Optimal epidemic suppression under an ICU constraint
OOptimal epidemic suppression underan ICU constraint ∗ Laurent Miclo † , Daniel Spiro ‡ and Jörgen Weibull § May 5, 2020
Abstract
How much and when should we limit economic and social ac-tivity to ensure that the health-care system is not overwhelmedduring an epidemic? We study a setting where ICU resources areconstrained while suppression is costly (e.g., limiting economic in-teraction). Providing a fully analytical solution we show that thecommon wisdom of “flattening the curve”, where suppression mea-sures are continuously taken to hold down the spread throughoutthe epidemic, is suboptimal. Instead, the optimal suppressionis discountinuous. The epidemic should be left unregulated in afirst phase and when the ICU constraint is approaching societyshould quickly lock down (a discontinuity). After the lockdownregulation should gradually be lifted, holding the rate of infectedconstant thus respecting the ICU resources while not unnecessar-ily limiting economic activity. In a final phase, regulation is lifted.We call this strategy “filling the box”. ∗ We wish to thank Tommy Andersson, Hannes Malmberg and Robert Östling forvaluable comments. † Toulouse Institute of Mathematics, Toulouse School of Economics and CNRS.Email: [email protected]. ‡ Department of Economics, Uppsala University. Email:[email protected]. § Department of Economics, Stockholm School of Economics. Email:[email protected]. a r X i v : . [ ec on . T H ] M a y Introduction
Amid the Covid-19 health and economic crisis one question stood atthe centre of professional opinion: How much and when should we limiteconomic and social activity to ensure that the health-care system isnot overwhelmed? This question embodies two simultaneous goals whenfighting a pandemic: (1) To ensure that each infected person gets thebest possible care, we need to ensure that the capacity of the health-care system (henceforth the ICU constraint) is never breached. UnderCovid-19 the ICU constraint is essentially the number of available respi-rators, indeed a scarce resource in most countries. It was perhaps bestepitomized by the UK slogan “Protect the NHS” and by the EmpiricalCollege report (Ferguson et al., 2020). (2) The more one is suppressingthe spread the costlier it is since, absent a vaccine, suppression boilsdown to keeping people away from each other thus limiting economicand social life.This paper extends the standard S.I.R. model (Kermack and McK-endrick, 1927) with those two extensions to provide an analytical answerto the above question. Our answer departs from the common wisdom.During the Covid-19 pandemic, authorities, news reporting and policymakers popularized the ideal policy as “flattening the curve”, i.e., im-posing continuous limitations to lower the number of simultaneouslyinfected in all time periods. This would ensure that the peak of thecurve never crossed the ICU constraint. We show that this policy issuboptimal. Instead, the optimal policy is characterized by what we call“filling the box” and a discontinuous suppression. More precisely, it pre-scribes (Theorem 1 and Figure 2) leaving the spread unregulated duringa first phase. As the number of infected approaches the ICU constraintwe enter a second phase where harsh suppression measures are imposedat once (a discontinuity) but afterwards gradually relaxed. The aim ofpolicy in this second phase is to stop the number of infected just belowthe ICU constraint and keep it constant at that level. The discontinu-ous tightening followed by gradual relaxation of suppression is optimalsince the underlying growth of infections is highest in the beginning ofthis phase. In a third phase, once the underlying growth of infectionssubsides, no suppression measures are taken.The logic behind this result is simple, but bears relevance for a diseasespreading such as Covid-19. When access to a vaccine is not realisticwithin a sufficiently near future and pinpointing each infectious personis not feasible, which is implicitly assumed in our model, full eradication See, e.g., the Empirical College report (Ferguson et al., 2020), Branswell (2020),Time (2020), Pueyo (2020), even Donald Trump (The Sun, 2020) and many more.
2s not possible. What remains then, is letting the infection spread in thepopulation but ensuring that each person gets best possible care, i.e.,ensuring the available respirators are sufficient at all times. But there isno point in leaving some of the respirators idle (if considering risk onecan view the ICU constraint as being the number of respirators with amargin). Hence, early suppression is unnecessary and costly. Once thenumber of infections reaches the ICU constraint, drastic suppression hasto be installed to keep it below. But also here it is unnecessarily costlyto suppress the whole curve as it leaves idle respirators. Hence, theaim during the second phase is to precisely fill the ICU capacity. Oncethe infection rate goes down so that the ICU constraint is no longerbinding – the third phase – suppression can be lifted. The number ofrespirators and the time axis can essentially be thought of as a box.“Filling the box” then simply means respecting the ICU constraint whilenot incurring costs to leave idle resources. In the concluding remarks wefurther discuss how various implicit assumptions may change this result.Apart from the policy implication our main contribution lies in themodel and the analysis itself. We develop and show how to fully ana-lytically solve an epidemic-economic model for the optimal suppressionpolicy. Importantly, the suppression policy is allowed to be fully timevarying. Our approach is thus clearly distinguished from a large numberof recent papers (not least in economics) that analyze policies numeri-cally (e.g., Wearing et al., 2005; Iacoviello and Liuzzi, 2008; Lee et al.,2011; Kar and Batabyal, 2011; Iacoviello and Stasio, 2013; Giamber-ardino and Iacoviello, 2017; Gollier, 2020; Wang, 2020; Farboodi et al,2020; Eichenbaum et al., 2020; Alvarez et al, 2020). In order to makeanalytical headway we abstract from many nuances that such, numer-ical, papers consider including the possibility of testing (Gollier, 2020;Wang, 2020, Berger et al., 2020), the arrival of a vaccine (Zaman et al.,2008; Iacoviello and Liuzzi, 2008; Lee et al., 2011; Kar and Batabyal,2011; Giamberardino and Iacoviello, 2017; Farboodi et al, 2020), treat-ment and education (e.g., Bakare et al., 2014) group heterogeneity (e.g.,Shim, 2013; Sjödin, 2020), contact tracing (see, e.g., Wearing et al., 2005;McCaw and McVernon, 2007; Britton and Malmberg, 2020; and refer-ences therein), time delays (Zaman et al., 2009), network effects (e.g.,Gourdin, 2011) and individual decision making (Farboodi et al, 2020;Eichebaum et al., 2020). Our exercise is a stepping stone for consideringalso such aspects in future work. To our knowledge ours it the first paperto at all consider optimal policy under an ICU constraint.In the epidemiology literature there exists a series of papers with an-alytical solutions for optimal policy. For a literature review on the early There is a much larger literature studying epidemics without controls, of course, The paper closest to ours is an elegant analysis byKruse and Strack (2020). They also look at optimal suppression withcosts which are increasing in suppression. They show existence of anoptimizer for a rather general health-cost function but only solve for theoptimizer in the special case where the health costs are linear in thenumber of currently infected. This is equivalent to assuming that thetotal number of deaths (over time) is proportional to the total number ofinfected (the linearity assumption implies bang-bang solutions for sup-pression) so it does not (directly) matter how many are infected at thesame time like is the focus of our paper (in that sense, it is similar toGrigorieva et al., 2016, and Grigorieva and Khailov, 2014). Our con-tribution is thus complementary to theirs since we study a health costwhich specifically captures the overwhelming of the health-care system.
Our model setup closely follows the canonical Susceptible-Infectious-Removed model (Kermack and McKendrick 1927; see also Brauer andCastillo-Chavez, 2011, for an excellent overview). At any time t ě , let x p t q be the population share of individuals who at time t are suscepti- see for instance Dickison et al (2012) and Brauer and Castillo-Chavez (2011), Pastor-Satorras et al. (2015) and references therein. Many papers analytically solve for a suppression policy while respecting a budgetconstraint (Hansen and Day, 2011; Bolzoni et al., 2019) or a time constraint (so thatthe suppression cannot be too long, Morris et al 2020) but disregarding that moresuppression within a time period is costlier than less suppression (Bolzoni et al., 2017;Piunovskiy et al., 2019). This is isomorphic to restricting the suppression policy tobe binary since, once there is suppression within a time period, it may as well beat full force. We allow the suppression policy to take any value within a period andchange in any way between time periods. Grigorieva et al. (2016) and Grigorieva andKhailov (2014) analyze an objective of minimizing the number of infectious duringor at some end period, but the control bears no cost. Abakus (1973) and Behncke(2000) analyzes an objective of minimizing the total (over time) number of infectedquarantine (see his Section 3) but the cost of putting a person in quarantine is onlytaken once so is independent of the length of quarantine. Finally, Gonzales-Eirasand Niepelt (2020) analyze an S.I. model, finding, just like some of the papers above,that the optimal control is binary. They also show existence of an optimizer for when a vaccine can arrive. y p t q be the population share of individualswho are infected at time t . All infected individuals are assumed to becontagious, and population shares are defined with respect to the initialpopulation size, N . Let λ p t q be the rate at time t of pairwise meet-ings between susceptible and infected, and let q p t q be the probability ofcontagion when an infected person meets a susceptible person at time t . Write b p t q “ λ p t q q p t q . Infected individuals are removed from thepopulation at rate α ą . This may be either because they get immuneor because they die. An important assumption is that those who areinfected never again become susceptible.The population dynamic is then defined by the following simple sys-tem of ordinary differential equations: " x p t q “ ´ b p t q y p t q x p t q y p t q “ b p t q y p t q x p t q ´ αy p t q (1)The initial condition is x p q “ ´ ε and y p q “ ε , for some ε Pp , q . That is, the infection enters the population at time zero ina population share ε ą . The state space of this dynamic is ∆ “ (cid:32) p x, y q P R ` : x ` y ď ( . The only difference from the standard S.I.R.model is that the propagation coefficient b p t q , instead of being a con-stant over time, say, b p t q “ β ą for all t ě , we here allow it to varyover time. Indeed, we will view b : R ` Ñ R ` as a function in the hands of asocial planner who strives to minimize the economic and social costs ofshutting down parts of the economy and social life in the population,while never letting the population share of infected individuals, y p t q ,exceed an exogenously given level γ . The latter is interpreted as the ca-pacity of the health-care system to treat infected patients. We refer toit as the ICU capacity or constraint . It is meant to capture a situation,such as under Covid-19, where if the number of simultaneously and se-riously ill exceeds the number of respirators implies instant death. Notbreaching the ICU capacity thus ensures that all get the best possiblecare. Implicitly this assumes that the duration of the infection in an individual is ani.i.d. exponentially distributed random variable with mean value { α . If z p t q denotes the population share of removed individuals in a standard S.I.R.model, then its dynamic is z p t q “ αy p t q , and x p t q ` y p t q ` z p t q “ at all times t ě . If, for example, on average 20% of those infected need intense care and thenumber of ICUs is C in a population of size N , then γ “ C { N . To allow for risk,the ICU constraint can of course also include a margin to the actual limit. In practice this of course is no guarantee against fatalities. We implicitly assumethat those that pass away despite getting the best care are not within the control
5e assume that the cost of keeping b p t q below its natural , or un-regulated level β is a linear function of the difference, while there is nocost of moving b p t q above β . The latter assumption is made to "tilt thetable" against us in the subsequent analysis, where we will show thateven under this assumption, it is suboptimal to enhance the propagationof the infection even if this can be done at no cost. Formally, the costfunction C : B Ñ R ` is defined by C p b q “ ş r β ´ b p t qs ` dt , and thesocial planner faces the optimization program min b P B γ C p b q , (2)where B is the class of piecewise continuous functions b : R ` Ñ R ` thathave finitely many points of discontinuity (including no discontinuity atall), and B γ , for any given γ ą , is the subset of functions in B forwhich y p t q ď γ at all times t ě . We focus on situations in which ε ă γ , that is, when the initialinfection level is below the ICU capacity constraint. Moreover, we as-sume that β ą α . Otherwise the population share of infected individualsdoes not increase from its initial value, which would imply herd immu-nity already from the outset, and thus the social planner’s optimizationprogram then has a trivial solution; laissez-faire , that is, b p t q ” β . We first comment on the set of policies allowed by the optimizationprogram. For functions b P B , it can be shown that (1) defines a uniquesolution trajectory through any given state p x p t q , y p t qq P ∆ and time t ě . Trivially all constant functions b : R ` Ñ R ` , with b p t q “ δ for some δ ą , belong to B . However, they do not all belong to B p γ q ,i.e., they may violate the ICU constraint. It is easy to show that suchconstant policies belong to B p γ q if δ is sufficiently low, for any given γ ą . Thus, to choose δ as high as possible, while keeping y p t q ď γ for all t ě , is a feasible policy (belongs to B p γ q ), and can be called flattening the curve . However, such a policy incurs an infinite cost since of the policy maker. See, e.g., Kruse and Strack (2020), Grigorieva et al. (2016)and Grigorieva and Khailov (2014) for models where the objective is to minimize thenumber of infected. To be more precise, we require that there is a finite set T Ă R ` such that thefunction b : R ` Ñ R ` is continuous at all other points, and that it is everywhereleft-continuous and has a right limit. We also require that b is positive everywhere,except on at most finitely many connected components. See Appendix for a proof. For this class of functions b , the time derivatives in (1)represent left derivatives. The solution trajectories p x p t q , y p t qq t ą are then uniquelydetermined and are continuous in t .
6t lasts forever, if and only if δ ă β . An alternative feasible controlfunction, with finite cost, is to only temporarily keep b p t q at a constantlevel δ ă β over a carefully chosen finite time interval, where δ is suchthat y p t q ď γ for all t ě . However, as will be shown below, also such“temporary constant shut down” policies are suboptimal. Before turningto the formal statement of our main result, we analyze some generalproperties of the dynamic induced by (1). Some well-known properties of the solutions to standard S.I.R. mod-els hold also here (see Brauer and Castillo-Chavez, 2011). A key suchproperty is that the population share of susceptible individuals, x p t q , isnon-increasing over time t . Roughly speaking, this follows from the firstequation in (1), since b p t q is always non-negative and y p t q is positiveat all times t ě . Being bounded from below by zero, x p t q necessarilyhas a limit value as t Ñ 8 , which we denote x . According to (1), alsothe sum y p t q ` x p t q is strictly decreasing over time t , and hence alsothis sum has a limit value, x ` y . By standard arguments, it is easilyverified that this implies that y “ . In other words, in the very longrun, the population share of infected individuals tends to zero. Denotingby z “ lim t Ñ8 z p t q the total population share of removed individualsduring the whole epidemic, we thus have z “ ´ x , and N z isapproximately (for large N ), the total number of infected individualsduring the epidemic.Let us now consider the solution to (1) through any through anygiven state p x p t q , y p t qq P ∆ and time t ě , where ă x p t q ă and ă y p t q ă . Dividing both sides of the second equation in (1) by x p t q ą and integrating, one obtains ln x p t q “ x p t q ´ ż tt b p s q y p s q ds @ t ě t .Moreover, integrating the sum of the two equations in (1), we obtain x p t q ` y p t q “ x p t q ` y p t q ´ α ż tt y p s q ds @ t ě t . In particular, if b p t q “ δ ą for all t ě t , for some δ ą , then for all t ě t : ln x p t q x p t q “ ´ δ ż tt y p s q ds “ δα r x p t q ´ x p t q ` y p t q ´ y p t qs , If y ą , then x p t q ` y p t q Ñ ´8 . y p t q “ y p t q ` αδ ln ˆ x p t q x p t q ˙ ´ x p t q ` x p t q @ t ě t . (3)This equation is well-known for S.I.R. models. Moreover, (3) impliesthat p x p t q , y p t qq Ñ p , x q P ∆ , where x by continuity solves (3) for y p t q “ , so x “ αδ ln x x p t q ` x p t q ` y p t q . (4)Since x p t q is strictly decreasing, x ă x p t q . It is easily verified thatthe fixed-point equation (4) has a unique solution x P p , x p t qq . The maximal population share of infected individuals, ˆ y “ sup t ě y p t q (still for b p t q ” δ for some δ ą ) is the peak infection level. It obtainswhen y p t q “ , or, equivalently (by (1)), when x p t q “ α { δ . From (3) weobtain ˆ y “ ` αδ ln ˆ αδ p ´ ε q ˙ ´ αδ . (5)This is thus the maximal population share of infected individuals whenthe suppression policy is held constant over time. Once the popula-tion share x p t q of susceptible individuals has fallen below the level α { δ ,achieved precisely when y p t q “ ˆ y , flock immunity is obtained; the pop-ulation share y p t q of infected individuals falls. In particular, the limitstate as t Ñ 8 is Lyapunov stable. That is, there is no risk of a secondinfection wave, since after any small perturbation of the limit populationstate p x , q P ∆ , obtained by exogenously inserting a small populationshare of infected individuals, the population share of infected individu-als will fall gradually back towards zero, while the population share ofsusceptible individuals gradually moves towards a somewhat lower, newlimit value.Equation (5) is particularly relevant for the case when δ “ β , thatis, under under laissez-faire. Because if the peak of the infection wavethen does not exceed the ICU capacity constraint, that is, if ` αβ ln ˆ αβ p ´ ε q ˙ ´ αβ ď γ, (6)then laissez-faire is optimal; b ˚ p t q ” β solves (2) at no cost. But if thepeak is above the ICU constraint, regulation has to be implemented.This is the topic of the next subsection. The right-hand sides of () is a continuous and strictly increasing functions f : p , x p t qq Ñ R of x . Moreover, f p x q Ñ ´8 as x Ó and f p x p t qq “ x p t q ` y p t q ą x p t q , f ą and f ă , so there exists a unique fixed point in p , x p t qq . .2 Optimization To the best of our knowledge, the optimization program (2) has not beenanalyzed before. We summarize below our main result, which treats allcases when laissez-faire is suboptimal. If (6) does not hold, then thesolution orbit (3) under laissez-faire intersects the capacity constraint y p t q “ γ twice. Let τ ą be the first such time and let x p τ q be thepopulation share of susceptible individuals at that time. Then τ “ min " t ě x p t q “ ´ γ ` αβ ln ˆ x p t q ´ ε ˙* (where x p t q is solved for according to (1) when b p t q ” β ), and x p τ q isthe larger of the two solutions to the associated fixed-point equation in x , x “ ´ γ ` αβ ln ˆ x ´ ε ˙ . (7)We note that x p τ q ą α { β . Let τ “ τ ` αγ ˆ x p τ q ´ αβ ˙ Theorem 1
Suppose that ε ă γ , α ă β and (6) does not hold. Thereexists a solution to program (2), one of which is the policy b ˚ P B p γ q defined by b ˚ p t q “ $&% β for t ď τ β ` αβγ p τ ´ t q for τ ă t ď τ β for t ą τ Every optimal policy b P B p γ q agrees with b ˚ on r , τ s and satisfies b p t q ě β for all t ą τ . We note that the optimal policy is laissez-faire both before time τ and after time τ . We also note that the optimal policy has exactly onediscontinuity, namely, a sudden shut-down at time τ ; then b ˚ p t q fallsfrom b ˚ p τ q “ β to lim t Ó τ b ˚ p t q “ β ` αβγ p τ ´ τ q “ αx p τ q . From time τ on, b ˚ p t q rises continuously until time τ , at whichpoint b ˚ p t q reaches the level β . In the mean-time, between times τ and This follows from the observation that the derivative of the right-hand side of(7) is less than unity at x “ x p τ q . , the population share y p t q of infected individual remains constant, atthe capacity level γ , while the population share x p t q falls linearly overtime to the level α { β , reached at time τ .One obtains the following expression for the minimized cost: C p b ˚ q “ β ż τ τ αβγ p τ ´ t q ` αβγ p τ ´ t q dt “ αγ ż αβγ p τ ´ τ q ˆ ´ ` s ˙ ds “ αγ r αβγ p τ ´ τ q ´ ln p ` αβγ p τ ´ τ qqs“ γ ˆ x p τ q ´ αβ ln p x p τ qq ˙ ` αβγ ˆ ln ˆ αβ ˙ ´ ˙ (8)The proof of Theorem 1 is mathematically involved, and is givenin the Appendix. It uses measure theory and views the minimizationas taking place in phase space (much in line with equation (3)). Fora rich enough measure space, existence of a solution to (2) is obtainedby topological arguments. Invoking the Picard-Lindelöf theorem, it isshown that the differential equations (1) indeed uniquely define solutions.The next step in the proof is to show that the minimizer measure isabsolutely continuous (with respect to Lebesgue measure). This bringsus back to functions b P B , now viewed as transforms of Radon-Nikodynderivatives of the measures in question. The rest of the proof consists inverifying that the above function, b ˚ , indeed corresponds to an optimalmeasure, and that it is unique in the sense stated. In particular, oneneeds to show that it is neither worthwhile to slow down nor speed upthe infection in its early phase (before time τ ). An early slow-downwould postpone the problem at a cost but without benefit, and an earlyspeed-up, although costless in our model, would imply that the capacityconstraint is reached sooner and at a higher speed, which would requirean even more drastic, and costly, shut down when the capacity constraintis reached.The result is illustrated in Figure 1, where the solid kinked curve isthe solution orbit induced by (1) under the optimal control function b ˚ .The dotted curve is the infection orbit under laissez-faire ( b p t q ” β ). Wenote that the limit share of susceptible individuals, x , is higher underthe optimal policy than under laissez-faire. Recalling that z “ ´ x is the total population share of infected individuals during the epidemic,we conclude that, the policy not only respects the ICU constraint, italso indirectly affects the total number that will ultimately have beeninfected at some point in time. 10 y Filling the boxUnregulated x Figure 1: The solution orbit (solid) in the p x, y q -plane under the optimal policy b ˚ , and the solution orbit under unregulated spread (dotted) . Parametervalues used: α “ . , β “ , γ “ . , and ε “ . . Figure 2 depicts the optimal policy as a function of time in com-parison to a strategy of flattening the curve, here assumed to take theform: keep b p t q at at the level δ ă β for which ˆ y “ γ (see (5)) untilthe infection wave has reached its peak, and then return to laissez-faire, b p t q “ β (outside the time range of the figure). The upper panel showsthe dynamics of infections and the lower panel the policy b p t q .As can be seen, and as expressed by the theorem, the optimal pol-icy is characterized by leaving the spread unregulated initially, then asudden shut-down of society (a discontinuity at τ ), followed by grad-ual (continuous) opening of society, until τ , from which onwards thepropagation is not regulated. The time axis and the ICU constraintcreate a square – a box. The economic logic behind the optimal policyis essentially to ensure that we do not close down society while leavingidle ICU resources – “filling the box”. This implies that whenever thenatural spread is not threatening the constraint, it should go unregu-lated. This holds in the early phase when only few have been infected,and in the last phase, when many have already been infected but mostof them also have recovered. It is only when the epidemic may breachthe ICU constraint – the second phase – that it should be regulated. Inorder to ensure that the constraint is not breached, strong suppressionhas to be imposed when reaching the ICU constraint – a sudden shut-11 y ( t ) Flattening the curveFilling the box
Time b ( t ) Figure 2: Upper panel: The share of infected over time under the optimalpolicy (solid) and flattening the curve (dotted). The horizontal dashed linerepresents the ICU constraint γ . Lower panel: Optimal suppression (solid) andflattening-the-curve suppression (dotted). The horizontal dashed representsthe baseline spread β . Parameter values used: α “ . , β “ , γ “ . , and ε “ . . down. The reason for the abruptness of this policy (the discontinuity)is that the natural infection is progressing very quickly at that point, soa sudden break is needed to stop it. This can be seen in the lower panelby the drop at τ . After that, b ˚ gradually increases. The reason forthis is that the suppression only needs to keep the infection just belowthe ICU constraint. Then since over time the number of susceptible ( x )is falling, the number of infected ( y ) is held endogenously constant andsince new infections depends on their product ( b p t q y p t q x p t q ) it followsthat b ˚ is increasing during the second phase. The policy as a functionof the population share simply is b ˚ p t q “ α { x p t q , i.e., recovery ( α ) de-termines what share of the susceptible population that can be allowedto be infected. A few further remarks about optimal policies are now inplace.It may be noted that the optimal policy never attempts to fullyeradicate the spread. In our model, like in all standard S.I.R. models,this is since y only asymptotically goes to zero. Hence, full eradication(a form of extreme corner solution) would imply locking down forever.We discuss this further in the conclusions.12urthermore, the optimal policy is unique during the first and thesecond phase but not during the third. The uniqueness during the firstphase is not obvious. To see this note that here there is no reason tohold back the spread. Then, given that b ą β has been assumed to becostless, why would accelerating the spread not be optimal? The answeris that, if one does that, then the ICU capacity is reached at a high speedof infection hence it would require hitting the breaks very hard. Thisis not optimal. The multiplicity of optimal strategies during the thirdis due to the same assumption – acceleration is free. Hence, not onlylaissez-faire is optimal, but also acceleration (of which we can think of asstimulus for economic interaction). The acceleration cannot be too fast,however, as it may then breach the ICU constraint. Naturally, shouldwe assume that there is a cost of acceleration (even the slightest) thismultiplicity disappears and a unique optimal policy emerges also in thethird phase – laissez-faire.Compared with the optimal policy, “flattening the curve” impliescosts that lead to idle resources. This is visible in the upper panelof Figure 2, where costs are incurred without the spread posing a threatto the health system – both before and after the peak, suppression costsare incurred for no reason. The additional cost of flattening the curve(instead of filling the box) can be seen in the lower panel by comparingthe rectangle between β and the dashed-dotted line on the one handwith the area between β and the solid line on the other. It is potentiallyvery large, in particular if the policy maker continues to flatten the curvelong after the peak. This paper has developed an economic SIR model to provide an analyt-ical answer to the question: What is the optimal time-varying suppres-sion policy to avoid a collapsed health-care system when suppressionis costly? We have shown that the general recommendation of “flat-tening the curve” is suboptimal. Instead the optimal policy essentiallyprescribes “filling the box”: in an initial phase the spread is unregulateduntil the number of infected approaches the ICU constraint whereby, in asecond phase, suppression discontinuously increases and then graduallydrops until, in a third phase, the spread is left unregulated again.A contribution of the paper is methodological, showing how to ob-tain a fully analytical solution to an S.I.R. model with economic costswhich are increasing in suppression. Another contribution is the policyimplication of “filling the box”. We discuss here the robustness of thispolicy to various perturbations.In our model attempting for a complete wipe out of the spread is13ever optimal. Technically this is since the number of infected onlyasymptotically goes to zero hence a wipe out would require suppressionto be in place for the infinite future. Naturally, if the rate of recoveryfrom infection happens quickly for everyone it could be optimal to gofor a full wipe out right away. We do not, however, find that feasible inmost cases since it is hard to practically identify all infected and sincein practice the cost of full suppression ( b “ ) is virtually infinite – afterall, people need to access food and medical services. Furthermore, unlesscountries are closed more or less indefinitely, a very costly prospect,under a pandemic such as Covid-19 one would be bound to import newcases.In the model we have assumed that the only medical harm is if vi-olating the ICU constraint. If two assumptions were added – medicalharm from the aggregate number of infected and existence of a vaccinewithin a reasonable time frame – then suppressing the spread more thanwhat our policy prescribes could be optimal. Likewise if the numberof simultaneously infected would cause harm more gradually. This, how-ever, does not seem to be case for Covid-19 where the bottleneck in mostcountries is the number of respirators – respecting the ICU constraintis the main issue. Another factor that could suggest early suppressionis if the ICU constraint can be expanded (for Covid-19 equivalent to anincreased number of respirators or development of a cure or improvedtreatment). Another possibility is that one learns about the parameters.However, for that to motivate regulation early, one has to assume thatthe suppression itself does not distort the signal. Finally, if the cost ofsuppression were convex, in particular so that small suppression is verycheap, then that would motivate some suppression early on. However,it would still most probably be optimal to discontinuously increase sup-pression to ensure that the number of infected is constant just below theICU constraint.This discussion highlights that there exist a number of questions thatcall for an analytical approach for full understanding of their impact.Our model and tools of analysis provide a stepping stone for doing so. References [1] Brauer and Castillo-Chavez (2011):
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Appendix: Proof of Theorem 1
This appendix provides a proof of Theorem 1, namely that b ˚ is a globalminimizer for the control problem of optimizing the functional C givenjust above (2), and that any other global minimizer coincides with b ˚ upto time when x hits α { β .Our strategy consists of the following steps: • The optimization problem is written in the phase space (cid:52) . • The new formulation admits a natural extension on a signed-mea-sure space. • Topological properties of this measure space and of the functionaldeduced from C imply the existence of a global minimizer. • A priori a global minimizer is a general signed measure, but itturns out to be an absolutely continuous, bringing us back to afunctional setting. • Calculus of variation arguments show that the minimizer is uniquelydetermined until the time when x reaches the level α { β , and thisleads to Theorem 1.To the best of our knowledge, there is no such result in calculusof variations or optimal control theory (see the books of Clarke [11] orLiberzon [34]). We therefore give a direct and self-contained proof (onlyrequiring a first knowledge of measure theory, as can be found e.g. inRudin [44]). For the optimization problem at hand, our extension tomeasure spaces seems natural, and we believe it is original. The Euler-Lagrange equations will not be satisfied and the constraints will playa more important role. This is related to the fact that we consider acost of suppression that is linear in downwards deviations, and zero forupwards deviations. If one is interested in more general costs of the form r C p b q (cid:66) ż `8 F p β ´ b p t qq dt (9)where the mapping F : R Ñ R ` is e.g. a strictly convex functionattaining its minimum at 0, then the Euler-Lagrange equations admitsolutions leading to optimal policies different from b ˚ (but b ˚ remainsa minimizer for certain functionals F , see Remark 8 at the end of thisappendix). When F is close to the mapping p¨q ` considered here, forinstance if F is given by @ x P R , F p x q (cid:66) " x ` (cid:15) , when x P R ` (cid:15) | x | ` (cid:15) , when x P R ´ (cid:15) ą is small, we expect that the corresponding solution will beclose to b ˚ . In particular a jump will still occur. We plan to investigatemore precisely this situation in future work.Let us now move toward the proof of Theorem 1 according to theabove strategy. We assume in the sequel that we are in the “interesting”case where y ă γ and where the laissez-faire policy b ” β leads y to takevalues strictly large than γ ( y “ ε in the main text). This hypothesiswill be referred to as the underlying assumption . (cid:52) We begin by rewriting the constrained control problem of minimizing C on B γ as a optimization problem in the associated phase space (cid:52) .Let us be more precise: B is the set of piecewise continuous func-tions b : R ` Ñ R ` with a finite number of discontinuities and suchthat t t ě b p t q “ u has a finite number of connected components.Let p x , y q P (cid:52) , with y P p , q , as well as b (cid:66) p b p t qq t ě P B be givenand consider p x, y q (cid:66) p x p t q , y p t qq t ě the maximal (over time) solution ofthe S.I.R. ODE (1) starting from p x p q , y p qq “ p x , y q (this is a slightgeneralization of the setting of Theorem 1, where x “ ´ y , the impor-tant hypothesis is that the underlying assumption holds). The existenceand uniqueness of this solution is a consequence of the Picard-Lindelöfor Cauchy-Lipschitz theorem, extended to a time-dependent vector fieldthat is left continuous with right limits (instead of continuous). The im-portant fact being that the r.h.s. of (1) is locally Lipschitz with respectto p x p t q , y p t qq . By B γ , we designate the set of b P B such that y alwaysremains below γ .We begin with a simple observation. Lemma 1
When y P p , q , the solution p x, y q is defined for all times t ě and p x p t q , y p t qq P p , q X (cid:52) . Proof
As already mentioned, the solution p x, y q is locally unique. From thisit follows that if x reaches , then it will stay there forever afterward,since the r.h.s. of (1) is zero if x p t q “ . Similarly, if y reaches , thenit will stay there forever afterward. It follows that x and y will staynon-negative. As a consequence, x is non-increasing and thus will staybelow x and never hit 1. From the identity x p t q ` y p t q “ ´ αy p t q ď we deduce that x ` y will stay below y ` x ď and in particular y willstay below 1. The first equation of (1) then implies that x p t q ě ´ b p t q x p t q x p t q ě x exp ´ ´ ş t b p s q ds ¯ so x remains positive. The inequality x ` y ď then insures that y neverreaches 1. Finally, the second equation of (1) then implies that y p t q ě ´ αy p t q so that y p t q ě y exp p´ αt q and y cannot reach 0 in finite time.Since p x, y q stays in the compact square r , s , the solution of (1) isdefined for all times. (cid:4) Remark 1
In particular, since for all t ě we have y p t q ą , a part ofthe population will always remain infectious, whatever the choice of thepolicy b : it is impossible to entirely eliminate the disease. This featureis due to the fact we are considering continuous populations, it wouldnot be true for approximating finite random populations. ˝ Introduce B ` the set of b P B that are everywhere positive (never-theless, if t is a discontinuity point of b , we can have b p t `q “ ).When b P B ` , Lemma 1 and the first equation of (1) imply that x isdecreasing, so x admits a limit x ě that it will never reach. Anotherconsequence is: Lemma 2
Assume that b P B ` . There exists a unique function ϕ : p x , x s Ñ p , q such that @ t ě , y p t q “ ϕ p x p t qq (10) The function ϕ is piecewise C , its left and right derivatives existeverywhere and differ only at a finite number of points. Denoting theright derivative by ϕ , we have @ r P p x , x s , ϕ p r q ą ´ Proof
As observed above, for b P B ` , x is decreasing from R ` to p x , x s . Since x it continuous, it is a homeomorphism between R ` to p x , x s . Denoteby τ its inverse, so that @ u P p x , x s , x p τ p u qq “ u (11)21et t P R ` be a time where b is continuous. Let u P p x , x s be suchthat τ p u q “ t . We can differentiate (11) at u to get that τ p u q “ x p τ p u qq “ ´ b p τ p u qq uy p τ p u qq Considering discontinuity time t of b , we see that the above relationalso holds, if τ p u q is seen as a right derivative (recall x is a left derivate).Furthermore, taking into account that τ is decreasing, we have the exis-tence of the left limit: lim v Ñ u ´ τ p v q “ ´ b p τ p u q`q uy p τ p u qq It leads us to define ϕ via @ u P p x , x s , ϕ p u q (cid:66) y p τ p u qq since this is indeed equivalent to (10). Its left and right derivatives existeverywhere as a consequence of the differentiability properties of y and τ .These left and right derivatives do not coincide only on a finite numberof points, those of the form x p t q , where t P R ` is a discontinuity time of b . Our conventions insure that (10) can be left differentiated everywhereand that @ t ě , y p t q “ ϕ p x p t qq x p t q (recall y is a left derivate), namely @ t ě , ϕ p x p t qq “ y p t q x p t q “ ´ ` αb p t q x p t q ą ´ Remark 2
From the knowledge of ϕ it is possible to reconstruct b , atleast when ϕ is Lipschitzian. Indeed, (1) can be written x p t q “ ´ b p t q x p t q ϕ p x p t qq ϕ p x p t qq x p t q “ b p t q x p t q ϕ p x p t qq ´ αϕ p x p t qq which implies that x p t q “ ´ ϕ p x p t qq x p t q ` αϕ p x p t qq i.e. x p t q “ α ϕ p x p t qq ` ϕ p x p t qq
22o when ϕ is Lipschitzian, we can solve this ODE to reconstruct x (cid:66) p x p t qq t ě . The trajectory y (cid:66) p y p t qq t ě is then obtained as p ϕ p x p t qqq t ě and b via the formula @ t ě , b p t q “ ´ x p t q x p t q y p t q ˝ This inequality can be translated into ϕ ą ´ on p x , x q . (cid:4) To any function ϕ as in the previous lemma, associate the quantity J p ϕ q (cid:66) ż x x L p ξ, ϕ p ξ q , ϕ p ξ qq dξ where for any p ξ, χ, χ q P p x , x s ˆ p , q ˆ p´ , `8q , L p ξ, χ, χ q (cid:66) βα ˆ ` χ χ ´ αβξχ ˙ ` The interest of these definitions is to enable us to write the costfunctional C in terms of ϕ : Lemma 3
For b P B ` and with the notations of Lemma 2, we have C p b q “ J p ϕ q Proof
Equation (12) enables us to recover b in terms of ϕ and x : @ t ě , b p t q “ αx p t qp ` ϕ p x p t qqq (12)and we deduce that C p b q “ ż ˆ β ´ αx p t qp ` ϕ p x p t qqq ˙ ` dt “ ´ ż ˆ β ´ αx p t qp ` ϕ p x p t qqq ˙ ` b p t q x p t q y p t q x p t q dt “ ż x x ˆ β ´ αu p ` ϕ p u qq ˙ ` b p τ p u qq uϕ p u q du where we used the change of variable t “ τ p u q , the mapping τ beingdefined in (11). 23et us remove the term b p τ p u qq in the latter integral. Replacing t by τ p u q , we get from (12) b p τ p u qq “ αu p ` ϕ p u qq so that C p b q “ ż x x ˆ β ´ αu p ` ϕ p u qq ˙ ` ` ϕ p u q αϕ p u q du “ βα ż x x ˆ ` ϕ p u q ϕ p u q ´ αβ uϕ p u q ˙ ` du “ J p ϕ q (cid:4) Let us now extend the above transformation to a policy b P B whichcan take the value 0. More precisely consider two times ď t ď t such that r t , t s or p t , t s is a connected component of the set t t ě b p t q “ u . We recall that this set is assumed to be finite union of suchintervals, if it is not empty.Let us first suppose that t ‰ t . On p t , t s , (1) is transformed into x p t q “ y p t q “ ´ αy p t q namely @ t P p t , t s , x p t q “ x p t q y p t q “ y p t q exp p´ α p t ´ t qq Since x remains constant and y is changing, one cannot represent y as a function ϕ of x . To circumvent this difficulty, we allow ϕ to jumpat x p t q , taking ϕ p x p t qq (cid:66) y p t q ϕ p x p t q´q (cid:66) y p t q “ y p t q exp p´ α p t ´ t qq This is illustrated in the following diagram: x y = 𝜑(𝑥) y(𝑡 )y(𝑡 ) x 𝑡 = x(𝑡 ) p t , t s to the cost C p b q is ż t t p β ´ q ` dt “ β p t ´ t q “ βα ln ˆ y p t q y p t q ˙ “ βα ln ˆ ϕ p x p t q ϕ p x p t q´q ˙ The above observations are also valid, but trivial, when t “ t .A priori, our definition of B does not exclude the fact that t “ `8 ,namely that b ends up vanishing identically after t . By convention inthis case ϕ p x p t q´q “ and the above formula gives an infinite contribu-tion to the cost, which is coherent with the fact that ş `8 t p β ´ b p t qq ` dt “ ş `8 t β dt “ `8 . Since this situation is not interesting for our optimiza-tion problem, we exclude it from our considerations and from now onthe connected components of t b “ u are assumed to be bounded.Repeating the above treatment to all the connected components of t b “ u and extending Lemmas 2 and 3 to the connected components of t b ą u , we can associate to any b P B a function ϕ satisfying(H1): ϕ is defined on p x , x s , takes values in p , q and ϕ p x q “ y ,(H2): ϕ has at most a finite number of discontinuity points and is rightcontinuous and admits a positive left limit at them,(H3): at any discontinuity point u , ϕ p u q ą ϕ p u ´q ,(H4): ϕ admits a right derivative ϕ , as well as a left derivative, outside afinite number of points (which includes the discontinuity points of ϕ , but also some points corresponding to the case t “ t describedabove, since due to (12), at these points ϕ may diverge to `8 ),(H5): ϕ ą ´ where it is defined,But the most important feature is that C p b q “ J p ϕ q where the functional is defined by J p ϕ q (cid:66) ż p x ,x q L p ξ, ϕ p ξ q , ϕ p ξ qq dξ ` βα ÿ u Pp x ,x s : ϕ p u q‰ ϕ p u ´q ln ˆ ϕ p u q ϕ p u ´q ˙ ( x is excluded from the last sum, since we don’t allow b to end upvanishing identically).There is a point which is not satisfactory in the above construction of ϕ : its definition domain p x , x s still depends explicitly on b via x . Itis possible to erase this drawback with the convention that ϕ vanishes on25 , x s Accordingly, L has to be extended to r , x s ˆ r , q ˆ p´ , `8q via the convention that L p ξ, χ, χ q “ if ξ P r , x s (or equivalently if χ “ ).Remarking that the condition y ď γ can be translated into ϕ ď γ ,we have embedded the problem of the global minimization of C over B γ into the problem of the global minimization of J over r F γ , the set offunctions ϕ that satisfy the requirements (H1a), (H2), (H3), (H4) and(H5), where (H1) has been replaced by(H1a): ϕ is defined on r , x s , takes values in r , γ s , ϕ p q “ , ϕ p x q “ y and if ϕ p u q ą for some u P p , x s , then ϕ p v q ą and ϕ p v ´q ą for all v P r u, x s .As it was explained in the main text, the most important part of theabove optimization problem concerns the contribution of the restrictionof the function ϕ to the interval r α { β, x s , since once x has reached α { β ,the laissez-faire policy is cost-free and induces y to be non-increasing.Let us furthermore assume the function ϕ satisfies ϕ p α { β q “ γ (13)This restriction will be justified a posteriori (see Subsection 5.6), butit can also be understood a priori according to the following heuristic.Let us come back to the temporal description given by p b, x, y q . To get x ě α { β is extremely costly and requires in fact C p b q “ `8 (think forinstance to the case where x “ x which asks that b ” ), so for theoptimization problem at hand, we can dismiss this possibility and assume x ă α { β . Then consider τ ` the first time x reaches α { β , we have y p τ ` q P p , γ s . As already observed, taking b “ β after time τ ` leadsto a non-increasing evolution of y on r τ ` , `8q . In particular, y p t q staysbelow γ for all times t P r τ ` , `8q , while this part of the trajectory doesnot participate positively to the cost. Due to our underlying assumption,the laissez-faire policy b ” β leads y to hit γ strictly before x reaches α { β . Then if y p τ ` q ă γ , it means that for some times t P r , τ ` q , wehave b p t q ă β and y p t q ă γ . Increasing a little b at such times, will havethe effect of increasing y p τ ` q while keeping y below γ . Furthermore thisoperation will decrease a little the cost. As a consequence, in order tofind a minimizing policy b for C , we can assume that y p τ ` q “ γ . Thisamounts to assuming (13).These considerations and Assumption (13) lead us to modify thefunctional set r F γ into F γ , replacing its first requirement (H1a) by(H1b): ϕ is defined on r α { β, x s , takes values in p , γ s , the left limits of ϕ are positive, ϕ p α { β q “ γ and ϕ p x q “ y
26f course the cost functional on such functions ϕ is given by J p ϕ q (cid:66) ż p α { β,x q L p ξ, ϕ p ξ q , ϕ p ξ qq dξ ` βα ÿ u Pp α { β,x s : ϕ p u q‰ ϕ p u ´q ln ˆ ϕ p u q ϕ p u ´q ˙ (14)Our ultimate goal is to prove that J admits a unique minimizer ϕ ˚ over F γ and that this minimizer is obtained from b ˚ by the operationdescribed above. Up to the justification of the reduction of (H1a) to(H1b), given at the end of Subsection 5.6, Theorem 1 will then be proven. J to measures To prove the existence of the minimizer ϕ ˚ of J on F γ , we begin bygeneralizing this optimization problem by replacing F γ by a set of mea-sures.To any given ϕ P F γ , we associate three measures µ , ψ and ν on I (cid:66) r α { β, x s via µ p dx q (cid:66) ϕ p x q ϕ p x q dx ` ÿ u Pp α { β,x s : ϕ p u q‰ ϕ p u ´q ln ˆ ϕ p u q ϕ p u ´q ˙ δ u p dx q ψ p dx q (cid:66) ϕ p x q dxν p dx q (cid:66) ϕ p x q ˆ ´ αβx ˙ dx (where δ u stands for the Dirac mass at u ).Note that ψ and ν are non-negative measures, but µ is a signedmeasure. Denote F µ the repartition function associated to µ via @ x P I, F µ p x q (cid:66) µ pr α { β, x sq Recall that ϕ , as well as its left limits, are positive on I , it followsthat ϕ is bounded below by a positive constant on I . This observationenables us to compute F µ : @ x P I, F µ p x q “ ln p ϕ p x qq ´ ln p ϕ p α { β qq “ ln p ϕ p x q{ γ q (15)We deduce that @ x P I, ϕ p x q “ γ exp p F µ p x qq ψ p dx q (cid:66) exp p´ F µ p x qq γ dxν p dx q (cid:66) ˆ ´ αβx ˙ exp p´ F µ p x qq γ dx On these expressions, it appears that ψ and ν only depend on µ (inaddition to the constants α, β, γ ), so they will be denoted ψ µ ν µ fromnow on.From (15) we get that the total weight µ p I q of µ is given by F µ x “ ln p y { γ q . Let us estimate the total variation } µ } tv of µ : Lemma 4
We have } µ } tv ď (cid:15) p x ´ α { β q ` ln p y { γ q where (cid:15) (cid:66) inf t ϕ p x q : x P I u “ min t ϕ p x q ^ ϕ p x ´q : x P I u Proof
Recall that any signed measure m on I can be decomposed into m ` ´ m ´ ,where m ´ and m ` are two non-negative measures mutually singular.The total variation is given by } m } tv “ m ´ p I q ` m ` p I q .Coming back to µ , we have µ ´ p dx q “ | ϕ p x q| ϕ p x q t ϕ p x qą u dx so that µ ´ p I q ď ż I ϕ p x q dx ď (cid:15) ż I dx “ x ´ α { β(cid:15) It follows that } µ } tv “ µ ´ p I q ` µ ` p I q “ µ ´ p I q ` µ p I q ď (cid:15) p x ´ α { β q ` ln p y { γ q (cid:4) The quantity (cid:15) ą associated to ϕ in the previous lemma can beestimated in terms of J p ϕ q : Lemma 5
We have (cid:15) ě y exp p´ β J p ϕ q{ α q x P I . We have J p ϕ q ě βα ż I L p u, ϕ p u q , ϕ p u qq du ě βα ż x x L p u, ϕ p u q , ϕ p u qq du “ βα ż x x ˆ ϕ p u q ϕ p u q ` αβϕ p u q ˆ ´ αβu ˙˙ ` du ě βα ż x x ϕ p u q ϕ p u q ` αβϕ p u q ˆ ´ αβu ˙ du ě βα ż x x ϕ p u q ϕ p u q du “ βα ln ˆ ϕ p x q ϕ p x q ˙ where in the last-but-one inequality, we took into account that ´ αβu ě for u ě x ě α { β . The above bound can be written ϕ p x q ě ϕ p x q exp p´ α J p ϕ q{ β q “ y exp p´ α J p ϕ q{ β q By taking the infimum over all x P I , we get the desired result. (cid:4) Consider an element ϕ of F γ such that J p ϕ q ă `8 , for instancethe function constructed from b ˚ as in the previous subsection and denote M (cid:66) p α p ` J p ϕ qq{ β q y p x ´ α { β q ` ln p y { γ q (16)The global minimization of J on F γ is equivalent of the global mini-mization of J on t ϕ P F γ : J p ϕ q ď J p ϕ qu . So according to Lemma 4,we can restrict our attention to measure µ satisfying(C1): } µ } tv ď M .Furthermore note that the belonging of ϕ to F γ implies three conditionson µ :(C2): µ p I q “ ln p y { γ q ,(C3): F µ ď ,(C4): µ ` ψ µ ě .Denote by M γ the set of signed measures µ on I which satisfy theconditions (C1), (C2), (C3) and (C4).Up to now, we did not use ν µ , its interest comes from the fact thatfor ϕ P F γ , the cost functional writes J p ϕ q “ αβ p µ ` ν µ q ` p I q µ P M γ , K p µ q “ αβ p µ ` ν µ q ` p I q since the global minimization of J on t ϕ P F γ : J p ϕ q ď J p ϕ qu can beembedded in the global minimization of K on M γ . K on M γ The successive extensions of our initial minimization problem workedout in the two previous subsections will be justified here by showing theexistence of a global minimizer, via abstract topological arguments. Itwon’t be clear that such a minimizer from M γ corresponds to an elementof B γ via the transformation of Subsection 5.1: this will be investigatedin the next subsections.Let us endow the set of (signed) measures on I with the weak topol-ogy, i.e. a sequence p µ n q n P N of such measures converges toward a measure µ if and only if for any continuous function g : I Ñ R , we have lim n Ñ8 µ n r g s “ µ r g s The existence of a global minimizer of K on M γ is a consequence ofa version of Weierstrass’ maximum theorem through the two followingresults. Proposition 3
The set M γ is compact. Proposition 4
The mapping K : M γ Ñ R is lower semi-continuous. Proof of Proposition 3
Consider the ball (with respect to the strong topology) B p M q consistingof the signed measures whose total variation is smaller or equal to M .It is well-known that B p M q is weakly compact. So it sufficient to showthat the sets S (cid:66) t µ P B p M q : µ p I q “ ln p y { γ qu S (cid:66) t µ P B p M q : F µ ď u S (cid:66) t µ P B p M q : µ ` ψ µ ě u are closed. ‚ Concerning S , this is obvious, since µ p I q “ µ r I s , where I is thecontinuous function on I always taking the value 1. ‚ For S , consider a sequence p µ n q n P N of measures from S convergingtoward a signed measure µ . We have to check that µ P S . Consider A µ , A is at most denumerable and for x P I z A , wehave lim n Ñ8 F µ n p x q “ F µ (17)We deduce @ x P I z A , F µ p x q ď It remains to use that F µ is right-continuous and that I z A is dense in I to extend the validity of the above inequality to the whole set I . ‚ For S , consider a sequence p µ n q n P N of measures from S convergingtoward a signed measure µ . We have to check that µ ` ψ µ ě . Consider g a continuous function on I . We begin by showing that lim n Ñ8 ψ µ n r g s “ ψ µ r g s (18)Indeed, for any n P N , we have ψ µ n r g s “ γ ż I g p x q exp p´ F µ n p x qq dx We have seen in (17) that F µ n is almost everywhere converging to F µ . Furthermore, we have for all n P N , @ x P I, | F µ n p x q| ď } µ n } tv ď M It follows that dominated convergence can be invoked to conclude to(18).We deduce that lim n Ñ8 p µ n ` ψ µ n qr g s “ p µ ` ψ µ qr g s Now assume furthermore that g ě . The above convergence impliesthat p µ ` ψ µ qr g s ě Since this is true for all non-negative continuous function g , we get that µ ` ψ µ ě , as desired. (cid:4) Proof of Proposition 4
31o get that the mapping K : M γ Ñ R is lower semi-continuous, it issufficient to write it as the supremum of (weakly) continuous functionson M γ . By definition, we have K p µ q “ αβ p µ ` ν µ q ` p I q “ sup g P C p I q : 0 ď g ď p µ ` ν µ qr g s where C p I q is the set of continuous functions on I .So it remains to check that for any fixed g P C p I q with ď g ď ,the mapping M γ Q µ ÞÑ p µ ` ν µ qr g s is continuous. The proof of this continuity is similar to the closure of S in the proof of Proposition 3: both the mappings M γ Q µ ÞÑ ψ µ and M γ Q µ ÞÑ ν µ are continuous (taking values in the set of non-negativemeasures on I endowed with the weak topology). (cid:4) Let µ ˚ be a minimizer of K on M γ . We will show here that µ ˚ isabsolutely continuous with respect to λ , the Lebesgue measure on I .Recall that any signed measure µ on I can be uniquely decomposedinto a sum µ a ` µ s ` µ c , where µ a is atomic, µ c is diffuse and singularwith respect to λ and µ c is absolutely continuous with respect to λ . Dueto Radon-Nikodym theorem, µ c admits a (signed) density f : I Ñ R with respect to λ , this relation will be denoted µ c “ f ¨ λ .Using this decomposition, the cost functional K can be written underthe form K p µ q “ βα ˆ µ a p I q ` µ s p I q ` ż I p f ` ν µ q ` dλ ˙ where we identified ν µ with its density with respect to λ , namely @ x P I, ν µ p x q “ γ ˆ ´ αβx ˙ exp p´ F µ p x qq Our goal here is to show that µ ˚ a “ “ µ ˚ s , by perturbative argu-ments. Both proofs will follow the same pattern, but the deduction of µ a “ is simpler, so for pedagogical reasons we will insist on this one. Proof of µ ˚ a “ The argument is by contradiction. Assume that µ ˚ a ‰ , then there exist x P I and (cid:15) ą such that µ ˚ a ě (cid:15)δ x . Since F µ ˚ p x q ´ F µ ˚ p x ´q ě (cid:15) and32hat F µ ˚ p x q ď , we have F µ ˚ p x ´q ď ´ (cid:15) and we can find x P p α { β, y q such that @ x P r x , x q , F µ ˚ p x q ď ´ (cid:15) It leads us to consider for r ą , the perturbation µ ˚ r defined by µ ˚ r (cid:66) µ ` r r x ,x q x ´ x ¨ λ ´ rδ x (where r x ,x q is the indicator function of r x , x q and the middle termis an absolutely continuous measure).Let us check that µ r P M γ for r ą small enough, namely that (C1),(C2), (C3) and (C4) are satisfied by µ ˚ r . ‚ } µ ˚ r } tv ď M :Since µ ˚ is also a minimizer of K on the space of signed measureson I , we have K p µ ˚ q ď L p ϕ q , where ϕ was defined above (16). Itfollows from Lemmas 4 and 5, taking into account the definition of M in(16), that } µ ˚ } tv ă M . Triangular inequality with respect to the totalvariation norm implies } µ ˚ r } tv ď } µ ˚ } tv ` ›››› r r x ,x q x ´ x ¨ λ ´ rδ x ›››› tv “ } µ ˚ } tv ` r insuring that for r ď p M ´ } µ ˚ } tv q{ , we have } µ ˚ } tv ď M . ‚ µ ˚ r p I q “ ln p y { γ q :The total weight of µ ˚ r is always the same as that of µ ˚ , since ˆ r x ,x q x ´ x ¨ λ ´ δ x ˙ p I q “ ´ “ ‚ F µ ˚ r ď :Note that outside p x , x q , F µ ˚ r coincides with F µ ˚ , so we just needto check that F µ ˚ r p x q ď for all x P p x , x q . Indeed, we have @ x P p x , x q , F µ ˚ r p x q “ F µ ˚ p x q ` r x ´ x x ´ x ď ´ (cid:15) ` r ď as soon as r ď (cid:15) { . ‚ µ ˚ r ` ψ µ ˚ r ě :Note again that outside p x , x s , we have µ ˚ r ` ψ µ ˚ r “ µ ˚ ` ψ µ ˚ , so wejust need to check that the measure µ ˚ r ` ψ µ ˚ r is non-negative on p x , x s .The diffusive singular part of µ ˚ r ` ψ µ ˚ r is the same as that of µ ˚ ` ψ µ ˚ andthe atomic ones only differ at x . Note that µ ˚ r pt x uq “ µ s pt x uq ´ r ě ´ r and this is non-negative, as soon as r ď (cid:15) . Concerning the absolutelycontinuous part, denote f ˚ (respectively f ˚ r ) the density of µ ˚ (resp. µ ˚ r )with respect to λ . We have a.e. for x P p x , x q , f ˚ r p x q “ f ˚ p x q ` rx ´ x (19)Identify ψ µ ˚ r with its density, so that we can write (a.e.) @ x P p x , x q , ψ µ ˚ r p x q “ exp p´ F µ ˚ r p x qq γ Note that @ x P p x , x q , ˇˇ F µ ˚ r p x q ´ F µ ˚ p x q ˇˇ “ ˇˇˇˇż xx rx ´ x dλ ˇˇˇˇ “ r p x ´ x q x ´ x ď r Let C (cid:66) exp p ` max x Pr x ,x s | F µ p x q|q γ For any r P r , s , we have @ x P p x , x q , ˇˇ ψ µ ˚ r p x q ´ ψ µ ˚ p x q ˇˇ ď Cr Comparing with (19), we get that if x has been chosen so that x ´ x ď C ´ , then a.e. for x P p x , x q , f ˚ r p x q ` ψ µ ˚ r p x q ě f ˚ p x q ` ψ µ ˚ p x q ě This ends the proof that µ r P M γ , as soon as r ą is small enoughand x is sufficiently close to x .Let us evaluate K p µ ˚ r q . We have K p µ ˚ r q “ K p µ ˚ q ´ r ` ż x x ˆ f ` rx ´ x ` ν µ ˚ r ˙ ` ´ p f ` ν µ q ` dλ (20)and for almost all x belonging to p x , x q , ˆ f p x q ` rx ´ x ` ν µ ˚ r p x q ˙ ` ´ p f p x q ` ν µ p x qq ` ď rx ´ x ` ν µ ˚ r p x q ´ ν µ p x q Note that for x P p x , x q , we have F µ ˚ r p x q ą F µ ˚ p x q , so that ν µ ˚ r p x q ă ν µ ˚ p x q . Thus we get ˆ f p x q ` rx ´ x ` ν µ ˚ r p x q ˙ ` ´ p f p x q ` ν µ p x qq ` ă rx ´ x
34t follows that that for r ą : ż x x ˆ f ` rx ´ x ` ν µ ˚ r ˙ ` ´ p f ` ν µ q ` dλ ă r and (20) implies the contradiction K p µ ˚ r q ă K p µ ˚ q . (cid:4) Remark 5
In Subsection 5.2, we have seen that the atomic part ofa measure corresponds to imposing the drastic policy b “ for sometime, as a partial attempt toward eradication of the disease (accordingto Remark 1 this goal cannot be fully attained). The significance of µ ˚ a “ is that such attempts are sub-optimal. From the above proof wesee that it is better to replace such attempts by future softer policies,replacing a (partial) Dirac mass at x by a density before x (recall that x and time go in reverse directions). ˝ Proof of µ ˚ s “ The pattern of the proof is the same as that for µ ˚ a “ , except that wehave to “thicken a little” x . Indeed, if µ ˚ s ‰ , we can find x P p α { β, x s and (cid:15) ą such that αβ ă x ´ (cid:15) µ ˚ s pp x ´ (cid:15) , x ` (cid:15) qq ą @ (cid:15) P p , (cid:15) s , µ ˚ s pp x ´ (cid:15) , x ` (cid:15) qq ě µ ˚ c pp x ´ (cid:15) , x ` (cid:15) qq In comparison with the previous proof, the restriction of µ ˚ s to p x ´ (cid:15) , x ` (cid:15) q plays the role of (cid:15)δ x with µ ˚ s pp x ´ (cid:15) , x ` (cid:15) qq playing the roleof (cid:15) .It is now possible to find x P p α { β, x ´ (cid:15) q such that for r ą smallenough, the measure µ ˚ r (cid:66) µ ` r r x ,x ´ (cid:15) q x ´ (cid:15) ´ x ¨ λ ´ rµ ˚ s pr x ´ (cid:15) , x ` (cid:15) sq p x ´ (cid:15) ,x ` (cid:15) q ¨ µ ˚ s belongs to M γ and K p µ ˚ r q ă K p µ ˚ q This is the desired contradiction. The details of the adaptation ofthe arguments of the proof of µ ˚ a “ are left to the reader. (cid:4)
35e have shown that any minimizer µ ˚ of K can be written in theform f ˚ ¨ λ . As a consequence, such a density f ˚ is a minimizer of asuitably formulated optimization problem, to which we now turn..For any integrable function f on I , let associate to f the notions thatwere previously associated to the measure f ¨ λ : for any x P I , F p x q (cid:66) F f ¨ λ p x q “ ż xα { β f p u q duψ F p x q (cid:66) γ exp p´ F p x qq ν F p x q (cid:66) ˆ ´ αβx ˙ exp p´ F p x qq We introduce D γ the set of integrable functions f such that F p x q “ ln p y { γ q , F ď , f ` ψ F ě and we consider on D γ the functional G p f q (cid:66) K p f ¨ λ q “ ż I p f ` ν F q ` dλ We have shown the optimization problem of G on D γ (which is anextension of the optimization problem of J on F γ ) admits global min-imizers: they are exactly the functions f ˚ such that f ˚ ¨ λ P M γ is aglobal minimizer of K . G on D γ In this subsection, a minimizer of G on D γ will be denoted f and ourgoal is to show that it is a.e. equal to the function f ˚ described below.Define x ˚ as the unique solution belonging to r α { β, x s of the equation x ˚ ´ αβ ln p x ˚ q “ ´ γ ´ αβ ln p x q (21)(the existence of this solution is due to our underlying assumption, notethat x ˚ coincides with x p τ q defined above (7)), and take @ x P r α { β, x s ,f ˚ p x q (cid:66) , if x ď x ˚ ´ ´ γ ´ ´ x ´ x ˚ ´ αβ ln p x { x ˚ q ¯¯ ´ ´ ´ αβx ¯ , if x ą x ˚ (22) Theorem 2
The function f ˚ is the unique minimizer of G on D γ , upto modifications on subsets of I with Lebesque measure zero. f , a minimizer of G on D γ , the notions defined atthe end of the previous subsection. In addition, we define v (cid:66) max t x P I : F p x q “ u G p f q (cid:66) ż vα { β p f ` ν F q ` dλ G p f q (cid:66) ż x v p f ` ν F q ` dλ Note that G p f q “ G p f q ` G p f q . We will investigate separately G p f q and G p f q . The computation of the former will be a consequence of: Lemma 6
The function F vanishes everywhere on r α { β, v s . Proof
We have G p f q ě ż vα { β f ` ν F dλ “ F p v q ` ż vα { β ν F dλ “ ż vα { β ν F dλ since by continuity of F , F p v q “ .The bound F ď implies that @ x P I, ν F p x q “ γ ˆ ´ αβx ˙ exp p´ F p x qq ě γ ˆ ´ αβx ˙ (23)and we deduce G p f q ě γ ż vα { β ´ αβx dx “ G p r f q where the function r f is given by @ x P I, r f p x q (cid:66) " , if x ď vf p x q , if x ą v Since F p v q “ , we get that for any x P r v, x s , that r F p x q (cid:66) ż xα { β r f p u q du “ ż xv r f p u q du “ ż xv f p u q du “ F p x q We deduce that r f ` ν r F coincides with f ` ν F on r v, x s and thus G p r f q “ G p f q , which implies G p r f q ď G p f q r f P D γ and that f is a minimizer of G on D γ , we must have G p r f q “ G p f q and G p r f q “ G p f q . In particular, (23)must be an equality a.e. on r α { β, v s . This means that F vanishes a.e. on r α { β, v s . The continuity of F then implies that F vanishes identicallyon r α { β, v s . (cid:4) As announced, we deduce the value of G p f q : Lemma 7 G p f q “ γ ˆ v ´ αβ ln p v q ˙ ` αβγ ˆ ln ˆ αβ ˙ ´ ˙ and the r.h.s. is increasing with respect to v . Proof
From Lemma 6, we deduce that f “ on r α { β, v s (a.e., as all statementsabout densities, in the sequel we will no longer mention it). Recallingthat ν F ě , we get G p f q “ ż vα { β ν F dλ “ ż vα { β ν dλ “ γ ż vα { β ´ αβx dx “ γ „ x ´ αβ ln p x q vα { β “ γ ˆ v ´ αβ ln p v q ˙ ` αβγ ˆ ln ˆ αβ ˙ ´ ˙ (cid:4) We now come to the study of G . Our main step in this direction is: Proposition 6
We have f ` ν F ď on r v, x s . Proof
Consider x P p v, x s . Since x can be arbitrary close to v , it is sufficientto show that f ` ν F ď on r x , x s . The advantage of considering sucha x is that we can find η ą such that @ x P r x , x s , F p x q ď ´ η (24)This property will be important for the perturbations of f we areto consider. More precisely they will be of the form f r (cid:66) f ` rg , with r ą sufficiently small and where g is an appropriate bounded functionon r α { β, x s and satisfying g “ on r α { β, x s .38efining for x P r α { β, x s F r p x q (cid:66) ż xα { β f r dλG p x q (cid:66) ż xα { β g dλ “ ż v _ xv g dλ we have F r “ F ` rG and it is clear from (24) that the inequality F r ď will be satisfied on r v, x s for r ą sufficiently small. Since F r and F coincide on r α { β, v s , we will get F r ď on I .It will be easy to impose that G p x q “ , to get F r p x q “ F p x q “ ln p y { γ q .It will be more tricky to insure that f r ` ψ F r ě (on r x , x s , sinceit is trivial on r α { β, x s where f r ` ψ F r coincides with f ` ψ F ) and wehave to be very careful about the choice of g . The construction of suchappropriate g is given as follows.First note that if x P r x , x s is such that f p x q ` ψ F p x q “ , then f p x q ` ν F p x q “ f p x q ` ψ F p x q ´ αβγx exp p´ F p x qq ď ´ (cid:15) with (cid:15) (cid:66) min " αβγx exp p´ F p x qq : x P r x , x s * Denote A (cid:66) t x P r x , x s : f p x q ` ν F p x q ď ´ (cid:15) { u B (cid:66) t x P r x , x s : f p x q ` ν F p x q ą u Let h be a bounded and measurable function defined on B . Considerthe two functions ξ and χ given on r x , x s by ξ p x q (cid:66) $&% , if x P Ah p x q , if x P B , otherwise χ p x q (cid:66) " ψ F p x q , if x P Aν F p x q , otherwiseSolve on r x , x s the weak ODE in G : G p x q “ G “ χG ` ξ (25)39this is always possible, even with irregular ξ and χ , see (26) below).Next extend G to r α { β, x s by imposing that G vanishes there (equiv-alently, keep solving (25) with ξ “ χ “ there) and define g (cid:66) G .Note that on A , we have g ´ ψ F G “ , i.e. g ´ γ ´ e ´ F G “ It follows that for r ě sufficiently small, say r P r , r q , with some r ą , we have g ´ γ ´ e ´ F r G ě { on A ( r depends on h through (25) via a bound on G ).The latter inequality can be written as B r f r ` B r ψ F r ě on A and we deduce that for r P r , r q , f r ` ψ F r ě on A Due to the definition of (cid:15) and A , f ` ψ F is bounded below by (cid:15) { on r x , x sz A . It follows that up to diminishing r , we can insure that f r ` ψ F r ě on r x , x s for all r P r , r q .Up to imposing G p x q “ , this is the type of perturbations f r weare to consider.Note that (25) can be solved explicitly: @ x P r x , x s , G p x q “ e H p x q ż xx e ´ H p u q ξ p u q du (26)where @ x P r x , x s , H p x q “ ż xx χ p u q du (27)Thus the condition G p x q “ writes ż x x e ´ H ξ dλ “ namely ż A e ´ H dλ ` ż B e ´ H h dλ “ (28)40nce this condition is satisfied, we have f r P D γ for r P r , r q . Itleads us to investigate G p f r q . Let us differentiate this quantity at r “ .First note that B r | r “ p f r ` ν F r q “ g ´ ν F G If x P r x , x s is such that f p x q ` ν F p x q “ , then x does not belongto A \ B , so g ´ ν F G “ . Taking into account the definition of B , weobtain by differentiation under the integral B r | r “ G p f r q “ ż B g ´ ν F G dλ “ ż B h dλ Since f is a global minimizer of G on D γ and that G p f r q ´ G p f q “ G p f r q ´ G p f q , we must have ż B h dλ ě (29)So we have shown that if h is such that (28) is true, then (29) holds.To finish the proof, it remains to see that this property implies that λ p D q “ .We proceed by contradiction, assuming λ p D q ą . Consider x Pp x , x q such that λ p D ´ q “ λ p D ` q (30)with D ´ (cid:66) D X r x , x q , D ` (cid:66) D X r x , x q Find a bounded and measurable function h on B such that ż B exp p´ H q h dλ “ ´ ż A exp p´ H q dλ (31)Consider next h “ h ` t exp p H q D ´ ´ t exp p H q D ` with t ě to be chosen later.Due to (30) and (31), (28) holds.However, as can be seen in (27), H is strictly increasing on B . Itfollows that @ x P D ´ , @ x P D ` , H p x q ă H p x q ż D h dλ “ ż D h dλ ` t ˆż D ´ e H dλ ´ ż D ` e H dλ ˙ diverges toward ´8 as t goes to `8 (the latter assertion follows from λ p D ´ q “ λ p D ` q “ λ p D q{ ą ). However, this leads to a contradictionwith (29), so we must have λ p D q “ , as desired. (cid:4) We can now come to the
Proof of Theorem 2
From Proposition 6, we deduce that G p f q “ and so G p f q “ G p f q .Lemma 7 tells us that G p f q will be minimal if v is as small as possible.From Proposition 6, we also get that for x P r v, x s , f p x q ď ´ γ ˆ ´ αβx ˙ exp p´ F p x qq (32)inequality which can rewritten under the form ddx exp p F p x qq ď ´ γ ddx ˆ x ´ αβ ln p x q ˙ (where d { dx corresponds to a weak derivative).Integrating this bound we obtain e F p x q ´ e F p v q ď ´ γ ˆ x ´ v ´ αβ ln p x { v q ˙ Recalling that F p v q “ and F p x q “ ln p y { γ q , we deduce y γ ´ ď ´ γ ˆ x ´ v ´ αβ ln p x { v q ˙ i.e. ´ γ ´ αβ ln p x q ď v ´ αβ ln p v q The r.h.s. is a quantity which is increasing with v . Thus we musthave v ě x ˚ , where x ˚ was defined in (21).The equality v “ x ˚ is realized if and only if (32) is an equality (a.e.),namely f p x q “ ´ γ ˆ ´ αβx ˙ exp p´ F p x qq (33)42ntegrating this equality as before, we get @ x P r x ˚ , x s , e F p x q ´ “ ´ γ ˆ x ´ x ˚ ´ αβ ln p x { x ˚ q ˙ Replacing this value of e F p x q in (33), we get the function announcedin (22). (cid:4) In the above arguments also enable to compute the minimal valueof G on D γ (which is also the minimal value of K on M γ according toSubsection 5.4). Corollary 1 min D γ G “ G p f ˚ q “ αβγ ˆ ln ˆ αβ ˙ ´ ` βα ´ ln p x q ˙ ´ Proof
In the proof of Theorem 2 we have seen that G p f ˚ q “ G p f ˚ q . Using theexpression given in Lemma 7, where v is replaced by x ˚ , we get G p f q “ γ ˆ x ˚ ´ αβ ln p x ˚ q ˙ ` αβγ ˆ ln ˆ αβ ˙ ´ ˙ It remains to take into account the characterization (21) of x ˚ . (cid:4) Remember this value of min D γ G is only valid under our underlyingassumption, otherwise this minimum is simply 0, as it is attained at the laissez-faire policy. It should be noted that min D γ G is decreasing withrespect to γ , as long as our underlying assumption is satisfied (recall (6)in the main text). This observation will be useful in the next subsection. Finally we come to the proof of Theorem 1. But first we have to re-turn to a rigorous justification of the restriction to (13), which was onlyheuristically discussed at the end of Subsection 5.1. We will also presentan extension of Theorem 1.In the setting of Subsection 5.1, consider a function ϕ : r α { β, x s Ñr , γ s with ϕ p α { β q ă γ , and satisfying (H2), (H3), (H4) and (H5). Due tothe fact that ϕ is right continuous and only jumps upward, this functionattains its maximum, say at x P r α { β, x s . Define a new function r ϕ via @ x P r α { β, x s , r ϕ p x q (cid:66) " ϕ p x q , if x ď x ϕ p x q , if x ą x r ϕ still satisfies (H2), (H3), (H4) and (H5). Recall thefunctional J defined in (14). Lemma 8
We have J p r ϕ q ď J p ϕ q and the inequality is strict if r ϕ ‰ ϕ . Proof
The argument is similar to the proof of Lemma 6 and is based on thefollowing computation: ż x α { β L p u, ϕ p u q , ϕ p u qq du ě βα ż x α { β ` ϕ p u q ϕ p u q ´ αβuϕ p u q du “ r ln p ϕ p u qs x α { β ` βα ż x α { β ϕ p u q ˆ ´ αβu ˙ du ě r ln p ϕ p u qs x α { β ` βαϕ p x q ż x α { β ´ αβu du “ ln p ϕ p x q{ ϕ p α { β qq ` βαϕ p x q „ u ´ αβ ln p u q x α { β ě βαϕ p x q „ u ´ αβ ln p u q x α { β If we replace ϕ by r ϕ in the above computations, all the inequalitiesbecome equalities, so the last term is in fact equal to ż x α { β L p u, r ϕ p u q , r ϕ p u qq du We have furthermore ÿ u Pp α { β,x s : ϕ p u q‰ ϕ p u ´q ln ˆ ϕ p u q ϕ p u ´q ˙ ě “ ÿ u Pp α { β,x s : r ϕ p u q‰ r ϕ p u ´q ln ˆ r ϕ p u q r ϕ p u ´q ˙ and the respective contributions of ϕ and r ϕ to the costs J p ϕ q and J p r ϕ q are the same on p x , x s . It follows that J p r ϕ q ď J p ϕ q The equality holds if in the above computation of the integral ş x α { β L p u, ϕ p u q , ϕ p u qq du , all inequalities are equalities and we get that ϕ p x q “ ϕ p x q for a.e. x P p α { β, x q , namely r ϕ “ ϕ .44 It follows that in the perspective of minimizing J , we can restrictour attention to functions ϕ attaining their maximum at α { β , i.e. wecan replace (H1a) by(H1c): ϕ is defined on r α { β, x s , takes values in r , γ s , ϕ p α { β q “ max r α { β,x s ϕ , ϕ p x q “ y and the left limits of ϕ are positive.To go further toward (H1b), let ϕ a function satisfying (H1c), (H2),(H3), (H4) and (H5), denote η (cid:66) ϕ p α { β q and assume that η ă γ . Fromthe development of Subsections 5.2 to 5.5, we get that J p ϕ q ě min D η G According to Corollary 1 (see also its following paragraph), we have min D η G ą min D γ G This observation ends up the justification of the replacement of (H1a)by (H1b), relatively to the search of a global minimizer of L .We can now come to the Proof of Theorem 1
One direct way would be to check that the procedure described inSubsection 5.1 transform b ˚ into f ˚ . There is even a faster way, as it issufficient to check that C p b ˚ q “ G p f ˚ q and this is a consequence of (8), on one hand, and of Corollary 1, on theother hand (recall that x ˚ “ x p τ q ).This argument shows that b ˚ is a global minimizer of C over B γ .To see that any global minimizer coincides with b ˚ on the time interval r , τ s , take into account Remark 2: since f ˚ is Lipschitzian, there isonly one policy leading to f ˚ (defined on r α { β, x s ). (cid:4) Let us end this subsection with several observations.
Remark 7
Corollary 1 provides in fact a quantitative formulation ofour underlying assumption, since it does correspond to min D γ G ą andwe get γ ă αβ ˆ ln ˆ αβ ˙ ´ ` βα ´ ln p x q ˙ ˝ Remark 8
Consider a cost functional of the form (9), where F coincideswith p¨q ` on R ` and is non-negative on p´8 , q . Then we get that r C ě C .Note nevertheless that r C p b ˚ q “ C p b ˚ q . It follows that b ˚ is also a globalminimizer of r C . In particular if F is positive on p´8 , q , then b ˚ is theunique minimizer of r C . ˝ Remark 9
Our extension of the optimization problem to measurespaces suggests that the S.I.R. ODE (1) could itself be generalized into dx “ ´ xy dBdy “ xy dB ´ αy dt (34)where B is a Radon signed measure on R ` (in (1), it is given by B pr , t sq “ ş t b p s q ds , for all t ě ). Equation (34) is to be understood in the Stieltjessense: for any t ě , x p t q “ x p q ´ ş r ,t s x p s q y p s q dB p s q y p t q “ y p q ` ş r ,t s x p s q y p s q dB p s q ´ ş r ,t s αy p s q ds (where x and y are themselves only right-continuous with left limits, infact they should be seen as repartition functions of measures). It wouldbe modeling very erratic policies and Theorem 2 would imply that evenamong them, b ˚ is a minimizer of the extension of C similar to J (as onewould guess). ˝˝