key: cord-0599944-3lvelbzx authors: Fliess, Michel; Join, C'edric; d'Onofrio, Alberto title: Feedback control of social distancing for COVID-19 via elementary formulae date: 2021-10-04 journal: nan DOI: nan sha: b897d87a37e35b71ee1dcdb90f18066d70bebd77 doc_id: 599944 cord_uid: 3lvelbzx Social distancing has been enacted in order to mitigate the spread of COVID-19. Like many authors, we adopt the classic epidemic SIR model, where the infection rate is the control variable. Its differential flatness property yields ele mentary closed-form formulae for open-loop social distancing scenarios, where, for instance, the increase of the number of uninfected people may be taken into account. Those formulae might therefore be useful to decision makers. A feedback loop stemming from model-free control leads to a remarkable robustness with respect to severe uncertainties and mismatches. Although an identification procedure is presented, a good knowledge of the recovery rate is not necessary for our control strategy. In two years an abundant mathematically oriented literature has been devoted to the worldwide COVID-19 pandemic. Some of the corresponding calculations had even a significant political impact (see, e.g., [1] , [61] ). Note that in the field of mathematical epidemiology of infectious diseases the role of modeling human behavior became increasingly important in the last 15 years. It gave birth to a novel research field named behavioral epidemiology of infectious diseases: see, e.g., [45] , [83] . A novel control technique for improving the social distancing is presented here. This fundamental topic has already been tackled by many authors: see, e.g., [2] , [3] , [4] , [7] , [8] , [9] , [10] , [12] , [13] , [16] , [18] , [19] , [17] , [21] , [27] , [28] , [29] , [32] , [35] , [39] , [48] , [51] , [52] , [53] , [54] , [57] , [60] , [59] , [63] , [70] , [73] , [79] . Most of those papers are based on the famous SIR (Susceptible-Infected-Recovered/Removed) model, which goes back to 1927 ( [37] ), or on some modifications of its compartments. This communication is also using the SIR model: • When, like in several papers, the infection rate is the control variable, the SIR model is (differentially) flat ( [26] ). Remember that flatness-based control is one of the most popular model-based control setting, especially with respect to concrete applications: see, e.g., [6] , [11] , [20] , [38] , [41] , [42] , [50] , [62] , [64] , [66] , [67] , [72] , [69] , [75] , [76] , [77] , [88] for some recent publications. Note that flatness has already been utilized by [31] for studying COVID-19 but with other purposes. • There are severe uncertainties: model mismatch, poorly known initial conditions, . . . We therefore close the loop around the reference trajectory via model-free control, or MFC, in the sense of [22] , [23] . MFC, which is easy to implement, has already been illustrated in a number of practical situations. Some new contributions are listed here: [30] , [33] , [34] , [40] , [43] , [46] , [44] , [49] , [55] , [56] , [65] , [68] , [71] , [74] , [80] , [81] , [84] , [85] , [86] , [89] , [90] , [91] . Let us single out here the excellent work by [78] on ventilators, which are related to COVID-19. In order to be more specific consider a flat system with a single input u and a single output y. Assume that y is a flat output. Our strategy (see also [82] , [24] ) may be summarized as follows: 1) To any output reference trajectory y corresponds at once thanks to flatness an open-loop control u . 2) Let z be some measured output. Write z the corresponding reference trajectory. Set u = u + ∆u, where ∆u is the control of an ultra-local local model ( [22] ). Its output ∆z = z − z is the tracking error. Closing the loop via an intelligent controller ( [22] ) permits to ensure local stability around z in spite of severe mismatches and disturbances. Our paper is organized as follows: • Section II shows that the SIR model, where the infection rate is the control variable, is flat and the population of recovered/removed individuals is a flat output; the recovery rate is identifiable in the sense of [25] . • Section III is devoted to a flatness-based control strategy, i.e., to a feedforward approach. Elementary closed-form of the control and state variables are easily derived. Various scenarios, where for instance the number of uninfected persons is increased, may thus be easily suggested to decision makers. • Closing the loop via an intelligent proportional regulator, stemming from model-free control, is the subject of Section IV. Computer simulations confirm an excellent robustness with respect to severe uncertainties. • A time-varying recovery rate is estimated in Section V via algebraic estimation methods ( [25] ). Techniques from Section IV show however good performances if this rate is wrongly assumed to be constant. • Some suggestions for future investigations and someconcluding remarks may be found in Section VI. A. The SIR model The SIR model (see, e.g., [87] for a nice introduction) reads: S, I and R, which are non-negative quantities, correspond respectively to the fractions of susceptible, infected and recovered/removed individuals in the population. We may set therefore β, 0 < β ≤ β ≤ β, which is here the control variable, 1 and the parameter γ > 0 are respectively the infection and recovery rates. Equations (1)- (2) show that System (1) is flat and that R is a flat output ( [26] ). The other system variables may be expressed as differential rational functions of R, i.e., as rational functions of R and its derivatives up to some finite order: Remark 1: If γ is not constant, but a differentiable function of time, Equations (3)-(4)-(5) remain valid: System (1) is still flat and R is still a flat output. Equation (5) shows however thatγ is needed. C. An addendum on the SEIR model The SEIR model (see, e.g., [14] ) is a rather popular extension of the SIR model: where α > 0 is an additional parameter. Equation (2) becomes Equations (6)- (7) show that the SEIR model is also flat and that R is a flat output: Identifiability of the recovery rate Equation (5) yields γ = βS −İ I γ is a differential rational function .of R and β: It is thus rationally identifiable ( [25] ). Remark 2: The above equation does not work for an identifiability purpose if γ is time-varying:γ is sitting on its right hand-side. If we assume that I and S are measured, Equation (4) yields γ is still rationally identifiable with respect to I, S, β. It will be useful in Section V. I reference (t) = I 0 e −λt where t ≥ 0, 0 ≤ I 0 ≤ 1, and λ ≥ 0 is some constant parameter. If we set R(0) = 0, it yields and the open-loop control The following inequalities are staightforward: λ < γ follows from β > 0; γI 0 < λ follows from Introduce the more or less precise quantity β accept , where β < β accept < β. It stands for the "harshest" social distancing protocols which are "acceptable" in the long run. Equation (9) yields therefore The positive root of the corresponding quadratic algebraic equation Equation (11) leads to the notation demonstrates that the proportion of uninfected people decreases if the social distancing obligations are relaxed. Set γ = 0.1, β accept = 0.22. Figure 1 displays the open-loop evolutions of β, I, S when λ = λ accept . Those behaviors are quite satisfactory. Set ∆I(t) = I(t) − I reference (t), β(t) = β flat (t) + ∆β(t). In order to take into account the various uncertainties, introduce the ultra-local model ( [22] ) • The function F , which is data-driven, subsumes the poorly known structures and disturbances. • The parameter a, which does not need to be precisely determined, is chosen such that the three terms in Equation (12) are of the same magnitude. • Fest = − 6 τ 3 t t−τ ((t − 2σ)∆I(σ) + aσ(τ − σ)∆β(σ)) dσ, where τ > 0 is "small", gives a real-time estimate, which in practice is implemented via a digital filter. Introduce ( [22] ) the intelligent proportional controller, or iP, where K P is a tuning gain. Equations (12) and (13) yield Local stability is ensured. Remark 3: When compared to classic PIs and PIDs (see, e.g., [5] ), the gain tuning of the iP is straightforward. The sampling time interval is 2 hours. In Equations (12) and (13), a = 0.1, K P = 1. where [İ] est is an estimate ofİ obtained along the lines developed by [47] and [58] for algebraic differentiators. VI. CONCLUSION [15] questions the relevance and usefulness of such control-theoretic considerations for non-pharmaceutical mitigation policies against COVID-19. We certainly do not claim to set aside those objections in this preliminary short study. The combination however of flatness-based and model-free controls presents nevertheless some major advantages as demonstrated here and by [82] and [24] . An extra theoretical effort must be made in order to design control strategy as close as possible to the real epidemic control enacted by Public Health authorities. Summarizing, we consider this results proposed in this work as a theoretical ideal framework, to be filled with a more realistic picture: an implementable non-pharmaceutical control strategy. Preliminary results, which we recently obtained, indicate that the methodology here proposed is in the right direction (see [36] ). 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