| Pardoux and Peng[73]were the first to introduce in 1990 nonlinear classical backward stochastic differential equations(BSDE).They proved the existence and the uniqueness of the solutions of nonlinear BSDE under Lipschitz assumptions.Since their pioneering works the theory of BSDE has been developing very quickly and dynamically,and this also thanks to its numerous applications,for example,in stochastic control,finance and the theory of second order partial differential equations.For stochastic control problems the stochastic maximum principle(SMP)and also Bellman’s dynamic programming principle(DPP)are both the most important approaches,in order to detect an optimal control strategy.For the use of Bellman’s DPP as approach we refer,for instance,to([41])and the references therein.There have been a lot of recent works on the stochastic maximum principle.For instance,Wu[87]discussed the SMP for the fully-coupled forward-backward stochastic equations(FBSDE)with a convex control state space,Hu[45]studied recursive stochastic optimal control problems with a control state space which does not need to be convex,Zhang,Li and Xiong[93]investigated a partially-observed optimal control problem.Related with,also the theory of fully coupled FBSDE has experienced a very dynamical development and attracted a lot of researchers,see[6,65,48,74,92].Since Kac[50]introduced the mean-field stochastic differential equation,also known as Mckean-Vlasov equation in 1956,the mean-field problem has received the attention of many scholars.Mean-field backward stochastic differential equations(mean-field BSDE)were introduced by Buckdahn,Djehiche,Li and Peng[18];for more properties of meanfield BSDE we refer to[20,58,27].There are several works which have been devoted to the investigation of the SMP involving this type of stochastic equations.For example,Buckdahn,Li and Ma[19]studied the optimal control problem for a class of general mean-field stochastic differential equations in which the coefficients depend nonlinearly on both the state process and its law.Li[59]considered general mean-field FBSDE with jumps whose generator depends also on the law of the solution.She also studied the associated integral partial differential equation.Without being exhaustive,let us also mention the works by Hao and Li[41],Buckdahn,Djehiche,Li[17]and Li[61],Buckdahn,Chen and Li[16].With the deepening of the research of optimal control theory in biomathematics,the public pays more attention to health in recent years and the rapid development of modern medical technology,the research on the compartmental model of controlled infectious diseases has attracted more and more attention.Compartmental models and the control issues associated to such dynamics have been extensively used for several decades now,see[4,9,40,69]),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 or infinite intervals.For example,Alvarez,Argente and Lippi[1]studied the SIR model for a problem of minimization of blockade costs and optimal blockade policy,and the recent pandemics have widely contributed to enriching the literature[1,54,53,14,37].Recent contributions have focused on state-constrained dynamics usually imposed by intensive-care units(ICU)logistics.Attempts to combine barrier theory or viability theory tools with optimal control problems are the object of[2](for viability-related techniques),[35]or[81](for barrier techniques)and the so-called"safety" issues arising in the process are often investigated.Based on the understanding of optimal control and related content,the present paper mainly discusses two important optimal control problems.In the first part,as discussed in chapter two,we consider a class of mean-field coupled forward-backward stochastic differential equation(FBSDE)and its optimal control problem.Under certain monotonicity assumptions,a theorem on the existence and the uniqueness of a solution of this kind of mean-field FBSDE is given.Then,the stochastic optimal control problem with convex control domain for mean-field coupled FBSDE is discussed,and Pontryagin’s maximum principle is obtained.Our equation is of more general mean field type,with coefficients of both the forward as well as the backward SDEs which depend not only on the controlled solution process(Xt,Yt,Zt)at the current time t,but also on the law of the path of(X,Y,v)of the solution process and the process by which it is controlled.In this case,a new variational inequality and the corresponding Hamiltonian function are obtained.In the second part,discussed in chapter three,the optimal control of an epidemic Compartmental model is studied.A unified theoretical structure for handling controlled compartmental models is constructed,an effective idea of describing the survival domain and solving the optimal control is given.We focus on the characterization of viability zones in compartmental models with varying population size,due to both deaths caused by epidemics and natural demography.A case study based on real data is conducted.Finally,the optimality of minimal("greedy")non-pharmaceutical interventions is completed by a viscosity approach.In what follows we will introduce the content and the structure of this thesis.In Chapter 1 we give the main problems studied in Chapter 2 to Chapter 3 as well as their research background.In Chapter 2,a class of mean-field coupled forward-backward stochastic differential equations and their optimal control are discussed.First of all,we prove a theorem(Theorem 2.2.1)for the existence of a solution for a general mean-field FBSDE(2.2.2)by using Malliavin derivatives and Schauder’s fixed point theorem,and the uniqueness theorem(Theorem 2.2.2)is obtained for a special mean-field FBSDE(2.2.25).Then,we introduce the derivatives with respect to the law(Definition 2.3.2).We extend there the notion of the derivative with respect to a measure from Euclidean spaces to general Banach spaces.Last but not least,we study the optimal control problem for meanfield FBSDE(2.4.1)with control,whose coefficients depend not only on the controlled solution process(Xt,Yt,Zt)at the current time t,but also on the law of the paths of the solution process(X,Y)and the process v by which the solution(X,Y,Z)is controlled.We prove the existence and the uniqueness of the solution of the related equations under suitable assumptions,and we deduce the stochastic maximum principle(Theorem 2.4.2),the main result of our paper.The novelty of this chapter is the following:We extend the mean field FBSDE,study a more general mean field case,and we prove the existence of the solution of FBSDE under just the continuity of the coefficient with respect to the law.At the same time,we extend the maximum principle when the control domain is convex,deduce the maximum principle for the mean-field case,and we get a necessary condition but also a sufficient one for the optimality of a control.In Chapter 3,the optimal control of epidemic compartments is studied.Firstly,we present some details about the dynamics associated to compartmental models and we fix the flow notations.We also define the notions of feasible and no-effort(safe)zone and the related viability vocabulary.Theoretical characterizations are then provided.These characterization are far from being new for control systems.However,we feel that the explanations in this section and the regular frameworks evoked there have an important impact on general compartmental models.This is especially true for Corollary 3.2.1 and the Practical Method 3.2.3.With the above theoretical support,we propose a systematic and unified treatment method for the compartmental model,i.e.,the Practical Method 3.2.3.Then,we discuss the classical SIR model and characterize the natural set on which the dynamics are posed as well as the regular feasible and no-effort zones.A raw SIR model with demography is then studied and a detailed discussion is the object of Theorem 3.3.2.In the population-expanding setting,an exponential normalization is proposed by Corollary 3.3.1.Finally,the population-size normalization is the object of Theorem 3.3.3.Focuses on the illustrations of these results on COVID-19 models in France,using data available from INSEE,comparison of models and their errors are equally provided.Based on the negligibility of epidemic-induced deaths illustrated in the previous section,the Appendix proposes a viscosity treatment of the optimality of greedy policies for simple confinement-induced costs.The novelty of this chapter is the following:This is the first paper to propose a systematic study of safety zones for compartmental models with possibly time-evolving total population balance.This study is complemented by a thorough analysis of the boundaries with respect to the balance of the involved parameters.In the same way,the last offers a variational(viscosity)treatment for the optimality of "greedy" policies(minimal intervention to guarantee feasibility)in a simple framework with state and target constraints.Although the method is strongly inspired by our recent contribution[38]to classical SIR dynamics,it is,again to our best knowledge,a first result for the optimality in a demographic SIRD+model.In Chapter 4 we give a summary and put forward possible future research directions.This dissertation includes the four chapters mentioned above.We now give an outline of the organisation and the main results of the thesis.Chapter 1 Introduction;Chapter 2 Mean-field coupled forward-backward stochastic differential equations and stochastic maximum principle;Chapter 3 A class of compartmental models with controls and time-varying population:normalizations,viability toolkit and comparison of models;Chapter 4 Summary and future works.Chapter 2:In this chapter we discuss mean-field coupled forward-backward stochastic differential equations and the stochastic maximum principle in the control case.Given a Brownian motion B=(Bt)t≥0,the filtration F generated by B and a finite time horizon T>0,consider the following mean-field coupled forward-backward stochastic differential equation(FBSDE):We obtain the existence of a solution of equation(0.0.22):Theorem 2.2.1 We assume(H2.1.1)and(H2.1.2)hold true.Then mean-field FBSDE(0.0.22)has an adapted solution(X,Y,Z)∈SF2 ×SF2 × MF2.(For the definition of the space SF2 and MF2,we refer to Section 2.1).In what follows we consider the following mean-field FBSDE,where X.∧t=(Xs∧t)s∈[0,T],Y.∨t=(Ys∨t)s∈[0,T].It is a special case of equation(0.0.22),and by the above Theorem 2.2.1 and Remark 2.2.1,the existence of the solution is proved.Let us discuss the uniqueness of the solution.Theorem 2.2.2 Under the assumptions(H2.1.2)and(H2.2.1),equation(0.0.23)has a unique solution(X,Y,Z)∈SF2×SF2×MF2.In order to investigate the stochastic maximum principle,we need to study the derivative with respect to the measure.The following results concerning Lion’s Lderivations of u:P2(K)→R extend easily from K=Rn(see Carmona and Delarue[25])to a general separable Banach space K with norm |·|K.Assume that there exists the linear functional derivative Dmu of u.The following Taylor expansion gives an explanation to the linear functional derivative:Proposition 2.3.1 Under our assumptions on u and Dmu we have the following first order Taylor expansion for u:P2(K)→R at m∈P2(K)holds true:u(m’)=u(m)+∫K Dmu(m,x)(m’(dx)-m(dx))+o(W2(m,m’)),as W2(m,m’)→0(m’∈P2(K)).(0.0.24)Here W2(·,·)denotes the 2-Wasserstein distance on the space P2(K)of probability measures over(K,P2(K))with finite 2nd-order moment.Moreover,we have Proposition 2.3.2 Given any ξ∈L2(Ω,F,P;K),we have u(Pξ+η)=u(Pξ)+E[<(?)x(Dmu)(Pξ,ξ),η>K’×K]+R(ξ,η),η∈L2(Ω,F,P;K),(0.0.25)where |R(ξ,η)|=o((E[|η|K2])1/2),as E[|η|K2]→0.Note that Dmu:P2(K)×K→R.Given μ∈P2(K),(?)x(Dmu)(μ,·)denotes the derivative of Dmu(μ,·):K→R.Observe that (?)x(Dmu):P2(K)×K→K’ takes its values in the dual space K’ of K.Based on the previous discussion,we give the definition of the derivative with respect to the measure:Definition 2.3.2 We define the L-derivative((?)μu(m,x))of u at(m,x)∈ P2(K)× K by putting:(?)μu(m,x)=(?)x(Dmu)(m,x),(m,x)∈ P2(K)× K.(0.0.26)Moveover,when we speak about the differentiability of u:P2(K)→R,we mean the L-differentiability.For our studies we are mainly interested in the case K=CTk×uT,where CTk:=C([0,T];Rk)and uT:=L2([0,T]).Remark that the dual space K’ of K is given by K’=BVTk×uT,where BVTk is the space of all càdlàg bounded variation functions on[0,T]with values in Rk.We will write<·,·>k=<·,·>K’×K for the duality product.Definition 2.3.3 For l=(l1,…,lk)∈BVTk,v∈uT,(f(=(f1,…,fk)),v)∈CTk×uT,the duality product<·,·>k is given byAfter giving the existence and the uniqueness theorem for the solution of the relevant FBSDE and the derivative of the solution with respect to the measure on Banach space,we begin to study the optimal control problem for our FBSDE.Here we study the stochastic maximum principle of Pontryagin type,supposing that when the control state space u is convex.Let u ∈uad=LF∞-(Ω,L2([0,T];u))be an admissible control.We consider the following controlled mean-field coupled forward-backward SDE:and we define the following cost functional:J(v)=E[∫0T L(t,Xtv,Ytv,Ztv,P(Xv,Yv,v))dt+φ(XTv,P(Xv,Yv,v))],v∈uad.Observe that in(0.0.27)the coefficients σ and f depend on the control v only through its law.This is related with the fact that we need the Malliavin calculus for the proof of the existence for(0.0.27).Let us suppose that there exists an optimal control u∈uad at which J(·):uad→R takes its minimum.Let v∈uad such that v+u∈uad.We put(X,Y,Z)=(Xu,Yu,Zu).Deriving formally(Xu+εv,Yu+εv,Zu+εv)with respect to ε ∈(0,1)at ε=0+ in L2-sense,we obtain the variational equation:Equation(0.0.28)is a linear mean-field FBSDE.We first prove the existence and the uniqueness of the solution of system(0.0.28).Lemma 2.4.1 Let the assumptions(H2.4.1)and(H2.4.2)be satisfied.Then the above linear mean-field FBSDE(0.0.28)has a unique solution(X1,Y1,Z1)∈SF2× SF2×MF2.After proving the existence and uniqueness for the variational equation,we show that its solution is the L2-derivative of ρ(?)(Xρ,Yρ,Zρ):=(Xu+ρv,Yu+ρv,Zu+ρv)inρ=0+.For this we begin with the following estimate:Lemma 2.4.2 We assume(H2.4.1)and(H2.4.2)hold.Then there is a constant Cp ∈ R+ such that,for v ∈uad,and p≥ 2,and Using the above estimates we shown:Lemma 2.4.3 We assume(H2.4.1)and(H2.4.2)hold.Then,(?)1/ρ(Xtρ-Xt)=Xt1,(?)1/ρ(Ytρ-Yt)=Yt1,(?)1/ρ(Ztρ-Zt)=Zt1,with convergence in SF2×SF2×MF2.Let us now study the so-called variational inequality.For this note that,because u is an optimal control,it holdsρ-1[J(u(·)+ρ(·))-J(u(·))]≥0,ρ∈(0,1).(0.0.31)Thus,thanks to the Lemmas 2.4.1 and 2.4.3 considering the limit in(0.0.31),as ρ→ 0,we getTheorem 2.4.1 We suppose(H2.4.1)and(H2.4.2)holds.Then,the following variational inequality holds true:where θt=(t,Xt,Yt,Zt,P(X,Y.∨t,v)).In order to derive the maximum principle and using the notations we introduced in(2.4.43),we consider the following adjoint FBSDE,Lemma 2.4.4 Under the assumptions(H2.4.1)and(H2.4.2)the adjoint equation(0.0.33)-(0.0.34)has a unique adapted solution(p,q,k)∈ SF2 ×SF2×MF2.The following lemma relates the adjoint with the variational equations.Lemma 2.4.5 Let p be the solution to the adjoint SDE(0.0.34),and(q,k)be the solution to the adjoint BSDE(0.0.34),and(X1,Y1,Z1)be that of(0.0.28).Then,Set ν=P(X,u),μ=P(X,Y,u).The terminal value YT=Φ(xT,P(X,u))and alsoφ(XT,P(X,Y,u))depend on the law.Which for their part,once derived,produce their own coefficients((?)μΦ)j*(t)[p(T)]and((?)μφ)i*(t),j=1,2,i=1,2,3,which we have to take into account in the definition of the Hamiltonian.It adds that the derivatives with respect to the measure depend on the whole solution process of the adjoint FBSDE.This makes that our Hamiltonian cannot be defined in the classical way.We define the Hamiltonian just as follows:H(t,x,y,z,ν,μ):=(-f(t,x,y,z,μ),σ(t,x,ν),L(t,x,y,z,μ),-Φ(·,ν),φ(·,μ)).With the notations introduced in Section 2.4,we get the following stochastic maximum principle:Theorem 2.4.2 Let u be an optimal control of the mean-field FBSDE control problem.Then,recalling the definition of DvH(t),we have the maximum principle:DvH(t)(v-u(t))≥0,for all v∈U,dtdP-a.e.where DvH(t)=-((?)μf)3*(t)[p]+((?)μσ)2*(t)[k]+((?)μL)3*(t)-((?)μΦ)2*(t)[p(T)]+((?)μφ)3*(t),and(p,(q,k))is the solution of the adjoint equation(0.0.33)-(0.0.34).Finally,we shows that the above SMP under the assumption of convexity of the Hamiltonian is also a sufficient one.Theorem 2.5.1 Let us suppose the convexity of the Hamiltonian(-f(t,x,y,z,μ)p(t),σ(t,x,ν)k(t),L(t,x,y,z,μ),-Φ(x’,ν)p(T),φ(x’,μ))in(x,x’,y,z,ν,μ)∈ R4 × P2(CT ×uT)× P2(CT2 ×uT),where(p,q,k)is the solution of the adjoint equation(2.4.44)-(2.4.45).Furthermore,we continue to suppose the standard assumptions(H2.4.1)and(H2.4.2)of the preceding section.Then,if an admissible control u ∈ uad satisfies(2.4.50),it is optimal:J(w)≥ J(u),for all w ∈ uad such that v:=w-u∈uad.Chapter 3:In this chapter,a unified research framework for different compartmental models is established to deal with the optimal control of the compartmental models of epidemics by using viability tools in a normalized way.We use this method to study the classical SIR Model,the birth-death SIR Model,and we focus on the statistical characteristics of the SIR Model with different population sizes,with considering deaths from epidemics and natural deaths.This is achieved by using a viscosity approach of viability.In addition,the methods of natural non-normalization evolution,exponential equilibrium normalization and population normalization are used.Then,based on the actual data of epidemic diseases in France in recent years,case study,data analysis and numerical simulation are carried out.This chapter is completed with a viscosity approach that minimizes("greedy")the optimality of non-pharmaceutical interventions.Given a family of interactions(between species,compartments,etc),a classical way of associating a dynamical system is the use of law of mass action.Whenever some of the interaction speed is subject to external interventions(e.g.confinement,vaccination or mere temperature variation for chemical reactions),this leads to a controlled vector dynamical system,generically referred to as dxu(t)=F(xu(t),u(t))dt,t≥0,(0.0.36)for every Borel measurable control u ∈ L0(R;U).The overall contribution of controls yields an affine dynamics,i.e.F(x,u)=F1(x)+ F2(x)u,where F1:RN →RN,F2:RN→RN×M.When distinguishing input components xin.and output components,for x=(xin,xout),one gets the following structureFor our readers’ sake,and in an effort of providing a self-contained material,we present here relevant mathematical notions such as viability,viability kernel,invariance,capture basin.Restrictions may be imposed on the variables(linked to available intensive care units,the overall number of deaths,etc.).As it is usually the case,the restrictions are given in the form gi(x)≤ 0,1≤i ≤ P,and they lead to a vector field G:RN→RP,G(x)=(g1(x),...,gP(x)).This defines a set TG:={x0∈T:gi(x)≤0,1≤i ≤P}.Definition 3.2.1 A set K is called viable with respect to(0.0.36)if,for every initial data x0∈K,there exists a control u ∈ L0(R;U)keeping the trajectory in K.It is called invariant if it is viable with every control u ∈ L0(R;U);We call no-effort or safe zone(or forward invariant set)the family A of all initial data x0∈RN such that,for every u ∈ L0(R;U),one has xx0,u(t)∈TG,for all t≥ 0;We call feasible(or viable,see next remark)zone the set B of all initial configurations x0∈RN for which there exists a control u ∈ L0(R;U)keeping the associated trajectory xx0,u(t)∈ TG,for all t≥ 0;The largest subset K(?)TG such that trajectories starting at x0 ∈K can be maintained in TG is called the viability kernel of TG.Let us begin to start with the theoretical analysis.With respect to the system of state equations(0.0.36)with affine structure,let us now introduce,for a regular functionφ∈ C1(RN;R),the linear operators L1φ(x):=(?)φ(x)F1(x)and L2φ(x):=(?)φ(x)F2(x),where the gradient is written as an horizontal 1 × N vector,while F1 and F2 are N × 1 and N × M matrices,respectively.Of course,in the particular case(0.0.37),one has Fi(x)=(Fi,in(xin),Fi,out(x)),for i=1,2 and x=(xin,xout).For the viability set and the invariant set we have the following theorem:Theorem 3.2.1 Let K be a closed subset of RN.The following assertions are equivalent.1.K is viable(resp.invariant)with respect to(0.0.37);2.every function φ∈C1(RN)with a local maximum on K at x satisfies3.the function v(x):=1-1K(x)is a viscosity supersolution of(3.2.3),i.e.,it satisfies the inequality in a viscosity sense,that is for every test function φ∈C1(RN)such that φ(x)-v(x)has a local maximum at x.Theorem 3.2.2 The feasible zone B with respect to(0.0.37)included in TG is characterized as the kernel B=ker(ψB):={x ∈RN:ψB(x)=0},where ψB is the smallest function such that:(a)It takes its values in[0,1]and is identically 1 on RN\TG;(b)it is lower semicontinuous;and(c)it is a viscosity supersolution to(0.0.38),with the inf formulation.The same kind of characterization applies to A,using the sup formulation in(0.0.38).Before applying the specific model,let us review the practical approach.Practical Method 3.2.31.In practice,when one is interested in one-dimensional controls u ∈ U:=[umin,umax]and searches for regular functions φ obeying(3.2.3)(or the more specific(3.2.5)),according to Remark 3.2.2,the set B is split into B1∪B2 (?) TG,where L2(φ)≥ 0,L1(φ)≤-uminL2(φ)on B1,L2(φ)<0,L1(φ)≤umaxL2(φ)on B2.2.For safe zones A,umin and umax swap place(these sets are,therefore,more restrictive).3.The inequalities on L1 are actually equalities in the interior of each of the two sets,according to(3.2.5).4.The same kind of argument applies to the search of invariant sets A with the mere swap between umin and umax in the system above.One can search simultaneously for invariant and viable sets Aj(?)Bj,j=1,2.5.One has to enforce a kind of "boundary" condition on RN\TG which isφ＞0.6.The actual boundary conditions are obtained by taking a look at the admissible part of the original constraints (?)TG,i.e.,by looking for each index 1≤i≤P at the saturated family gi(x)=0.7.Depending on the type of problem,instead of searching the domain given by an implicit equation φ(x)=0 characterized in Theorem 3.2.2,one searches for an explicit one x1=(≤)φ(x1,...,xN).According to Corollary 3.2.1,the maximality of the viable domain now becomes a search for maximal φ.After the previous theoretical and practical discussion,we discuss and solve different compartment models,including SIR model,SIRD+ model.We consider a maximally admissible level of the infections imax(linked to Intensive Care Units capacities)and the ensuing restriction on the i component of the state variable x.g(x)=i-imax≤0.First,we discuss the classical SIR Model 3.3.1.After our standardized discussion,we come to the following conclusion:Theorem 3.3.11.A basic invariant set is given by T2:={(s(t),i(t),r(t))∈R+3:r(t)=1-s(t)i(t),s(t)+i(t)≤1,t≥0}.2.The largest invariant safe set is A={(s(t),i(t),r(t))∈T2:s(t)≤φ((i(t)),i(t)≤imax,t≥0},where φ is the unique solution of the problem-βi-βφ(i)+γlog φ(i)=-βimax-βsmax+γ log smax,φ(i)≥ smax with smax:=γ/β.3.The largest viable set is B={(s(t),i(t),r(t))∈T2:s(t)≤φ(i(t)),i(t)≤imax,t≥0},where φ is the unique solution of the problem-βumini-βuminφ(i)+γlogφ(i)=-βuminimax-βuminsmin+γlog smin,φ(i)≥smin,with smin:=γ/βumin.4.The functionsφ in 2.and 3.are smooth,more precisely,φ∈C([0,imax])∩C∞(0,imax)).Let us now turn to the birth-death SIR model(3.3.4),for which we have obtainedTheorem 3.3.2.Making the T2ext-Induced Normalization for the SIRD+equation,i.e.,equation(3.3.13),we get the following result:Corollary 3.3.11.The largest safe(invariant)set w.r.t,(3.3.4)of the form A={(t,s(t),i(t),r(t))∈R+4:s(t)≤e(μ+-μ-)tφ(e-(μ+-μ-)ti(t),i(t)≤imaxe(μ+-μ-)t,t≥ 0},with smooth C1-function φ,if exists,is obtained for-φ(e-(μ+-μ-)ti)+(γ+ν)/β log(βφ(e-(μ+-μ-)t i)-μ+)-e-(μ+-μ-)ti=-(γ+μ++ν)/β+(γ+ν)/βlog(γ+ν)-imax.2.The largest feasible(viable)set B={(t,s(t),i(t),r(t))∈R+4:s(t)≤e(μ+-μ-)tφ(e-(μ+-μ-)ti(t)),i(t)≤imaxe(μ+-μ-)t,t≥ 0},with smooth C1 φ,if exists,is obtained for-φ(e-(μ+-μ-)ti)+(γ+ν)/βumin log(βuminφ(e-(μ+-μ-)t i)-μ+)-e-(μ+-μ-)ti=-(γ+μ++ν)/βumin+(γ+ν)/βuminlog(γ+ν)-imax.Moreover,we investigate the population-normalized SIRD+equation.After our technical discussion,we get the following result for the population-normalized model(3.3.20):Theorem 3.3.31.The largest safe(invariant)set A={(s(t),i(t))∈R+2:s(t)≤φ(i(t)),i(t)≤imax,t≥ 0},with smooth C1-function φ,if exists,is obtained for smax:=(μ++γ+ν-νimax)/βand φ(i)≥φ(imax)=smax,as follows.(a)If ν=0,then-φ(i)-i+γ/β log(βφ(i)-μ+)=-smax-imax+γ/β log(βsmax-μ+).(b)If ν=β thenⅰ.if μ+>0,then log(i+φ(i)-(γ+ν)/ν)-ν/μ+φ(i)=log(imax+smax-(γ+ν)/ν)-ν/μ+smax;ⅱ.if μ+=0,then φ(i)=smax-i+imax.(c)If 0<ν≠β,then(1+φ’(i))(μ++(ν-β)φ(i))-νφ’(i)(i+φ(i)-(γ+ν)/ν)=0,φ’(i)≤-1 and φ(imax)=smax.2.Similar assertions are true for the viable zone B by merely replacing β with βumin and smax with smin:=(μ++γ+ν-νimax)/βumin.Then,we consider numerical examples and make the comparison of different models,by using aggregated data from the INSEE open database on the COVID-19 epidemic in France between January 2020 and March 2022 in Section 3.4.Furthermore,we consider the confinement cost-related functional l(u):=1-u and the related performance criterion:Minimize J(s0,i0,u):=∫0∞ l(u(t))dt over u ∈L0(R;U)such that(ss0,i0,u,is0,i0,u)∈B.The formal Hamilton-Jacobi equation satisfied by V is The value of this minimization is denoted by V(s0,i0).Under the assumption ν≈0,we give a "greedy" control strategy Finally,the viscosity approach is used to prove that the greedy control strategy is our optimal control strategy.The value function related to this strategy is the following:Proposition 3.5.1 The value associated with the policy(0.0.40)denoted by W(s0,i0):=J(s0,i0,ugreedy)is explicitly given by Here,s1(s0,i0)is the point at which the 1-controlled trajectory issued from(s0,i0)saturates the constraint i=imax,i.e.,satisfies(3.5.6).Similarly,(s2(s0,i0),i2(s0,i0))is the point at which the 1-controlled trajectory issued from(s0,i0)touches the boundary of B,i.e.,the solution of(3.5.7).Finally,B0 is the domain delimited by the backward in time trajectory reaching smin=(μ++γ)/βumin with constant control 1,i.e.,it is described by the curveφB0(i)+i-γ/β log(βφB0(i)-μ+)=smin+imax-γ/βlog((μ++γ)/umin-μ+)with φB0(i)≥(γ+μ+)/βumin.To check the optimality of ugreedy,one proceeds as in[38,Section 3.3]by verifying that the associated cost W satisfies(0.0.39). |
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