An Optimal Control for a Two-Dimensional Spatiotemporal SEIR Epidemic Model

In this paper, we present an application of optimal control theory on a two-dimensional spatial-temporal SEIR (susceptible, exposed, infected, and restored) epidemic model, in the form of a partial differential equation. Our goal is to minimize the number of susceptible and infected individuals and to maximize recovered individuals by reducing the cost of vaccination. In addition, the existence of the optimal control and solution of the state system is proven. +e characterization of the control is given in terms of state function and adjoint. Numerical results are provided to illustrate the effectiveness of our adopted approach.


Introduction
In the literature, there are numerous books and articles [1][2][3][4][5][6][7] that deal with epidemic mathematical models. It is well established that human mobility plays an important role in the spread of an epidemic [8][9][10][11][12][13]. Mathematical modelling of the spread of infectious diseases has an important influence on disease management and control [14][15][16]. In general, after the initial infection, a host remains in a latency period before becoming infectious, so the population can be divided into four categories: susceptible (S), exposed (E), infected (I), and recovered (R). In this contribution, we treat a model of epidemic type SEIR in which the model takes into account the total population size as a refrain for the transmission of the disease, and it is assumed that it is constant over time.
e approach used is based on the work of El Alami Laaroussi et al. [17,18], which was applied on a SIR model. So, our goal is to characterize optimal control in the form of a vaccination program, maximizing the number of people reestablished and minimizing the number of susceptible, infected people and the cost associated with vaccination over an infinite space and in a time domain. e theory of semigroups and optimal control makes it possible to show the existence of state system solutions and optimal control and to obtain the optimal characterization of this control in terms of state functions and adjoint functions. To illustrate the solutions, based on the numerical results, we find that the use of the vaccine control strategy in the spatial region helps to fight the spread of the epidemic in this region over a period of 60 days. e structure of this article is as follows. Section 2 is devoted to the basic mathematical model and the associated optimal control problem. In Section 3, we prove the existence of a strong global solution for our system. e existence of the optimal solution is proved in Section 4. e necessary optimality conditions are defined in Section 5. As an application, the numerical results associated with our control problem are given in Section 6. Finally, we conclude the paper in Section 7.
where β(SI/N) is the total number of infection per unit of time, N is the total population (N(t) � S(t) + E(t) + I(t) + R(t) � N(0) � N), μ is the rate of deaths from causes unrelated to the infection, incidence rate, ω is the rate of losing immunity, β is the transmission constant, and σ − 1 and c − 1 are, respectively, the average duration of latent and infective periods. e positive constants d S , d E , d I , and d R denote the corresponding diffusion rate for susceptible, exposed, infectious, and recovered individuals. We denote by Ω a fixed and bounded domain in IR 2 with smooth boundary zΩ and η is the outward unit normal vector on the boundary. e initial conditions and no-flux boundary conditions are given by

e Optimal Vaccination.
Eligible controls are contained in the ensemble where v(x, t) represents the vaccination rate at time and position x. We seek to minimize the functional objective for some positive constant v max . K 1 , K 2 , and K 3 are constant weights. e cost of vaccination is a nonlinear function of v, and we choose a quadratic function indicating the additional costs associated with high vaccination rates. e parameter (α/2), with the units (Population/km 2 )/ vaccin 2 , balances the cost squared of the vaccine with the cost associated with the infected population. Our objective is to find control functions such that (i) We put H(Ω) � (L 2 (Ω)) 4 ; we denote by W 1,2 ([0, T], H(Ω)) the space of all absolutely continuous functions y :

The Existence of the Optimal Solution
In this section, we will prove the existence of an optimal control for problem (5) subject to reaction diffusion system (1)-(3) and (v) ∈ U ad . e main result of this section is the following theorem.
Proof. From eorem 1, we know that, for every v ∈ U ad , there exists a unique solution y to system (1)- (3). Assume that where (y n 1 , y n 2 , y n 3 , y n 4 ) is the solution of system (1)-(3) corresponding to the control (v n ) for n � 1, 2, . . .. at is, By eorem 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], e Ascoli-Arzela theorem (see [21]) 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 in relation to t. From the boundedness of Δy n i in L 2 (Q), which implies it is weakly convergent in L 2 (Q) on a subsequence denoted again by Δy n i , for all distribution φ, which implies that Δy n i ⟶ Δy * i weakly in L 2 (Q), i � 1, 2, 3, 4. In addition, the estimates (29) lead to International Journal of Differential Equations We put N(y) � (β/(y 1 + y 2 + y 3 + y 4 )); we now show that y n 1 y n 3 ⟼ y * 1 y * 3 and N(y n )y n 1 y n 3 ⟼ N(y * )y * 1 y * 3 strongly in L 2 (Q), and we write and we make use of the convergences y n i ⟶ y * i strongly in L 2 (Q), i � 1, 3, and of the boundedness of y n 1 , y n 3 in L ∞ (Q), and then y n 1 y n 3 ⟶ y * 1 y * 3 and N(y n )y n 1 y n 3 ⟼ N(y * )y * 1 y * 3 strongly in L 2 (Q).
Since v n is bounded, we can assume that v n ⟶ v * weakly in L 2 (Q) on a subsequence denoted again by v n . Since U ad is a closed and convex set in L 2 (Q), it is weakly closed, so v * ∈ U ad .
We now show that v n y n 1 + y n 2 + y n (33) and making use of the convergences y n i ⟶ y i * strongly in is shows that J attains its minimum at (y * , v * ), and we deduce that (y * , v * ) verifies problem (1)-(3) and minimizes the objective functional (5). e proof is complete.

Hence, system (38)-(40) can be written in the form
and its solution can be expressed as

S(t − s)H(s)Y(s)ds
By (43) and (48), we deduce that us, 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 Let v * be an optimal control of (1)-(4), y * � (y * 1 , y * 2 , y * 3 , y * 4 ) be the optimal state, Z be the matrix defined by Z � the adjoint matrix associated to Z, H * be the adjoint matrix associated to H, and p � (p 1 , p 2 , p 3, , p 4 ) be the adjoint variable; we can write the dual system associated to system (1)- (4): Proof. Similar to eorem 1, by making the change of variable s � T − t and the change of functions q i (s, x) � p i (T − s, x) � p i (t, x), (t, x) ∈ Q, i � 1, 2, 3, 4, we can easily prove the existence of the solution to this lemma.
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. □ Theorem 3. Let α > 0, v * be an optimal control of (1)- (4) and let y * and p ∈ W 1,2 ([0, T]; H(Ω)) with y i * and p i ∈ L(T, Ω) for i � 1, 2, 3, 4.p is the adjoint variable, and y * is the optimal state. y * is the solution to (1)-(4) with the control v * . en, Proof. We suppose v * is an optimal control and y * � (y * 1 , y * 2 , y * 3 , y * 4 ) � (y 1 , y 2 , y 3 , y 4 )(v * ) are the corresponding state variables. Consider v ε � v * + εh ∈ U ad and corresponding state solution y ε � (y ε 1 , y ε 2 , y ε 3 , y ε 4 ) � (y 1 , y 2 , y 3 , y 4 )(v ε ); we have International Journal of Differential Equations We use (37) and (50), and we have (53) 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 We take h = u − v * and we use (52) and (53); then, By standard arguments varying u, we obtain αv * � − L * p. (54) As (56) □

Numerical Results
In this section, we give the results obtained by the numerical resolution of the optimality system ((1)-(3), (50), (51)) using forward-backward sweep method (FBSM) [22]. 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. e upper limits of the optimality condition are considered to be v max � 1 [23]. e constant weighting values in the objective function are K 1 � 1, Figures 1-4 show the results without vaccination, and we can see clearly the spread of the disease throughout the domain, for both situations of the onset of the disease.
In Figure 3, in the absence of control and for the two situations at the beginning of the epidemic, it can be seen that the number of infected individuals increases by I 0 (x, y) = 0 for (x, y)/∉ Ω i , i = 1; 2, to about 9 people infected. To validate our vaccination strategy, we consider two ways to do it. e first is to inject vaccination after 30 days of the onset of the disease, while for the second, vaccination begins on the first day of the epidemic. In Figures 5 and 6, when injecting the 8 International Journal of Differential Equations vaccine after 30 days, it is easy to see the effectiveness of our control strategy in slowing the spread of the epidemic, since in Figure 6, after 60 days, the number of infected individuals has decreased to about 6 infected individuals, which is a gain by comparing it with the uncontrolled case. Another benefit of our control strategy is illustrated in Figure 7 for recovered individuals, as the number of individuals has increased to reach 4 recovered individuals. In the second case, when the vaccination against the disease starts from the first day (t � 1), the effectiveness of our vaccination strategy to control the spread of the epidemic is clear, since the disease disappears quickly (Figures 8-10). e comparison of these results with those obtained when vaccination is introduced at 30 days allows to conclude the influence of the vaccination from the first days for the elimination of the epidemic.   E 0 (x, y) 3 for (x, y) ∈ Ω i i � 1, 2 0 for (x, y) ∉ Ω i Initial exposed population (people/km 2 ) International Journal of Differential Equations 9 (1) (1) (2) (1)

Conclusion
In this work, we proposed an effective vaccination strategy for a two-dimensional spatiotemporal SEIR model, in order to minimize the number of susceptible and infected individuals and to maximize the number of individuals recovered. To achieve this goal, we have based our mathematical work on the use of semigroup theory and optimal control to show the existence of solutions for our state system, and these solutions are positive and related. In addition, we have proven the existence and characterization of optimal control that achieves both our goal and reduce the cost of vaccination. e characterization of the control was made in terms of state functions and adjoint functions. A numerical simulation was given to validate our control strategy.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

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