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

Abstract

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. The 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.

1. Introduction

In the literature, there are numerous books and articles [1–7] that deal with epidemic mathematical models. It is well established that human mobility plays an important role in the spread of an epidemic [8–13]. Mathematical modelling of the spread of infectious diseases has an important influence on disease management and control [14–16]. 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. The 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. The 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. The 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. The existence of the optimal solution is proved in Section 4. The 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.

2. The Basic Mathematical Model

2.1. The Model

In this paper, we consider the following SEIR epidemic model (susceptible (S), exposed (E), infected (I), and recovered (R)):where 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 γ⁻¹ are, respectively, the average duration of latent and infective periods. The 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 IR² 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

2.2. The Optimal Vaccination

Eligible controls are contained in the ensemblewhere represents the vaccination rate at time and position x. We seek to minimize the functional objectivefor some positive constant .

K₁, K₂, and K₃ are constant weights. The cost of vaccination is a nonlinear function of , and we choose a quadratic function indicating the additional costs associated with high vaccination rates.

The parameter , with the units , 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 ; we denote by the space of all absolutely continuous functions having the property that .(ii).

3. Existence of Solution

We study in this section the existence of a global strong solution, positivity, and boundedness of solutions of problem for (1)–(3). Let the solution of system (1)–(3) with . A denotes the linear operator defined as follows:where the domain of A is defined by

Theorem 1. Let Ω be a bounded domain from ; with the boundary smooth enough, on Ω( with i = 1, 2, 3, 4), the problem (1)–(3) has a unique (global) strong solution such that with i = 1, 2, 3, 4. In addition y₁, y₂, y₃, and y₄ are nonnegative. Furthermore, there exists C > 0 (independent of ) for all :

Proof. To prove the existence of a (global) strong solution for system (1)–(3), now we write system (1)–(3) as shown in (7) (see Appendix). LetSystem (10) represents the nonlinear term of (1) and we consider the function ; then, we can be rewrite system (1)–(3) in the space H(Ω) as follows:It is clear that function is Lipschitz continuous in uniformly with respect to .
As the operator A defined in (7) and (8) is dissipating, self-adjoint and generates a C₀-semi-group of contractions on [19]. Therefore, Theorem A.1 (see Appendix) assures problem (1)–(3) which admits a unique strong solution withIn order to show that for i = 1, 2, 3, 4, we denote and is the C₀-semigroup generated by the operator , where By₁ = d₁Δy₁ and . It is clear that the function satisfies the system:Note that this system has a solution given byAs and , we have . Similarly, the function satisfies .
Then,and analogously, we haveThus, we have proved thatBy the first equation of (1), we obtainUsing the regularity of y₁ and Green’s formula, we can writeThen,Since for i = 1, 2, 3, 4 are bounded independently of and , we deduce thatWe make use of (12), (13), and (21) in order to getand conclude that the inequality in (9) holds for i = 1 similarly for y₂, y₃ and y₄.
In order to show the positiveness of y_i for i = 1, 2, 3, 4, we write system (1) in the form:It is easy to see that the functions , and are continuously differentiable satisfying , , for all y₁, y₂, y₃, y₄ ≥ 0 (see [20]). This completes the proof.

4. 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 . The main result of this section is the following theorem.

Theorem 2. Under the hypotheses of Theorem 1, the optimal control problem (1)–(5) admits an optimal solution .

Proof. From Theorem 1, we know that, for every , there exists a unique solution y to system (1)–(3). Assume thatLet be a minimizing sequence such thatwhere is the solution of system (1)–(3) corresponding to the control for n = 1, 2, …. That is,By Theorem 1, using the estimate (9) of the solution , there exists a constant C > 0 such that for all , is compactly embedded in , so we deduce that is compact in .
Let us show that is equicontinuous in . As is bounded in , this implies that for all ,The Ascoli–Arzela theorem (see [21]) implies that is compact in . Hence, selecting further sequences, if necessary, we have in , uniformly with respect to t and analogously, we have for in for i = 2, 3 4, uniformly in relation to t.
From the boundedness of in , which implies it is weakly convergent in on a subsequence denoted again by , for all distribution φ,which implies that weakly in . In addition, the estimates (29) lead to weakly in , weakly in , weakly in , i = 1, 2, 3, 4.
We put ; we now show that and strongly in , and we writeand we make use of the convergences strongly in , i = 1, 3, and of the boundedness of , in , and then and strongly in .
Since is bounded, we can assume that weakly in on a subsequence denoted again by . Since U_ad is a closed and convex set in , it is weakly closed, so .
We now show thatWriting with i = 1, 2, 3, 4,and making use of the convergences strongly in and weakly in , for i = 1, 2, 3, 4, one obtains that weakly in .
By taking n ⟶ ∞ in (26)–(28), we obtain that y^∗ is a solution of (1)–(3) corresponding to . Therefore,This shows that J attains its minimum at , and we deduce that verifies problem (1)–(3) and minimizes the objective functional (5). The proof is complete.

5. Necessary Optimality Conditions

Let and  =  +  ∈ U_ad; in this section, we show the optimality conditions to problem (1)–(3), and we find the characterization of optimal control. First, we need the Gateaux differentiability of the mapping . For this reason, denoting by and the solution of (1)–(3) corresponding to and respectively.

Proposition 1. The mapping with for i = 1, 2, 3, 4 is Gateaux differentiable with respect to . For all direction , is the unique solution in with of the following equation:

Proof. Put for , , and .
We denote S^ɛ system (1) corresponding to and S^∗ system (1) corresponding to , subtracting system S^ɛ from S^∗; so, we have:with the homogeneous Neumann boundary conditions:We prove that are bounded in uniformly with respect to ɛ. For this end, denoting by ,Then, (38) is rewritten asLet be the semigroup generated by A; then, the solution of (42) can be expressed asOn the other hand, the coefficients of the matrix H^ɛ are bounded uniformly with respect to ɛ; using Gronwall’s inequality, we havewhere . Then,Hence, in , i = 1, 2, 3, 4. Denotewhere , , and .
Hence, system (38)–(40) can be written in the formand its solution can be expressed asBy (43) and (48), we deduce thatThus, all the coefficients of the matrix H^ɛ tend to the corresponding coefficients of the matrix H in . An application of Gronwall’s inequality yields to in as ɛ ⟶ 0, for i = 1, 2, 3, 4.
Let be an optimal control of (1)–(4), be the optimal state, Z be the matrix defined by , , Z^∗ be the adjoint matrix associated to Z, H^∗ be the adjoint matrix associated to H, and be the adjoint variable; we can write the dual system associated to system (1)–(4):

Lemma 1. Under hypotheses of Theorem 1, if is an optimal pair, then there exists a unique strong solution to system (50) with for i = 1, 2, 3, 4.

Proof. Similar to Theorem 1, by making the change of variable s = T − t and the change of functions , 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, be an optimal control of (1)–(4) and let with 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 . Then,

Proof. We suppose is an optimal control and are the corresponding state variables. Consider  =  + ɛh ∈ U_ad and corresponding state solution ; we haveWe use (37) and (50), and we haveSince J is Gateaux differentiable at and U_ad is convex, as the minimum of the objective functional is attained at , it is seen that for all u ∈ U_ad.
We take h = u − and we use (52) and (53); then, . We conclude that equivalent to for all u ∈ U_ad. By standard arguments varying u, we obtainThen,As   U_ad, we have

6. 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. The upper limits of the optimality condition are considered to be [23]. The constant weighting values in the objective function are K₁ = 1, K₂ = 1, K₃ = −1, α = 2.

Table 1 

Initial conditions and parameter values.

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.

Figure 1 

Susceptible behavior within Ω without control.

Figure 2 

Exposed behavior within Ω without control.

Figure 3 

Infected behavior within Ω without control.

Figure 4 

Recovered behavior within Ω without control.

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 I₀(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.

The 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 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).

Figure 5 

Susceptible behavior within Ω without control (vaccine strategy after 30 days).

Figure 6 

Infected behavior within Ω without control (vaccine strategy after 30 days).

Figure 7 

Recovered behavior within Ω with control (vaccine strategy after 30 days).

Figure 8 

Susceptible behavior within Ω with control (vaccine strategy starts from the first day).

Figure 9 

Infected behavior within Ω without control (vaccine strategy starts from the first day).

Figure 10 

Recovered behavior within Ω with control (vaccine strategy starts from the first day).

The 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.

7. 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. The characterization of the control was made in terms of state functions and adjoint functions. A numerical simulation was given to validate our control strategy.

Appendix

First, recall a general existence result which we use in the sequel (Proposition 1.2, p. 175, [24]; see also [19, 25]). Consider the initial value problemwhere A is a linear operator defined on a Banach space X, with the domain D(A) and  : [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 .

Theorem A.1. Let X be a real Banach space, A : D(A) ⊆ X ⟶ X be the infinitesimal generator of a C₀−semigroup of linear contractions S(t), t ≥ 0 on X, and  : [0, T] × X ⟶ X be a function measurable in t and Lipschitz continuous in x ∈ X, uniformly with respect to t ∈ [0, T].(i)If z₀ ∈ X, then problem (A.1) admits a unique mild solution, i.e., a function z ∈ C([0, T]; X) which verifies the equality .(ii)If X is a Hilbert space, A is self-adjoint and dissipative on X, and z₀ ∈ D(A), then the mild solution is in fact a strong solution and

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.