Toward More Realistic Social Distancing Policies via Advanced Feedback Control

Abstract

A continuously time-varying transmission rate is suggested by many control-theoretic investigations on non-pharmaceutical interventions for mitigating the COVID-19 pandemic. However, such a continuously varying rate is impossible to implement in any human society. Here, we significantly extend a preliminary work (M. Fliess, C. Join, A. d’Onofrio, Feedback control of social distancing for COVID-19 via elementary formulae, MATHMOD, Vienna, 2022), based on the combination of flatness-based and model-free controls with respect to the classic parsimonious SIR model. Indeed, to take into account severe uncertainties and perturbations, we propose a feedback control where the transmission rate, i.e., the control variable, is piecewise constant. More precisely, the transmission rate remains constant during an appreciable time interval, which is not too large. Strict extended lockdowns may therefore be avoided. The poor knowledge of fundamental quantities such as the rate of infection hinders a precise calibration of the transmission rate. Thus, the results of our approach ought therefore not to be regarded as rules of action to follow accurately but as a guideline for a wise behaviour.

1. Introduction

The social distancing strategies and in particular the severe lockdown measures due to the worldwide COVID-19 pandemic (see, e.g., [1]) have stimulated a huge number of mathematically oriented investigations among which we select control-theoretic publications: See, e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Most diverse viewpoints have been developed. Those studies however do not seem to exert any influence on policy-makers (see, e.g., [36] for some explanations). Our aim is to start drawing a roadmap in order to change this state of affairs.

Like many authors in the above–mentioned works, we select the classic SIR compartmental model [37] (see also, e.g., [38,39,40]). An excellent justification for employing such a simple model has been presented by Sontag [33]: The social and political use of epidemic models must take into account their degree of realism. Good models do not incorporate all possible effects, but rather focus on the basic mechanisms in their simplest possible fashion. Not only it is difficult to model every detail, but the more details the more the likelihood of making the model sensitive to parameters and assumptions, and the more difficult it is to understand and interpret the model as well as to play what-if scenarios to compare alternative containment policies. It turns out that even simple models help pose important questions about the underlying mechanisms of infection spread and possible means of control of an epidemic. In addition, the rate of infection and other fundamental quantities are difficult, if not impossible, to know precisely (see, e.g., [41,42,43]). This epistemological hindrance to mathematical epidemiology provides further legitimacy for using a parsimonious modeling.

The transmission rate $\beta$, which corresponds to the social interactions and infection probability per contact, is chosen as the control variable, like in most papers which are quoted above. Thus, our work can be framed in the field of behavioural epidemiology of the infectious disease (see, e.g., [44]).

The SIR model happens then to be (differentially) flat. This concept [45,46] (see also the books [47,48,49,50]) has given rise, as is commonly known, to numerous concrete applications mainly in engineering (see, e.g., [51] for a recent excellent publication about cranes), but also in other domains (see, e.g., [52] in quantum physics). Also of particular interest here is its use [53] for COVID-19 predictions. Take a flat system with a single control variable $u\left(t\right)$ and a flat output variable $y\left(t\right)$. From a suitable reference trajectory $y^{*}\left(t\right)$, i.e., a suitable time-function, the corresponding open-loop nominal control variable $u^{*}\left(t\right)$ is derived at once from the flatness property. Severe uncertainties, like model mismatch, poorly known initial conditions, external disturbances, ..., prompt us to mimic what has been already done by [18,54,55], i.e., to close the loop via model-free control in the sense of [56,57]. Among the numerous remarkable concrete applications of this approach let us cite some recent ones in different domains (see, e.g., [58,59,60,61,62]), and, especially here, mask ventilators for COVID-19 patients [63]. Take another output variable $z\left(t\right)$ and its corresponding reference trajectory $z^{*}\left(t\right)$. The feedback loop, which relates $\mathsf{\Delta }\beta =\beta -{\beta }^{*}$ and the tracking error $\mathsf{\Delta }z=z-z^{*}$, is expressed as an intelligent Proportional, or iP, controller [56]. This is much easier to implement than traditional PI and PID controllers (see, e.g., [64]) and ensures local stability around $z^{*}$ with a remarkable level of robustness. Inspired by techniques in [65,66], which were performed in practice for a greenhouse and ramp metering on highways, we close the loop such that the control variable $u=u^{*}+\mathsf{\Delta }u$

• takes only a finite number of numerical values,
• remains constant during some time interval, two weeks here, in our computer simulations.

These features, which are new to the best of our knowledge, imply a limited number of different non-pharmaceutical interventions which moreover are not too severe. They might therefore be socially acceptable. Only low computing capacity is necessary for conducting numerous in silico experiments, i.e., computer experiments.

Our paper is organized as follows. The flatness property of the SIR model is shown in Section 2. An open-loop strategy is easily derived in Section 2.2. Section 3 introduces closed-loop control via model-free control. Several computer simulations, which considerably improve [18], are displayed in Section 4. Section 5 is devoted to a discussion of the possible implications of our approach.

2. SIR and Open-Loop Control

2.1. Flatness

The well known SIR model which studies the populations of susceptible, whose fraction is denoted as S, infectious, whose fraction is denoted as I, and recovered or removed, whose fraction is denoted as R), reads:

$$\left\{\begin{array}{c}\dot{S}=-\beta IS\hfill \\ \dot{I}=\beta IS-\gamma I\hfill \\ \dot{R}=\gamma I\hfill \end{array}\right.$$

(1)

The transmission rate $\beta$ and the recovery/removal rate $\gamma$ are positive. Equation (1) yields that $S+I+R$ is constant. We may set

$$S+I+R=1$$

Straightforward calculations yield:

$$\left\{\begin{array}{cc}\hfill I& =\frac{\dot{R}}{\gamma }\hfill \\ \hfill S& =1-R-I=1-R-\frac{\dot{R}}{\gamma }\hfill \\ \hfill \beta & =-\frac{\dot{S}}{IS}=\frac{\gamma \dot{R}+\ddot{R}}{\dot{R}\left(1-R-\frac{\dot{R}}{\gamma }\right)}\hfill \end{array}\right.$$

(2)

The system variables I, S and $\beta$ may be expressed as rational differential functions of R, i.e., as rational functions of R and its derivatives up to some finite order. In other words, System (1) is, as already observed [18], flat, and R is a flat output.

Remark 1.

The SEIR (Susceptible-Exposed-Infected-Recovered/Removed) model (see, e.g., [38,39]) is a rather popular extension of the SIR model:

$$\left\{\begin{array}{c}\dot{S}=-\beta IS\hfill \\ \dot{E}=\beta IS-\alpha E\hfill \\ \dot{I}=\alpha E-\gamma I\hfill \\ \dot{R}=\gamma I\hfill \end{array}\right.$$

(3)

where $\alpha >0$ is an additional parameter. Now

$$S+E+I+R=1.$$

(4)

Equations (3) and (4) show that the SEIR model is flat and that R is again a flat output:

$$\left\{\begin{array}{cc}\hfill I& =\frac{\dot{R}}{\gamma }\hfill \\ \hfill E& =\frac{\dot{I}+\gamma I}{\alpha }=\frac{\ddot{R}+\gamma \dot{R}}{\gamma \alpha }\hfill \\ \hfill S& =1-R-I-E=1-R-\frac{\dot{R}}{\gamma }-\frac{\ddot{R}+\gamma \dot{R}}{\gamma \alpha }\hfill \\ \hfill \beta & =-\frac{\dot{S}}{IS}\hfill \end{array}\right.$$

2.2. Elementary Formulae for Open-Loop Control

Select the following decreasing exponential reference trajectory, where

• $\lambda >0$,
• contrarily to [18] we do not start with $R^{*}\left(0\right)=0$,

$$R^{*}\left(t\right)=\mathrm{\Lambda }exp(-\lambda t)+R^{*}(\infty )$$

(5)

The quantities $\mathrm{\Lambda }$, $R^{*}(\infty )={lim}_{t\to +\infty }{R}^{*}\left(t\right)$ are given below. Formulae (2) and (5) yield

$$I^{*}\left(t\right)=-\frac{\mathrm{\Lambda }\lambda exp(-\lambda t)}{\gamma }$$

(6)

Formula (6) implies of course that $\mathrm{\Lambda }<0$. The fundamental constraint

$$\lambda <\gamma$$

follows at once from the obvious differential inequality $\dot{I}>-\gamma I$. Moreover

$$S^{*}\left(t\right)=1-R^{*}(\infty )-\mathrm{\Lambda }(1-\frac{\lambda }{\gamma })exp(-\lambda t)$$

(7)

The corresponding open loop control reads

$${\beta }^{*}\left(t\right)=\frac{\gamma \left(\gamma -\lambda \right)}{(1-R^{*}(\infty ))\gamma +\mathrm{\Lambda }(\lambda -\gamma )exp(-\lambda t)}$$

Note that ${\beta }^{*}\left(t\right)>0$. Thus

$${\beta }^{*}(\infty )=\underset{t\to +\infty }{lim}{\beta }^{*}\left(t\right)=\frac{\gamma -\lambda }{1-R(\infty )}$$

(8)

The parameter $\mathrm{\Lambda }$ in Equation (5) may be expressed thanks to Equation (6):

$$\mathrm{\Lambda }=-\frac{\gamma }{\lambda }{I}^{*}\left(0\right)$$

Thus

$$R^{*}(\infty )=R^{*}\left(0\right)-\frac{\gamma }{\lambda }{I}^{*}\left(0\right)$$

In order to avoid discontinuities at time $t=0$, set $R^{*}\left(0\right)=R\left(0\right)$, $I^{*}\left(0\right)=I\left(0\right)$. The following expressions for $\lambda$ and $R^{*}(\infty )$ will be used:

$$\mathrm{\Lambda }=-\frac{\gamma }{\lambda }I\left(0\right)$$

$$R^{*}(\infty )=R\left(0\right)-\frac{\gamma }{\lambda }I\left(0\right)$$

$${\beta }^{*}(\infty )=\frac{\lambda (\gamma -\lambda )}{(1+R(0\left)\right)\lambda +\gamma I\left(0\right)}$$

(9)

Introduce the more or less precise quantity ${\beta }_{\mathrm{accept}}$: It is the “harshest” social distancing protocols which is “acceptable” in the long run. Equation (9) yields

$${\beta }_{\mathrm{accept}}=\frac{\lambda (\gamma -\lambda )}{(1+R(0\left)\right)\lambda +\gamma I\left(0\right)}$$

and an algebraic equation of degree 2 for determining $\lambda$

$${\lambda }^2-\lambda \left((R^{*}\left(0\right)-1){\beta }_{\mathrm{accept}}+\gamma \right)-\gamma I^{*}\left(0\right){\beta }_{\mathrm{accept}}=0$$

(10)

The two roots of Equation (10) are real. The only positive one

$$\frac{(R\left(0\right)-1){\beta }_{\mathrm{accept}}+\gamma +\sqrt{{[(R\left(0\right)-1){\beta }_{\mathrm{accept}}+\gamma ]}^2+4\gamma I\left(0\right){\beta }_{\mathrm{accept}}}}{2}$$

is the value of $\lambda$ we are looking for. The corresponding reference trajectory and nominal open-loop control follow at once.

3. Closed-Loop Control

In order to take into account the poor modeling via Equation (1), introduce the ultra-local model [56]

$$\frac{d}{dt}\mathsf{\Delta }I=\mathcal{F}+\mathfrak{a}\mathsf{\Delta }\beta$$

(11)

where

• $\mathsf{\Delta }I=I-I^{*}$, $\mathsf{\Delta }\beta =\beta -{\beta }^{*}$,
• the constant parameter $\mathfrak{a}$, which does not need to be precisely determined, is chosen such that the three terms in Equation (11) are of the same magnitude.
• $\mathcal{F}$ subsumes the poorly known internal structure and the external disturbances.
• An estimate ${\mathcal{F}}_{\mathrm{est}}$ of $\mathcal{F}$ is given [56] by the integral

  $${\mathcal{F}}_{\mathrm{est}}=-\frac{6}{{\tau }^3}{\int }_{t-\tau }^t\left((t-2\sigma )\mathsf{\Delta }I\left(\sigma \right)+\mathfrak{a}\sigma (\tau -\sigma )\mathsf{\Delta }\beta \left(\sigma \right)\right)d\sigma$$

  which in practice may be computed via a digital filter.

An intelligent proportional, or iP, controller [56] reads

$$\mathsf{\Delta }\beta =-\frac{{\mathcal{F}}_{\mathrm{est}}+K_P\mathsf{\Delta }I}{\mathfrak{a}}$$

(12)

where $K_P$ a classic tuning gain and ${\mathcal{F}}_{\mathrm{est}}$ an estimate of $\mathcal{F}$. Combining Equations (11) and (12) yields

$$\frac{d}{dt}\mathsf{\Delta }I+K_P\mathsf{\Delta }I=\mathcal{F}-{\mathcal{F}}_{\mathrm{est}}$$

(13)

If the estimate ${\mathcal{F}}_{\mathrm{est}}$ is “good”, i.e., ${\mathcal{F}}_{\mathrm{est}}\approx \mathcal{F}$, then Equation (13) shows that we are led to a pure integrator. Taking $K_P<0$ yields at once

$$\underset{t\to +\infty }{lim}\mathsf{\Delta }I\left(t\right)\approx 0$$

Thus local stability around 0 is ensured in spite of mismatches and external disturbances.

4. Computer Simulations

Set in Equation (1) $\gamma ={\gamma }_{\mathrm{model}}=\frac{1}{7}$. The rate $I\left(t\right)$ of infected people is assumed to be counted every 2 h. This time lapse, which is of course too short in practice, has been chosen in order to have enough points for our numerical analysis. The iP (12) is employed in all the scenarios below, with $\mathfrak{a}=0.01$ and $K_P=15\mathfrak{a}$. Contrarily to [18], the scenarios below do not necessarily start at the beginning of the epidemic.

4.1. Unrealistic Scenarios

A naïve application of Section 2.2 leads to a continuous evolution of the control variable $\beta$, i.e., of the social distancing. This is impossible to implement in real life.

4.1.1. Scenario 1

Let us first assume that $I\left(0\right)$ and $R\left(0\right)$ are perfectly known. This initial time 0 is set after 35 or 45 days of epidemic spreading, where $\beta =3.6{\gamma }_{\mathrm{model}}$. Thus $I\left(0\right)$ after 35 days is less than after 45 days. Figure 1 and Figure 2 display excellent results, where ${\beta }_{\mathrm{acccept}}=0.95{\gamma }_{\mathrm{model}}$. Note that, even here, closed-loop control is necessary in order to counteract the unavoidable rounding errors.

Figure 1. Scenario 1: Open-loop control. (a) 35 days later: $\beta$, ${\beta }^{*}$ (- -, blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black); (b) 45 days later: $\beta$, ${\beta }^{*}$ (- -, blue), $\gamma$ (- ., green) and ${\beta }_{\mathrm{accept}}$ (- -, black).

Figure 2. Scenario 1: States. (a) 35 days later: S and $S^{*}$ (- -); (b) 35 days later: I and $I^{*}$ (- -); (c) 45 days later: S and $S^{*}$ (- -); (d) 45 days later: I and $I^{*}$ (- -).

4.1.2. Scenario 2

The initial time is set after 35 days of epidemic spreading. Introduce some mismatches:

• ${\gamma }_{\mathrm{real}}=1.05{\gamma }_{\mathrm{model}}$ (Figure 3a and Figure 4a,b),
  

  Figure 3. Scenario 2: Control. (a) $0.95{\gamma }_{model}$: $\beta$, ${\beta }^{*}$ (- -,blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black); (b) $1.05{\gamma }_{model}$: $\beta$, ${\beta }^{*}$ (- -,blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black).
  

  Figure 4. Scenario 2: States. (a) $0.95{\gamma }_{model}$: S and $S^{*}$ (- -); (b) $0.95{\gamma }_{model}$: I and $I^{*}$ (- -); (c) $1.05{\gamma }_{model}$: S and $S^{*}$ (- -); (d) $1.05{\gamma }_{model}$: I and $I^{*}$ (- -).
• ${\gamma }_{\mathrm{real}}=0.95{\gamma }_{\mathrm{model}}$ (Figure 3b and Figure 4c,d).

It is obvious then that the tracking of $I^{*}$ yields to deviate from $R^{*}$ and ${\beta }_{\mathrm{accept}}^{*}$.

4.2. Less Unrealistic Scenarios

Take the same conditions as in Section 4.1.1: no model mismatch, perfectly known initial conditions, the initial time 0 is set after 35 or 45 days of epidemic spreading.

4.2.1. Scenario 3

Allow only a finite number of numerical values of the control variable $\beta$. The control variable $\beta$ takes 50 uniformly distributed values between $0.5{\gamma }_{\mathrm{model}}$ and $1.2{\gamma }_{\mathrm{model}}$. Figure 5 and Figure 6 (35 days) and Figure 7 and Figure 8 (45 days) display excellent results.

Figure 5. Scenario 3: Control. $\beta$, ${\beta }^{*}$ (- -,blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black).

Figure 6. Scenario 3: States. (a) S and $S^{*}$ (- -); (b) I and $I^{*}$ (- -).

Figure 7. Scenario 4: Control. $\beta$, ${\beta }^{*}$ (- -,blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black).

Figure 8. Scenario 4: States. (a) S and $S^{*}$ (- -); (b) I and $I^{*}$ (- -).

4.2.2. Scenario 4

Allow $\beta$ now to take any value between $0.5{\gamma }_{\mathrm{model}}$ and $1.2{\gamma }_{\mathrm{model}}$ but to change only every 14 days. Figure 9 and Figure 10 (35 days) and Figure 11 and Figure 12 (45 days) display rather violent alternations in the social distancing rules. The tracking of $I^{*}$ remains good.

Figure 9. Scenario 5: Control. $\beta$, ${\beta }^{*}$ (- -,blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black).

Figure 10. Scenario 5: States. (a) S and $S^{*}$ (- -); (b) I and $I^{*}$ (- -).

Figure 11. Scenario 5: Control. $\beta$, ${\beta }^{*}$ (- -, blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black).

Figure 12. Scenario 5: States. (a) S and $S^{*}$ (- -); (b) I and $I^{*}$ (- -).

4.3. Scenarios 5–6: A More Realistic Policy

We follow Section 4.2 and combine the conditions on the control variable $\beta$ of Section 4.2.1 and Section 4.2.2. Thus $\beta$ takes only a finite number of values and change every 14 days. Correct results are provided in Figure 13 and Figure 14 (35 days) and Figure 15 and Figure 16 (45 days). The tracking of $I^{*}$ is still excellent.

Figure 13. Scenario 6: Control. $\beta$, ${\beta }^{*}$ (- -,blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black).

Figure 14. Scenario 6: States. (a) S and $S^{*}$ (- -); (b) I and $I^{*}$ (- -).

Figure 15. Scenario 6: Control. $\beta$, ${\beta }^{*}$ (- -,blue), $\gamma$ (- ., green) and ${\beta }_{acceptable}$ (- -, black).

Figure 16. Scenario 6: States. (a) S and $S^{*}$ (- -); (b) I and $I^{*}$ (- -).

5. Conclusions

We have designed a disease mitigation strategy, which is based on recent advances in control synthesis, where

• continuous manipulations of non-pharmaceutical interventions, which are an obstacle to the implementation of the vast majority of theoretical control strategies in epidemiology, even for the remarkably different problem of vaccination awareness campaigns (see [67]), are avoided,
• severe and long lockdowns are replaced by more subtle alternations of more or less strict social distancing measures.

Today empirical control strategies are adopted in practice. However, they are based on the principle of trial and error, which is very risky in epidemic context. Thus, the introduction of more rigorous but realistically constrained approaches might be of interest to policymakers. Another point to be stressed is that our overall results are also robust in presence of uncertainties in key parameters.

Several other questions arise:

• How should one modify the above approach and simulations when vaccinations and variants are taken into account? See, e.g., [68,69,70,71,72,73] for some preliminary modeling issues.
• Another interrogation is about the interpretation of the numerical values of the control variable $\beta$. What is, for instance, the influence of closing nightclubs as done in France and elsewhere? Available estimation techniques would suffer from the poor knowledge of I, and therefore of R and S. This inefficiency includes the techniques employed in [18], where the assumed knowledge of I is unrealistic.