Verification Results for Age-Structured Models of Economic-Epidemics Dynamics
aa r X i v : . [ ec on . T H ] A ug VERIFICATION RESULTS FOR AGE-STRUCTURED MODELSOF ECONOMIC-EPIDEMICS DYNAMICS
GIORGIO FABBRI, FAUSTO GOZZI, AND GIOVANNI ZANCO
Abstract.
In this paper we propose a macro-dynamic age-structured set-up forthe analysis of epidemics/economic dynamics in continuous time.The resulting optimal control problem is reformulated in an infinite dimen-sional Hilbert space framework where we perform the basic steps of dynamicprogramming approach.Our main result is a verification theorem which allows to guess the feedbackform of optimal strategies. This will be a departure point to discuss the behaviorof the models of the family we introduce and their policy implications.
Keywords : COVID-19, macro-dynamic models, epidemiological dynamics,Hilbert spaces, verification theorem.
JEL Classification : E60, I10, C61. Introduction
The outbreak of the COVID-19 pandemic represents, in addition to an epidemio-logical historical event, an exceptional economic shock. Data from the OECD (2020)suggest that in many countries the loss of GDP due to the presence of the virus andthe consequent containment measures will be at least 10%. For this reason, to-gether with the obvious upsurge in medical scientific production on the subject, thephenomenon has had a great echo in economic literature with a strong pressure tomerge economic and epidemiological models.A specific effort has been made to integrate epidemiological compartmental mod-els (SIR, SEIR, SEI...) into a macroeconomic dynamic context, see for examplethe contributions of Alvarez et al., (2020), Eichenbaum et al., (2020), Jones et al.,(2020) and Krueger et al.(2020).These articles focus on a series of questions essential to health and economic pol-icy and they look, often numerically, at the trade-off between measures capable ofcontaining contagion and those capable of avoiding economic collapse. However,they model the spread of the epidemic with age homogeneous epidemiological com-partmental models so they cannot take into account one of the characteristic traits of the current epidemic, i.e. the great difference in the effects of the disease amongpeople of different ages .In order to address this limitation Acemoglu et al. (2020), Gollier (2020) andFavero et al. (2020) introduce models where the population is divided into a finitenumber of homogeneous “risk groups” and they study joint economic and epidemi-ological effects of introducing group-specific policies. Nonetheless in their formula-tions there is no possibility to move from one group to another and then this kindof approach can take into account the different effects of the disease on different agegroups only if it is assumed that the duration of the epidemic is negligible comparedto the age range contained in each group. However, this hypothesis is not very likelyin the case of an epidemic lasting several years and it is inadequate in the case ofdiseases that become endemic in the population .Instead of using age-homogeneous epidemiological compartmental models or epi-demiological compartmental models with closed risk groups, it is possible, as wedo in the present work, to describe more accurately the joint dynamics of the epi-demic and of the age structure of the population by using explicit age-structuredcompartmental models, i.e. age-specific epidemiological models with ageing processmodeled `a la Mc Kendrick (1925). This type of models was initially introducedby Anderson and May (1985) and Dietz and Schenzle (1985) and later adapted tonumerous contexts and applications, see the books by Iannelli (1995), Iannelli andMilner (2017) and Martcheva (2015) for a structured and modern description of thematter.The more accurate description of the ageing-epidemics diffusion dynamics comesat a price and, indeed, one of the features of the continuous time compartmentalage-structured models is to describe the epidemiological dynamics through transporttype partial differential equations (PDEs). This means that, if one wants to studyan associated optimal control problem through dynamic programming, its dynamicsneeds to be seen in an infinite-dimensional set-up.In this paper we present verification type results for a class of macro-dynamicmodels that incorporates an epidemiological dynamics which generalizes the bench-mark age-structured SIR model. In fact, the probability of aggravation of COVID-19 infection and mortality varies very signifi-cantly with age. Salje et al. (2020) find for example that less than 1% of people under 40 years ofage who contract the disease need hospital care against more than 10% of people over 70 years ofage and that mortality in the two groups is respectively less than 0.02% and more than 2%. These limitations are obviously justified by the need to produce policy indications in a shorttime in order to contrast the spread of the current pandemic.
ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 3
In the model, the planner optimizes an optimal social functional by choosingpolicies to reduce the spread of the virus, taking into account (i) their effectiveness interms of human lives saved (ii) their impact on the labor supply and therefore on thelevel of production and consumption reached by the country (iii) their direct costs.As in standard growth models, the planner also decides the level of consumption,which can be age-specific, and consequently, the level of investments that determinesthe dynamics of capital accumulation.The class of models that we study in the abstract form is rather general and isable, in the context of the epidemiological dynamic described by an age-structuredSIR, to reproduce as special cases several of the settings proposed by the recentarticles mentioned above (more details can be found in Section 2). Specific traits ofthe model are:- The epidemiological model is general and can be set with a wide varietyof age-dependent parameters: mortality rate (due to epidemics and alsoto other causes), chance of being hospitalized if infected, birth rate andprobability of contagion among cohorts. Moreover, the age-specific mortalityrate can take into account the saturation of hospitals and health systems, aphenomenon that has been repeatedly observed in the areas most affectedby COVID-19 (see for instance Moghadas, 2020)- The planner has two different policy levers: on the one hand, as in most ofthe models mentioned, she can reduce the mobility of people and partiallystop the economic activity (lock-down), on the other hand she can implementsome costly action to reduce the diffusion of the virus, for instance by testingthe population extensively to try to quarantine individuals fast once theyhave contracted the virus. Both policies can be age-specific (in particulartargeted lock-downs suggested by Acemoglu et al., 2020 are among possiblepolicies)- The time horizon can be either infinite (as in benchmark growth models) orfinite if it is considered (e.g. Gollier, 2020) that the spread of the virus stopsat some point due to the discovery of a vaccine or a cure- Labor productivity is age-specific (this fact is important for policies: tar-geted lock-downs for less productive people impact less the production).The production function (as a function of aggregated labor and capital) isgeneral as well as the optimal social function that can be specified to takeinto account cost-benefit analysis, strictly humanitarian or economic targets,standard (Benthamite and Millian for instance) social welfare functionals.Since our results are proven for the abstract model they hold for any possible spec-ification.
G.FABBRI, F.GOZZI, AND G. ZANCO
The contribution of this work is (i) to propose a general fully age-structuredmacro-dynamic set-up in continuous time for analysis of epidemics and economicdynamic (Section 2); (ii) to provide a suitable Hilbert space environment whereone can rewrite the problem and perform dynamic programming (Section 3); (iii)to prove verification type results (Section 5), see in particular Theorem 5.2 andCorollaries 5.3 and 5.4.We must be clear on the fact that we do not solve the problem explicitly, nornumerically. Here our main goal is to provide a general ground which can be thedeparture point to attack special cases of our general model. In particular our maincontribution is the proof of the verification type results of Section 5. These arenontrivial to obtain in our general infinite dimensional setting and they are crucialto find the optimal policies in a closed-loop form depending on the derivatives ofthe value function. These type of theorems are the object of various papers (see e.g.the papers Faggian and Gozzi (2010), Fabbri et al. (2010)) or of book chapters (e.g.Chapter 5 of Yong and Zhou, 1999 or Chapter 4 of Li and Yong, 1995) but none oftherm applies to our case. The main reasons are the following. First, due to theage-structured nature of the problem and the presence of the mortality forces, wehave to work with semigroups in weighted infinite dimensional spaces which do nothave regularizing properties (which are very useful and which are usually true whenthe state equation is of (nondegenerate) second order). Second, the presence of thenonlinear equation for capital rules out the standard regularity assumptions thatare used e.g. in Chapter 4 of Li and Yong (1995) and that we treat using ad-hoc arguments. Third, in our case we have state constraints, which makes much moredifficult to deal with the problem. We use the approach of weakening the constraintswhich has been used, up to now, only in case when explicit solutions of the HJBequation are available (see e.g. Fabbri and Gozzi, 2008, Boucekkine et al., 2019).The paper is organized as follows. In Section 2 we introduce the structure of themodel: epidemiological dynamics, policies, structure of the economy and welfarefunctional. In Section 3 we show how to reformulate the model and the relatedoptimal control problem in a suitable Hilbert space setting. Section 4 is devoted todynamic programming while in Section 5 we provide the verification results. Section6 concludes. 2.
The model
Epidemics dynamics.
We denote by s ( a, t ) the density of susceptible indi-viduals of age a ∈ [0 , ¯ a ] (being ¯ a > t ≥
0. Similarly i ( a, t ) (respectively r ( a, t )) denotes the density of infected/infectious individuals(respectively recovered individuals) of age a at time t . Hence the total numbers of ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 5 susceptible, infected, and recovered individuals are S ( t ) = Z ¯ a s ( a, t ) da, I ( t ) = Z ¯ a i ( a, t ) da, R ( t ) = Z ¯ a r ( a, t ) da. The age-dependent density of the total population n ( a, t ) is then given by n ( a, t ) = s ( a, t ) + i ( a, t ) + r ( a, t )and the total population is N ( t ) = Z ¯ a n ( a, t ) da = S ( t ) + I ( t ) + R ( t ) . In modeling the mortality we generalize the standard age-structure SIR framework(see Martcheva, 2015, Chapter 12). First we defineΞ( t ) := Z ¯ a i ( a, t ) ξ ( a ) da the number of people at time t in “critical conditions” i.e. people who have to usethe services of hospital/healthcare facilities to treat themselves at the risk of satu-rating them. In the case of the COVID-19 epidemic, the emphasis is, for example,on people needing to be hospitalized in an ICU (see for instance Moghadas, 2020).Ξ( t ) depends on the number of sick people per cohort multiplied by the prevalence ξ ( a ) of people in need of specific care for each age group . In a context of saturationof hospital services, the mortality of the infected will be increased. Let us thereforeassume that the mortality rate for infected individual µ I is not only a function of theage of the individuals but it also en increasing function of Ξ. We use then the nota-tion µ I ( a, Ξ( t )). We suppose, for simplicity, that the (age-specific) mortality rates ofsusceptible and recovered individuals, respectively µ S ( a ) and µ R ( a ) do not depend on Ξ( t ). Finally γ ( a ) and β ( a ) denote respectively the (age-specific) recovery andbirth rates.The age-specific force of infection λ ( a, t ) depends on the distribution of infectedindividuals as follows(1) λ ( a, t ) = 1 N ( t ) Z ¯ a m ( a, τ ) i ( τ, t ) dτ. In this expression the joint-distribution m ( a, τ ) measures the different probability ofcontagion between cohorts (for instance virus diffusion can be easier among childrenfor childhood diseases). It is the continuous version of the social contact matrixacross age classes used by Gollier (2020). In the case of COVID-19 for example, in the data of Salje (2020), 2.9% of infected individualsare hospitalized ranging from 0.1% in people under 20 years to approximately 30% in individualswith 80 years of age or older. Indeed it is possible to incorporate this dependence in the model without big problems.
G.FABBRI, F.GOZZI, AND G. ZANCO
All in all, the laissez faire benchmark population dynamics, that is the epidemicsdynamics without policy intervention (omitting the initial conditions at time t = 0)is the following:(2) ∂s ( a,t ) ∂t + ∂s ( a,t ) ∂a = − λ ( a, t ) s ( a, t ) − µ S ( a ) s ( a, t ) , ∂i ( a,t ) ∂t + ∂i ( a,t ) ∂a = λ ( a, t ) s ( a, t ) − ( µ I ( a, Ξ( t )) + γ ( a )) i ( a, t ) , ∂r ( a,t ) ∂t + ∂r ( a,t ) ∂a = γ ( a ) i ( a, t ) − µ R ( a ) r ( a, t ) s (0 , t ) = R ¯ a β ( a ) n ( a, t ) dai (0 , t ) = 0 r (0 , t ) = 0 . This system is the standard age-structure SIR model (see Martcheva, 2015, Chap-ter 12) except for the fact that µ I depend on Ξ. In the particular case where µ I ( a, Ξ( t )) = ˜ µ I ( a ) we are exactly in the standard setting. Note that, since λ andΞ depend linearly on i , the system (2) is non linear in the variables ( s, i, r ).We now introduce two of the three policy levers that the planner has in our model(the third is the choice of consumption and will be described in the next subsection).We suppose that the planner can deal with the epidemic in two ways:(i) partially stopping economic activity and people mobility then reducing thecontagion frequency among individual (lockdown);(ii) implementing some costly action to reduce the diffusion of the virus, for in-stance by testing the population extensively to try to quarantine individualsfaster once they have contracted the virus.More precisely(i) We suppose that the planner can reduce mobility and then the probability ofinfecting and being infected of cohort a at time t by a factor θ ( a, t ) ∈ [0 ,
1] atthe cost of reducing the contribution of the concerned individuals to work orby reducing their work productivity (for example resorting to teleworking).This is the type of intervention which is modeled in almost all the macro-dynamic models we mentioned in the introduction, for instance in Alvarez etal., (2020) and Eichenbaum et al., (2020) where, by the way, age-structurepolicies are not possible since there is no age structure of the population.Taking different values of θ for different a correspond to target lock-downsof Acemoglu et al. (2020).(ii) We suppose, as in some of the mentioned papers, that the planner can reduceby a factor η ( a, t ) ∈ [0 ,
1] the probability that infected individuals of cohort
ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 7 a at time t contaminate other people. This is done at the cost(3) D η ( t ) := D (cid:18)Z ¯ a η ( a, t ) i ( a, t ) e ( a ) da (cid:19) . where e ( a ) is an age-specific relative cost and D is a concave (as, for instancein Piguillem and Shi, 2020) or linear (as in Gollier, 2020) function which rep-resents some form of congestion (e.g. shortage of tests on the internationalmarket or shortage of suitable medical personnel to administer the tests).The evolution of the epidemics is then again described by (2) but, instead of λ ( a, t ) written in (1) we have now the following age-specific force(4) λ θ,η ( a, t ) = θ ( a, t ) N ( t ) Z ¯ a m ( a, τ ) θ ( τ, t ) η ( τ, t ) i ( τ, t ) dτ. Hence we get the following state equation (still omitting the initial conditions attime t = 0):(5) ∂s ( a,t ) ∂t + ∂s ( a,t ) ∂a = − λ θ,η ( a, t ) s ( a, t ) − µ S ( a ) s ( a, t ) , ∂i ( a,t ) ∂t + ∂i ( a,t ) ∂a = λ θ,η ( a, t ) s ( a, t ) − ( µ I ( a, Ξ( t )) + γ ( a )) i ( a, t ) , ∂r ( a,t ) ∂t + ∂r ( a,t ) ∂a = γ ( a ) i ( a, t ) − µ R ( a ) r ( a, t ) s (0 , t ) = R ¯ a β ( a ) n ( a, t ) dai (0 , t ) = 0 r (0 , t ) = 0 . Of course if the authority fixes θ ( a, t ) ≡ η ( a, t ) ≡ Production and capital accumulation.
We suppose that labor supply isperfectly inelastic to wage, that infected people do not work and that labor produc-tivity is age-specific and proportional to a certain parameter α ( a ) (we can specifyfor instance α ( a ) = 0 for children or for individuals older than a fixed retirementage). Total labor supply in efficiency units in the laissez faire benchmark is thengiven by R ¯ a ( s ( a, t ) + r ( a, t )) α ( a ) da . In the controlled case we suppose that gettinga factor θ ( a, t ) ∈ [0 ,
1] in the expression of the age-specific force of diffusion impactsthe productivity of cohort a reducing the productivity to ϕ ( θ ( a, t )) so that total We abstract from other reasons of productivity heterogeneity among population and from het-erogeneity of tasks.
G.FABBRI, F.GOZZI, AND G. ZANCO labor supply in efficiency units is now(6) L ( t ) = Z ¯ a ( s ( a, t ) + r ( a, t )) α ( a ) ϕ ( θ ( a, t )) da. We suppose that ϕ : [0 , → [0 ,
1] is an increasing function with ϕ (1) = 1.As for the production we stick to the standard structure of neoclassical growthmodels and we suppose that the total production at time t is described by anaggregated production function F of the two factors: labor L ( t ) and capital K ( t ): Y ( t ) = F ( K ( t ) , L ( t )) . This formulation is more general than that used by other macro-dynamic papers wementioned. Indeed in all of them except Favero et al. (2020) which uses a Cobb-Douglas production function, the authors use production functions which are linearfunction of the of labor (or effective labor) or even do not model production.We abstract from international trade (closed economy) and from governmentalexpenditure so the planner can choose at any time t ≥ Y ( t ) among total investment M ( t ), consumption of various cohortsand costs for testing people, which is defined in (3) above. If we denote by c ( a, t )the per-capita consumption of individuals of age a at time t we get the followingbudget constraint: Y ( t ) = M ( t )+ C ( t )+ D η ( t ) := M ( t )+ Z ¯ a c ( a, t ) n ( a, t ) da + D (cid:18)Z ¯ a η ( a, t ) i ( a, t ) e ( a ) da (cid:19) . Supposing to have an exponential capital depreciation `a la
Jorgenson we get thedynamic accumulation law for capital:(7)˙ K ( t ) = F ( K ( t ) , L ( t )) − Z ¯ a c ( a, t ) n ( a, t ) da − δK ( t ) − D (cid:18)Z ¯ a η ( a, t ) i ( a, t ) e ( a ) da (cid:19) . where δ > A similar approach is considered for instance by Jones et al. (2020) which introduce an “effectivelabor supply”.
ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 9
Choosing the target.
For the functional to maximize there are several inter-esting choices in the literature. It is not easy to include them all in an abstract formthat leaves the problem tractable, so in this section we introduce several functionalsthat will be discussed later in the article.The first functional we introduce is a standard welfare functional. Observe that,even if the model we study here is not directly an endogenous fertility model, thefact of having an endogenous mortality (depending on the choice of θ ) makes it defacto an endogenous population size model. Therefore we have to choose carefullythe structure of the social utility that we describe. We implicitly fix the utility ofdead people (and non-born people through the initial condition R ¯ a β ( a ) n ( a, t ) da )equal to 0 and we consider the following social utility functional:(8) Z ∞ Z ¯ a e − ρt n ν ( a, t ) u ( c ( a, t ) , θ ( a, t )) da dt. To assure, for the same per capita (age-dependent) consumption, the instantaneousutility to be increasing in the number of living people and therefore the planner beingaverse to death of agents, the per-capita utility function u needs to be positive. Stillobserve that in this model formulation there is room for a dilution effect: the largerthe population the lower the percapita consumption so the instantaneous utilitydoes not need to always be increasing in the population size.The per-capita utility function u depends both on the individual consumptionand on the mobility freedom θ . We suppose that u is (positive and) an increasingfunction in both the variables. The dependence of utility on θ is not standard but therelevance of this choice can easily be argued by looking at the various side effects oflock down (see for example Clemens, 2020). In any case, as a special case, of courseone can specify u so that it does not depend on theta.The form of this first functional is the age-structured version of a standard func-tional often appearing in the optimal population literature. The parameter ν whichappears in its expression measures the degree of altruism towards individuals of fu-ture cohorts (see Palivos and Yip, 1993). The case ν = 1 corresponds to the classicaltotal utilitaristic (or “Benthamite”) case where the planner target is to maximizethe sum of individuals’ utility.The functional (8) is infinite horizon and implicitly suggests that no exogenouselement impedes the spread of the virus. Another possibility, as suggested by Gollier(2020), is to consider a final time T at which an event (a cure or more probably thediscovery of a vaccine) stops the epidemics. We describe some possible targets inthis context.The trade-off of virus containment policies is: reducing the number of deaths VSeconomic losses. Some of the functional aspects can be dwelt on only one of theseaspects. For instance one can decide to focus on economic activity and to maximize the final production capacity:(9) F ( K ( T ) , ˜ L ( T ))where ˜ L ( T ) is defined as(10) ˜ L ( T ) = Z ¯ a n ( a, T ) α ( a ) da (once the vaccine is in and the outbreak is over, everyone is cured and everyone isproductive) or even more simply, to maximize final capital level(11) K ( T )or to maximize the flow of production(12) Z + ∞ e − ρt Y ( t ) dt. or its finite counterpart(13) Z T e − ρt Y ( t ) dt. Conversely one can focus on humanitarian aspects and decide to maximize the num-ber of deaths due to the virus:(14) Z T Z ¯ a µ I ( a, Ξ( t )) i ( a, t ) dadt. It is also possible to simply take into account both the economic and humanitarianaspects by taking a weighted sum of the (12) (or 11) or (13)) and (9), this is thechoice of Acemoglu et al. (2020).3.
Infinite dimensional formulation of the model
In this section we introduce a convenient infinite-dimensional formulation forsystem (5) coupled with equation (7) and for the control problem of maximizingthe target given in (8) (or in (12), (9), (11), (13)). We make the following set ofassumptions, which also includes those already stated in the previous sections. Theseassumptions will be always true in the remainder of the paper without mentioningthem.
Hypothesis 3.1 ( i ) µ S and µ R are positive, belong to L loc (0 , ¯ a ) and Z ¯ a µ S ( a ) da = Z ¯ a µ R ( a ) da = + ∞ ; ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 11 ( ii ) µ I : [0 , ¯ a ] × R → R + is measurable. Moreoves it is Lipschitz continuous inthe second variable, uniformly with respect to the first one. Finally it isincreasing in the second variable and Z ¯ a µ I ( a, κ ) da = + ∞ , ∀ κ ∈ R ;( iii ) F ( · , L ) is Lipschitz for every L ∈ R + , with Lipschitz constants uniformlybounded in L ; ( iv ) ϕ : [0 , → [0 , is increasing and ϕ (1) = 1 ; ( v ) α, β, γ, e, ξ : [0 , ¯ a ] → R + are in L (0 , ¯ a ) ; ( vi ) D : R → R is positive and concave; ( vii ) δ > , ν ∈ [0 , ; ( viii ) u : R × [0 , → R is positive, continuous and increasing in both variables; We now start rewriting the system (5) but first we introduce an important nota-tional standard.
Notation 3.2
In the system (5) the three state trajectories, s ( · , · ), i ( · , · ), r ( · , · )are seen as function of two variables, i.e.( s, i, r )( · , · ) : [0 , + ∞ ) × [0 , ¯ a ] → R , ( t, a ) → ( s ( t, a ) , i ( t, a ) , r ( t, a ))However now it is convenient to see such trajectories as functions from t ∈ R + to asuitable infinite dimensional Hilbert space H of functions in the variable a ∈ [0 , ¯ a ]with values in R . H can be seen also as the product of three Hilbert spaces offunctions with values in R and its generic element will be denoted by h = ( h , h , h )or, if no confusion is possible, by ( s, i, r ). To avoid misunderstandings we will denotethe state trajectories putting a hat over the original name, i.e.(ˆ s, ˆ i, ˆ r ) : R + → H t → (cid:16) ˆ s ( t ) , ˆ i ( t ) , ˆ r ( t ) (cid:17) Sometimes we write ˆ h for (ˆ s, ˆ i, ˆ r ) and, when we want to underline that they arefunctions we write ˆ h ( · ) or (ˆ s ( · ) , ˆ i ( · ) , ˆ r ( · )). Now observe that, for every t ≥ s ( t ) , ˆ i ( t ) , ˆ r ( t ) are functions of a . We will denote their value at a given a ∈ [0 , ¯ a ]with ˆ s ( t )[ a ] , ˆ i ( t )[ a ] , ˆ r ( t )[ a ] so to emphasize the different role of the two variables.Clearly we will haveˆ s ( t )[ a ] = s ( t, a ) , ˆ i ( t )[ a ] = i ( t, a ) , ˆ r ( t )[ a ] = r ( t, a ) . The same will be done for the controls strategies c ( · , · ), θ ( · , · ), η ( · , · ). More preciselywe will fix a control space Z of functions in the variable a ∈ [0 , ¯ a ] with values in R .Also Z can be seen as the product of three Hilbert spaces of functions with values This assumption is not verified for the Cobb-Douglas type functions which, however, can betreated ad hoc in this framework. in R and its generic element will be denoted by z = ( z , z , z ) or, if no confusionis possible, by ( c, θ, η ). Also here, to avoid misunderstandings, we will denote thecontrol trajectories putting a hat over the original name, i.e. we call ˆ c, ˆ θ, ˆ η thefunctions (ˆ c, ˆ θ, ˆ η ) : R + → Z t → (cid:16) ˆ c ( t ) , ˆ θ ( t ) , ˆ η ( t ) (cid:17) such that ˆ c ( t )[ a ] = c ( t, a ) , ˆ θ ( t )[ a ] = θ ( t, a ) , ˆ η ( t )[ a ] = η ( t, a ) . Sometimes we write ˆ z for (ˆ c, ˆ θ, ˆ η ) and, when we want to underline that they arefunctions we write ˆ z ( · ) or (ˆ c ( · ) , ˆ θ ( · ) , ˆ η ( · )).We are now ready to introduce the spaces H , Z , and the space Z of basic controlstrategies (this is not the space of admissible control strategy which will have to takeaccount of the state constraints that we will introduce below).Define the probability of surviving to age a for a susceptible individual as π S ( a ) = exp (cid:18) − Z a µ S ( τ ) dτ (cid:19) and, similarly, define the probability of surviving to age a for a recovered individualas π R ( a ) = exp (cid:18) − Z a µ R ( τ ) dτ (cid:19) . Consider the set H = (cid:26) h ∈ L (0 , ¯ a ; R ) : h π S ∈ L (0 , ¯ a ) , h ∈ L (0 , ¯ a ) , h π R ∈ L (0 , ¯ a ) (cid:27) .H is a Hilbert space when endowed with the inner product h h, g i H = h h π S , g π S i L + h h , g i L + h h π R , g π R i L =: h h , g i π S + h h , g i L + h h , g i π R . Remark 3.3
The choice of the space H is different from the standard one made,e.g., in Iannelli and Martcheva, 2003. Indeed it would be standard to put the weightalso on the second component. however this is not possible since, in our model, wehave the new and important feature that the mortality force µ I is state dependent.With our choice the space H is bigger than the usual one but it is still possible toformulate the problem there. We finally observe that this choice will reflect also inthe form of the adjoint operator in Proposition 3.4. ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 13
It is useful, for later purpose to introduce the positive cone in H as follows H + = { h ∈ H : h i ≥ , ¯ a ] } ⊂ H. The control space Z is given as: Z = (cid:8) z = ( z , z , z ) : z i ∈ L (0 , ¯ a ) , i = 1 , , z ( a ) ≥ , z ( a ) , z ( a ) ∈ [0 , , ∀ a ∈ [0 , ¯ a ] (cid:9) .H is a Hilbert space when endowed with the inner product h z, w i Z = h z , w i L + h z , w i L + h z , w i L . Finally the space Z of all basic control strategies is, coherently with the require-ments of Section 2, a space of functions from R + to Z and is given as follows Z := L ( R + ; Z );For coherence with the Notation 3.2 introduced above we will call z = ( c, θ, η ) thepoints of Z . and ˆ z = (ˆ c, ˆ θ, ˆ η ) the points of Z .Now we reformulate system (5) providing also existence and uniqueness of thesolution. We need to introduce some operators which comes from the various termof the system.First we introduce the unbounded linear operator A : D ( A ) ⊂ H → H defined as A = − ∂∂a − µ S − ∂∂a − γ γ − ∂∂a − µ R with domain D ( A ) = (cid:26) h ∈ H : h π S , h , h π R ∈ W , (0 , ¯ a ) ,h (0) = Z ¯ a β ( a )( h + h + h )( a ) da, h (0) = h (0) = 0 (cid:27) , corresponding to the linear part of system (5). It can be shown as in Iannelli andMartcheva (2003) that A generates a strongly continuous semigroup T ( t ) on H such that T ( t )( H + ) ⊂ ( H + ) for every t ≥ H → R (15) ¯Ξ( h ) = Z ¯ a h ( a ) ξ ( a ) da. (16) For every control point z = ( c, θ, η ) ∈ Z we define the nonlinear operators (depend-ing only on the components θ and η of the control point)Λ θ,η : H + \ { } → L ∞ (0 , ¯ a )Λ θ,η ( h )( a ) = θ ( a ) R ¯ a ( h + h + h )( τ ) dτ Z ¯ a m ( a, τ ) θ ( τ ) η ( τ ) h ( τ ) dτ and B θ,η : H + \ { } → H,B θ,η ( h ) ( a ) = − Λ θ,η ( h )( a ) h ( a )Λ θ,η ( h ) h ( a ) − µ I ( a, Ξ( h )) h ( a )0 . Now by Hypothesis 3.1(ii) and by the fact that (see the definition of Z ) we have θ, η ∈ L ∞ (0 , ¯ a ), the operator B θ,η is Lipschitz continuous on H + \ { } and thereexists a positive constant α such that αB θ,η ( h ) + h ∈ H + for every h ∈ H + (seeIannelli and Martcheva, 2003).We can consequently write system (5) as the evolution equation for the unknownˆ h : [0 , + ∞ ) → D ( A ) with control strategies ˆ θ and ˆ η :(17) ddt ˆ h ( t ) = A ˆ h ( t ) + B ˆ θ ( t ) , ˆ η ( t ) (ˆ h ( t )) . Given control strategies ˆ θ ( · ) , ˆ η ( · ) and an initial condition h ∈ H + we look for mildsolutions of the above systems in H + , i.e., for functions [0 , + ∞ ) ∋ t ˆ h ( t ) ∈ H + that satisfy ˆ h ( t ) = T ( t ) h + Z t T ( t − s ) B ˆ θ ( s ) , ˆ η ( s ) (ˆ h ( s )) ds Thanks to the fact that A generates a strongly continuous semigroup that leaves H + invariant and thanks to the properties of B , there exists (see e.g. Bensoussanet al. (2007) a unique function ˆ h ( · ) that satisfies (17) and such that ˆ h (0) = h andˆ h ( t ) ∈ H + for every t ∈ [0 , + ∞ ). Such solution will be denoted by ˆ h ˆ θ, ˆ η ; h or byˆ h ˆ z ; h .We now add the equation for K to the system, see (7). For control points z =( c, θ, η ) ∈ Z we define the functionals on HL θ ( h ) = Z ¯ a ( h ( a ) + h ( a )) α ( a ) ϕ ( θ ( a )) da,C c ( h ) = Z ¯ a c ( a )( h + h + h )( a ) da,D η ( h ) = D (cid:18)Z ¯ a η ( a ) h ( a ) e ( a ) da (cid:19) . ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 15
For any control strategy ˆ z ( · ) ∈ Z , any h ∈ H + and any K ∈ R we are thenconsidering the Cauchy problem(18) ˆ h ′ ( t ) = A ˆ h ( t ) + B ˆ θ ( t ) , ˆ η ( t ) (ˆ h ( t )) K ′ ( t ) = − δK + F (cid:16) K ( t ) , L ˆ θ ( t ) (ˆ h ( t )) (cid:17) − C ˆ c ( t ) (ˆ h ( t )) − D ˆ η ( t ) (ˆ h ( t )) , ˆ h (0) = h ,K (0) = K . Observe that the first equation does not depend on K . Hence, once we know themild solution h ˆ z ; h of the first equation we can plug it into the second one. Since F isLipschitz in K , uniformly in the second variable , we know that the second equationhas a solution K ˆ z ; K ,h (note that it depends also on h since the trajectory ˆ h appearsin the second equation). We then conclude that, for every ˆ z = (ˆ c, θ, η ) ∈ Z theabove system admits a unique solution (cid:0) h ˆ z ; h , K ˆ z ; K ,h (cid:1) such that h ˆ z ; h is the mildsolution of the first equation with initial datum h and h ( t ) ∈ H + for every t ≥ ddt (ˆ h, K )( t ) = e A (ˆ h ( t ) , K ( t )) + e B ˆ z ( t ) (cid:16) ˆ h ( t ) , K ( t ) (cid:17) , t ≥ h, K )(0) = ( h , K ) ∈ H + × R where e A : D ( A ) × R → H × R is the linear operator defined by e A ( h, K ) = ( Ah, − δK )and, for z = ( c, θ, η ) ∈ Z , e B z = e B c,θ,η : H × R → H × R is given by e B z ( h, K ) = (cid:16) B θ,η ( h ) , F ( K, L θ ( h )) − C c ( h ) − D η ( h ) (cid:17) . The following proposition follows from basic material in Iannelli and Martcheva(2003), Iannelli (1995), Bensoussan et al. (2007).
Proposition 3.4
The linear operator e A generates a strongly continuous semigroup e T ( t ) on H × R that leaves H + × R invariant, while the operator B z is Lipschitz.The Cauchy problem (19) admits a unique mild solution, that coincides with that of(18).The adjoint operator of e A with respect to the inner product h ( h, K ) , ( p, Q ) i H × R := h h, p i H + KQ , that is the linear operator e A ∗ : ( D ( A ∗ ) × R ) → H × R given by e A ∗ ( p, Q ) = ( A ∗ p, − δQ ) , The fact the two equations are not fully coupled can be exploited to cover also the case when F is a Cobb-Douglas function by using a Bernoulli-type change of variable. where A ∗ = ∂∂ a + µ s ∂∂ a − γ γπ R ∂∂ a + µ R on D ( A ∗ ) = (cid:26) p = ( p , p , p ) : p π S , p , p π R ∈ W , (0 , ¯ a ) ,p π S (¯ a ) = p (¯ a ) = π π R (¯ a ) = p (0) = 0 (cid:27) . Now we are in position to define precisely the set of admissible control strategiesand to rewrite the target functionals. First of all, due to the presence of positivityconstraints both on ˆ h and K the set of admissible control strategies depends on theinitial data and is the set(20) Z ad ( h , K ) = n ˆ z ( · ) ∈ Z : (cid:16) h ˆ z ; h , K ˆ z ; K ,h (cid:17) ( t ) ∈ H + × R + for a.e. t ∈ [0 , + ∞ ) o . We now rewrite the target functionals starting by (8). Define the function J : H + × R × Z → R ,(21) J ( h , K ; ˆ z ) = Z ∞ e − ρt Z ¯ a (cid:16) ˆ h ˆ z ; h ( t )[ a ] + ˆ h ˆ z ; h ( t )[ a ] + ˆ h ˆ z ; h ( t )[ a ] (cid:17) γ u (cid:16) ˆ c ( t )[ a ] , ˆ θ ( t )[ a ] (cid:17) dadt. Then for every given initial datum ( h , K ) ∈ H + × R + the problem of maximizing(8) in Section 2 translate precisely in maximizing the function J ( h , K ; · ) given(21) over the set Z ad ( h , K ). It will be useful, as a shorthand, to define the function U : H + × Z → R ,(22) U ( h ; z ) = U ( h ; c, θ ) = Z ¯ a ( h ( a ) + h ( a ) + h ( a )) γ u ( c ( a ) , θ ( a )) da , so that(23) J ( h , K ; ˆ z ) = Z ∞ e − ρt U (cid:16) h ˆ z ; h ( t ); ˆ c ( t ) , ˆ θ ( t ) (cid:17) dt. The other infinite horizon problem of Section 2 have the same set of admissiblestrategies, hence to define it precisely it is enough the rewrite the correspondingtarget functional. To do this we simply have to change the function J . The target(12) can be rewritten defining the functional J in the form (23) simply substituting U from (32) with U defined as follows (recall the definition of L ( · ) given in (6))(24) U ( h, K ; θ ) = F (cid:18) K, Z ¯ a ( h ( a ) + h ( a )) α ( a ) ϕ ( θ ( a )) da (cid:19) , ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 17
The value functions of the two maximization problems described above are definedas(25) V i ( h, K ) : = sup ˆ z ∈Z ad ( h,K ) J i ( h, K ; ˆ z ) , i = 1 , . The other four functionals of Section 2.3 are taken with finite horizon
T >
0. Itis useful, to apply the dynamic programming approach, to let also the initial timevary. Hence, when studying these targets the initial condition of the state equation(19) is taken at a generic time t ∈ [0 , T ]:(26) ddt (ˆ h, K )( t ) = e A (ˆ h ( t ) , K ( t )) + e B ˆ z ( t ) (cid:16) ˆ h ( t ) , K ( t ) (cid:17) , t ∈ [ t , T ](ˆ h, K )( t ) = ( h , K ) ∈ H + × R This is the state equation in this case and its solution (which exists and is uniquethanks to Proposition 3.4) is denoted by (cid:0) h ˆ z ; t ,h , K ˆ z ; t ,K ,h (cid:1) . The set of admissiblecontrol is also a bit different:(27) Z ad ( t , h , K ) = n ˆ z ( · ) ∈ Z : (cid:16) h ˆ z ; t ,h , K ˆ z ; t ,K ,h (cid:17) ( t ) ∈ H + × R + for a.e. t ∈ [ t , T ] o . For the same reason also the lower extremum of the integral of the target is takenat a generic time t ∈ [0 , T ]. It follows that the target functional and the valuefunction also depend on t .To rewrite target (9) we then define(28) J ( t , h , K ; ˆ z ) = F (cid:18) K ˆ z ; t ,h ,K ( T ) , Z ¯ a (cid:16) ˆ h ˆ z ; t ,h ( T )[ a ] + ˆ h ˆ z ; t ,h ( T )[ a ] + ˆ h ˆ z ; t ,h ( T )[ a ] (cid:17) α ( a ) da (cid:19) The target (11) can be rewritten as(29) ˆ J ( t , h , K ; ˆ z ) = K ˆ z ; t ,h ,K ( T )While the above two targets only contain a final reward, the next two containonly a current reward. To rewrite target (11) we set(30) J ( h , K ; ˆ z ) = Z T e − ρt U (cid:16) h ˆ z ; h ( t ); ˆ c ( t ) , ˆ θ ( t ) (cid:17) dt. where U is defined in (24) Finally, to rewrite target (14) we define(31) J ( t , h , K ; ˆ z ) = Z Tt Z ¯ a µ I (cid:16) a, Ξ(ˆ h ˆ z ; t ,h ( t )) (cid:17) ˆ h ˆ z ; t ,h ( t )[ a ] dadt. or simply J ( t , h , K ; ˆ z ) = R Tt U (ˆ h ˆ z ; t ,h ) dt , where we set(32) U ( h ) = Z ¯ a µ I ( a, Ξ( h )) h ( a ) da . Note that here above Ξ : L (0 , ¯ a ) → R is the linear functional given by Ξ( h ) = R ¯ a h ( a ) ξ ( a ) da , as from (15). The value functions in the above four finite horizoncases are defined as: V i ( t, h, K ) : = sup ˆ z ∈Z ad ( t,h,K ) J i ( t, h, K ; ˆ z ) , i = 3 , , , . Dynamic Programming and HJB equations
The starting point of the dynamic programming approach to the problems of thispaper is the Dynamic Programming Principle, which we call DPP from now on, (seee.g. Theorem 1.1, p. 224 of Li and Yong, 1995, for a statement and a proof whichapply to this case) which is a functional equation for the value function. Once DPPis established the standard path is to write the differential form of DPP, the HJBequation, find a solution v of it, and prove a Verification Theorem i.e. a sufficientcondition for optimality in terms of the function v (which can be then proved tobe the value function) and its derivatives. Both steps may be very complicated,depending on the features of the problem; this is particularly true when one dealswith problems in infinite dimension. Indeed, while for finite dimensional problemsthe theory of HJB equations and of the corresponding verification results is quite wellestablished with many regularity results, this is not the case for infinite dimensionalproblems. Indeed only few results are available and each case must be treated adhoc. One can see, for example, Theorem 5.5, p.263 of Li and Yong (1995) and thepapers Faggian and Gozzi (2010), Fabbri et al. (2010).Here we abstract away from the existence and uniqueness of regular solutions ofthe HJB equation (which is a challenging subject and which will be next step of ourwork) and we concentrate on Verification Theorems and their consequences.Since we formulated various different problems with different targets, here weconcentrate on the targets (8) and (14) simply observing that the results can beeasily extended to the other cases.Consider first the problem of maximizing, for every initial datum ( h , K ) ∈ H + × R + the target (8) over all ˆ z ∈ Z ad ( h , K ). Formally, the Hamilton-Jacobi-Bellman (HJB) equation associated to such control problem is (the unknown hereis v : H + × R + → R )(33) ρv ( h, K ) = sup z ∈ Z H CV ( h, K, D h v ( h, K ) , D K v ( h, K ); z )where the so-called Current Value Hamiltonian is defined as H CV : (( D ( A ) ∩ H + ) × R ) × ( H × R ) × Z → R (34) H CV ( h, K, p, Q ; z ) = h e A ( h, K ) , ( p, Q ) i H × R + h e B z ( h, K ) , ( p, Q ) i H × R + U ( h ; z ) ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 19
However this form of the HJB equation is not very convenient for two main reasons. • First of all the unknown is defined only in H + × R + , because U is definedin H + . This is a serious problem since the set H + has empty interior in H and this creates problems in defining properly the Fr´echet derivative D h v .To overcome this problem we observe that U can be immediately extendedto the half space (here is the function with constant value (1 , ,
1) on H )(35) H := { h ∈ H : h h, i ≥ } . Indeed the interior part of this set in H is simply IntH := { h ∈ H : h h, i > } . Note that in this way we are enlarging the positivity constraint on the vari-ables ( s, i, r ) so the resulting equation is the one associated to a differentproblem with a greater value function which we call V . However, as ex-plained, e.g., in the appendix of Boucekkine et al. (2019), this would allowto solve also the original one if the resulting optimal strategies satisfies theoriginal constraints (i.e. the corresponding state trajectory ˆ h stays in H + ). • Second, the term e A ( h, K ) creates problem since it requires h ∈ D ( A ) whichis not in general satisfied when we take the mild solution ˆ h of the equation(17). Hence it is better to bring the operator e A on the other side of theinner product. The drawback of this is that we need to require an additionalregularity for the solution v : that D h v belong to D ( A ∗ ) (see Definition 4.1below).We then consider the unknown v defined on H × R + and modify the CurrentValue Hamiltonian as follows (we keep the same name of it since we will be usingonly the following one from now on) H CV : (cid:0) H × R (cid:1) × (( D ( A ∗ ) × R ) × Z → RH CV ( h, K, p, Q ; z ) = h ( h, K ) , e A ∗ ( p, Q ) i H × R + h e B z ( h, K ) , ( p, Q ) i H × R + U ( h ; z )= h h , ∂p ∂a + µ S p i L πS + h h , ∂p ∂a − γp + γπ R p i L + h h , ∂p ∂a + µ R π i L πR − δKQ − h Λ θ,η ( h ) h , p i L πS + h Λ θ,η ( h ) h , p i L − h µ I ( · , Ξ( h )) h , p i L + F ( K, L θ ( h )) Q − C c ( h ) Q − D η ( h ) Q + h ( h + h + h ) γ ( · ) , u ( z ( · )) i L . We denote by H CV the part of the Hamiltonian that depends on the controls, i.e. H CV ( h, K, p, Q ; z ) = −h Λ θ,η ( h ) h , p i L πS + h Λ θ,η ( h ) h , p i L + F ( K, L θ ( h )) Q − C c ( h ) Q − D η ( h ) Q + h ( h + h + h ) γ ( · ) , u ( z ( · )) i L . and set H = H CV − H CV (36) = h h , ∂p ∂a + µ S p i L πS + h h , ∂p ∂a − γp + γπ R p i L + h h , ∂p ∂a + µ R π i L πR − δKQ − h µ I ( · , Ξ( h )) h , p i L so thatsup z ∈ Z H CV ( h, K, p, Q ; z ) = H ( h, K, p, Q ) + sup z ∈ Z H CV ( h, K, p, Q ; z ) . Finally we call H ( h, K, p, Q ) := sup z ∈ Z H CV ( h, K, p, Q ; z ) so the HJB equation (33)rewrites as(37) ρv ( h, K ) = H ( h, K, p, Q ) + H ( h, K, p, Q )Now we give the definition of classical solution of (37) in the interior of ourenlarged state space H × R . Here we abstract away from the boundary conditionsas they will not be crucial for our purposes. Clearly they will become a key pointwhen we want to prove results on existence/uniqueness/regularity of solutions of(37). Definition 4.1
We say that a function v : IntH × (0 , + ∞ ) −→ R is a classical solution of the HJB equation (33) if( i ) v is continuously Fr´echet differentiable in IntH × (0 , + ∞ ) ;( ii ) the derivative D h v ( h, K ) belongs do D ( A ∗ ) for every ( h, K ) ∈ IntH × (0 , + ∞ ) and A ∗ D h v is continuous in IntH × (0 , + ∞ ) ;( iii ) v satisfies equation (33) for every ( H, k ) ∈ IntH × (0 , + ∞ ) . We have the following result, which generalizes, e.g., Proposition 1.2, p. 225 ofLi and Yong (1995).
Theorem 4.2
Consider the problem of optimizing the target functional (21) overthe set of control strategies (38) Z ad ( h , K ) = n ˆ z ( · ) ∈ Z : (cid:16) h ˆ z ; h , K ˆ z ; K ,h (cid:17) ( t ) ∈ H × R + for a.e. t ∈ [0 , + ∞ ) o , where H is defined as in (35) . Suppose that the value function V of this “en-larged” problem is continuously Fr´echet differentiable in IntH × (0 , + ∞ ) and that D h V ( h, K ) ∈ D ( A ∗ ) for every ( h, K ) ∈ IntH × (0 , + ∞ ) . Then V is a classicalsolution of the Hamilton-Jacobi-Bellman equation (33) in IntH × (0 , + ∞ ) . ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 21
Proof.
For simplicity, in this proof, we will write V for V ∇ V for the vector( D h V , D K V ). By Theorem 1.1, p. 224 of Li and Yong (1995) V satisfies thedynamic programming principle, that is, for every ( h , K ) ∈ IntH × (0 , + ∞ ) andevery t ≥ V ( h , K ) = sup ˆ z ∈Z ad ( h ,K ) (cid:26)Z t e − ρs U (cid:16) ˆ h ˆ z ; h ( s ) , ˆ z ( s ) (cid:17) ds + e − ρt V (cid:16) ˆ h ˆ z ; h ( t ) , K ˆ z ; h ,K ( t ) (cid:17)(cid:27) From now on for simplicity we will write h ˆ z ( t ) for (cid:16) ˆ h ˆ z ; h ( t ) , K ˆ z ; h ,K ( t ) (cid:17) . Using thechain rule for mild solutions (see for example Proposition 5.5 Li and Yong, 1995) wehave, for every ( h , K ) ∈ IntH × (0 , + ∞ ), every ˆ z = (ˆ c, ˆ θ, ˆ η ) ∈ Z ad ( h , K ) andevery t ≥ V (cid:16) h ˆ z ( t ) (cid:17) − V ( h , K ) = Z t h h ˆ z ( s ) , e A ∗ ∇ V ( h ˆ z ( s )) i H × R ds + Z t h e B ˆ z ( h ˆ z ( s )) , ∇ V ( h ˆ z ( s )) i H × R ds. Therefore, using also (39),0 ≥ Z t e − ρs U ( h ˆ z ; h ( s ); ˆ c ( s ) , ˆ θ ( s )) ds + e − ρt V (cid:16) h ˆ z ( t ) (cid:17) − V ( h , K )= Z t e − ρs U ( h ˆ z ; h ( s ); ˆ c ( s ) , ˆ θ ( s )) ds + e − ρt V (cid:16) h ˆ z ( t ) (cid:17) − e − ρt V ( h , K ) + ( e − ρt − V ( h , K )= Z t e − ρs U ( h ˆ z ; h ( s ); ˆ c ( s ) , ˆ θ ( s )) ds + e − ρt Z t h h ˆ z ( s ) , e A ∗ ∇ V ( h ˆ z ( s )) i H × R ds + Z t h e B ˆ z ( h ˆ z ( s )) , ∇ V ( h ˆ z ( s )) i H × R ds + ( e − ρt − V ( h , K ) . Now, since we are in an open set we know that the control strategies can be takenconstant (so ˆ z ( t ) = ˆ z (0) for all t ≥ t and take the limit as t →
0; finding(40) 0 ≥ U ( h ; ˆ c (0) , ˆ θ (0)) + h ( h , K ) , e A ∗ ∇ V ( h , K ) i H × R + h e B π ( h , K ) , ∇ V ( h , K ) i H × R − ρV ( h , K ) . Therefore we obtain0 ≥ sup z ∈ Z H CV ( h , K , D h V ( h , K ) , D K V ( h , K ); z ) − ρV ( h , K ) . To prove the reverse inequality we fix again ( h , K ) ∈ IntH × (0 , + ∞ ); by def-inition of the value function, for any positive ǫ and any positive t we can find an admissible control ˆ z ǫ ( · ) ∈ Z ad ( h , K ) such that − ǫt ≤ Z t e − ρs U (cid:16) h ˆ z ǫ ( s ); ˆ c ǫ ( s ) , ˆ θ ǫ ( s ) (cid:17) ds + e − ρt V ( h ˆ z ǫ ( t )) − V ( h , K ) . Using the equation satisfied by h hatz ǫ we get − ǫt ≤ e − ρt (cid:16) V ( h ˆ z ǫ ( t ) − V ( h , K ) (cid:17) + Z t e − ρs U (cid:16) h ˆ z ǫ ; h ( s ); ˆ c ǫ ( s ) , ˆ θ ǫ ( s ) (cid:17) ds + ( e − ρt − V ( h , K )= e − ρt h e T ( t )( h , K ) − ( h , K ) , ∇ V ( h , K ) i H × R + e − ρt h Z t e T ( t − s ) e B π ǫ ( h π ǫ ( s )) , ∇ V ( h , K ) i H × R ds + Z t e − ρs U (cid:16) h ˆ z ǫ ; h ( s ); ˆ c ǫ ( s ) , ˆ θ ǫ ( s ) (cid:17) ds + o ( t ) + ( e − ρt − V ( h , K )We can then find a continuous function σ : [0 , + ∞ ] → [0 , + ∞ ] such that σ (0) = 0and σ ( ǫ ) ≤ t e − ρt h ( e T ( t ) − Id)( h , K ) , ∇ V ( h , K ) , i H × R + 1 t Z t h e B π ǫ ( h , K ) , ∇ V ( h , K ) i H × R ds + 1 t Z t e − ρa U (cid:16) h ; ˆ c ǫ ( s ) , ˆ θ ǫ ( s ) (cid:17) ds + o (1) + e − ρt − t V ( h , K ) ≤ sup z ∈ Z H CV ( h , K , D h V ( h , K ) , D K V ( h , K ); z ) + e − ρt − t V ( h , K ) + o (1) , which implies, taking the limit as t → σ ( ǫ ) ≤ sup z ∈ Z H CV ( h , K , D h V ( h , K ) , D K V ( h , K ); z ) − ρV ( h , K ) . Letting now ǫ go to 0 we get the result. Remark 4.3
The above Theorem (4.2) holds in a completely similar way for theother problems where the target is changed. Of course, in case of finite horizonproblems the HJB equation is different and, for example, in the case of target (29),is − ∂v ( t, h, K ) ∂t = sup z ∈ Z H CV ( h, K, D h v ( h, K ) , D K v ( h, K ); z )for t ∈ [0 , T ], ( h, K ) ∈ IntH × (0 , + ∞ ) and with the final condition v ( T, h, K ) = K .Note finally that in all such cases we would take the enlarged constraint h ∈ H instead of the one h ∈ H + . ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 23 Verification theorems
We first recall the definition of optimal strategy for our starting problem and forthe “enlarged” one.
Definition 5.1
For ( h , K ) ∈ H + × R + (respectively ( h , K ) ∈ IntH × (0 , + ∞ ) ), an admissible control strategy ˆ z ∗ ∈ Z ad ( h , K ) (respectively ˆ z ∗ ∈Z ad ( h , K ) ) is called optimal at ( h , K ) if V ( h , K ) = J ( h , K ; ˆ z ∗ ( · )) , respectively V ( h , K ) = J ( h , K ; ˆ z ∗ ( · )) , that is, if it is a maximizer for J . The corresponding solution ( h π ∗ ; h , K π ∗ ; K ) of(19) is called an optimal state trajectory . The following result is the so-called Verification Theorem which provides sufficientoptimality conditions.
Theorem 5.2
Let v be a classical solution of the HJB equation (33) , with theadditional property that for every ˆ z ∈ Z ad ( h , K )(41) lim T → + ∞ e − ρT v (cid:16) ˆ h ˆ z ; h ( T ) , K ˆ z ; h ,K ( T ) (cid:17) = 0; then V ( h , K ) ≤ v ( h , K ) for every ( h , K ) ∈ IntH × (0 , + ∞ ) . Moreover, ifan admissible control ˆ z ∗ ∈ Z ad ( h , K ) is such that sup z ∈ Z H CV ( h ˆ z ∗ ( t ) , ∇ v ( h ˆ z ∗ ( t )) , z ) = H CV ( h ˆ z ∗ ( t ) , ∇ v ( h ˆ z ∗ ( t )) , ˆ z ∗ ( t ))(42) then ˆ z ∗ is optimal at ( h , K ) and V ( h , K ) = v ( h , K ) .Proof. We write ∇ v for ( D h v, D K v ). Moreover, as in the proof of Theorem 4.2 wewrite for simplicity h ˆ z ( t ) in place of ( h ˆ z ; h ( t ) , K ˆ z ; h ,K ( t )). We first prove that, forevery ˆ z ∈ Z ad ( h , K ) we have the fundamental identity(43) v ( h , K ) = J ( h , K ; ˆ z ( · ))+ Z ∞ e − ρt (cid:20) sup z ∈ Z H CV ( h ˆ z ( t ) , ∇ v ( h ˆ z ( t )) , z ) − H CV ( h ˆ z ( t ) , ∇ v ( h ˆ z ( t )) , ˆ z ( t )) i dt. Indeed, differentiating the function t e − ρt v ( h ˆ z ( t )) and integrating on [0 , T ] wefind v ( h , K ) = e − ρT v ( h ˆ z ( T )) + Z T e − ρt ρv ( h ˆ z ( t )) dt − Z T e − ρt h ( h ˆ z ( t )) , e A ∗ ∇ v ( h ˆ z ( t )) i H × R dt − Z T e − ρt h e B ˆ z ( t ) ( h ˆ z ( t )) , ∇ v ( h ˆ z ( t )) i H × R dt. We can then add and subtract the term R T e − ρt U ( h ˆ z,h ( t ) , ˆ z ( t )) dt on the right handside and use the fact that v solves the HJB equation (33) to obtain v ( h , K ) = e ρT v ( h ˆ z ( T )) + Z T U ( h ˆ z ; h ( t ) , ˆ z ( t )) dt + Z T e − ρt (cid:20) sup z ∈ Z H CV ( h ˆ z ( t ) , ∇ v ( h ˆ z ( t )); ˆ z ( t )) − H CV ( h ˆ z ( t ) , ∇ v ( h ˆ z ( t )); ˆ z ( t )) i dt. The fundamental identity then follows taking the limit as T → ∞ and using (41).Since the last integral is always non-negative, and eventually taking the supremumover all admissible controls on the right hand side, we get the first claim. The secondclaim follows observing that, for such ˆ z ∗ we have, from (43) and the first claim, V ( h , K ) ≤ v ( h , K ) = J ( h , K ; ˆ z ∗ )which implies that V ( h , K ) = v ( h , K ) = J ( h , K ; ˆ z ∗ ) and so the claim. Corollary 5.3
Let v be a classical solution of the HJB equation (33) and assumethat the set valued map ( h, K ) → arg max z ∈ Z H CV ( h, K, ∇ v ( h, K ); z ) admits a measurable selection G : IntH × (0 , + ∞ ) → Z . Let ( h , K ) ∈ IntH × (0 , + ∞ ) and assume that the closed loop equation (44) ddt (ˆ h, K )( t ) = e A (ˆ h ( t ) , K ( t )) + e B G (ˆ h ( t ) ,K ( t )) (cid:16) ˆ h ( t ) , K ( t ) (cid:17) , t ≥ h, K )(0) = ( h , K ) , admits a solution (ˆ h G ; h , K G ; h ,K ) such that the control strategy ˆ z ∗ ( t ) = G (cid:16) ˆ h G ; h ( t ) , K G ; h ,K ( t ) (cid:17) belongs to Z ad ( h , K ) . then ˆ z ∗ is optimal. ERIFICATION RESULTS FOR AGE-STRUCTURED MODELS... 25
Proof.
It immediately follows from the previous Theorem 5.2 and from the funda-mental identity (43).
Corollary 5.4
Suppose that the value function V is a classical solution of theHamilton-Jacobi-Bellman equation (33) , that ˆ z ∗ ∈ Z ad ( h , K ) is optimal at ( h , K ) and that lim T → + ∞ e − ρT V (cid:16) h π ∗ ; h ( t ) , K π ∗ ; K ( t ) (cid:17) = 0 . Then ˆ z ∗ satisifes (42) .Proof. Proceeding as in the proof of Theorem 5.2 we can show that the value function V satisfies the fundamental identity (43). Since ˆ z ∗ ( · ) is optimal the integral termon the right hand side of (43) must be 0, and the claim follows. Remark 5.5
The above results allow, if we can find V , at least numerically, tosolve the problems with the enlarged constraints. To pass to our initial controlproblem we have to show that, for some ( h , K ) ∈ H + × R + , the optimal control ofthe enlarged problem is als admissible for the initial problem. This has been donee.g. in Boucekkine et al (2019) and the same idea may work in some special casesof our set-up. 6. Conclusion
Given the strong differences in the effects of some epidemics (and particularly thatof COVID-19) as individuals vary in age, it is important, in trying to understandthe economic impact of the contagion and in evaluating the policies to combat it,to model it as precisely as possible.In the previous contributions which integrate the epidemiological dynamics inmacro-dynamic models, the stratification by age of population is often absent and,when introduced, it is modeled using a finite number of groups with no possibilityto move from one group to another.In this paper we propose a general fully age-structured time continuous set-up formacro analysis of epidemics and economic dynamic.After rewriting the problem using a suitable Hilbert space reformulation of theassociated infinite dimensional optimal control problem, we provide verification typeresults which, given our general infinite dimensional setting cannot be derived fromprevious results in the literature.
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Acknowledgements
The work of Giorgio Fabbri is supported by the French National Research Agencyin the framework of the “Investissements d’avenir” program (ANR-15-IDEX-02) andin that of the center of excellence LABEX MME-DII (ANR-11-LABX-0023-01).The work of Fausto Gozzi and Giovanni Zanco is supported by the Ital-ian Ministry of University and Research (MIUR), in the framework of PRINprojects 2015233N54 006 (Deterministic and stochastic evolution equations) and2017FKHBA8 001 (The Time-Space Evolution of Economic Activities: Mathemat-ical Models and Empirical Applications)
Univ. Grenoble Alpes, CNRS, INRA, Grenoble INP, GAEL, Grenoble, France.
E-mail address : [email protected] Dipartment of Economics and Finance, LUISS University, Rome, Italy.
E-mail address : [email protected] Dipartment of Economics and Finance, LUISS University, Rome, Italy.
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