, 16] . However, MPC requires accurate model data and measurements of all state variables for feasibility. As shown in [12] , uncertainty in the data and measurements can be compensated to a certain extent by using e.g. inter-J o u r n a l P r e -p r o o f Journal Pre-proof val predictions, however it is not possible to prove a priori that MPC does not exceed the available ICU capacity. This is one of the advantages of the approach presented here.

We Simulations illustrate that the proposed controller (6) is able to achieve the control objective and that a relax- these measures may be included in the model by additional control values u i ∈ (0, 1) as suggested e.g. in [12] .

Another topic of future research is the combination of dierent models for dierent countries or regions, where dierent control values (due to government policies) are active. In particular, it needs to be investigated how the interactions between dierent regions, based on migration ows, inuence the spread of the disease. Such an approach will possibly reveal which social distancing measures must be taken in neighboring regions with dierent outbreak levels.

Last but not least, we like to note that the approach presented here is not restricted to the model (1) or to the COVID-19 pandemic specically, but the feedback controller (6) can be applied, mutatis mutandis, to any epidemiological model available in the literature (modeling any epidemic), appended by the dynamics for the population response.

J o u r n a l P r e -p r o o f Journal Pre-proof for almost all t ∈ [0, ω). Then Grönwall's lemma implies that, for all t ∈ [0, ω),

We show (iv). Set z(t) := IA(t) IS (t) and y(t) := ψ(t)S(t) N −D(t) , then we may observe thaṫ z(t) = −pβ A z(t) 2 +((1−p)β A −pβ S )z(t)+(1−p)β S y(t)

Therefore, for any C > 0 it follows that z(t) ≤ max z(0),

for all t ∈ [0, ω). Since for C = B we nd that

for A as in (5), the claim follows.

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Proof. The proof of (i) relies on the well-known concept of integrating factors, see e.g. for almost all t ∈ [0, ω), and hence Grönwall's lemma implies z(t) ≥ e −αAt z(0). Therefore, we have I A (t) ≥ e −αAt z(0) + 1−p p I S (t) ≥ 1−p p I S (t), where the last inequality follows from the assumption I A (0) ≥ 1−p p I S (0) in (A2). To show (iii), set β max := max{β S , β A } and α min := min{α S , α A } and recall N − D(t) ≥ R(t) ≥ R 0 , theṅ