Toward more realistic social distancing policies via advanced feedback control

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 of the classic 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. 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 behavior.

Like many authors in the above-mentioned works, we select the classic SIR compartmental model [Kermack & McKendrick(1927)] (see also, e.g., [Brauer & Castillo-Chavez(2012)], [Hethcote(2000)], [Murray(2002)]).An excellent justification for employing such a simple model has been presented by Sontag [Sontag(2021)]: 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., [Havers et al.(2020)], [Pérez-Rechel et al.(2021)], [Perkins et al.(2020)]).This epistemological hindrance to mathematical epidemiology provides further legitimacy for using a parsimonious modeling.
The transmission rate β, which corresponds to the social interactions and to 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 behavioral epidemiology of infectious disease (see, e.g., [Manfredi & d'Onofrio(2013)]).
The SIR model happens then to be (differentially) flat.This concept [Fliess et al.(1995)], [Fliess et al.(1999)] (see also the books [Sira-Ramírez & Agrawal(2004)], [Lévine(2009)], [Rigatos(2015)], [Rudolph(2021)]) has given rise, as is commonly known, to numerous concrete applications mainly in engineering (see, e.g., [Bonnabel & Clayes(2020)] for a recent excellent publication about cranes), but also in other domains (see, e.g., [Guéry-Odelin et al.(2019)] in quantum physics).Also of particular interest here is its use [Hametner et al.(2022)] for COVID-19 predictions.Take a flat system with a single control variable u(t) and a flat output variable y(t).From a suitable reference trajectory y * (t), i.e., a suitable time-function, the corresponding open-loop nominal control variable u * (t) 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 [Villagra & Herrero-Pérez(2012)], [Fliess et al.(2021)], [Fliess et al.(2022)], i.e., to close the loop via model-free control in the sense of [Fliess & Join(2013)], [Fliess & Join(2021)].Among the numerous remarkable concrete applications of this approach let us cite two recent ones in different domains (see, e.g., [Kuruganti et al.(2021)], [Lv et al.(2022)], [Sancak et al.(2021)], [Wang et al.(2021)]), and, especially here, mask ventilators for COVID-19 patients [Truong et al.(2021)].Take another output variable z(t) and its corresponding reference trajectory z * (t).The feedback loop, which relates ∆β = β − β * and the tracking error ∆z = z − z * , is expressed as an intelligent Proportional, or iP, controller [Fliess & Join(2013)].This is much easier to implement than traditional PI and PID controllers (see, e.g., [ Åström & Murray(2008)]) and ensures local stability around z * with a remarkable level of robustness.Inspired by techniques in [Lafont et al.(2015)] and [Join et al.(2022)], which were performed in practice for a greenhouse and ramp metering on highways, we close the loop such that the control variable u = u * + ∆u • takes only a finite number of numerical values, • remains constant during some time interval. 1hese 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. 2ur paper is organized as follows.The flatness property of the SIR model is shown in Section II.An open-loop strategy is easily derived in Section II-B.Section III introduces closed-loop control via model-free control.Several computer simulations are displayed in Section IV. 3 Section V is devoted to a discussion of the possible implications of our approach.

II. SIR AND OPEN-LOOP CONTROL A. Flatness
The well known SIR model,4 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: The transmission rate β and the recovery/removal rate γ are positive.Equation (1) yields that S + I + R is constant.We may set The system variables I, S and β 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 [Fliess et al.(2022)], flat, and R is a flat output.
Introduce the more or less precise quantity β accept : It is the "harshest" social distancing protocols which is "acceptable" in the long run.Equation ( 9) yields and an algebraic equation of degree 2 for determining λ The two roots of Equation ( 10

III. CLOSED-LOOP CONTROL
In order to take into account unavoidable mismatches and disturbances, introduce the ultra-local model [Fliess & Join(2013)] where • the constant parameter a, which does not need to be precisely determined, is chosen such that the three terms in Equation ( 11) are of the same magnitude.• F subsumes the unknown internal structure and the external disturbances.
• An estimate F est of F is given [Fliess & Join(2013)] by the integral which in practice may be computed via a digital filter.An intelligent proportional, or iP, controller [Fliess & Join(2013)] reads where K P a classic tuning gain and F est an estimate of F. Combining Equations ( 11) and ( 12) yields If K P < 0, and if the estimate F est is "good", i.e., F est ≈ F, then Thus local stability around 0 is ensured in spite of mismatches and external disturbances.

IV. COMPUTER SIMULATIONS 7
Set in Equation ( 1) γ = γ model = 1 7 .The rate I(t) of infected people is assumed to be counted every 2 hours. 8The iP (12) is employed in all the scenarios below, with a = 0.01 and K P = 15a.

A. Unrealistic scenarios
A naïve application of Section II-B leads to a continuous evolution of the control variable β, i.e., of the social distancing.This is obviously impossible to implement in real life.
1) Scenario 1: Let us first assume that I(0) and R(0) are perfectly known.This initial time 0 is set after 35 or 45 days of epidemic spreading, where β = 3.6γ model .Thus I(0) after 35 days is less than after 45 days.Figures 1 and 2 display excellent results, where β acccept = 0.95γ model .Note that, even here, closed-loop control is necessary in order to counteract the unavoidable rounding errors.

B. Scenario 2
The initial time is set after 35 days of epidemic spreading.Introduce some mismatches: • 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 an 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 for policy-makers.Another point to be stressed is that our overall results are also robust in presence of uncertainties in key parameters.