Chiraz Trabelsi
email: chiraz.trabelsi@umontpellier.fr

Rim Amami
email: rabamami@iau.edu.sa

Walid Ben Arbi

Hani Abidi
email: abidiheni@gmail.com

Walid Ben Aribi
email: ben.aribi.walid89@gmail.com

Constrained Optimal Control Problem Applied to Vaccination for COVID-19 Epidemic

Keywords: Covid-19 Epidemic, SEAIR, ordinary differential equations, Optimal control problem, penality method. MSC Classification: 34H05, 49J15

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Introduction

Appearing in 2019 in Wuhan, COVID-19 very quickly turned out to be a serious health problem around the world, with catastrophic consequences for the evolution of humankind. Our modern world has never faced a disease of this magnitude. All statistics data about coronavirus COVID-19 comes from the World Health Organization, Johns Hopkins CSSE, and Worldometers. Charts include the number of infected, deaths, and recovered people.

There is now no medicine or specific treatment for COVID-19, and most countries have been relying on non-pharmaceutical interventions, such as wearing face masks, washing hands taking care of personal hygiene, physical distancing, rapid-test and even more partial or total lockdown in order to curtail the spread of the disease. Therefore the study of the novel coronavirus has relatively attracted some importance in mathematical epidemiology due to it seriousness and the way its spreads worldwide.

For instance, several models have been proposed to provide insight into the effect that inoculation of a certain portion of the population will have on the dynamics of the COVID-19 pandemic.

The authors in [START_REF] Perkins | Optimal Control of the COVID-19 Pandemic with Nonpharmaceutical Interventions[END_REF] apply optimal control theory to determine optimal strategies for the implementation of non-pharmaceutical interventions to control COVID-19. The studied model was calibrated to data from the USA and focused the analysis on optimal controls from May 2020 through December 2021.

The importance of the isolation strategy was emphasized in [START_REF] Danane | Mathematical analysis and simulation of a stochastic COVID-19 Levy jump model with isolation strategy[END_REF] to reduce the infection of COVID-19. The authors prove the existence and uniqueness of a global positive solution for a COVID-19 stochastic model with an isolation strategy. Some numerical simulations are presented to illustrate the theoretical results.

Many mathematical models conclude that lockdown is the best way to reduce the spread of COVID-19 effectively among all the aforementioned control strategies [START_REF] Lin | He A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action[END_REF]. However, partial or total lockdown strategies are very risky for a country's economic stability, therefore some suggested mathematical models focused on analyzing the effect of COVID-19 rapid-test as an alternative to suppress the spread of COVID-19 [START_REF] Aldila | Optimal control problem arising from COVID-19 transmission model with rapid-test[END_REF][START_REF] Tanne | Covid-19: how doctors and healthcare systems are tackling coronavirus worldwide[END_REF]. However, these measures have been not an effective protection to mitigate the pandemic globally. To bring this pandemic to an end, a large share of the world needs to be immune to the virus. The safest way to achieve this is with a vaccine. Within less than 12 months after the beginning of the COVID-19 pandemic, several research teams rose to the challenge and developed vaccines that protect from SARS-CoV-2.

Among others [START_REF] Gaff | Optimal control applied to vaccination and treatement strategies for various epidemiological models Mathematical biosciences and engineering[END_REF] the optimal control theory is applied to suggest the most effective mitigation strategy to minimize the number of individuals who become infected in the course of an infection while efficiently balancing vaccination and treatment applied to the models. The SIR model analysis is presented here including the proof of uniqueness and existence of the optimal control solutions.

The motivation of this study is derived from the work [START_REF] Djidjou Demasse | Optimal COVID-19 epidemic control until vaccine deployment[END_REF], who adopt a more modelling approach based on optimal control theory to determine the best strategy to implement until vaccine deployment. Therefore, in the present study, we incorporate the vaccination component to the model in [START_REF] Djidjou Demasse | Optimal COVID-19 epidemic control until vaccine deployment[END_REF], to derive an extended SEAIR model to examine the effectiveness of the COVID-19 jabs which are currently being deployed to many countries to help combat the raging pandemic situation. In the same area, the authors in [START_REF] Perez | An extended SEIARD model for COVID-19 vaccination in Mexico: analysis and forecast[END_REF] propose and analyze an extended SEIARD model with vaccination to examine the effectiveness of the COVID-19 jabs which are currently being deployed to many countries to help combat the raging pandemic situation. Some numerical simulations are achieved using reported data on COVID-19 infections and vaccination in Mexico. This paper is composed of four sections. We first present in Section 2, the structure of our epidemiological model, precisely we present the equations and assumptions of the extended SEAIR model with vaccination. In Section 3 we introduce the objective function and a reformulation of the optimal control problem to minimize the incidence satisfying the constraints of the total and maximum daily vaccine administration and we derive the optimality condition. We use the penalty method to approximate this constrained optimization problem and derive an optimality system that characterizes the optimal control. In Section 4, we carry out numerical simulations using reported data on COVID-19 infections and vaccination in France, and conclude with a summary.

Model formulation

The model describes the epidemic dynamics of COVID-19 in a population after vaccine deployment and is an extension of the standard SEAIR model incorporating a temporary protection vaccine compartment to the model [START_REF] Djidjou Demasse | Optimal COVID-19 epidemic control until vaccine deployment[END_REF], regardless of the disease severity whether mild or severe infections. To derive the mathematical model, first, we subdivide our population into two subpopulations, the unvaccinated population and the vaccinated population (individuals who have received one, two, or three doses of the vaccine). The model classifies individuals into principle key compartments of:

• Susceptible (S).

• Latent (E): infected but asymptomatic and not infectious.

• Asymptomatic infectious (A).

• Symptomatic infectious (I).

• Immune (R) or Death (D).

• Temporary protection vaccine (V). Now, we introduce the model 1 and describe its parameters. Figure 1 illustrates the different compartments for the extended SEAIR model with vaccination. We will denote by N (t) the total population size at time t which is given by

N (t) = S(t) + E(t) + A(t) + I(t) + R(t) + V S(t) + V E(t) + V A(t) + V I(t) + V R(t) = S(t) + E(t) + A(t) + I(t) + R(t) + V (t)
We denote by c 1 the control effort which represents the percentage of reduction in transmission due to public health measures at time t, and β A , β I are asymptomatic and symptomatic transmission rates respectively. Susceptible individuals become exposed by contact with asymptomatic infectious individuals at a rate β A and by contact with symptomatic infectious individuals at a rate β I . The force of infection is defined by λ = (1 -c 1 (t))(β A A + β I I) [START_REF] Djidjou Demasse | Optimal COVID-19 epidemic control until vaccine deployment[END_REF]. A proportion p of exposed individuals (λS) move to the asymptomatic infection class at a rate ϵ. We assume that asymptomatic infectious individuals become symptomatic infectious at rate σ.

Infective individuals leave the compartment at the rate (1 -α[I]) with the fraction f recovering from disease, whereas the rest dying of infection, we assume the disease-induced mortality rate α to be a step function as follows:

α[I] = α min if I < I * α max if I ≥ I *
and I * is the total number of infected hosts in the healthcare system, or simply the healthcare capacity and α bound disease-induced mortality rate.

Since an antibody test is not required or recommended before getting the COVID-19 vaccination, we assume the time-dependent control function c 2 (t) measures the rate at which Susceptible, Latent, Asymptomatic infectious and recovered individuals are vaccinated with vaccine efficacy ψ. As the vaccine does not completely remove the infection, we also assume that a proportion b, b ≪ p, of exposed vaccine individuals (λV S) move to asymptomatic infection at the same rate ϵ.

A proportion δ of vaccine asymptomatic infectious individuals become recovered at the rate σ, while the remainder (1 -δ) move to vaccine Symptomatic infection class at the same rate σ and recovered at rate δ(1 -α ′ [I]) with α ′ ≪ α, denotes the disease-induced mortality for vaccinated individuals, whereas the rest dying of infection.

For the whole population level, we assume also that natural mortality increases because of hospital saturation. We capture this using the following step function for the mortality rate µ,

µ[I] = 0 if I < I * µ if I ≥ I *
with µ the natural mortality rate with hospital saturation.

The principal parameters used throughout this paper and their interpretation are as follows:

• µ : natural mortality rate with hospital saturation.

• α, α ′ : bounds disease induced mortality rate with α ′ ≪ α.

• c 1 ∈ [0, 1] control effort: the percentage of reduction in transmission due to public health measure at time t.

• c 2 ∈ [0, 1] control effort. The time-dependent control function c 2 (t) measures the rate at which susceptible individuals are vaccinated.

• force infection: λ = (1-c 1 (t))(β A A+β I I) where β I = symptotic transmission rate β A = asymptotic transmission rate

• ψ : susceptible, latent, asymptomatic, and recovered individuals are vaccinated with vaccine efficacy ψ.

• p : proportion of infections unvaccinated individuals.

• b : proportion of infections vaccinated individuals (It can be considered too as the loss of vaccine protection).

• ϵ : waiting rate to viral shedding.

• σ : waiting rate to symptom onset.

• f : recovery rate from infections unvaccinated individuals.

• γ : recovery rate from infections vaccinated individuals.

• δ : proportion of recovered among vaccine asymptomatic infectious individuals.

Hence, our model 1 is described by the following system of ODEs:

S ′ (t) = -λpS(t) -µ[I]S(t) -ψc 2 (t)S(t) (1) E ′ (t) = λpS(t) -(ϵ + µ[I])E(t) -ψc 2 (t)E(t) (2) A ′ (t) = ϵE(t) -(σ + µ[I])A(t) -ψc 2 (t)A(t) (3) I ′ (t) = σA(t) -(f + µ[I])I(t) (4) R ′ (t) = (1 -α[I])f I(t) -µ[I]R(t) -ψc 2 (t)R(t) (5) V S ′ (t) = ψc 2 (t)S(t) -(λb + µ[I])V S(t) (6) V E ′ (t) = ψc 2 (t)E(t) + λbV S(t) -(ϵ + µ[I])V E(t) (7) V A ′ (t) = ψc 2 (t)A(t) + ϵV E(t) -(σ + µ[I])V A(t) (8) V I ′ (t) = (1 -δ)σV A(t) -(γ + µ[I])V I(t) (9) V R ′ (t) = ψc 2 (t)R(t) + δσV A(t) + (1 -α ′ [I])γV I(t) -µ[I]V R(t) (10) D ′ (t) = α[I]f I(t) + α ′ [I]γV I(t) + µ[I]N (11)
with the following initial conditions

S(0) = S 0 = N 0 -(I 0 + E 0 + A 0 + D 0 ) ≥ 0 E(0) = E 0 ≥ 0, A(0) = A 0 ≥ 0, I(0) = I 0 ≥ 0, R(0) = R 0 ≥ 0 V S(0) ≥ 0, V E(0) ≥ 0, V A(0) ≥ 0, V I(0) ≥ 0, V R(0) ≥ 0 (12) D(0) = D 0 ≥ 0.
Remark 2.1 We should mention that the previous equations ( 6), ( 7), ( 8), [START_REF] Ledzewicz | Schattler On optimal control for a general SIR-model with vaccination ans treatment Discrete and continuous Dynamical systems[END_REF], and ( 10) can be replaced by the following global equation:

(13) V ′ (t) = ψc 2 (t)(S(t) + E(t) + A(t) + R(t)) -µ[I]V (t).
In summary, using the global equation ( 13) the previous nonlinear system of ODEs describing the COVID-19 dynamics under the initial conditions ( 12) can be written as follows :

(14)                    S ′ (t) = -λpS(t) -µ[I]S(t) -ψc 2 (t)S(t) E ′ (t) = λpS(t) -(ϵ + µ[I])E(t) -ψc 2 (t)E(t) A ′ (t) = ϵE(t) -(σ + µ[I])A(t) -ψc 2 (t)A(t) I ′ (t) = σA(t) -(f + µ[I])I(t) R ′ (t) = (1 -α[I])f I(t) -µ[I]R(t) -ψc 2 (t)R(t) V ′ (t) = ψc 2 (t)(S(t) + E(t) + A(t) + R(t)) -µ[I]V (t) D ′ (t) = α[I]f I(t) + α ′ [I]γV I(t) + µ[I]N.
with the initial conditions :

S(0) = S 0 = N 0 -(I 0 + E 0 + A 0 + D 0 ) ≥ 0 E(0) = E 0 , A(0) = A 0 , I(0) = I 0 , R(0) = R 0 , V (0) = V 0 ≥ 0 (15) D 0 = D 0 ≥ 0.

Optimal control problem

In the present section, the optimal control theory is applied to suggest the most effective mitigation strategy to minimize the number of the infectious people, while also minimizing the effort of vaccinating the population and the effort of the public health measures during a fixed time period.

We recall that the control variable c 1 (t) is used for the reduction in transmission due to public health measures at time t and to analyze vaccine efficacy enhancement, time-dependent control c 2 (t) is introduced. The purpose of introducing these time-dependent controls is to analyze the effect of its variations with time on the dynamic of COVID-19.

Objective function

Let us first define the objective function and then derive the necessary optimality condition. Our goal is to minimize the number of people who become infected, and thus the number of people who die due to the COVID-19 infection at a minimal efforts. Thus, we seek to minimize the following objective functional :

(16) J(c) = T 0 I(t) + B 1 c 2 1 (t) + B 2 c 2 2 (t) dt,
where the control effort pair c = (c 1 , c 2 ). We define the set of admissible controls to be

U = {(c 1 , c 2 ) are Lebesgue measurable functions: (c 1 (t), c 2 (t)) ∈ [0, 1] 2 , ∀t ∈ [0, T ]}.
B 1 , B 2 are constants that can be chosen to balance the relative costs of the public health restriction and vaccination. Quadratic terms c 2 1 and c 2 2 are introduced to account for nonlinear costs potentially arising at high intervention levels and since implementation of any public health intervention and the expense of vaccination does not have a linear cost, see, for example [START_REF] Ledzewicz | Schattler On optimal control for a general SIR-model with vaccination ans treatment Discrete and continuous Dynamical systems[END_REF].

The first term in the objective function (I(t)) corresponds to the total number of infected individuals by the COVID-19 epidemic. The second term (B 1 c 2 1 (t) + B 2 c 2 2 (t)) represents the total cost associated with the implementation of the control measures. It's a quadratic expression to find a known solution (for more details see [START_REF] Lin | An optimal control theory approach to nonpharmaceutical interventions[END_REF]). The main goal of our observations is to search for optimal control variables c * i , f or i = 1, 2 associated with public health restrictions and efficacy of vaccination respectively, mainly to find a function c * = (c * 1 , c * 2 ) such that

J(c * ) = J(c * 1 , c * 2 ) = min U J(c).
To find the optimal control pair c * (t), that minimizes J(c), we follow standard results from optimal control theory applied to systems of ordinary differential equations.

In general, when the world has faced its most dangerous pandemic, the vaccination coverage (proportion of vaccinated people in a population at a given time) and the maximum daily vaccine administration are limited. Therefore, we assume there are practical limitations in our optimal control problem.

Realistic restrictions

We define positive constants c max as the maximum daily vaccination and c total as the vaccine coverage, and our optimal control problem integrates these realistic constraints using state variable inequality constraints, or stated mathematically

(17)    c 2 (t)(S(t) + E(t) + A(t) + R(t)) ≤ c max T 0 c 2 (t)(S(t) + E(t) + A(t) + R(t))dt ≤ c total
The problem we are now facing is to minimize the number of infected individuals using a limited total vaccination. This can be stated [START_REF] Tanne | Covid-19: how doctors and healthcare systems are tackling coronavirus worldwide[END_REF] minimize J(c) subject to [START_REF] Perez | An extended SEIARD model for COVID-19 vaccination in Mexico: analysis and forecast[END_REF] and (17).

Pontryagin's Maximum Principle cannot be used to deal with this problem as stated due to the constraint on vaccination coverage, though we can use a simple trick to convert this problem to a more familiar form. We introduce a new state variable, we denote z(t) and set

z(t) = t 0 c 2 (s)(S(s) + E(s) + A(s) + R(s))ds.
Then, it follows

(19)    z ′ (t) = c 2 (t)(S(t) + E(t) + A(t) + R(t)) z(0) = 0 z(T ) ≤ c total
Thus, the constrained minimization problem ( 18) is transformed into: minimize J(c) subject to [START_REF] Perez | An extended SEIARD model for COVID-19 vaccination in Mexico: analysis and forecast[END_REF] and to (20)

           z ′ (t) = c 2 (t)(S(t) + E(t) + A(t) + R(t)) z(0) = 0 z(T ) ≤ c total 0 ≤ c 1 (t) ≤ 1, and 0 ≤ c 2 (t) ≤ 1 c 2 (t)(S(t) + E(t) + A(t) + R(t)) ≤ c max
are constraints that are required to be satisfied.

Using the monotonically increasing property of z(t), we get z(T ) ≤ c total is equivalent to z(t) ≤ c total , for all 0 ≤ t ≤ T.

Finally, our transformed constrained minimization problem takes the following form

(21) minimize J(c) = T 0 I(t)dt + T 0 B 1 c 2 1 (t) + B 2 c 2 2 (t) dt, subject to (22)
S ′ (t) = -λpS(t) -µ[I]S(t) -ψc 2 (t)S(t) E ′ (t) = λpS(t) -(ϵ + µ[I])E(t) -ψc 2 (t)E(t) A ′ (t) = ϵE(t) -(σ + µ[I])A(t) -ψc 2 (t)A(t) I ′ (t) = σA(t) -(f + µ[I])I(t) R ′ (t) = (1 -α[I])f I(t) -µ[I]R(t) -ψc 2 (t)R(t) V ′ (t) = ψc 2 (t)(S(t) + E(t) + A(t) + R(t)) -µ[I]V (t) D ′ (t) = α[I]f I(t) + α ′ [I]γV I(t) + µ[I]N z ′ (t) = c 2 (t)(S(t) + E(t) + A(t) + R(t)),
and S(0

) = S 0 = N 0 -(I 0 + E 0 + A 0 + D 0 ), E 0 , A 0 , R 0 , V 0 , D 0 ≥ 0, z(0) = 0, z(T ) ≤ c total , 0 ≤ c 1 (t) ≤ 1, 0 ≤ c 2 (t) ≤ 1, c 2 (t)(S(t) + E(t) + A(t) + R(t)) ≤ c max .
Our main goal is to approximate the solution of this constrained optimization problem.

Existence of an optimal control pair

Now, we focus on establishing a criterion for the existence of optimal solutions to the constrained minimization problem (21) subject to (22). We begin by examining the conditions of the Filippov-Cesari existence theorem [START_REF] Cesari | Optimization Theory and Applications: Problems with Ordinary Differential Equations[END_REF]. Here we first establish the following theorem on the existence of optimal control. Theorem 3.1 There exists an optimal control pair c * 1 (t), c * 2 (t), such that the objective functional J(c) subject to ( 22) is minimized over U .

For this purpose, let us recall the Filippov-Cesari existence theorem [START_REF] Cesari | Optimization Theory and Applications: Problems with Ordinary Differential Equations[END_REF]. Theorem 3.2 Let x(t) = (x 1 (t), ..., x n (t)) ∈ R n be a state vector and u(t) = (u 1 (t), ..., u r (t)) ∈ R r be a control vector associated to the following optimal control problem (23) min

x,u

t 1 t 0 F (t, x(t), u(t))dt, with (24)
.

x = g 1 (t, x(t), u(t)), x(t 0 ) = x 0 , with the terminal conditions

x i (t 1 ) ≥ x 1 i , i = 1, . . . , m -1 x i (t 1 ) free , i = m, . . . , n, it is meant that the value of x i (t 1 ) is unrestricted, (25)
and for u(t) ∈ U, with U is a fixed set in R r we have the following constraints

(26) g 2 (t, x(t), u(t)) ≥ 0,
where the function

F : R × R n × R r → R, g 1 : R × R n × R r → R n and g 2 : R × R n × R r → R
) : y = g 1 (t, x, u), y n+1 ≥ F (t, x, u), g 2 (t, x, u) ≥ 0, u ∈ U } is convex for all (t, x) ∈ [t 0 , t 1 ] × R n .
3. There exists a number θ 1 > 0 such that ||x(t)|| < θ 1 for all admissible pairs (x(t), u(t)), and all t ∈ [t 0 , t 1 ].

4.

There exists an open ball B(0, θ 2 ) ⊂ R r which contains the set Ω(t, x) = {u ∈ U : g 2 (t, x, u) ≥ 0} for all x ∈ B(0, θ 1 ).

Then there exists an optimal pair (x ⋆ (t), u ⋆ (t)) to the problem (24),( 25),(26) with u ⋆ (t) measurable.

Proof of Theorem 3.1.

The previous nontrivial requirements from Filippov-Cesari's theorem are verified below in order to establish the existence of a solution to the optimal control problem.

We therefore introduce the state variable x(t) = (S(t), E(t), A(t), I(t), R(t), V (t), D(t), z(t)) T , and control vector u(t) = (c 1 (t), c 2 (t)) ∈ U and F (t, x(t), u(t)) = x 4 (t) + B 1 c 2 1 (t) + B 2 c 2 2 (t), the dynamics of the system is of the form,

(27) g 1 (t, x(t), u(t)) =             -λpS(t) -µ[I]S(t) -ψc 2 (t)S(t) λpS(t) -(ϵ + µ[I])E(t) -ψc 2 (t)E(t) ϵE(t) -(σ + µ[I])A(t) -ψc 2 (t)A(t) σA(t) -(f + µ[I])I(t) (1 -α[I])f I(t) -µ[I]R(t) -ψc 2 (t)R(t) ψc 2 (t)(S(t) + E(t) + A(t) + R(t)) -µ[I]V (t) α[I]f I(t) + α ′ [I]γV I(t) + µ[I]N c 2 (t)(S(t) + E(t) + A(t) + R(t))             and g 2 (t, x(t), u(t)) = c max -c 2 (t)(x 1 (t) + x 2 (t) + x 3 (t) + x 5 (t)).
For ease of notation we omit the dependence on time of the variables. For the finite time interval, the variables S, E, A, I, R and V remain nonnegative if the initial values are nonnegative, (decrease only proportional to their present sizes, respectively). The variables z and D are also nonnegative since the changes in these variables are nonnegative.

To derive the upper bounds for the solutions, we use the approach explained in [START_REF] Gaff | Optimal control applied to vaccination and treatement strategies for various epidemiological models Mathematical biosciences and engineering[END_REF] and [START_REF] Kim | Constrained optimal control applied to vaccination for influenza[END_REF] to show that the total population size N is bounded above N (0). Since none of S, E, A, I, R and V can be negative the upper bound of N is also an upper bound for S, E, A, I, R and V . The boundedness of the auxiliary state variable z follows from the boundedness of the control c 2 , S, E, A, R and V . Note that F, g 1 , and g 2 are of class C 1 , and g 1 is bounded. Thus, there exists a solution for the system (22) which guarantees an admissible pair (x(t), u(t)).

The control set U = [0, 1] 2 is closed then the condition 1 is trivial, and as is also a compact set then condition 4 holds. Now, we verify the condition 2. We first note that the control set U is closed and convex, the integrand of the objective functional F (t, x, .) and the constraint control function g 2 (t, x, .) are convex on U. The function g 1 (t, x, .) in the system (27) can be written as linear function of the control variable with coefficients depending on the state variables and time therefore

g 1 (t, x, u) = k(t, x) + l(t, x)u,
thus g 1 is also a convex function on U. Let ỹ1 , ỹ2 ∈ N , for ω ∈ [0, 1], we can prouve easily that the affine combination ω ỹ1 + (1 -ω)ỹ 2 belongs to N for all (t, x) ∈ [t 0 , t 1 ] × R 8 due to the convexity of the functions F, g 1 and g 2 , which shows that N is convex set.

Finally, condition 3 follows from the boundedness of solutions to the system (22) for a finite time interval. Filippov-Cesari's conditions theorem are verified.

Penalty Method

Several methods have been proposed for handling constrained optimal control problems. The most common approach is to convert them into unconstrained optimization problems, often via the use of a penalty method. Namely, the unconstrained problems are formed by adding a term, called penalty function, to the objective function that consists of a penalty parameter multiplied by a measure of violation of the constraints.

There are different ways to construct the penalty term, a widely used one is the quadratic penalty. An appropriate quadratic penalty function associated with the minimization problem (20) is:

J p (c) = T 0 I(t) + B 1 c 2 1 (t) + B 2 c 2 2 (t) + µ 1 max 0, h 1 (t) 2 + µ 2 max 0, h 2 (t) 2 dt
where B 1 , B 2 are constants, µ 1 , µ 2 are penalty coefficients that change the relative severity of the constraints violation and h 1 , h 2 are inequality constraints given by

h 1 (t) = c 2 (t)(S(t) + E(t) + A(t) + R(t)) -c max and h 2 (t) = z(t) -c total .
Thus, the penalty function J p (c) can be written as follows

J p (c) = T 0 I(t) + B 1 c 2 1 (t) + B 2 c 2 2 (t) + µ 1 (c 2 (t)(S(t) + E(t) + A(t) + R(t)) -c max ) 2 H 1 (c 2 (t)(S(t) + E(t) + A(t) + R(t)) -c max ) + µ 2 (z(t) -c total ) 2 H 2 (z(t) -c total ) dt
where H 1 and H 2 denote the Heaviside step functions, given by :

H 1 (c 2 (t)(S(t)+E(t)+A(t)+R(t))-c max ) = 0 if c 2 (t)(S(t) + E(t) + A(t) + R(t)) ≤ c max 1 if c 2 (t)(S(t) + E(t) + A(t) + R(t)) > c max H 2 (z(t) -c total ) = 0 if z(t) ≤ c total 1 if z(t) > c total
Then, to minimize the solution of the constrained optimization problem (21), we should minimize J p (c), mainly to find the control c * so as to (28) minimise J p (c) subject to the nonlinear system of ODEs (22)

z ′ (t) = c 2 (t)(S(t) + E(t) + A(t) + R(t)), z(0) = 0, 0 ≤ c 1 (t), c 2 (t) ≤ 1.
Pontryagin's Maximum principle is used to derive the optimality system which provides the necessary conditions of the optimal solutions of (28). Introduce a piecewise differentiable vector-valued functions λ(t) = (λ 1 (t), λ 2 (t), . . . , λ 8 (t)) where each λ i is the adjoint variable corresponding to x i , (with 8 states we will need 8 adjoints). Setting ẋ(t) = dx/dt, we introduce the augmented Hamiltonian for the constraints control as follows:

H(t, x, u, λ) = < λ(t), ẋ(t) > +I(t) + B 1 c 2 1 (t) + B 2 c 2 2 (t) + µ 1 (z ′ (t) -c max ) 2 H 1 (z ′ (t) -c max ) + µ 2 (z(t) -c total ) 2 H 2 (z(t) -c total ) = λ 1 (t) -p(1 -c 1 (t))(β A A + β I I)S(t) -µ[I]S(t) -ψc 2 (t)S(t) + λ 2 (t) p(1 -c 1 (t))(β A A + β I I)S(t) -(ϵ -µ[I])E(t) -ψc 2 (t)E(t) + λ 3 (t) ϵE(t) -(σ + µ[I])A(t) -ψc 2 (t)A(t) + λ 4 (t) σA(t) -(f + µ[I])I(t) + λ 5 (t) (1 -α[I])f I(t) -µ[I]R(t) -ψc 2 (t)R(t) + λ 6 (t) ψz ′ (t) -µ[I]V (t) + λ 7 (t)z ′ (t) + λ 8 (t) α[I]f I(t) + α ′ [I]γV (t) + µ[I]N + I(t) + B 1 c 2 1 (t) + B 2 c 2 2 (t) + µ 1 (z ′ (t) -c max ) 2 H 1 (z ′ (t) -c max ) + µ 2 (z(t) -c total ) 2 H 2 (z(t) -c total ) - 2 i=1 ω i1 (t)c i (t) - 2 i=1 ω i2 (t)(1 -c i (t)),
where z ′ (t) = c 2 (t)(S(t)+E(t)+A(t)+R(t)) and ω ij (t) ≥ 0 are the penalty multipliers satisfying

ω i1 (t)c i (t) = ω i2 (t)(1 -c i (t)) = 0 at c i (t) = c * i (t) for i = 1, 2 with c * i (t) = (c * 1 (t), c * 2 (t))
is the optimal control pair should be found. On differentiating the augmented Lagrangian H with respect to state variables and setting the result to zero, we get the following adjoint system:

λ ′ 1 (t) = - ∂H ∂S = λ 1 (t)(-p(1 -c 1 (t))(β A A + β I I) + µ[I]λ 1 (t) + ψc 2 (t)λ 1 (t) -p(1 -c 1 (t))(β A A + β I I)λ 2 (t) -ψc 2 (t)λ 6 (t) -c 2 (t)λ 7 (t) -µ[I]λ 8 (t) -2c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ) = (λ 1 (t) -λ 2 (t))(p(1 -c 1 (t))(β A A + β I I)) + (λ 1 (t) -λ 8 (t))µ[I] + (λ 1 (t) -λ 6 (t))ψc 2 (t) -c 2 (t)λ 7 (t) -2c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ). λ ′ 2 (t) = - ∂H ∂E = (ϵ + µ[I])λ 2 (t) + ψc 2 (t)λ 2 (t) -ϵλ 3 (t) -ψc 2 (t)λ 6 (t) -c 2 (t)λ 7 (t) -µ[I]λ 8 (t) -2c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ) = ϵ(λ 2 (t) -λ 3 (t)) + (λ 2 (t) -λ 8 (t))µ[I] + (λ 2 (t) -λ 6 (t))ψc 2 (t) -c 2 (t)λ 7 (t) -2c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ). λ ′ 3 (t) = - ∂H ∂A = β A λ 1 (t)(p(1 -c 1 (t))S(t) -β A λ 2 (t)(p(1 -c 1 (t))S(t) + (σ + µ[I])λ 3 (t) + ψc 2 (t)λ 3 (t) -σλ 4 (t) -ψc 2 (t)λ 6 (t) -c 2 (t)λ 7 (t) -µ[I]λ 8 (t) -2c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ) = β A (p(1 -c 1 (t))S(t)(λ 1 (t) -λ 2 (t)) + σ(λ 3 (t) -λ 4 (t)) + µ[I](λ 3 (t) -λ 8 (t)) + ψc 2 (t)(λ 3 (t) -λ 6 (t)) -c 2 (t)λ 7 (t) -2c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ). λ ′ 4 (t) = - ∂H ∂I = β I (p(1 -c 1 (t))S(t)(λ 1 (t) -λ 2 (t)) + (f + µ[I])λ 4 (t) -(1 -α[I])f λ 5 (t) -ψc 2 (t)λ 6 (t) -c 2 (t)λ 7 (t) -α[I]f λ 8 (t) -µ[I]λ 8 (t) -1 = β I (p(1 -c 1 (t))S(t)(λ 1 (t) -λ 2 (t)) + f (λ 4 (t) -λ 5 (t)) + µ[I](λ 4 (t) -λ 8 (t)) + α[I]f (λ 5 (t) -λ 8 (t)) -ψc 2 (t)λ 6 (t) -c 2 (t)λ 7 (t) -1. λ ′ 5 (t) = - ∂H ∂R = µ[I]λ 5 (t) + ψc 2 (t)λ 5 (t) -ψc 2 (t)λ 6 (t) -c 2 (t)λ 7 (t) -µ[I]λ 8 (t) -c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ) = µ[I](λ 5 (t) -λ 8 (t)) + ψc 2 (t)(λ 5 (t) -λ 6 (t)) -c 2 (t)λ 7 (t) -c 2 (t)µ 1 (z ′ (t) -c max )H 1 (z ′ (t) -c max ). λ ′ 6 (t) = - ∂H ∂V = µ[I]λ 6 (t) -α ′ [I]γλ 8 (t) -µ[I]λ 8 (t). λ ′ 7 (t) = - ∂H ∂z = -2µ 2 (z(t) -c total )H 2 (z(t) -c total ). λ ′ 8 (t) = - ∂H ∂D = 0.
with the transversality conditions λ i (T ) = 0, for i = {1, . . . , 8}. Now, we differentiate the augmented Lagrangian H with respect to c = (c 1 , c 2 ):

∂H ∂c 1 = λ 1 (t)p(β A A + β I I)S(t) -λ 2 (t)p(β A A + β I I)S(t) + 2B 1 c 1 (t) -ω 11 (t) + ω 12 (t). ∂H ∂c 2 = -ψS(t)λ 1 (t) -ψE(t)λ 2 (t) -ψA(t)λ 3 (t) -ψR(t)λ 5 (t) + ψ(S(t) + E(t) + A(t) + R(t))λ 6 (t) + (S(t) + E(t) + A(t) + R(t))λ 7 (t) + 2B 2 c 2 (t) + 2µ 1 (S(t) + E(t) + A(t) + R(t))(z ′ (t) -c max )H 1 (z ′ (t) -c max ) -ω 21 (t) + ω 22 (t).
Recall that z ′ (t) = c 2 (t)(S(t) + E(t) + A(t) + R(t)).

To sum up, we find the optimal control by solving the state system with initial conditions and adjoint equations, we obtain c * 1 , c * 2 defined as:

c * 1 = -λ 1 (t)p(β A A + β I I)S(t) + λ 2 (t)p(β A A + β I I)S(t) + ω 11 (t) -ω 12 (t) 2B 1 c * 2 = ψ S(t)λ 1 (t) + E(t)λ 2 (t) + A(t)λ 3 (t) + R(t)λ 5 (t) 2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 + -(S(t) + E(t) + A(t) + R(t)) ψλ 6 (t) -λ 7 (t) + 2µ 1 c max )H 1 + ω 21 (t) -ω 22 (t) 2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 .
To find an explicit expression for the optimal control without the penalty multipliers ω ij , where i, j = 1, 2, we start the discussion for the first component c * 1 with c 1 ∈ [0, 1) to avoid the case where the force infection λ = 0 (case where c 1 = 1). For that, we consider the following cases: Case 2 : On the set {t|c * 1 (t) = 0}, we have ω 12 (t) = 0 and ω 11 (t) ≥ 0. Hence

c * 1 = -λ 1 (t)p(β A A + β I I)S(t) + λ 2 (t)p(β A A + β I I)S(t) + ω 11 (t) 2B 1 , which implies that c * 1 = -λ 1 (t)p(β A A + β I I)S(t) + λ 2 (t)p(β A A + β I I)S(t) 2B 1 ≤ 0.
Combining these two cases, the optimal control c * 1 is characterized as

(29) c * 1 = min 1, max 0, (λ 2 (t) -λ 1 (t))p(β A A + β I I)S(t) 2B 1 .
We proceed with the same reasoning to find c * 2 , where c 2 ∈ [0, 1] and we add the following third case:

Case 3: On the set {t|c * 2 (t) = 1}, we have ω 21 (t) = 0 and ω 22 (t) ≥ 0. Hence

1 = c * 2 = ψ S(t)λ 1 (t) + E(t)λ 2 (t) + A(t)λ 3 (t) + R(t)λ 5 (t) 2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 + -(S(t) + E(t) + A(t) + R(t)) ψλ 6 (t) -λ 7 (t) + 2µ 1 c max )H 1 -ω 22 (t) 2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 which implies that c * 2 = ψ S(t)λ 1 (t) + E(t)λ 2 (t) + A(t)λ 3 (t) + R(t)λ 5 (t) 2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 + -(S(t) + E(t) + A(t) + R(t)) ψλ 6 (t) -λ 7 (t) + 2µ 1 c max )H 1 2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 ≥ 1.
To conclude, we get c * 2 = min 1, max 0, ψ S(t)λ 1 (t) + E(t)λ 2 (t) + A(t)λ 3 (t) + R(t)λ 5 (t)

2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 + -(S(t) + E(t) + A(t) + R(t)) ψλ 6 (t) -λ 7 (t) + 2µ 1 c max )H 1 2B 2 + 2µ 1 (S(t) + E(t) + A(t) + R(t)) 2 H 1 . (30)

Numerical simulations and discussion

In this section, we perform numerical simulations for a previous model to estimate the evolution of the COVID-19 outbreak in France. The simulations focus on solving a constrained optimal control problem applied to vaccination.

The values of the model parameters are obtained from available local data and previous studies.

The simulations consider the period before the vaccination program in France, as well as the period after vaccination. Figures are presented to illustrate the simulation results, including the number of infected individuals and the number of deaths. The impact of control measures and vaccination on the spread of the infection and the reduction of deaths is analyzed.

Overall, the simulations provide insights into the effectiveness of vaccination and control measures in managing the COVID-19 outbreak in France.

For simplicity and straightforward analysis of causality, we take initial conditions

I 0 = 6458 E 0 = 2 * 6458 A 0 = 1.5 * 6458 S 0 = N 0 -(I 0 + E 0 + A 0 + D 0 ) R 0 = V S 0 = V E 0 = V A 0 = V I 0 = V P 0 = V R 0 = 0 D 0 = 34
On the following Figure 2, each color corresponds to one of compartment of the model (Susceptible, Latent, Asymptomatic, Infected, Recovered and Death). We can see that in the absence of any control the number of infected individuals grows exponentially fast at the beginning of the epidemic (curve in black color) and the epidemic reaches its peak, equilibrium point as the intersection of the curves for those still susceptible and those immunized following recovery, 110 days after the beginning. This is a natural behavior of an epidemic. However, waiting for a large part of the population to become infected in order to mitigate the epidemic is certainly not the best strategy, especially when the disease presents a high mortality due to the severity of the infection or to the saturation of the healthcare system.

In the following section, we will simulate the solutions of the extended SEAIR model (1) to estimate the impact of the vaccination program in France.

Vaccination effect in France

We will now simulate the solution to the model (1) to assess the impact of the vaccination program to combat the infectious diseases caused by COVID-19. The first dose of COVID-19 vaccine in France was received for the first time on December 27, 2020. Since that, the number of doses has multiplied rapidly to protect the population as quickly as possible. In this optimal case, the improved control measures not only impact the number of infected individuals but also have an influence on the number of deaths, illustrated in Figure 4. By implementing the optimal control measures in scenario 1, the number of deaths is expected to be reduced to 205 compared to the other scenarios. Figure 5 shows the progression of optimal controls c * 1 and c * 2 over time. For all four scenarios, we observe that control c * 1 , defined by the percentage reduction in transmission due to a public health measure or sanitary control at time t, always takes the maximum that is defined in equation ( 29) this shows the necessity of sanitary restraint to the extinction of the epidemic.

Whereas for the c * 2 control, which measures the rate at which susceptible individuals are vaccinated, we observe that the curves for the different scenarios are characterized by low and almost constant growth throughout the vaccination period (between 1.778e -04 and 1.785e -04), indicating that the optimal vaccination strategy is to vaccinate all individuals simultaneously when total coverage is possible.

The following figures 6 describe the evolution of the λ i , i ∈ {1, . . . , 8} weights of the Lagrangian to control the relative importance of constraints compared to the objective function J p during the optimization problem-solving.

Conclusion

In this work, a SEAIR mathematical model for the spread of infectious diseases such as COVID-19 is developed. Then, an optimal control problem aiming at minimizing the number of infected people with minimal effort is developed and studied.

The main goal of our studies is to search for optimal control variables c * = (c * 1 , c * 2 ) associated respectively with public health restrictions and vaccination effectiveness. We formulated an optimal control problem and used the conditions of the Filippov-Cesari existence theorem to characterize optimal control. Unconstrained problems are formed by adding to the objective function a penalty function.

Then, we solved numerically the model for the COVID-19 pandemic in France using parameters that have been developed in the literature. The optimal strategy was verified under different scenarios: health restrictions limited to 20, 50, 75, and 100 percent.

In fact, the results suggest starting vaccination at the same time, whatever the current constraints.

Our results show that higher maximum c * 1 restriction rates reduce the number of severely infected people in all countries, but do not lead to a slight reduction in infection. We therefore recommend the mandatory introduction of parallel vaccination for all populations, as the only way to reduce mortality.

In conclusion, this study provides valuable information on the optimal vaccination control strategy and public health measures. The results may help in making decisions aimed at reducing the number of people infected and mortality caused by infectious diseases.

Figure 1 :

 1 Figure 1: flow diagram for the SEAIR model with vaccination compartments

Case 1 :* 1 becomes c * 1 =

 11 On the set {t|0 < c * 1 (t) < 1}, we have ω 11 (t) = ω 12 (t) = 0. Hence the optimal control c -λ 1 (t)p(β A A + β I I)S(t) + λ 2 (t)p(β A A + β I I)S(t) 2B 1 .

Figure 2 :

 2 Figure 2: The model simulation before the vaccination program in France

Figure 4 :

 4 Figure 4: Comparaison between death people in France before and after vaccination

Figure 5 :

 5 Figure 5: Progression of the optimal Controls c * 1 and c * 2 over time.

Figure 6 :

 6 Figure 6: Adjoint Functions

Acknowledgments. The authors would like to thank Professor Monique Pontier for his constructive comments and suggestions to improve the quality of our work.

Model state variables and parameters

Table 1 provides a summary of the model's state variables and parameters. The state variables include the densities of susceptible, vaccine susceptible, latent, vaccine latent, asymptomatic infectious, vaccine asymptomatic infectious, symptomatic infectious, vaccine symptomatic infectious, recovered individuals, vaccine recovered individuals and total deaths. The parameters include the basic reproduction number, proportion of infections, proportion of infections vaccinated, proportion of recovery among vaccinated, transmission rates, reduced transmission factor of infections, control effort, immigration rate, waiting rates to viral shedding and symptom onset, recovery rates from infections, natural mortality rate, disease-induced mortality rates, healthcare capacity, cost weight, and initial conditions. The best-fit values of these parameters were obtained from [START_REF] Djidjou Demasse | Optimal COVID-19 epidemic control until vaccine deployment[END_REF]

SEAIR model before vaccination

First, we consider the model ( 1) before the vaccine deployment, during the period from December 2020 through October 2021 in France. We assume that the vaccinated sub-populations, V S, V E, V A, V I and V R are equal to zero. We use Runge Kutta scheme to solve the set of differential equations.

Vaccines used

Efficiency The table 2 provides an overview of the distribution of vaccines in France, highlighting their efficacy rates.

In the numerical simulations, we compare the optimal intervention strategies under different settings of a proportion of infections, waiting rate to viral shedding, symptom onset, and the infection rate among vaccines. In France, these results are given for the period from February 15, 2021 to April 30, 2021 when the number of cases go past 50000 infected cases (Figure 3). Vaccination is highly concentrated. From Figure 3, we can observe that the number of infected individuals is influenced by the control parameter of the measures taken to control the spread of diseases c 1 . In the absence of control measures, the number of infected individuals exceeds 16 million (1.6e7). Under the optimal vaccination, the spread of infection is significantly reduced in Scenario 1 when c 1 and c 2 range between 0 and 0.95, resulting in a lower rate of new infections. Furthermore, they show a clear correlation between an increase in the maximum vaccination threshold and a significant reduction in the peak of critical infections. This underlines the importance of achieving high vaccination coverage rates to mitigate the spread of infectious diseases and protect vulnerable populations


Camille Roubille
email: roubille@chu-montpellier.fr

Benjamin Eduin

Cyril Breuker

Laetitia Zerkowski

Simon Letertre

Cédric Mercuzot

Jonathan Bigot

Du Guilhem

François Cailar

Pierre Roubille

Fesler

Guilhem Du Cailar

• François Roubille

Pierre Fesler

Predictive risk factors for death in elderly patients after hospitalization for acute heart failure in an internal medicine unit

Keywords: Acute heart failure, Internal medicine, Hypoalbuminemia, Death, Prognosis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Acute heart failure (AHF) is a major cause of unscheduled hospitalization in elderly patients and is associated with increased mortality [START_REF] Metra | Heart failure[END_REF].

The incidence of HF in Europe is approximately 5/1000 person-years in adults [START_REF] Brouwers | Incidence and epidemiology of new onset heart failure with preserved vs reduced ejection fraction in a community-based cohort: 11-year follow-up of PREVEND[END_REF]. The prevalence of HF is estimated to be approximatively 1-2% in adults in developed countries [START_REF] Gbd | Disease and injury incidence and prevalence col-laborators global, regional, and national incidence, prevalence, and years lived with disability for 354 diseases and injuries for 195 countries and territories, 1990-2017: a systematic analysis for the global burden of disease study 2017[END_REF] and increases with age, with an estimated increase of more than 10% in those aged 70 years or older [START_REF] Van Riet | Epidemiology of heart failure: the preva-lence of heart failure and ventricular dysfunction in older adults over time. A systematic review[END_REF]. Although the prognosis of HF has improved in recent decades, it remains a public health issue, with a high risk for re-hospitalization following an admission for AHF [START_REF] Ambrosy | The global health and economic burden of hospitalizations for heart failure: lessons learned from hospitalized heart failure registries[END_REF]. Indeed, HF is characterized by episodes of acute decompensation, which may lead to repeated and prolonged hospitalizations [START_REF] Mcdonagh | ESC guidelines for the diagnosis and treatment of acute and chronic heart failure[END_REF], especially in the elderly and particularly in non-cardiologic units such as internal medicine departments. Moreover, due to the aging of the population in European countries, hospitalization for AHF in internal medicine units mainly involve multimorbid elderly patients.

Despite the improved survival of patients with HF, over-all mortality remains a concern, with an estimated overall mortality rate of approximatively 50% at 5years, especially in elderly patients aged over 80 years [START_REF] Gerber | A contemporary appraisal of the heart failure epidemic in Olmsted County, Min-nesota, 2000 to 2010[END_REF]. Mortality rate at 5 years has also been reported to be higher, about 65% following diagnosis of HF [START_REF] Tsao | Temporal trends in the incidence of and mortality associated with heart failure with pre-served and reduced ejection fraction[END_REF].

Moreover, HF reduces the quality of life and increases the risk of hospitalization. In France, a crosssectional survey reported a prevalence of HF of 2.3%, increasing to about 15% in people over 85 years of age, with 52.9% of people having an impaired quality of life [START_REF] De Peretti | Préva-lences et statut fonctionnel des cardiopathies ischémiques et de l'insuffisance cardiaque dans la population adulte en France : apports des enquêtes déclaratives « Handicap-Santé[END_REF]. In a French national cohort of 70,657 patients aged 65 years or older who survived after hospitalization for HF in 2015, 31.8% had at least one readmission for HF at 1 year [START_REF] Logeart | Current aspects of the spectrum of acute heart failure syn-dromes in a real-life setting: the OFICA study[END_REF].

Therefore, one of the challenges in the management of HF remains the identification of factors associated with poor prognosis and mortality. The aim of our study was to deter-mine the predictive risk factors of mortality after hospitalization for AHF in an internal medicine unit.

Methods

Study population

The cohort included 180 patients who have been consecutively hospitalized in our medicine department between 2016 and 2017 for AHF. Of the 180 patients, 164 were included in the analysis after excluding those without any available data after hospitalization (n = 5) and those who died during or immediately after hospitalization (n = 11) (e.g., palliative care), to explore predictive risk factors associated with medium-and long-term mortality after hospitalization for AHF.

The diagnosis of HF was made on the basis of clinical signs of congestion, left ventricular ejection fraction (LVEF) less than 50% or signs of congestion on transthoracic echocardiography, with an elevated Nt-proBNP level without chronic renal failure [START_REF] Mcdonagh | ESC guidelines for the diagnosis and treatment of acute and chronic heart failure[END_REF]. AHF may be either acute de novo HF or acute exacerbation of chronic HF. According to the recognized terminology [START_REF] Mcdonagh | ESC guidelines for the diagnosis and treatment of acute and chronic heart failure[END_REF], HF patients with preserved ejection fraction (HFpEF) comprised HF patients with a LVEF ≥ 50%.

Study outcomes

Demographic and clinical characteristics were assessed during hospitalization, including age, gender, weight (at admission and discharge), body mass index (BMI), and conditions being responsible for chronic HF such as valvular disease, ischemic heart disease, atrial fibrillation. Various comorbidities were collected at admission: hypertension, diabetes, dyslipidemia, chronic obstructive pulmonary disease, chronic kidney diseases, sleep apnea-hypopnea syndrome, dementia, stroke, and peripheral artery disease. Comorbidities were declarative and defined as "presence" or "absence" at inclusion and were assessed by the Charlson Comorbidity Index [START_REF] Charlson | Validation of a combined comorbidity index[END_REF]. At discharge, chronic heart failure treatment was collected by a pharmaceutical team during the medication reconciliation process, following a previously described validated protocol [START_REF] Breuker | Patients with diabetes are at high risk of serious medication errors at hospital: interest of clinical pharmacist intervention to improve healthcare[END_REF].

Regarding biological characteristics, the C-reactive protein (CRP) level, hemoglobin, ferritin, NT-proBNP, troponin-hs, creatin kinase, uric acid, protid, albumin, HDL, LDL, HbA1C, creatinine levels and transferrin saturation (TSAT) were determined at admission. Renal function was assessed by CKD-EPI (ml/min/1.73 m 2 ). The primary endpoint was the occurrence of all-cause death, based on data from the national database [START_REF]Fichier des personnes décédées (Décès)-data[END_REF]. We completed data collection by calling the patient's family physician. The protocol of this study was approved by the Institutional Review Board of Montpellier University Hospital (IRB-MTP_2021_04_202100801).

Statistical analysis

Continuous variables are expressed in mean ± standard deviation (SD) or median and interquartile range (IQR) in case of skewed distribution and were compared using Student's t test (Table 1). Qualitative variables are expressed as frequency and compared using Fisher's exact test. Cox proportional hazards regression model [expressed as hazard ratio (HR) with corresponding 95% confidence interval (CI) and p values] was used to assess the association between each clinical or biological outcome and the occurrence of death, adjusting on age and CKD-EPI, which are established risk factors of HF mortality. HR are expressed per SD for each continuous variable as computed in Table 1. Due to skewed distribution, CRP and NT-proBNP were log-transformed prior to analysis. The statistics were performed using IBM SPSS Statistics 25 software. p value < 0.05 was considered to be statistically significant.

Results

Study population

Baseline clinical and biological characteristics, including comorbidities and treatment at discharge, are shown in Table 1. The study population comprised 164 patients, including 106 women, mean age of 82.9 ± 10.3 years, a mean BMI of 28.4 ± 7.7, a mean LVEF of 47.2 ± 9%, and mostly HFpEF (86%). As regards comorbidities, 127 (77%) had hypertension, 60 (37%) had diabetes mellitus, 45 (27%) had dyslipidemia, and 64 (39%) had chronic kidney disease. Mean Charlson comorbidity index was 6.5 ± 2.5. Most of the patients (58%) received ACE-I or ARB, 62% received beta-blockers and 84% received loop diuretics.

Outcomes

After a median follow-up period of 17.5 months (IQR 6-38), 109 (66.4%) patients died within a mean time of 14 months (IQR 3-29). Forty-nine patients (30%) died within the first year. In univariate analysis, patients who died during follow-up were significantly older, had lower BMI, lower LVEF, and higher Charlson comorbidity index. These patients had a lower renal function, higher levels of NT-proBNP (median of 4944 ng/l [2370-14403] compared to 1740 ng/l [1119-3503], p < 0.001), lower TSAT, and lower levels of HDL and LDL cholesterol compared with patients who survived (Table 1). Of note, 36 (22%) patients had both diabetes and CKD, of whom 24 died and 12 remained alive.

In multivariate analysis (Table 2), after adjustment for age and renal function, risk factors for death were weight at admission and discharge (respectively HR 0.76 [0.59-0.96] p = 0.022 and HR 0.75 [0.59-0.96] p = 0.020), BMI (HR 0.69, CI [0.53-0.90], p = 0.005), albuminemia (HR 0.77 [0.63-0.94], p = 0.009), ferritinemia (HR 1.38 [1.08-1.76], p = 0.010), uricemia (HR 1.28 [1.02-1.59], p = 0.030), NT-proBNP level (HR 2.46 [1.65-3.67], p < 0.001) and longer hospital stay (HR 1.25 [1.056-1.49] p = 0.013). We did not find any statistically significant association between mortality and the treatment for heart failure at discharge.

Discussion

The present study explores, in a real-life setting, the long-term outcomes of elderly patients hospitalized for AHF in an internal medicine department. Patients hospitalized in France for AHF are now increasingly older, with preserved LVEF, and increasingly hospitalized in non-cardiologic units. In the OFICA (Observatoire Français de l'Insuffisance Cardiaque Aigue) nationwide survey, representing a single-day snapshot of all unplanned hospitalizations because of AHF in 170 French hospitals, 18% of the 1658 patients, nearly 50% of whom were octogenarians, were admitted to an internal medicine unit [START_REF] Logeart | Current aspects of the spectrum of acute heart failure syn-dromes in a real-life setting: the OFICA study[END_REF]. First, our study population, mostly female (n = 106, 64.6%), elderly (82.9 years ± 10.0), with a preserved LVEF (86%), has similar characteristics as other published cohorts of patients hospitalized for AHF in internal medicine departments [START_REF] Martínez | Multimorbidity and acute heart failure in internal medi-cine[END_REF][START_REF]Grupo de Trabajo de Insuficiencia Cardiaca de la Sociedad Espanola de Medicina Interna (SEMI) [Heart failure in internal medicine departments[END_REF]. For instance, Martinez et al.'s study population included primarily elderly women, with preserved ejection fraction and with multiple comorbidities, the main ones being cardiovascular risk factors, especially hyper-tension, chronic kidney diseases, respiratory diseases, and chronic anaemia [START_REF] Martínez | Multimorbidity and acute heart failure in internal medi-cine[END_REF]. In the present study, 22% of patients were diabetic with chronic renal failure, which are known to be at a very high cardiovascular risk [START_REF] Tonelli | Alberta kidney disease network risk of coronary events in people with chronic kidney disease compared with those with diabetes: a population-level cohort study[END_REF] and for whom comprehensive risk factors control has been reported to improve cardiovascular outcomes and overall mortality [START_REF] Sasso | Efficacy and durability of multifactorial intervention on mortality and MACEs: a randomized clinical trial in type-2 diabetic kidney disease[END_REF]. AHF patients who are hospitalized in the internal medicine department are frequently older and affected by many comorbidities compared with patients hospitalized in cardiology units. As regards HF treatments, the French cohort study ODIN (Observatoire De l'Insuffisance Cardiaque) [START_REF] Juillière | Prescription of cardiovascular drugs in the French ODIN cohort of heart failure patients according to age and type of chronic heart failure[END_REF] reported that HT medications were less frequently prescribed with advancing age and with dosages lower than recommended. In the present study, we found a low rate of patients on ACE and ARB therapy (57%), without any reason other than age and comorbidities.

Second, we found a high mortality rate, with 65% of patients dying 5 years after hospitalization for AHF. A study based on data from the French national health insurance sys-tem in 2009, showed a 2year mortality rate of 40% [START_REF] Tuppin | Two-year outcome of patients after a first hospitalization for heart failure: a national observational study[END_REF]. In the ESC-HF long term registry, which is a prospective observational study of 6,629 AHF patients enrolled from 2011 to 2015 in 21 European countries, Chioncelet al. reported a readmission rate of 25.9% at 1 year and an all-causes 1-year mortality rate of 26.7%, but their study population was much younger with only 34% over 75 years of age [START_REF] Chioncel | Clinical phenotypes and outcome of patients hos-pitalized for acute heart failure: the ESC heart failure long-term registry[END_REF] compared with 61.6% in our study.

Indeed, various studies have shown that elderly patients with HF have a higher risk of poor prognosis [START_REF] Komajda | Management of octogenarians hospital-ized for heart failure in euro heart failure survey I[END_REF][START_REF] Mahjoub | Long-term survival in patients older than 80 years hospitalised for heart failure. A 5-year prospective study[END_REF]. In another French study, the 5-year survival in elderly patients older than 80 years was reported at 19%, dramatically lower than the survival of the age-and sex-matched general population of 48% [START_REF] Mahjoub | Long-term survival in patients older than 80 years hospitalised for heart failure. A 5-year prospective study[END_REF], consistently with our results. Female gender, hypertension, atrial fibrillation and comorbidities were reported to be more frequent in the elderly, as well as underuse and under dosage of medications recommended for HF management which may contribute to such a poor prognosis [START_REF] Mahjoub | Long-term survival in patients older than 80 years hospitalised for heart failure. A 5-year prospective study[END_REF][START_REF] Komajda | Contemporary management of octogenarians hospital-ized for heart failure in Europe: euro heart failure survey II[END_REF].

As regards prognostic factors, especially in the elderly, albumin levels have been associated with mortality in the older patients with HF [START_REF] Metra | Acute heart failure in the elderly: differences in clinical characteristics, outcomes, and prognostic factors in the VERITAS study[END_REF][START_REF] Horwich | Albumin levels predict survival in patients with systolic heart failure[END_REF][START_REF] Kinugasa | A sim-ple risk score to predict in-hospital death of elderly patients with acute decompensated heart failure hypoalbuminemia as an addi-tional prognostic factor: hypoalbuminemia as an additional prog-nostic factor[END_REF]. In one study including 438 patients, hypoalbuminemia predicted 1-year mortality (HR adjusted 2.05, 95% CI 1.10-3.81, p = 0.001) [START_REF] Uthamalingam | Serum albumin and mortality in acutely decompensated heart failure[END_REF]. Higher serum albumin levels (HR 0.67, 95% CI 0.53-0.85) were associated with longer survival in 1,120 patients with HFpEF [START_REF] Chien | Malnutrition in acute heart failure with preserved ejection fraction: clinical correlates and prognostic implications[END_REF]. Albumin level is a well-known marker of nutritional status, of utmost importance in elderly subjects. Hypoalbuminemia in HF patients is likely multifactorial, related to malnutrition due to decreased oral intake, as well as to increased catabolic activity, or chronic inflammation [START_REF] Chien | Malnutrition in acute heart failure with preserved ejection fraction: clinical correlates and prognostic implications[END_REF]. It may contribute to pulmonary edema and exacerbation of HF, due to decreased osmotic pressure. Hypoalbuminemia may provide additional prognostic criteria in elderly HF patients.

Furthermore, we found an inverse relationship between BMI and 5-year death. A higher BMI has been associated with reduced mortality in established HF patients, consistent with the "obesity paradox" [START_REF] Lavie | Impact of obesity and the obesity paradox on preva-lence and prognosis in heart failure[END_REF]. This inverse relationship between obesity and prognosis in HF may be explained by the limitations of using BMI to define body composition [START_REF] Lavie | Body composition and survival in stable coronary heart disease: impact of lean mass index and body fat in the "obesity paradox[END_REF]. On the other hand, cachexia has been associated with poor prognosis in HF [START_REF] Anker | Prognostic importance of weight loss in chronic heart failure and the effect of treatment with angioten-sinconverting-enzyme inhibitors: an observational study[END_REF], with a known association between lean body mass, muscle strength, and subsequent survival [START_REF] Lavie | Body composition and survival in stable coronary heart disease: impact of lean mass index and body fat in the "obesity paradox[END_REF]. Of note, the Charlson comorbidity index was not independently associated with mortality in our population. More than the number of comorbidities per se, it may be more the profile of comorbidities that may have a different impact on the prognosis of elderly patients. Therefore, beyond the effect of the number of comorbidities, it may be more relevant to identify subsets of patients who are at higher risk of severe outcomes. In this line of thought, different frailty phenotypes have been associated with different impacts on mortality, as shown in the REPOSI study [START_REF] Marcucci | REPOSI investigators defining aging phenotypes and related outcomes: clues to recognize frailty in hospitalized older patients[END_REF]. In the present study, our data highlight the importance of malnutrition over other comorbidities in the prognosis of elderly patients after AHF.

We also found an association between increased levels of uric acid and ferritin, and mortality. Consistently with our results, hyperuricemia has been associated with a higher risk of HF rehospitalization or death at 6 months [START_REF] Palazzuoli | Prognostic significance of hyperuricemia in patients with acute heart failure[END_REF]. This point seems important, given that some studies revealed a greater prognostic value of hyperuricemia in patients with-out chronic renal failure, suggesting that hyperuricemia may be predictive of poor outcome when it is a marker of increased xanthine oxidase activity, but not when it is related to impaired renal excretion of uric acid [START_REF] Filippatos | Hyperuri-caemia, chronic kidney disease, and outcomes in heart failure: potential mechanistic insights from epidemiological data[END_REF]. However, the association we found between ferritin levels and mortality is not in line with the previously reported association between iron deficiency and mortality in HF [START_REF] Mcdonagh | ESC guidelines for the diagnosis and treatment of acute and chronic heart failure[END_REF][START_REF] Klip | Iron deficiency in chronic heart failure: an international pooled analysis[END_REF]. It is possible that in our cohort, hyperferritinemia is rather related to the biological inflammatory syndrome. Indeed, the mean CRP level was elevated in our population, partly due to coexisting infection in several patients, but was not associated with mortality. Regarding TSAT, we found that deceased patients had a lower TSAT, with an association with mortality in univariate analysis but not in multivariate analysis. This is consistent with recent studies recommending iron supplementation in these patients, especially the AFFIRM-HF study [START_REF] Ponikowski | Ferric carboxymaltose for iron deficiency at discharge after acute heart failure: a multicentre, double-blind, randomized, con-trolled trial[END_REF]. Further studies are needed to clarify this last point, especially in the acute phase of HFpEF.

Limitations

The present study has several limitations. First, this is a retrospective analysis of a cohort of consecutive elderly patients admitted for AHF, based on the experience of a single internal medicine unit. However, as a tertiary center, this population could be considered representative of the large region (2 million inhabitants). Another limitation is a relatively small sample size. which may have implied relatively low power, and may not have identified significant associations. For example, the lack of association between discharge treatments and mortality should be taken with caution. Nevertheless, the sample size appears to be similar to other studies conducted in internal medicine departments, for example the study of Martinez et al. [START_REF] Martínez | Multimorbidity and acute heart failure in internal medi-cine[END_REF]. The exclusion of patients who died during or immediately after hospitalization may have created a bias. However, we believe that it allowed us to analyze the factors and comorbidities involved only in the medium-and long-term mortality of patients hospitalized for AHF. Moreover, we do not have information about muscle mass and body composition, which would have been of interest to further explore the association we found between malnutrition, BMI, and death. However, this appears to be related to the design of our study, which aimed to explore the association between non-selected comorbidities and death, without specifically focusing on malnutrition outcomes. Furthermore, all the criteria investigated here are widely available in clinical routine, which enables translation in clinical practice. In summary, in this retrospective study of a cohort of elderly patients with AHF hospitalized in an internal medicine unit, BMI, albuminemia, ferritinemia and uricemia may provide additive prognostic values after hospital dis-charge, given their association with mortality. These results highlight the need to consider nutritional support in elderly patients with HFpEF admitted for acute HF and paves the way for interventional studies in the field.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

Data could be available on reasonable request.

Author contributions

All authors had full access to the data and reviewed and approved the manuscript for publication; CR, BE and PF take responsibility for the integrity of the data and the accuracy of the data analysis; concept and design, all authors; acquisition, analysis, or interpretation of data, all authors; drafting of the manuscript, CR; critical revision of the manuscript, all authors; agreement to be accountable for all aspects of the work, all authors; statistical analysis, PF. All authors have read and agreed to the published version of the manuscript.

Declarations

Conflict of interest C.R.: reports personal fees from Servier and Pfizer unrelated to this work; L.Z.: reports personal fees from Servier unrelated to this work; S.L.: reports personal fees from Servier unrelated to this work; C.M.: reports personal fees from Servier unrelated to this work; F.R.: reports grants, personal fees and non-financial sup-port from Air Liquide, grants and personal fees from Abbott, Astrazeneca and Novartis, personal fees from Vifor, Servier, Abiomed, Zoll, Medtronic, Resmed, LVL, Eole, Pfizer, outside the submitted work; P.F.: has received travel and accommodation fees from Actelion, John-son and Johnson, BMS, Pfizer, and Servier unrelated to this work. The remaining authors declare no conflict of interest.

Institutional review board

The protocol of this study was approved by the Institutional Review Board of Montpellier University Hospital (IRB-MTP_2021_04_202100801).


Maxime Lefrançois
email: maxime.lefrancois@emse.fr

SAREF4SYST: a SAREF Reference Ontology Pattern for Representing Systems and their Interconnections

Keywords: Ontology Design Pattern, SAREF, Topology, Systems, Connections

published or not. The documents may come

Introduction

The Smart Applications REFerence ontology (SAREF) [START_REF] García-Castro | The ETSI SAREF Ontology for Smart Applications: A Long Path of Development and Evolution[END_REF][START_REF] Etsi | Smart Applications; Reference Ontology and oneM2M Mapping[END_REF] is a reference ontology for the Internet of Things (IoT) developed by the European Telecommunication Standards Institute (ETSI) SmartM2M Technical Committee (TC) in close interaction with the industry. SAREF contains core concepts that are common to several IoT domains and, to be able to handle specific data elements for a certain domain, dedicated extensions of SAREF have been created for the domains of energy, building, environment, smart city, industry and manufacturing, agrifood, automotive, eHealth / ageing well, wearables, water, smart lifts [START_REF] Etsi | Extension to SAREF; Part 1: Energy Domain[END_REF][START_REF]Extension to SAREF; Part 3: Building Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 2: Environment Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 4: Smart Cities Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 5: Industry and Manufacturing Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 6: Smart Agriculture and Food Chain Domains[END_REF][START_REF]Extension to SAREF; Part 7: Automotive Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 8: eHealth/Ageing-well Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 9: Wearables Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 10: Water Domain[END_REF][START_REF] Etsi | Extension to SAREF; Part 11: Lift Domain[END_REF]. SAREF serves as the means to connect the extensions in different domains.

SAREF and its different extensions were developed quite independently by different teams of experts, sometimes in parallel. Sometimes different modelling decisions were made, with the result that SAREF extensions having important discrepancies. TC SmartM2M started to identify ontology patterns that may be used to homogenise the structure of SAREF extensions. Specialist Task Force (STF) 653 "SAREF Patterns" aims at homogenising and facilitating the use of SAREF and existing 11 SAREF extensions by grounding them using a set of SAREF reference ontology patterns, which consist of combination of an ontological definition, and specification of how to apply it to different domains. The ultimate goal is to help ensuring a homogeneous structure of the overall SAREF ontology, speed up the development of extensions, and improve semantic interoperability.

In this paper we introduce the first such SAREF reference ontology pattern SAREF4SYST, which defines systems, connections between systems, and connection points to which systems can be connected. SAREF4SYST has been published as ETSI Technical Specification (TS) 103 548 in 2019 as part of previous STF 556 "Consolidation of SAREF and its community of users, based on the experience of the EUREKA ITEA -12004 SEAS" [START_REF] Etsi | SAREF consolidation with new reference ontology patterns, based on the experience from the SEAS project[END_REF], but it has not yet been presented to the community of researchers on pattern-based ontology engineering.

The rest of this paper is organised as follows. Section 2 provides general application scenarios for the pattern. Then Section 3 overviews the ontological description of the pattern and how it may be used in different domains. Finally Section 4 discusses some of its influences and uptake.

Motivating Scenarios for SAREF4SYST

SAREF4SYST defines Systems, Connections between systems, and Connection Points at which systems may be connected. These core concepts can be used generically to define the topology of features of interest, which is knowledge relevant for use cases in various application domains of Cyber-Physical Systems.

SAREF4SYST can be used for example to describe zones inside a building (systems), that share frontiers of different types (connections). Properties of systems are typically state variables (e.g. temperature, volume, energy, mass), while those of connection points are typically interaction capacities and flows (e.g. thermal transmittance, energy flow, matter flow, impedance).

The following paragraphs illustrate the applicability of SAREF4SYST in three different domains. Note that modeling properties of features of interest is not part of SAREF4SYST.

Power Energy. In the Smart Energy domain, electric power systems can exchange electricity with other electric power systems. The electric energy can flow both ways in some cases (from the Public Grid to a Prosumer), or in only one way (from the Public Grid to a Load). Electric power systems can be made up of different sub-systems. Generic sub-types of electric power systems include producers, consumers, storage systems, transmission systems.

Electric power systems may be connected one to another at electrical connection points. An Electric power system may have multiple connection points (Multiple Winding Transformer generally have one single primary winding with two or more secondary windings). Generic sub-types of electrical connection points include plugs, sockets, direct-current, single-phase, three-phase, connection points.

An Electrical connection may exist between two Electric power systems through two of their respective connection points. Generic sub-types of electrical connections include Single-phase Buses, Three-phase Buses. A single-phase electric power system can be connected using different configurations at a three-phase bus (phase 1-to-neutral, phase 2-to-neutral, phase 3-to-neutral).

Smart Building. Buildings, Storeys, Spaces, are different sub-types of Zones. Zones can contain sub-zones. Zones can be adjacent or intersect with other zones. Two zones may share one or more connections. For example some fresh air may be created inside a storey if it has two controllable openings to the exterior at different cardinal points. Lifts have one or more openings, and evolve in a shaft. Storeys may have openings to the shaft. Both the lift openings and the storey openings may have doors that may be in an open or closed state.

Communication networks. Smart devices contain microcontrollers with Input/Output ports and RadioFrequency (RF) communication modules of different kinds. Wired communication may be established between two devices directly, and between two or more devices through some bus. RF communication from a sender to receivers may be established at a certain radiofrequency, with each party powering its RF module at a certain level, and receivers having a relative measure of the Received Signal Strengh Indicator (RSSI) and Signal-to-Noise Ratio (SNR).

Ontological Description and Application Guidelines

SAREF4SYST consists both of a core ontology, and guidelines to create ontologies applying the SAREF4SYST ontology pattern. The core ontology is a lightweight OWL-DL ontology that defines 3 classes and 9 object properties. It uses namespace https://saref.etsi.org/saref4syst/ prefixed as s4syst: and is available as open source at https://saref.etsi.org/sources/saref4syst/ under the ETSI BSD-3 license, and is submitted as Architectural ODP on the ontologydesignpatterns.org website as http://ontologydesignpatterns.org/wiki/Submissions:SAREF4SYST.

Figure 1 provides a general overview of these classes and properties. In this figure, rectangles are used to denote classes. Plain arrows are used to represent Object Properties between classes. The origin of the arrow is the domain of the property, and the target of the arrow is the range of the property. Dashed arrows with identifiers between stereotype signs (i.e. "« »") refer to OWL axioms that are applied to some property. Four pairs of properties are inverse one of the other; the property s4syst:connectedTo is symmetric, and properties s4syst:hasSubSystem and s4syst:subSystemOf are transitive. A symbol =1 near the target of an arrow denotes a cardinality of 1. A symbol ∃ denotes a local existential restriction.

Systems and sub-systems

A s4syst:System is defined as a part of the universe that is virtually isolated from the environment. The system properties are typically state variables (e.g. consumed or stored energy, agent population, temperature, volume, humidity). Figure 2 illustrates classes and properties that can be used to define connected systems and their sub-systems. This module reuses the PartOf ODP. A connection between two s4syst:Systems, modelled by s4syst:connectedTo, describes the potential interactions between connected s4syst:Systems. For example s4lift:canConnectToNetwork, from the SAREF extension for Smart Lifts (SAREF4LIFT [START_REF] Etsi | Extension to SAREF; Part 11: Lift Domain[END_REF]) describes a relationship between a system and a network it can connect to. s4syst:connectedTo is symmetric. Sub-properties, however, are not necessarily symmetric, and can be defined for example to model that some matter only flows from one system to the other. For example property s4lift:connectedToEmergencyBattery links a s4lift:ElectricPowerSystem system to another that is its emergency battery.

These core entities are defined as follows:

s4syst:System. The class of systems, i.e., systems virtually isolated from the environment, whose behaviour and interactions with the environment are modeled. Systems can be connected to other systems. Connected systems interact in some ways. Systems can also have subsystems. Properties of subsystems somehow contribute to the properties of the supersystem. s4syst:connectedTo. Links a system to a system it is connected to. Connected systems interact in some way. The exact meaning of "interact" is defined by sub properties of s4syst:connectedTo. Property s4syst:connectedTo is symmetric. This property can be qualified using class s4syst:Connection, which connects the two systems. If there is a connection between several systems, then one may infer these systems are pairwise connected. s4syst:subSystemOf. Links a system to its super system. Properties of subsystems somehow contribute to the properties of the super system. The exact meaning of "contribute" is defined by sub properties of s4syst:subSystemOf. Property s4syst:subSystemOf is transitive.

Connections between systems

A connection can be qualified using class s4syst:Connection. Figure 3 illustrates classes and properties that can be used to qualify connections between s4syst:Systems. These core entities are defined as follows:

s4syst:Connection. The class of connections between systems. This class qualifies property s4syst:connectedTo. A connection describes potential interactions between systems. Any two connected systems are connected through a connection. A connection can connect more than two systems at the same time. s4syst:connectsSystem. Links a connection to one of the systems it connects. s4syst:connectedThrough. Links a system to one of its connections to other systems.

Connections points of systems

A s4syst:System connects to other s4syst:Systems at connection points. A connection point belongs to one and only one s4syst:System, and can be described using the class s4syst:ConnectionPoint.

Figure 4 illustrates the classes and the properties that can be used to describe connection points of a s4syst:System. One can then associate a s4syst:ConnectionPoint with properties (saref:Property) that describe it (e.g. position and speed, voltage and intensity, thermal transmittance).

These core entities are defined as follows:

s4syst:ConnectionPoint. The class of connection points of systems, at which they may be connected to other systems. This class qualifies properties s4syst:connectsSystem and s4syst:connectedThrough. A connection point belongs to exactly one system. Any system connected through a connection is connected at one of its connection points to the connection. The system of a connection point that is connected through a connection is itself connected through the connection. s4syst:connectionPointOf. Links a connection point to the one and only one system it belongs to. s4syst:connectsAt. Links a system to one of the connection points at which it connects. s4syst:connectsSystemThrough. Links a connection point to one of the connections through which it connects its system. s4syst:connectsAt. Links a connection to one of the connection points at which it connects a system.

Furthermore, the following sub-property chain axioms are defined:

s4syst:connectsSystem ⊑ s4syst:connectsSystemAt 𝑜 s4syst:connectionPointOf (1) s4syst:connectedThrough ⊑ s4syst:connectsAt 𝑜 s4syst:connectsSystemThrough (2)
These allow to infer that a connection at a connection point necessarily connects the system of that connection point (Axiom 1), and that if a connection point connects its system through a connection, then the system is connected through this connection (Axiom 2).

Applications of the SAREF4SYST Pattern

ETSI TS 103 548 [START_REF] Etsi | SAREF consolidation with new reference ontology patterns, based on the experience from the SEAS project[END_REF] not only describes the terms and the axioms in the SAREF4SYST ontology, but also provides many examples of how the classes and properties may be extended. In addition, it devises guidelines for applying the pattern to verticals. Applications of the SAREF4SYST pattern are subsets of ontologies that define sub-classes and/or sub-properties of the classes and properties of SAREF4SYST. In particular, extensions should preserve the connective words in the name of properties (Of, At, Through).

The label of the sub-property should use the same morpho-syntactic structure as its super-property. ([14, Clause 5.

3.3.2])

The technical specification also guides the users in choosing the type of axioms that may be appropriate for the pattern application.

Conclusion

SAREF4SYST is the first SAREF Reference Ontology pattern. It has proven to be useful to model various types of system, connection through which these systems interact in some way, and the connection point at which these systems connect. More patterns for SAREF are currently being discussed in the context of STF 578 "SAREF Patterns", and should be described beside SAREF4SYST in a new version of ETSI TS 103 548.

Figure 1 :

 1 Figure 1: Overview of the SAREF4SYST ontology pattern (adapted from [14])

Figure 2 :

 2 Figure 2: Systems, sub-systems

Figure 3 :

 3 Figure 3: Connections between systems through which systems connect.

Figure 4 :

 4 Figure 4: Connection points of systems, where other systems connect

Figure 5 :

 5 Figure 5: Overview of the SAREF4LIFT classes for systems and connections (Source: [13])

https://saref.etsi.org/saref4lift/v1.1.1/example/TS103735_Clause_6_4_SLInstallation.ttl

https://saref.etsi.org/saref4lift/v1.1.1/example/TS103735_Clause_6_5_SLConfiguration.ttl

A sub-property of the s4syst:connectedThrough property may have an inverse property. If defined, the English-tagged labels and comments of these two properties shall reflect this aspect. ([14, Clause 5.

3.3.2])

A sub-class of the s4syst:System class may have an existential restriction on a sub-property of s4syst:connectedThrough to some sub-class of s4syst:Connection ([14, Clause 5.3.3.2])

Acknowledgments

This work has been partly supported the ETSI Specialist Task Forces 556 "Consolidation of SAREF and its community of users, based on the experience of the EUREKA ITEA -12004 SEAS", and CoSWoT projet (grant ANR-19-CE23-0012 from Agence Nationale de la Recherche, France).

Influences and Uptake

SAREF4SYST is the result of incorporating the seas:SystemOntology ontology into SAREF, which is one of the core modules of the Smart Energy Aware Systems (SEAS) ontology [START_REF] Lefrançois | Planned ETSI SAREF extensions based on the W3C&OGC SOSA/SSNcompatible SEAS ontology patterns[END_REF]. SAREF4SYST relaxes the SEAS Systems pattern as seas:subSystemOf was defined as a functional, and s4syst:subSystemOf is not.

The SAREF extension for Smart Lifts (SAREF4LIFT [START_REF] Etsi | Extension to SAREF; Part 11: Lift Domain[END_REF]) is the first extension that extensively apply the SAREF4SYST pattern. Figure 5 illustrates how the core three classes have been specialized. SAREF4LIFT furthermore uses the SAREF4SYST patterns to define sup-properties:

Examples Smart Lift Installation 1 and Smart Lift General Configuration 2 illustrate the use of the SAREF4SYST pattern and its different classes and properties. Sub-properties of s4syst:hasSubSystem: s4lift:hasAlarmSOSSystem Sub-properties of s4syst:connectedTo: s4lift:canConnectToNetwork, s4lift:connectedToEmergencyBattery Sub-properties of s4syst:connectsAt: s4lift:hasMainPowerSupply, s4lift:hasStandardPowerSupply Sub-properties of s4syst:connectedThrough: s4lift:hasCarService Several works have closely or remotely adopted the patterns behind SAREF4SYST. The TUBES System Ontology (TSO) and the Flow System Ontology (FSO) align to SAREF4SYST for the description of flow systems in buildings. TSO aims to explicitly define the hierarchical, structural, and functional aspects of interconnected building service systems in the Architecture Engineering, and Construction (AEC) industry and their relationships to spatial entities throughout their whole life cycle. FSO aims to describe the matter and energy flow between systems and components, and the composition of such systems [START_REF] Kukkonen | An ontology to support flow system descriptions from design to operation of buildings[END_REF][START_REF] Pauen | A roadmap toward a unified ontology for building service systems in the aeco industry: Tso and fso[END_REF].

SAREF4SYST is used by Bjørnskov and Jradi [START_REF] Bjørnskov | An ontology-based innovative energy modeling framework for scalable and adaptable building digital twins[END_REF] to model interconnected systems in the Digital Twin of a building, and simulate its behaviour.

Roxin et. al. in [START_REF] Roxin | Investigating potential alignments between modelica standard library and saref ontologies[END_REF] investigates potential alignment between SAREF ontologies, including SAREF4SYST, and the Modelica system modeling language for modeling of cyber-physical systems supporting acausal connection of components governed by mathematical equations. SAREF4SYST naturally aligns to the system modeling in Modelica.


Mohsen Hadavi
email: moh3nhadavi@gmail.com

Maxime Lefrançois
email: maxime.lefrancois@emse.fr

The urdflib Library for MicroPython: Manipulating RDF on Constrained Devices

Keywords: Semantic Web of Things, MicroPython, Programming the Semantic Web, Internet of Things

This demonstration paper introduces the urdflib library for MicroPython, which facilitates the development of RDF manipulation programs for embedded devices that run MicroPython, and additionally ensures the API is compatible with the widely used RDFLib Python library. The library is openly available on GitHub, a demo is presented, and the performances of the library are evaluated on the Unix and the ESP32 ports of MicroPython. urdflib performs better than RDFLib on the Unix port.

Introduction

In recent years, the world of embedded systems and microcontroller programming has witnessed a significant transformation, with an ever-increasing demand for efficient and versatile solutions. As electronic devices continue to permeate every aspect of modern life, the need for simplified and accessible programming tools becomes more pronounced. MicroPython [START_REF] George | Micropython-python for microcontrollers[END_REF], an implementation of Python tailored for microcontrollers, has emerged as a game-changer in this domain, bridging the gap between traditional embedded systems programming and the world of high-level languages.

It has been proven that Python (and its variant for microcontrollers MicroPython) is easier to learn and use than C [START_REF] Fangohr | A comparison of c, matlab, and python as teaching languages in engineering[END_REF]. Recent efforts to define compressed protocols and syntaxes for the Web contribute to the vision of a Semantic Web of Things, where CoAP is used instead of HTTP, and compressed RDF syntaxes such as CBOR-LD are employed. New compression mechanisms such as SCHC [START_REF] Minaburo | Rfc 8724: Schc: Generic framework for static context header compression and fragmentation[END_REF] help bring IPv6, UDP, CoAP, and CBOR to LPWAN (Low-Power Wide Area Network) and LoWPAN (Low-Power Wireless Personal Area Networks) networks.

Typically, embedded programming is done in C/C++, potentially using the Arduino framework. Sord 1 and Serd 2 are lightweight C libraries to work with RDF data and store RDF statements in memory, and are a perfect fit for high-performance or resource-limited applications.

In this demonstration paper, we contribute to the vision of the Semantic Web of Things by providing a simple and accessible, yet lightweight library for RDF processing, based on MicroPython. Our library, named urdflib, uses Sord and Serd as the backend, and provides an API that conforms to the widely used RDFLib Python library [START_REF] Krech | [END_REF], in such a way that simple RDFLib programs could be adapted to constrained devices with a simple change of the module import declaration: i m p o r t u r d f l i b a s r d f l i b

In the following sections, we will thoroughly examine the critical components of our research. Section 2 will extensively cover the design and implementation decisions made for urdflib, along with an availability statement. Section 3 will provide an initial assessment of the library, emphasizing its compatibility with Unix and ESP32 ports of MicroPython. Moving on to Section 4, we will investigate other relevant research in this domain and underscore the unique attributes of urdflib when compared to these works. Lastly, Section 5 will be dedicated to summarizing our findings, drawing conclusions, and discussing the resulting implications.

Design, Implementation, Availability, Demonstration

Compatibility with RDFLib. urdflib is intended to provide users with a significant interpretation of RDFLib, ensuring both ease of use and compatibility with existing code. This goal was achieved by running tests using the RDFLib repository, thus allowing users to stick to the RDFLib APIs. However, it is necessary to acknowledge that the underlying implementation in the urdflib module is written in the C language. As of today, urdflib implements a coherent subset of the RDFLib API, leaving out namespaces, syntax parsers and serializers, and other storage and querying plugins.

Integration with the MicroPython Firmware. Three methods for creating a module in MicroPython are available: Core module, external module, and port module. The implementation of urdflib utilizes the Sord, Serd, and Zix C libraries, which required modifications in their code. As a result, these libraries were merged into a single repository and added to MicroPython as an external module. Consequently, urdflib now operates as a core module, utilizing Serd and Sord to establish an API that facilitates RDF manipulation on resource-constrained devices. Two MicroPython ports have been tested successfully: the Unix and the ESP32 ports.

Decoupling with the C backend library. The dependency of urdflib to Serd is only scoped to the separate middleware folder. To migrate to a new RDF C library backend, one would only need to change the files in this folder.

Resource availability statement. The source code for urdflib is available from GitHub 3under the open MIT license. urdflib is registered on Zenodo, with a permanent DOI and canonical citation. 4 Additionally, comprehensive documentation and practical examples illustrating how to create MicroPython modules like urdflib can be found on GitHub 5 .

Demonstration A demonstration of the library running on an ESP32 via REPL (Read, Evaluate, Print, Loop), is available online. 6

Performance Evaluation

When the urdflib module is incorporated into MicroPython, it only adds 27 KB to the firmware size for the ESP32 port, which represents 1.5 % of the 1.5 MB original MicroPython firmware for ESP32. The approximate memory overhead during firmware writing is 2 KB.

Table 1 summarizes the results of an experiment conducted to assess the process of adding triples on three different platforms, having different resource capabilities. The time taken and memory allocations for operations were measured. The platforms used in the experiment were: (py-rdflib) A MacBook Air with RDFLib package installed, a 1,6GHz Dual-Core Intel Core i5 processor, with 16GB 2133 MHz LPDDR3 memory; (mpy-unix-urdflib) The same MacBook Air running the MicroPython Unix port with the urdflib module; and (mpy-esp32-urdflib) An ESP32 device with a memory capability of 128KB, running MicroPython with the urdflib module. As illustrated in Table 1, urdflib demonstrates significantly higher speed and lower memory usage than RDFLib. Step-by-step instructions to compile the code and reproduce the experiments are available online.7

Related Work

This section lists some related work from the literature on RDF for the IoT (Internet of Things). RDF4Led [START_REF] Le-Tuan | Rdf4led: An rdf engine for lightweight edge devices[END_REF] is an RDF engine for edge devices. It is compared against Virtuoso and Jena TDB on three types of hardware with 256-512 MB of RAM. The constrained device urdflib targets are microcontrollers such as the ESP32 with just a few MB of RAM. Cowl [START_REF] Bilenchi | Cowl: A lightweight owl library for the semantic web of everything[END_REF] is a lightweight implementation of OWL 2 that aims to target devices with severe processing and memory limitations. A similar work as ours could be led to use it as the backed for a lightweight version of a Python implementation of OWL, such as Owlready2 8 . LiRoT [START_REF] Bento | Do arduinos dream of efficient reasoners?[END_REF] use Serd and Sord to propose a lightweight incremental reasoner that can be embedded in constrained objects, so that reasoning on them in a fog architecture becomes possible. It demonstrates lower reasoning time for small numbers of triples, which makes it suitable for the Semantic Web of Things.

Regarding the lightweight RDF syntaxes, [START_REF] Charpenay | Towards a binary object notation for rdf[END_REF] proposes to use a CBOR equivalent of JSON-LD as a lightweight syntax for RDF. RDF/CBOR is another proposed syntax for RDF based on CBOR 9 .

[9] compares HDT and CBOR for exchanging RDF.

Conclusion and Future Work

The urdflib module facilitates the development of RDF manipulation programs for embedded devices that run MicroPython and additionally ensures the API is compatible with the widely used RDFLib Python library. urdflib may have an impact in contributing to the adoption of Semantic Web technologies by easing its use for IoT learners and hackers. While this library represents a significant advancement in working with RDF on constrained devices, there are further steps to be taken in the future. These include: (i) use compact data structure for storing IRIs, (ii) use native datatypes for storing literals instead of their lexical form, (iii) implement serializers and parsers for compressed syntaxes that minimize memory usage because most RDF syntaxes are verbose and not suitable for IoT communication scenarios, and (iv) provide API for lightweight reasoner LiRoT or other features due to the intended goals.

Table 1

 1

	Performance Evaluation
	Platform	# triples Initial Time (us) Adding Time (us) Memory Allocation (B)
	py-rdflib	100	173302	6544	289640
	py-rdflib	1000	180661	65505	2319151
	py-rdflib	10000	180499	449035	22944084
	mpy-unix-urdflib	100	17	1222	60160
	mpy-unix-urdflib	1000	20	15857	578560
	mpy-unix-urdflib	10000	32	115643	1909248
	mpy-esp32-urdflib	50	12276	90363	18880
	mpy-esp32-urdflib	100	10924	179512	34880
	mpy-esp32-urdflib	180	8001	330541	60480

https://github.com/moh3nhadavi/micropython-urdflib, or https://gitlab.com/coswot/micropython-urdflib

https://zenodo.org/record/8342624

https://github.com/moh3nhadavi/micropython-usermod/tree/update

https://ci.mines-stetienne.fr/urdflib/demo

https://github.com/moh3nhadavi/micropython-urdflib-iswc2023

Acknowledgments

This work is supported by grant ANR-19-CE23-0012 from Agence Nationale de la Recherche, France, for project CoSWoT 10 .