key: cord-0575667-64iepup4 authors: DIM, Emilio Molina; LJLL,; MISTEA,; Rapaport, Alain title: An optimal feedback control that minimizes the epidemic peak in the SIR model under a budget constraint date: 2022-03-11 journal: nan DOI: nan sha: b361a5307970b8d442974f317969f3bf16ecf214 doc_id: 575667 cord_uid: 64iepup4 We give the explicit solution of the optimal control problem which consists in minimizing the epidemic peak in the SIR model when the control is an attenuation factor of the infectious rate, subject to a L 1 budget constraint. The optimal strategy is given as a feedback control which consists in an singular arc maintaining the infected population at a constant level until the immunity threshold is reached, and no intervention outside the singular arc. Since the pioneer work of Kermack and McKendrick [16] , the SIR model has been very popular in epidemiology, as the basic model for infectious diseases with direct transmission (see for instance [22, 18] as introductions on the subject). It retakes great importance nowadays due to the recent coronavirus pandemic. In face of a new pathogen, non-pharmaceutical interventions (such as reducing physical distance in the population) are often the first available means to reduce the propagation of the disease, but this has economical and social prices... In [20, 19] , the authors underline the need of control strategies for epidemic mitigation by "flattering the epidemic curve", rather than eradication of the disease that might be too costly. Several works have applied the optimal control theory considering interventions as a control variable that reduces the effective transmission rate of the SIR model, and studied optimal strategies with criteria based on running and terminal cost over fixed finite interval or infinite horizon [4, 7, 8, 15, 21, 5, 9, 12, 17, 6] . However, the highest peak of the epidemic appears to be the highly relevant criterion to be minimized (especially when there is an hospital pressure to save individuals with severe forms of the infection). In [20] , the authors studied the minimization of the peak of the infected population under the constraint that interventions occur on a single time interval of given duration. In the present work, we consider the same criterion, but under a budget constraint on the control (as an integral cost) that we believe to be more relevant as it takes into account the strength of the interventions and does not impose an a priori single time interval of given length for the interventions to take place (although we have been able to prove that the optimal solution consists indeed in having interventions on a single time interval but with a control strategy different that the one obtained in [20] ). Let us also mention a more recent work [1] that considers a kind of "dual" problem, which consists in minimizing an integral cost of the control under the constraint that the epidemic stays below a prescribed value and an additional constraint on the state at a fixed time. The structure of the optimal strategy given by the authors in [1] is similar to the one we obtained without having to fix a time horizon and a terminal constraint. All the cited works rely on numerical methods to provide the effective control. Here, we give an explicit analytical expression of the optimal control. Let us stress that optimal control problems with maximum cost are not in the usual Mayer, Lagrange or Bolza forms of the optimal control theory [10] , for which the necessary optimality conditions of Pontryagin's Principle apply, but fall into the class of optimal control with L ∞ criterion, for which characterizations have been proposed in the literature mainly in terms of the value function (see for instance [3] ). Although necessary optimality conditions and numerical procedures have been derived from theses characterizations (see for instance [2, 11] ), these approaches remain quite difficult and numerically heavy to be applied on concrete problems. On another hand, for minimal time problems with planar dynamics linear with respect to the control variable, comparison tools based on the application of the Green's Theorem have shown that it is possible to dispense with the use of necessary conditions to prove the optimality of a candidate solution [14] . Although our criterion is of different nature, we show in the present work that it is also possible to implement this approach for our problem. The paper is organized as follows. In the next section, we posit the problem of peak minimization to be studied. In Section 3, we define a class of feedback strategies that we called "NSN", and give some preliminary properties. Section 4 proves that the existence of an NSN strategy which is optimal for our problem, and makes it explicit. Finally, Section 5 illustrates the optimal solutions on numerical simulations and discusses about the optimal strategy. We consider the SIR model where S, I and R denotes respectively the proportion of susceptible, infected and recovered individuals in a population of constant size. The parameters β and γ are the transmission and recovery rates of the disease. The control u, which belongs to U := [0, 1], represents the efforts of interventions by reducing the effective transmission rate. For simplicity, we shall drop in the following the R dynamics. Throughout the paper, we shall assume that the basic reproduction number R 0 is larger than one, so that an epidemic outbreak may occur. Assumption 1. For a positive initial condition (S(0), I(0)) = (S 0 , I 0 ) with S 0 +I 0 ≤ 1, we consider the optimal control problem which consists in minimizing the epidemic peak under a budget constraint where U denotes the set of measurable functions u(·) that take values in U and satisfy the L 1 constraint +∞ 0 u(t)dt ≤ Q Remark 1. From equations (1), one can easily check that the solution I(t) tends to zero when t to +∞ whatever is the control u(·), so that the supremum of Equivalently, one can consider the extended dynamics. with the initial condition (S(0), I(0), C(0)) = (S 0 , I 0 , Q) and the state constraint A solution of (3) is admissible if the control u(·) takes its values in U and the condition (4) is fulfilled. Let us denote the immunity threshold Note that S(·) is a non increasing function and that one hasİ ≤ 0 when S ≤ S h , whatever is the control. If S 0 ≤ S h , the maximum of I(·) is thus equal to I 0 for any control u(·), which solves the optimal control problem. We shall now consider that the non-trivial case. Under this assumption, we thus know that for any admissible solution, the maximum of I(·) is reached for S ≥ S h . For the control u = 0, one can easily check that following property is fulfilled and the maximum of I(·) is then reached for the value We define the "NSN" (for null-singular-null) strategy as follows. We denote the L 1 norm associated to the NSN control where u ψĪ (·) is the control generated by the feedback (6). This control strategy consists in three phases: 1. no intervention until the prevalence I reachesĪ (null control), 2. maintain the prevalence I equal toĪ until S reaches S h (singular control), Remark 2. There is no switch of the control between phases 2 and 3, because u(t) tends to zero when S(t) tends to S h , according to expression (6). One can check straightforwardly the the following properties are fulfilled. , the maximal value of the control u ψĪ (·) is given by Moreover, any solution given by the NSN strategy verifies We first show that the function L can be made explicit. Proof. Note first that whatever isĪ, S(·) is decreasing with the control (6). One can then equivalently parameterize the solution I(·), C(·) by As long as I <Ī, one has u = 0 which gives Remind, from the definition of I h , that the solution I(·) with u = 0 reaches I h in finite time. Therefore, one can define the numberσ One then obtains and with (8) one can write which finally gives the expression (7). Then, the best admissible NSN control can be given as follows. for which the solution with the NSN strategy is admissible is given by the valueĪ (Q) := I h QβS h + 1 (9) and one has L(Ī (Q)) = Q We give now our main result that shows that the NSN strategy is optimal. Proposition 2. Let Assumptions 1 and 2 be fulfilled. Then, the NSN feedback is optimal with whereĪ (Q) is defined in (9), andĪ is the optimal value of problem (2). Proof. When Q ≥ I h −I0 βS h I0 , the NSN strategy is admissible and the corresponding solution verifies max t≥0 I(t) = I 0 which is thus optimal. Consider now Q < I h −I0 βS h I0 . Let (S (·), I (·), C (·)) be the solution generated by the NSN strategy with I =Ī (Q), and denote u (·) the corresponding control. Let One can straightforwardly check with equations (3) that the solution is Remind, from Corollary 1, that one has C (t h ) = 0 by equation (10)). Clearly, one has (S(T ),Ĩ(T )) = (S h , I(t h )) andC(T ) < 0. We consider now in the (S, I) plane the simple closed curve Γ which is the concatenation of the trajectory (S(·),Ĩ(·)) on forward time with the trajectory (S(·), I(·)) in backward time: Then one has By the Green's Theorem, one obtains where D is the domain bounded by Γ (see Figure 1 as an illustration). This implies C(t h )