Control of the Spread of COVID-19 by the Sentinel Method and Numerical Simulation of the Studied Model Solution

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Introduction
In practice, this study is taking place within the framework of the fight against COVID-19. The COVID-19 (coronavirus disease 2019) pandemic is a global public health problem, the first cases of which were first reported in Wuhan, China, on 31st December 2019. Our work is important in the strategy fight against COVID-19. COVID-19 is a respiratory infection caused by a Coronovirus called SARS-Cov6-2. A first strategy to control the spread of COVID-19 in the world consisted in confining the population, this made it possible to reduce the number of reproduction. In the theoretical framework, to estimate the number of reproduction, few authors have used mathematical models with spatial diffu-sion. However, to predict the evolution of the epidemic, researchers have used mathematical models of propagation such as the SIR epidemiological model (see, for example, [1]). In our work, we considered a SCIRD-type model of COVID-19 transmission in which we include mortality terms and spatial diffusion. Indeed, let us assume the most realistic situation of geographical spread of epidemics; moreover, we will consider the case of disturbances and incomplete data. The prediction of unknown data with the confrontation of real parameters is sometimes necessary, and our objective is to show that by using the sentinel method of JL Lions, a mathematical answer can nevertheless be given. More precisely, let Ω be an open and bounded domain of ℝ N . For the time T > 0, set Q = ð0, TÞ × Ω, Σ = ð0, TÞ × Γ. We denote by ν the outer normal on Γ. Then, consider the following problem: The function Sðt, xÞ represents susceptible persons at risk of contacting COVID-19 at time t at the point x ∈ Ω, the function Cðt, xÞ represents carriers (dead corpse) that transmit the COVID-19 at time t at the point x ∈ Ω, the function Iðt, xÞ describes infective persons capable of transmitting the COVID-19 to persons at risk at time t at the point x ∈ Ω, the function Rðt, xÞ represents recovered persons who have been treated of COVID-19, and the function Dðt, xÞ gives the total number of deaths at time t at the point x ∈ Ω.
It should be noted that this type of mathematical model without spatial diffusion has already been studied by Atangana [2], it uses Italian data to propose a mathematical model of COVID-19 transmission. The COVID-19 problem have been studied by several authors. In [3] The parameters of the considered model are presented in Table 1.
In a biological explanation of the model, let N be the density of the total population N ≕ S + C + I + R + D. As there is a term of spatial diffusion −γΔS, the total population N is not constant, and therefore, ∂ t N ≠ 0.
The term λÎ represents the disturbance around the infected population whose density is not completely known; τŜ 0 is the missing term of the population likely to be infected. We suppose that kÎk L 2 ðQÞ ≤ 1 and also that kŜ 0 k L 2 ðΩÞ ≤ 1. It is the real parameters λ and τ which are unknown; the other parameters are known. If we add the first five equations of the model (1) we are studying, we get the simplified system: where N 0 ≕ S 0 + C 0 + I 0 + R 0 + D 0 . Therefore, in the following, system (3) is considered and we assume the following: (H1): μ ∈ Cð½0, T × ΩÞ μðt, xÞ ≥ 0 a:e in Q: (H2): ∇μ ∈ ½L ∞ ðQÞ n: Remark 1. It is assumed that the initial data S 0 , C 0 , I 0 , R 0 , and D 0 are known function ant belong to L 2 ðΩÞ. Under the assumptions (H1) and (H2), according to Lions and Magenes [6], system (3) of parabolic type admits a unique solution such that Now, the problem is as follows: is there a method to get information about the term λÎ (so-called pollution term) from the infected population, insensitive to the missing term from the susceptible population τŜ 0 (so-called perturbation term)? In other words, we are interested in identifying the unknown parameters in model (3) which has incomplete data; more precisely, we are interested in the identification of pollution in the COVID-19 phenomenon, in the sense that we know little about the initial data as well as certain missing terms. The position (internal) and the nature of pollution (COVID-19) are known, but we do not know its amplitude (number of people infected by COVID-19). Our goal is to find a method to obtain information on these amplitudes. A partial answer can be obtained using the least squares method which is to take the unknowns, fλÎ, τŜ 0 g = fw, vg as control variables, and we can not clearly separate v and w.
The sentinel method of Lions [2] provides the right answer to this type of problem. In this paper, we construct sentinels when the supports of the observation function and of the control function are included in two different open subsets of ℝ N (see Nakoulima [7]).

Abstract and Applied Analysis
Several authors studied the sentinel problem. We refer to [7][8][9] and the references therein. In [10], Ainseba et al. used the method of sentinels to identify parameters of pollution in a river. Bodart and Demeestere applied it in [11] to identify an unknown boundary. In [12], Soma and Sawadogo studied the boundary sentinels with given sensitivity in population dynamics problem. In [9], Mophou and Nakoulima studied the problem of sentinels with given sensitivity. Sawadogo introduced in [13] the distributed sentinels into the equation of the dynamics of populations to study a population subject to a migratory phenomenon. Recently in [14], Hamadi et al. have studied the existence of construction of a sentinel for the epidemiological model SIR with spatial diffusion. In this paper, we apply the sentinel method to identify parameter in a mathematical model of COVID-19 transmission with spatial diffusion and incomplete data which takes into account the possibility of transmission from dead populations to susceptible populations. The problem is as follows: given h 0 ∈ L 2 ðU × OÞ, find a control w in L 2 ðU × ωÞ such that if N = Nðλ, τÞ is solution of (3) and S′ is defined by (6); then (7) and (8) hold.
The remainder of this paper is as follows: in Section 2, we briefly explain the concept of the sentinel In Section 3, we establish the equivalence between sentinel problem and null controllability problem. Sections 4 and 5 are devoted, respectively, to preliminary results and proof of main result. In Section 6, we formulate the information given by sentinel. In Section 7, we highlight the numerical simulation of the considered model solution.

Sentinel Method
The sentinel concept relies on the following three objects: some state equation (for instance), some observation function, and some control function w to be determined.
A state equation represented here by (3) and we denote by N = Nðt, x, λ, τÞ = Nðλ, τÞ depends on two parameters λ and τ the unique solution of (3).
An observation on nonempty open subset O ⊂ Ω is called the observatory set. The observation is N in O, for the time T. We denote by N obs this observation A function S′ = S′ðλ, τÞ called "sentinel". Let h 0 ∈ L 2 ðð0, TÞ × OÞ. Let on the other hand ω be some open and nonempty subset of Ω such that ω ≠ O.
For a control function w ∈ L 2 ðð0, TÞ × ωÞ, we define the functional We say that S ′ defines a sentinel for the problem (1) if there exists w such that S ′ is insensitive (at first order) with respect the to missing terms τN 0 , which means where here ð0 ; 0Þ corresponds to λ = τ = 0 and w is of minimal norm in L 2 ðð0, TÞ × ωÞ. That is,

Null Controllability Problem
We show in this section that the existence of the sentinel comes to null controllability property. We begin by transforming the insensibility condition (7). Set Then, the function Problem (10) is linear and has a unique solution N τ . The insensibility condition (7) holds if and only if We can transform (11) by introducing the classical adjoint state. More precisely, we define the function q = qðt ; xÞ as the solution of the backward problem: Since h 0 ; ∈L 2 ðð0, TÞ × OÞ and w ∈ L 2 ðð0, TÞ × ωÞ, the adjoint problem admits a unique solution given by q ∈ L 2 ðQÞ ∩ Cð½0, T ; H −1 ðΩÞÞ. The function q depends on the control w that we shall determine: Indeed, if we multiply the first equation in (12) by N τ and we integrate by parts over Q, we obtain So, condition (7) (or (11)) is equivalent to Thus, sentinel problems (6), (7), and (8) are equivalent to the following null controllability problem: given h 0 ∈ L 2 ðð0, TÞ × OÞ, find a control w in L 2 ðð0, TÞ × ωÞ such that if q is the solution of (12), then (8) and (14) hold.
In the following, we set

Existence of a Sentinel
For the study of the existence of a sentinel, we use a method developed in [15]. We begin with some observability inequality. Using (15), we have the following: Proposition 2. Let be ρ ∈ V ; then, there exists a positive con- where θ ′ ∈ C 2 ðQÞ positive with 1/θ ′ bounded Proof. See Fursikov and Imanualov [16].
According to the RHS of (16), we consider the space V endowed with the bilinear form að:;:Þ defined by According to Proposition 2, this symmetric bilinear form is a scalar product on V .
Let V be the completion of V with respect to the norm Then, V is a Hilbert space for the scalar product aðρ ; b ρÞ and the associated norm.
Remark 3. We can precise the structure of the elements of V. Indeed, let H θ ′ ðQÞ be the weigthed Hilbert space defined by endowed with the natural norm Now if h 0 ∈ L 2 ðQÞ and θ ′ h 0 ∈ L 2 ðQÞði:e:,h 0 ∈ L 2 θ′ ðQÞÞ, then from (16) and the Cauchy-Schwartz inequality, we deduce that the linear form defined on V by is continuous. Therefore, from the Lax-Milgram theorem, there exists a unique u ∈ V solution of the variational equation: Proposition 4. Assume that h 0 ∈ L 2 θ′ ðQÞ, and let ρ be the unique solution of (21). We set and Then, the pair ðw ; qÞ is such that (12)-(14) hold (i.e., there is some insensitive sentinel defined by (6) and (7)). 4 Abstract and Applied Analysis Moreover, we have where C is a constante and L 2 ðQ ω Þ = ð0, TÞ × ω.

Construction of the Sentinel
Having shown the existence of a sentinel, our goal now is to build this sentinel for the (1). Note that the existence of this sentinel justifies the existence of the optimal control.

Existence of the Optimal Control.
For the following, we will consider the following optimization problem: Theorem 5. There is a unique couple ðŵ,qÞ solution of the problem ðPÞ.
Proof. By Proposition 4, the domain E is nonempty. On the other hand, the map w ↦ kwk L 2 ðQ ω Þ is continuous, coercive, and strictly convex; then, we deduce that there is a unique solution for the problem (P)

Penalization Method.
In this subsection, we are concerned with the optimality system for ðŵ,qÞ. Since a classical way to derive this optimality system is the method of penalization due to Lions [8], here, we use this method. For this, we introduce the penalized cost function: where ε > 0. We consider the following problem ðP ε Þ: The following proposition gives the existence of the solution for ðP ε Þ. Proposition 6. It is assumed that the assumptions of the previous section are satisfied. Then, there is a couple ðw ε , z ε Þ a as unique solution of the problem ðP ε Þ.
Proof. Since E ⊂ U and E ≠ ∅, consequently, U is nonempty and is closed. Moreover, J ε is continuous, coercive, and strictly convex. Then, the problem ðP ε Þ has a unique solution denoted by ðw ε , z ε Þ. Now, we study the convergence of the couple ðw ε , z ε Þ when ε ⟶ 0. For this, we have the following result: Proposition 7. Let ðw ε , z ε Þ the unique solution of the problem ðP ε Þ. Then, Proof. As ðw ε , z ε Þ is the solution of ðP ε Þ, then, especially for ðŵ,qÞ ∈ E ⊂ U. Inequality (30) is written which implies Thus, From (33), we conclude that w ε is bounded in L 2 ðQ ω Þ. So

Abstract and Applied Analysis
On the other hand, because we have ðw ε , z ε Þ ∈ U and according to the inequality (34), we have With Given z ε solution of (36), then z ε ∈ H 2,1 ðQÞ. Moreover, we have Since kh ε k L 2 ðQÞ ≤ C ffiffi ε p , then, where C is a constant. Therefore, there is a subsequence denoted by ðz ε Þ ε , such that z ε ⇀ z 0 in H 2,1 ðQÞ. As the injection of H 2,1 ðQÞ into L 2 ðQÞ is compact and by passing to the limit, we finally find On the other hand, J ε is convex and continuous; then, Using in (41), the estimate We get Finally, as ðŵ,qÞ is the unique solution of ðPÞ, then, w 0 =ŵ. On the other hand z 0 is solution of (40) and by uniqueness of the solution for the heat equation, we deduce that z 0 =q.
The following proposition gives the optimality system for the pair ðw ε , z ε Þ. Proposition 8. The problem ðP ε Þ admits an optimal solution ðw ε , z ε Þ if and only if, there exists a unique function ρ ε ∈ L 2 ðQÞ such that fw ε , z ε , ρ ε g is the solution of the following optimality system: with and with Proof. Since ðw ε , z ε Þ is the unique solution of ðP ε Þ then, by applying the Euler-Lagrange, optimality conditions, we find or From the definition of the functional J ε and the linearity of the operator L * , J ε ðw ε + λw, z ε Þ is written: So Abstract and Applied Analysis There remains Passing to the limit, we get A calculation analogous to that above in equation (48) We introduce the following adjoin state defined by So (53) and (54) become, respectively, and We deduce From where Thus, for z in DðQÞ, we deduce that On the other hand, ρ ε ∈ L 2 ðQÞ with Lρ ε ∈ L 2 ðQÞ; then by application of the Lions-Magenes theorem, the trace function exists.
We multiply (60) by z ∈ C ∞ ð QÞ, and we integrate by part over Q; we get In particular for z ∈ C ∞ ð QÞ such that z = 0 on Σ, zðTÞ = zð0Þ = 0 in Ω and the fact that we have (54), then (61) becomes Finally,

Optimality System (OS).
In this subsection, we will state an important result which gives the optimality system for the problem ðPÞ, characterizing the optimal control, which also implies the construction of the sentinel.
Theorem 9. The pair ðŵ,qÞ is the unique solution of ðPÞ, if and only if there exists a function b ρ such that the triplet fŵ,q, b ρg is the solution of the following optimality system: Proof. From the following equality, 7 Abstract and Applied Analysis and from the fact w ε satisfies the inequality (33), it comes and since we deduce that According to (66) and (68), we conclude that ðρ ε Þ ε is bounded in V. Therefore, there are a subsequence ðρ ε Þ ε and a function b ρ in V such that Thus, On the other hand, from the previous proposition, we have wε ⇀ŵ weakly in L 2 Q ω ð Þ: ð71Þ By uniqueness of the limit, we deduce that

Information Given by the Sentinel
We assume that the population density N is observed on O, so Because of (7), we can write In (6), Sðλ ; τÞ is observed and using (3) So (7) becomes where y 0 = yðλ = 0, τ = 0Þ. From (6), we have where here χ O and χ ω denote the characteristic functions of O and ω, respectively. The derivative N λ = ð∂N/∂λð0, 0ÞÞ only depends onÎ and other known data. Consequently, the estimate (76) contains the information on λÎ (see for details Remark 10 below).  Abstract and Applied Analysis Remark 10. The knowledge of the optimal control w provides informations about the pollution term λÎ, and let N λ = ∂N/∂λð0, 0Þ be the solution of Multiplying (78) by q and integrating by parts over Q, we get So from (16) and (77), we deduce ð

Numerical Simulation of the Studies Model Solution
In this part, the idea is to highlight the numerical simulation of the solution of model (1), for λ = τ = 0. The purpose is to present the method of lines (MOL) solution of the COVID-19 modeling equations, a system of five partial differential equations (PDEs), describing the interaction resulting between susceptible persons at risk of contacting COVID-19, carriers (dead corpse), infective persons, recovered  Abstract and Applied Analysis persons, and number of deaths. The method of lines is a semidiscrete method which involves reducing an initial boundary value problem to a system of ordinary differential equations (ODEs) in time through the use of a discretization in space. The resulting ODE system is solved by applying solver ode15s of MATLAB. This method provides very accurate numerical solution for linear or nonlinear PDE's in comparison with other existing methods. For initial conditions, let us look at the case of Ouagadougou city, Burkina Faso, where population is estimated to 3 million. Notice that Burkina Faso population is around 20 million. Towards the beginning of March 2020, the number of people infected was around 3. In order to make our model fit that data, the initial conditions (day zero) are Sð0Þ = 3000000, Cð0Þ = 2, Ið0Þ = 3; Rð0Þ = 0; Dð0Þ = 0. One can note that parameter values are not for Burkina Faso excepted rate of natural death parameter. One must note that for all the following figures that the legend is data 1 for t = 0, data2 for t = 10, data 3 for t = 20, and data 4 for t = 30 Using the values of the parameters in Table 1, we obtain the following graphs. Figure 1 shows that susceptible persons' number decreases progressively with a peak at t = 10 [day]. Figure 2 shows that the peak of the pandemic is reached on the tenth day, before beginning to decrease.
According to the numerical simulation of Figure 3, the number of infected persons evolves in the same manner as those of susceptible persons and carriers. Figure 4 shows that number of people cured increases.

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Abstract and Applied Analysis Figure 5 show that The number of deaths linked to COVID6-19 is decreasing.
The results are due to the fact that one does not have accurate data on COVID-19 death rate.
Remark 11. According to observations made after the numerical simulation of the COVID-19 model, it is very important to have real data that will allow us to do a good simulation with a good interpretation of the results in order to make them useful for decision makers.

Conclusion
In this paper, using a mathematical model of COVID-19 transmission, we have determined the number of people infected with COVID-19 by the sentinel method. Indeed, we first studied the existence and the construction of a sentinel for the system (3). We have shown that the search for a sentinel is indeed equivalent to the study of controllability. The null controllability is ensured by an observability inequality obtain through Carleman's estimates. Then, we used the constructed sentinel to obtain information on the number of people infected with COVID-19. Finally, we made a numerical simulation of the studied model solution.
In perspective, in the concrete applications of singular nonlinear problems in epidemiology, the problem of controllability takes on its full meaning for epidemiological models. For example, consider the following system: as well as the functional and we will then seek to determine the control v in such a way that it brings the number of infected to zero after a time T.

Data Availability
No underlying data was collected or produced in this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.