CONTROL OF A REACTION-DIFFUSION SYSTEM: APPLICATION ON A SEIR EPIDEMIC MODE

1Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University Casablanca, BP 7955, Sidi Othman, Casablanca, Morocco 2Laboratory of Applied Sciences and Didactics, Higher Normal School Tetouan , Abdelmalek Essaadi University, Tetouan, Morocco 3Laboratory of Systems Engineering, High School of Technology Fkih Ben Saleh, Sultan Moulay Slimane University, Beni Mellal, Morocco


INTRODUCTION
In epidemiology, mathematical modeling has become an important tool for analyzing the causes, dynamics and spread of epidemics. Indeed, mathematical models provide a better understanding of the mechanisms underlying the spread of emerging infectious diseases, and allow authorities to make decisions about effective control strategies. One of the most basic procedures in disease modeling is to use a model, in which the population is divided into different groups depending on the stage of infection, with assumptions about the nature and rate of transfer time from compartment to another.Several diseases that confer immunity against reinfection have been modeled using SIR,SIS, SEIR ... etc (see [1][2][3][4][5][6][7][8][9][10]). With S is the class of the susceptible population, which represents individuals not yet infected with the disease and who are susceptible to contracting the disease, I is the class of infected persons who represents the individuals infected with the disease, and who can transmit the disease to sensitive people. E is the class of people exposed and R is the class of recovered that represents people immunized against the disease. In the literature, several studies have been carried out on the SEIR models, for example, Greenhalgh [18] has examined the SEIR models integrating density dependence into the mortality rate. Cooke and van den Driessche [19] presented and studied the SEIRS models with two delays. Li and Muldowney [20] and Li et al. [21] investigated the overall dynamics of SEIR models with non-linear incidence. Li ,Smith H L,and al. [22] analyzed the overall dynamics of a SEIR model with vertical transmission and bilinear incidence. Zhang.al [25] studied the overall dynamics of a SEIR model with immigration from different compartments.
The impact of the mobility of individuals plays an important role in the transmission of the disease, which makes it necessary to introduce the spatial factor in the SEIR models, in order to give a more realistic study [11][12][13][14][15][16][17] [30]. Based on the SEIR model presented by Zhang.al, in which we introduce the spatial factor, and adopting a strategy in the form of a control problem, we formulate a spatial-temporal SEIR epidemiological model, as that system of parabolic partial differential equations coupled with no-flux boundary conditions. The main objective of this work is to set up an optimal control approach based on a combination of minimization of the number of latent and infected individuals and the therapeutic treatment, for the Reaction-Diffusion SEIR model. To achieve this goal, we characterize an optimal control pair in the form of an effort that reduces contact between infectious and susceptible individuals, and a treatment envisioned to combat the spread of the disease. We prove the existence of state system solutions and the existence of optimal control. Optimal control theory provides the characterization of the optimal control pair in terms of state and adjoint functions. The optimality system is solved numerically, using a forward-backward sweep method (FBSM) [23]. The numerical simulations of our control strategy show the effectiveness of the approach we have adopted. The document is organized as follows: In Section 2, we present the mathematical model and the associated optimal control problem. We prove the existence of a strong global solution for our system in section 3. In section 4, we show the existence of the optimal solution. The necessary optimality conditions are defined in section 5. As an application, the numerical results associated with our problem control are given in section 6. Finally, we conclude the document in section 7.

THE BASIC MATHEMATICAL MODEL AND OPTIMAL CONTROL PROBLEM
The model (1) used by zhang consists of four compartments SEIR: susceptible, exposed, infected and recovered. We note their densities at time t and at position x by.
S(t, x), I(t, x), E(t, x), R(t, x), respectively: The probability for an individual to take part in a contact.  (1), in order to take into account the effect of the spatial factor, we introduce the In addition, we include two controls v 1 and v 2 which represent respectively the effort that reduces the contact between infectious and susceptible individuals and the rate of treatment of infectious individuals. We assume that v 2 (t, x)I individuals are removed from the infected class and added to the recovred class. The mathematical system with controls is given by the nonlinear differential equations We denote by Ω a fixed and bounded domain in IR 2 with smooth boundary ∂ Ω and η is the outward unit normal vector on the boundary. The initial conditions and no-flux boundary conditions are given by Eligible controls are contained in the ensemble We seek to minimize the functional objective for some positive constant v max .
For the parameter ρ 1 2 , ρ 2 2 , our objective is to find control functions such

THE EXISTENCE OF THE OPTIMAL SOLUTION
In this section, we will prove the existence of an optimal control for the problem (6) subject to reaction diffusion system (2)-(4) and (v 1 , v 2 ) ∈ U ad . The main result of this section is the following theorem.
Proof. From Theorem 1, we know that, for every (v 1 , v 2 ) ∈ U ad , there exists a unique solution y to system (2)(3)(4) . Assume that where y n 1 , y n 2 , y n 3 , y n 4 is the solution of system (2)(3)(4) corresponding to the control v n 1 , v n 2 for n = 1, 2, .... That is and By theorem (1) using the estimate (9) of the solution y n i , there exists a constant C > 0 such that for all n ≥ 1,t ∈ [0, T ] The Ascoli-Arzela Theorem(See [24]) implies that y n 1 is compact in C [0, T ] : L 2 (Ω) . Hence, selecting further sequences, if necessary, we have y n 1 −→ y * 1 in L 2 (Ω), uniformly with respect to t and analogously, we have for y n i −→ y * i in L 2 (Ω) for i = 2 , 3 4, uniformly with respect to t.
Proposition 3. The mapping y : U ad → W 1,2 ([0, T ] ; H (Ω )) with y i ∈ L (T, Ω ) for i = 1, 2, 3, 4 is Gateaux differentiable with respect to v * . For all direction v ∈ U ad , y (v * ) v = Y is the unique solution in W 1,2 ([0, T ] ; H (Ω )) with Y i ∈ L (T, Ω ) of the following equation for i = 1, 2, 3, 4.F (y 1 , y 2 , y 3 , y 4 ) = y 1 y 3 y 1 + y 2 + y 3 + y 4 , We denote S ε the system (2 ) corresponding to v ε and S * the system (2) corresponding to v * , subtracting system S ε from S * , we have with the homogeneous Neumann boundary conditions We prove that Y ε i are bounded in L 2 (Q) uniformly with respect to ε . For this end, denoting by Then (24) given by ,t ≥ 0)be the semi-group generated by A, then the solution of (27) can be expressed as On the other hand the coefficients of the matrix H ε are bounded uniformly with respect to ε, using Gronwall's inequality, we have Hence y ε i → y * i in L 2 (Q), i = 1, 2, 3.
Denoting by H = . Hence, then system (24)(25)(26) can be written in the form and its solution can be expressed as By (28) and (32) we deduce that Thus all the coefficients of the matrix H ε tend to the corresponding coefficients of the matrix H in L 2 (Q), An application of Gronwall's Inequality yields to Y ε i → Y i in L 2 (Q) as ε → 0, for i = 1, 2, 3, 4.
Let v * be an optimal control of (2-5), y * = y * 1 , y * 2 , y * 3 , y * 4 be the optimal state, Z is the matrix , K = (0, K 1 , K 2 , 0), Z * is the adjoint matrix associated to Z, H * is the adjoint matrix associated to H and p = (p 1 , p 2 , p 3, , p 4 ) is the adjoint variable,we can write the dual system associated to system (2-5): To obtain the necessary conditions for the optimal control problem, applying standard optimality techniques, analyzing the objective functional and utilizing relationships between the state and adjoint equations,we obtain a characterization of the control optimal.
We use (23) and (34), we have Since J is Gateaux differentiable at v * and U ad is convex, as the minimum of the objective functional is attained at v * it is seen that J (v * ) (u − v * ) ≥ 0 for all u ∈ U ad .
We take h = u−v * and we use (37)-(38) then u ∈ U ad . By standard arguments varying u, we obtain ρv * = −L * p Then

NUMERICAL RESULTS
In this section, we present the results obtained by the numerical resolution of the optimality system ((2-4),(34)(35)), based on a discrete iterative scheme that converges following a test appropriate to the forward-backward scanning method (FBSM) [23]. We adopt two situations for the resolution: the first is that the disease starts with the middle of domain Ω (1) , and in the second situation the disease begins with the lower corner Ω (2). A rectangular area of 30 km × 40 km is considered, and the parameter values and the initial values are given in Table 1   -In the first scenario, we illustrate the dynamics of the system without intervention.
-In the second scenario, from the 48 day of onset of the disease, we simultaneously apply the strategy that reduces contact between infectious and susceptible individuals, and that of treatment.
-In the third scenario, we repeat the second scenario, but from the first days of the onset of the epidemic.   infected to 2 infected (Fig 6), and the number of recovered increased by a density of 1 recovered to 5 recovered (Fig 7). But despite these results, the disease is not completely gone, as there are still infected individuals, which can be a major source of the spread of the epidemic.

CONCLUSION
In this article, we present a theoretical work that can be used in the study of several infectious diseases in the form of SEIR model, we have also expanded this model to take into account the spatial spread of the disease, we studied a couple Optimal control for this SEIR spatio-temporal model, the first control has the role of reducing contact between susceptible and infected, and the second is in the form of a therapeutic treatment. Theoretically, we demonstrate the existence of optimal controls and the solution of the state system. The characterization of the optimal control torque is determined in terms of state functions and adjoint functions. The numerical resolution is based on the forward / backward scanning method (FBSM). The numerical results have shown that the introduction of control from the first day of the disease, which reduces the contact between the susceptible and the infected, and that of the treatment plus the corresponding cost, constitutes the best optimal strategy for obtaining better results. This strategy has made it possible to block the spread of the epidemic.

APPENDIX
First recall a general existence result which we use in the sequel (Proposition 1.2, p. 175, [28]; see also [27,29]. Consider the initial value problem where A is a linear operator defined on a Banach space X, with the domain D(A) and g : [0, T ] × X → X is a given function. If X is a Hilbert space endowed with the scalar product (·, ·), then the linear operator A is called dissipative if (Az, z) ≤ 0, (∀z ∈ D(A)).