A Spatiotemporal SIR Epidemic Model Two-dimensional with Problem of Optimal Control

abstract: In the context of a more realistic model, in this work, we are interested in studying a spatiotemporal two-dimensional SIR epidemic model, in the form of a system of partial differential equations (PDE). A distribution of a vaccine in the form of a control variable is considered to force immunity. The purpose is to characterize a control that minimizes the number of susceptible, infected individuals and the costs associated with vaccination over a finite space and time domain. In addition, the existence of the solution of the state system and the optimal control is proved. The characterization of the control is given in terms of state function and adjoint function. The numerical resolution of the state system shows the effectiveness of our control strategy.


Introduction
Mathematical modeling in the field of epidemiology has become an important tool, since it gives an approximate idea of the causes, dynamics and spread of epidemic. In addition, it can provide useful control measures to make decisions about effective control strategies [1]. SIR is among the elementary models in the mathematical modeling of diseases, it consists of divided the population into different class, depending on the stage of infection. The susceptible class (S) includes individuals who may contract the disease but are not yet infectious. The infectious class (I) includes those who have the disease and can transmit it. The recovered compartment (R) includes persons who have recovered from the disease with permanent immunity. In the literature, there are a great deal of mathematical studies of diseases that give an interesting insight into the use of mathematical models in epidemiology. For example, Baily et al. [2], Anderson et al. [3], Hethcote [4], Brauer and Castillo-Chavez [5], Keeling and Rohani [6] , Huppert and Katriel [7] and [8,9,10,11,12,13,25,26]. In this contribution, we consider an epidemic SIR model, spatiotemporal in two dimensions in the work of Lotfi. et al [14] and Hattaf. et al [15], in this system we introduce a vaccine in the form of a control variable,in order to minimise susceptible, infected individuals and the costs associated with vaccination. The existence of the state system solution and the optimal control is proved, and the characterization of the optimal control in terms of state function and adjoint is given. For the validation of our strategy, we present the numerical results obtained.

The model without controls
In this paper, we consider the following SIR epidemic model: With f (S, I, R) = β 1+α1S+α2I+α3SI is the incidence rate, such as α 1 , α 2 , α 3 ≥ 0 are constants . Λ is the recruitment rate of the population, µ is the natural death rate of the population,d is the death rate due to disease, r is the recovery rate of the infective individuals, β is the infection coefficient. The positive constants d S , d I ,and d R denote the corresponding diffusion rate for susceptible, infectious, and recovered individuals. 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 In this step, the numerical results obtained by using the finite difference method of the system (1) without control are given. We have adopted two situations: In the first, the disease starts from the middle (1) and in the second, the disease starts at the corner (2). Figures 1, 2, and 3 present numerical results for susceptible, infected, and recovered individuals.Results show that in both situations, susceptible individuals become infected after an incubation period, and after a period of time, the disease spreads throughout the population.In order to fight against the spread of the disease we adopted a strategy based on the introduction of a vaccine in the form of a control variable.

The model with controls
The controlled model is the following: With f (S, I, R) = β 1+α1S+α2I+α3SI and initial conditions and no-flux boundary conditions are given by v(x, t) represents the vaccination rate at time and position x. We seek to minimize the functional objective Eligible controls are contained in the ensemble for some positive constant v max . Where ρ 1 , ρ 2 are constant weights. The cost of vaccination is a nonlinear function of v, we choose a quadratic function indicating the additional costs associated with high vaccination rates.
The parameter α 2 , with the units P opulation/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
Step 1: This step studies the local existence of positive solutions to system (2.1)-(2.6) in view of Theorem (8.1) (see appendix). We use a truncation procedure for g. For a fixed positive integer k > 0, let us define the function sets D 1 = {z| z > k}, D 2 = {z| |z| < k} ,D 3 = {z| z < −k}and consider the following auxiliary problem: where g k t, y k = g k 1 t, y k , g k 2 t, y k , g k 3 t, y k . Here, for each index i, g k i t, y k are defined as follows: where [y i ] Dsi means that y i ∈ Ds i , and As the operator A defined in (3.1)-(3.2) is dissipating, self-adjoint and generates a C 0 -semi-group of contractions on H (Ω) [23], it is clear that function g k t, y k becomes Lipschitz continuous in y k uniformly with respect to t ∈ [0, T ]. Therefore, theorem (8.1) (see appendix) assures problem (2.1-2.6) admits a unique strong solution y k ∈ W 1,2 ([0, T ] , H (Ω)) with In order to show that and where Note that this system has a solution given by and analogously, we have Thus we have proved that By the first equation of (2.1), we obtain Using the regularity of y k 1 and the Green's formula, we can write Since y k i L ∞ (Q) for i = 1, 2, 3 are bounded independently of v and y 0 1 ∈ H 2 (Ω) , we deduce that We make use of (3.6), (3.7), and (3.10), in order to get y k 1 ∈ L (T, Ω) ∩ L ∞ (Q) and conclude that the inequality in (3.3) holds for i = 1, similarly for y k 2 and y k 3 . In order to show the positiveness of y k i for i = 1, 2, 3, we start by demonstrating that y k 2 is positivive to be, we set y k we multiply the second equation of (2.4) and we integrate on Ω we obtain: We put b = f y k y k 1 and Gronwall's inequality leads to then y k− 2 = 0,on deduces that y k 2 (t, x) ≥ 0, To demonstrate the positivity of y k 1 and y k 3 , we write the 1st and 3rd equations of (2.1) in the form It is obvious to see that the functions F k 1 y k 1 , y k 2 , y k 3 and F k 3 y k 1 , y k 2 , y k 3 , are continuously differentiable satisfying Since initial data of system (3.11) are nonnegative, we deduce the positivity of y k 1 , y k 2 and y k 3 (see [24]). Now we particularize k > 0 large enough such that For example, we can take k > 2max y L ∞ (Ω) , i = 1, 2, 3 . Let θ ∈ (0, T ) be maximal with property × Ω and i = 1, 2, 3. So, g k (t, y 1 , y 2 , y 3 ) coincides with g(t, y 1 , y 2 , y 3 ) for(t, x) ∈ [0, θ] × Ω, and consequently y k = y k 1 , y k 2 , y k 3 is a local solution for (2.4)-(2.6) defined on [0, θ] × Ω.
Step  In this section, we will prove the existence of an optimal control for the problem (2.7) subject to reaction diffusion system (2.4)-(2.6) and (v) ∈ U ad . The main result of this section is the following theorem.
Proof. From Theorem 3.1, we know that, for every v ∈ U ad , there exists a unique solution y to system (2.4-2.6) . Assume that where (y n 1 , y n 2 , y n 3 ) is the solution of system (2.4-2.6) corresponding to the control (v n ) for n = 1, 2, .... That is The Ascoli-Arzela Theorem(See [22]) 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 , uniformly with respect to t.
From the boundedness of ∆y n i in L 2 (Q), which implies it is weakly convergent in L 2 (Q) on a subsequence denoted again △y n i then for all distribution ϕ Which implies that △y n i → △y * i weakly in L 2 (Q),i = 1, 2, 3, In addition, the estimates (4.4) leads to ∂y n i ∂t → ∂y * i ∂t weakly in L 2 (Q), i = 1, 2, 3 y n i → y * i weakly in L 2 0, T ; H 2 (Ω) , i = 1, 2, 3 y n i → y * i weakly star in L ∞ 0, T ; H 1 (Ω) , i = 1, 2, 3 We now show that y n 1 y n 2 → y * 1 y * 2 and f (y n ) y n 1 y n 2 → f (y * ) y * 1 y * 2 strongly in L 2 (Q), we write and we make use of the convergences y n i −→ y * i strongly in L 2 (Q), i = 1, 2, and of the boundedness of y n 1 , y n 2 in L ∞ (Q), then y n 1 y n 2 → y * 1 y * 2 and f (y n ) y n 1 → f (y * ) y * 1 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 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 → v * y * 1 weakly in L 2 (Q) Writing v n y n 1 − v * y * 1 = (y n 1 − y * 1 ) v n + (v n − v * ) y * 1 and making use of the convergences y n 1 −→ y * 1 strongly in L 2 (Q), and v n −→ v * weakly in L 2 (Q), one obtains thatv n y n 1 → v * y * 1 weakly in L 2 (Q). By taking n → ∞ i in (4.1-4.3),, we obtain that y * is a solution of (2.4-2.6) corresponding to (v * 1 ) ∈ U ad . Therefore This shows that J attains its minimum at (y * , v * ) , we deduce that (y * , v * ) verifies problem (2.4-2.6) and minimizes the objectif functional (2.7). The proof is complet ✷

Necessary optimality conditions
Let v ∈ U ad and v ε = v * + εv ∈ U ad , in this section, we show the optimality conditions to problem (2.4-2.6), and we find the characterization of optimal control. First , we need the Gateaux differentiability of the mapping v → y(v). For this reason, denoting by y ε = (y ε 1 , y ε 2 , y ε 3 ) = (y 1 , y 2 , y 3 ) (v ε ) and y * = (y * 1 , y * 2 , y * 3 ) = (y 1 , y 2 , y 3 ) (v * ) the solution of (2.4-2.6) corresponding to v ε and v * respec- Proposition 5.1. The mapping y : We denote S ε the system (2.4 ) corresponding to v ε and S * the system (2.4) 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 , t ≥ 0)be the semi-group generated by A, then the solution of (5.5) 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 where Γ > 0 (i = 1, 2, 3). Then . Hence, then system (5.2-5.4) can be written in the form and its solution can be expressed as By (5.6) and (5.10) 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. ✷ Let v * be an optimal control of (2.4-2.8), y * = (y * 1 , y * 2 , y * 3 ) be the optimal state, D is the matrix defined 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.

Numerical results
We present the results obtained, by numerical resolution using the forward-backward sweep method (FBSM) [21], of our optimality system, which is formulated by state equations with initial and boundary conditions (2.4-2.6), adjoint equations with transversality conditions(5.12), and optimal control characterization (5.13).our strategy is to apply two types of treatment respectively to susceptible and infected individuals, in order to fight the spread of the disease. We will keep the same situations described previously in section 2.1: the first, the disease starts from the middle of the domain Ω (1) and in the second, the disease begins in the lower left corner of Ω (2). In this work, we take the density of 45 in order to model a situation of high contacts. Concerning the choice of the domain Ω, we take a rectangular grid of size 30 km × 40 km: The parameter values and the initial values are given in table 1. These values are extracted from [14].Moreover, the upper limits of the optimality condition are considered to be v max = 1 [19] and the constant weighting values in the objective function are ρ 1 = 1, ρ 2 = 1, α = 2, taken from [20] .

Optimal control simulation
To validate our vaccination strategy, we will proceed in two different ways: 1-Start vaccination against the disease after 20 days.
2-Vaccination against the disease starts from the first day.
In the first case, when introducing vaccination after 20 days, it can be seen in Figures 4 and 5 that the number of susceptible and infected individuals decreases rapidly.on the other hand, in Figure 6, we can clearly see the increase in the number of individuals recovered.

Conclusion
In this contribution, we presented an SIR model in the form of a system of partial derivative equations with initial and boundary conditions.We have shown the existence of the solution of our state system and optimal control, so we have given a characterization of this control.Numerical simulation has proven the positive impact of our vaccination strategy.In fact Figures 1, 2 and 3 in the absence of the vaccine, have shown the spread of the disease in the entire domain, especially when the disease begins in the middle of the domain.However, when we introduced the vaccine, we observed that the number of infected individuals has decreased and the number of recovered has increased, which is very beneficial and reflects the importance of our control strategy.It should be noted that it is preferable to apply the vaccine during the first days of the onset of the disease, in order to block the spread in the population (see Figures 7 and 8 ).

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