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

Authors: Khalid Adnaoui; Adil El Alami Laaroussi
Journal: International Journal of Differential Equations
 (2020)

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## 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.


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## Body

## 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)):(1)∂S∂t=dSΔS−μS+ωR−βSIN+μN1−vx,t,∂E∂t=dEΔE+βSIN−μ+σE,∂I∂t=dIΔI−μ+γI+σE,∂R∂t=dRΔR−μ+ωR+γI+μNvx,t,t,x∈Q=0,T×Ω,where βSI/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 γ−1 are, respectively, the average duration of latent and infective periods. The positive constants dS, dE, dI, and dR 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)∂S∂η=∂E∂η=∂I∂η=∂R∂η=0,t,x∈Σ=0,T×∂Ω,(3)S0,x=S0≥0,E0,x=E0≥0,I0,x=I0≥0,R0,x=R0≥0.


### 2.2. The Optimal Vaccination

Eligible controls are contained in the ensemble(4)Uad=v∈L∝Q0≤v≤vmax≤1,where vx,t represents the vaccination rate at time and position x. We seek to minimize the functional objective(5)Jv=∫0T∫ΩK1St,x+K2It,x−K3Rt,xdxdt+α2vL2Q2,for some positive constant vmax.K
1, K2, and K3 are constant weights. The cost of vaccination is a nonlinear function of v, and we choose a quadratic function indicating the additional costs associated with high vaccination rates.The parameterα/2, with the units Population/km2/vaccin2, balances the cost squared of the vaccine with the cost associated with the infected population. Our objective is to find control functions such that(6)Jv∗=minJv,v∈Uad.(i)
We putHΩ=L2Ω4; we denote by W1,20,T,HΩ the space of all absolutely continuous functions y:0,T⟶HΩ having the property that ∂y/∂t∈L20,T,HΩ.(ii)
ℒT,Ω=L20,T,H2Ω∩L∞0,T,H1Ω.


## 2.1. The Model

In this paper, we consider the following SEIR epidemic model (susceptible (S), exposed (E), infected (I), and recovered (R)):(1)∂S∂t=dSΔS−μS+ωR−βSIN+μN1−vx,t,∂E∂t=dEΔE+βSIN−μ+σE,∂I∂t=dIΔI−μ+γI+σE,∂R∂t=dRΔR−μ+ωR+γI+μNvx,t,t,x∈Q=0,T×Ω,where βSI/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 γ−1 are, respectively, the average duration of latent and infective periods. The positive constants dS, dE, dI, and dR 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)∂S∂η=∂E∂η=∂I∂η=∂R∂η=0,t,x∈Σ=0,T×∂Ω,(3)S0,x=S0≥0,E0,x=E0≥0,I0,x=I0≥0,R0,x=R0≥0.


## 2.2. The Optimal Vaccination

Eligible controls are contained in the ensemble(4)Uad=v∈L∝Q0≤v≤vmax≤1,where vx,t represents the vaccination rate at time and position x. We seek to minimize the functional objective(5)Jv=∫0T∫ΩK1St,x+K2It,x−K3Rt,xdxdt+α2vL2Q2,for some positive constant vmax.K
1, K2, and K3 are constant weights. The cost of vaccination is a nonlinear function of v, and we choose a quadratic function indicating the additional costs associated with high vaccination rates.The parameterα/2, with the units Population/km2/vaccin2, balances the cost squared of the vaccine with the cost associated with the infected population. Our objective is to find control functions such that(6)Jv∗=minJv,v∈Uad.(i)
We putHΩ=L2Ω4; we denote by W1,20,T,HΩ the space of all absolutely continuous functions y:0,T⟶HΩ having the property that ∂y/∂t∈L20,T,HΩ.(ii)
ℒT,Ω=L20,T,H2Ω∩L∞0,T,H1Ω.


## 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 y=y1,y2,y3,y4=S,E,I,R the solution of system (1)–(3) with y0=y10,y20,y30,y40=S0,E0,I0,R0. A denotes the linear operator defined as follows:(7)A:DA⊂HΩ⟶HΩ,Ay=dSΔy1,dEΔy2,d1Δy3,dRΔy4∈DA,∀y=y1,y2,y3,y4∈DA,where the domain of A is defined by(8)DA=y∈H2Ω4,∂y1∂η=∂y2∂η=∂y3∂η=∂y4∂η=0,a.ex∈∂Ω.Theorem 1.
Let Ω be a bounded domain fromℝ2; with the boundary smooth enough, yi0≥0 on Ω( with i = 1, 2, 3, 4), the problem (1)–(3) has a unique (global) strong solution y∈W1,20,T:HΩsuch thatyi∈ℒT,Ω∩L∞Q with i = 1, 2, 3, 4. In addition y1, y2, y3, and y4 are nonnegative. Furthermore, there exists C > 0 (independent of v) for all t∈0,T:(9)∂yi∂tL2Q+yiL20,T;H2Ω+yiH1Ω+yiL∞Q≤C,fori=1,2,3,4.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). Let(10)g1yt=−μy1+ωy4−βy1y3N+μN1−vx,t,g2yt=βy1y3N−μ+σy2,t∈0,T,g3yt=−μ+γy3+σy2,g4yt=−μ+ωy4+γy3+μNvx,t.
System (10) represents the nonlinear term of (1) and we consider the function gyt=g1yt,g2yt,g3yt,g4yt; then, we can be rewrite system (1)–(3) in the space H(Ω) as follows:(11)∂y∂t=Ay+gyt,t∈0,T,y0=y0.
It is clear that functiong is Lipschitz continuous in y=y1,y2,y3,y4 uniformly with respect to t∈0,T.
As the operatorA defined in (7) and (8) is dissipating, self-adjoint and generates a C0-semi-group of contractions on HΩ [19]. Therefore, Theorem A.1 (see Appendix) assures problem (1)–(3) which admits a unique strong solution y∈W1,20,T,HΩ with(12)y1,y2,y3y4∈L20,T,H2Ω.
In order to show thatyi∈L∞Q for i = 1, 2, 3, 4, we denote M=maxg1L∞Q,y10L∞Ω and St,t≥0 is the C0-semigroup generated by the operator B:DB⊂L2Ω⟶L2Ω, where By1 = d1Δy1 and DB=y1∈H2Ω,∂y1/∂η=0,a.e∂Ω. It is clear that the function U1t,x=y1−Mt−y10L∞Ω satisfies the system:(13)∂U1∂tt,x=dS△U1+g1t,yt−M,t∈0,T,U10,x=y10−y10L∞Ω.
Note that this system has a solution given by(14)U1t=Sty10−y10L∞Ω+∫0tSt−sg1s,ys−Mds.
Asy10−y10L∞Ω≤0 and g1s,ys−M≤0, we have U1t,x≤0,∀t,x∈Q. Similarly, the function U2t,x=y1+Mt+y10L∞Ω satisfies U2t,x≥0,∀t,x∈Q.
Then,(15)y1t,x≤Mt+y10L∞Ω,∀t,x∈Q,and analogously, we have(16)yit,x≤Mt+yi0L∞Ω,∀t,x∈Q,fori=2,3,4.
Thus, we have proved that(17)yi∈L∞Q∀t,x∈Q,fori=1,2,3,4.
By the first equation of (1), we obtain(18)∫0t∫Ω∂y1∂s2dsdx+dS2∫0t∫ΩΔy12dsdx−2dS∫0t∫Ω∂y1∂sΔy1dsdx=∫0t∫Ω−μy1+ωy4−βy1y3N+μN1−vx,t2dsdx.
Using the regularity ofy1 and Green’s formula, we can write(19)2∫0t∫Ω∂y1∂sΔy1dx=−∫0t∂∂s∫Ω∇y1k2dxds=−∫Ω∇y12dx+∫Ω∇y102dx.
Then,(20)∫0t∫Ω∂y1∂s2dsdx+dS2∫0t∫ΩΔy12dsdx+dS∫Ω∇y12dx−dS∫Ω∇y102dx=∫0t∫Ω−μy1+ωy4−βy1y3N+μN1−vx,t2dsdx.
SinceyiL∞Q for i = 1, 2, 3, 4 are bounded independently of v and y10∈H2Ω , we deduce that(21)y1∈L∞0,T,H1Ω.
We make use of (12), (13), and (21) in order to get(22)y1∈ℒT,Ω∩L∞Q,and conclude that the inequality in (9) holds for i = 1 similarly for y2, y3 and y4.
In order to show the positiveness ofyi for i = 1, 2, 3, 4, we write system (1) in the form:(23)∂y1∂t=d1Δy1+H1y1,y2,y3,y4,∂y2∂t=d2Δy1+H2y1,y2,y3,y4,∂y3∂t=d3Δy3+H3y1,y2,y3,y4,∂y4∂t=d4Δy3+H4y1,y2,y3,y4,t,x∈Q.
It is easy to see that the functionsH1y1,y2,y3,y4,H2y1,y2,y3,y4,H3y1,y2,y3,y4, and H4y1,y2,y3,y4 are continuously differentiable satisfying H10,y2,y3,y4=ωy4+μN1−vx,t≥0, H2y1,0,y3,y4=βy1y3/N≥0,H3y1,y2,0,y4=σy2≥0,H4y1,y2,y3,0=γy3+μNvx,t≥0, 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 v∈Uad. The main result of this section is the following theorem.Theorem 2.
Under the hypotheses of Theorem1, the optimal control problem (1)–(5) admits an optimal solution y∗,v∗.Proof.
From Theorem1, we know that, for every v∈Uad, there exists a unique solution y to system (1)–(3). Assume that(24)infv∈UadJv>−∞.
Letvn⊂Uad be a minimizing sequence such that(25)limn⟶∝Jvn=infv∈UadJv,where y1n,y2n,y3n,y4n is the solution of system (1)–(3) corresponding to the control vn for n = 1, 2, …. That is,(26)∂y1n∂t=d1Δy1n−μy1n+ωy4n−βy1ny3ny1n+y2n+y3n+y4n+μy1n+y2n+y3n+y4n1−vn,∂y2n∂t=d2Δy2n+βy1ny3ny1n+y2n+y3n+y4n−μ+σy2n,∂y3n∂t=d3Δy3n−μ+γy3n+σy2n,t,x∈Q,∂y4n∂t=d4Δy4n−μ+ωy4n+γy3n+μy1n+y2n+y3n+y4nvn,(27)∂y1n∂η=∂y2n∂η=∂y3n∂η=∂y4n∂t=0,t,x∈Σt,x∈Σ,(28)yin0,x=yi0fori=1,2,3,4x∈Ω.
By Theorem1, using the estimate (9) of the solution yin, there exists a constant C > 0 such that for all n≥1,t∈0,T,(29)∂yin∂tL2Q≤C,yinL20,T;H2Ω≤C,yinH1Ω≤C,i=1,2,3,4.
H
1
Ω is compactly embedded in L2Ω, so we deduce that y1nt is compact in L2Ω.
Let us show thaty1nt,n≥1 is equicontinuous in C0,T:L2Ω. As ∂y1n/∂t is bounded in L2Q, this implies that for all s,t∈0,T,(30)∫Ωy1n2t,xdx−∫Ωy1n2s,xdx≤Kt−s.
The Ascoli–Arzela theorem (see [21]) implies that y1n is compact in C0,T:L2Ω. Hence, selecting further sequences, if necessary, we have y1n⟶y1∗ in L2Ω, uniformly with respect to t and analogously, we have for yin⟶yi∗ in L2Ω for i = 2, 3 4, uniformly in relation to t.
From the boundedness ofΔyin in L2Q, which implies it is weakly convergent in L2Q on a subsequence denoted again by Δyin, for all distribution φ,(31)∫QφΔyin=∫QyinΔφ⟶∫Qyi∗Δφ=∫QφΔyi∗,which implies that Δyin⟶Δyi∗ weakly in L2Q,i=1,2,3,4. In addition, the estimates (29) lead to ∂yin/∂t⟶∂yi∗/∂t weakly in L2Q, i=1,2,3,4,yin⟶yi∗ weakly in L20,T;H2Ω, i=1,2,3,4,yin⟶yi∗ weakly in L∞0,T;H1Ω, i = 1, 2, 3, 4.
We putNy=β/y1+y2+y3+y4; we now show that y1ny3n⟼y1∗y3∗ and Nyny1ny3n⟼Ny∗y1∗y3∗ strongly in L2Q, and we write(32)y1ny3n−y1∗y3∗=y1n−y1∗y3n+y3n−y3∗y1∗,Nyn=βy1n+y2n+y3n+y4n,Ny∗=βy1∗+y2∗+y3∗+y4∗,Nyny1ny3n−Ny∗y1∗y3∗=Nyny1ny3n−y1∗y3∗+y1∗y3∗Nyn−Ny∗,Nyn−Ny∗=βy1n+y2n+y3n+y4n−βy1∗+y2∗+y3∗+y4∗,and we make use of the convergences yin⟶yi∗ strongly in L2Q, i = 1, 3, and of the boundedness of y1n, y3n in L∞Q, and then y1ny3n⟶y1∗y3∗ and Nyny1ny3n⟼Ny∗y1∗y3∗ strongly in L2Q.
Sincevn is bounded, we can assume that vn⟶v∗ weakly in L2Q on a subsequence denoted again by vn. Since Uad is a closed and convex set in L2Q, it is weakly closed, so v∗∈Uad.
We now show that(33)vny1n+y2n+y3n+y4n⟶v∗y1∗+y2∗+y3∗+y4∗weakly inL2Q.
Writing withi = 1, 2, 3, 4,(34)vnyin−v∗yi∗=yin−yi∗vn+vn−v∗yi∗,and making use of the convergences yin⟶yi∗ strongly in L2Q and vn⟶v∗ weakly in L2Q, for i = 1, 2, 3, 4, one obtains that vny1n+y2n+y3n+y4n⟶v∗y1∗+y2∗+y3∗+y4∗ weakly in L2Q.
By takingn ⟶ ∞ in (26)–(28), we obtain that y∗ is a solution of (1)–(3) corresponding to v∗∈Uad. Therefore,(35)Jy∗,v∗=K1∫0T∫Ωy1∗t,xdxdt+K2∫0T∫Ωy3∗t,xdxdt−K3∫0T∫Ωy4∗t,xdxdt+α2v∗L2Q2≤limn⟶∝infK1∫0T∫Ωy1nt,xdxdt+K2∫0T∫Ωy3nt,xdxdt−K3∫0T∫Ωy4nt,xdxdt+α2vnL2Q2=limn⟶∝K1∫0T∫Ωy1nt,xdxdt+K2∫0T∫Ωy3nt,xdxdt−K3∫0T∫Ωy4nt,xdxdt+α2vnL2Q2=infv∈UadJy,v.
This shows thatJ attains its minimum at y∗,v∗, and we deduce that y∗,v∗ verifies problem (1)–(3) and minimizes the objective functional (5). The proof is complete.


## 5. Necessary Optimality Conditions

Letv∈Uad and vε = v∗ + εv ∈ Uad; 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 v⟶yv. For this reason, denoting by yε=y1ε,y2ε,y3ε,y4ε=y1,y2,y3,y4vε and y∗=y1∗,y2∗,y3∗,y4∗=y1,y2,y3,y4v∗ the solution of (1)–(3) corresponding to vε and v∗ respectively.(36)H=−βy3∗y2∗+y3∗+y4∗y1∗+y2∗+y3∗+y4∗2−μv∗μ−μv∗−βy1∗y1∗+y2∗+y4∗y1∗+y2∗+y3∗+y4∗2+μ−μv∗μ−μv∗+ωβy3∗y2∗+y3∗+y4∗y1∗+y2∗+y3∗+y4∗2−μ−σβy1∗y1∗+y2∗+y4∗y1∗+y2∗+y3∗+y4∗200σ−μ−γ0μv∗μv∗μv∗+γ−μ−ω+μv∗,L=−μy1∗+y2∗+y3∗+y4∗00μy1∗+y2∗+y3∗+y4∗.Proposition 1.
The mappingy:Uad⟶W1,20,T;HΩ with yi∈ℒT,Ω for i = 1, 2, 3, 4 is Gateaux differentiable with respect to v∗. For all direction v∈Uad,y′v∗v=Y is the unique solution in W1,20,T;HΩ with Yi∈ℒT,Ω of the following equation:(37)∂Y∂t=AY+HY+Lv,t∈0,T,Y0=0.Proof.
PutYiε=yiε−yi∗/ε for i=1,2,3,4.Fy1,y2,y3,y4=βy1y3/y1+y2+y3+y4, M1ε=Fy1ε,y2ε,y3ε,y4ε−Fy1∗,y2,ε,y3ε,y4ε/y1ε−y1∗, and M3ε=Fy1∗,y2,ε,y3ε,y4ε−Fy1∗,y2∗,y3∗,y4∗/y3ε−y3∗.
We denoteSɛ system (1) corresponding to vε and S∗ system (1) corresponding to v∗, subtracting system Sɛ from S∗; so, we have:(38)∂Y1ε∂t=dSΔY1ε+−M1ε−μvεY1ε+μ−μvεY2ε+−M3ε+μ−μvεY3ε+μ−μvε+ωY4ε−μvy1∗+y2∗+y3∗+y4∗,∂Y2ε∂t=dEΔY2ε+M1εY1ε−μ+σY2ε+M3εY3ε,x,t∈Q,∂Y3ε∂t=dIΔY3ε+σY2ε−μ+γY3ε,∂Y4ε∂t=dRΔY4ε+μvεY1ε+μvεY2ε+μvε+γY3ε+−μ−ω+μvεY4ε+μvy1∗+y2∗+y3∗+y4∗,with the homogeneous Neumann boundary conditions:(39)∂Y1ε∂η=∂Y2ε∂η=∂Y3ε∂η=∂Y4ε∂η=0,x,t∈Σ,(40)Yiε0,x=0,x∈Ω,fori=1,2,3,4.
We prove thatYiε are bounded in L2Q uniformly with respect to ɛ. For this end, denoting by Yε=Y1ε,Y2ε,Y3ε,Y4ε,(41)Hε=−M1ε−μvεμ−μvε−M3ε+μ−μvεμ−μvε+ωM1ε−μ−σM3ε00σ−μ−γ0μvεμvεμvε+γ−μ−ω+μvε,L=−μy1∗+y2∗+y3∗+y4∗00μy1∗+y2∗+y3∗+y4∗.
Then, (38) is rewritten as(42)∂Yε∂t=AYε+HεYε+Lv,t∈0,T,Yε0=0.
LetSt,t≥0 be the semigroup generated by A; then, the solution of (42) can be expressed as(43)Yεt=∫0tSt−sHεsYεsds+∫0tSt−sLvsds.
On the other hand, the coefficients of the matrixHɛ are bounded uniformly with respect to ɛ; using Gronwall’s inequality, we have(44)YiεL2Q≤Γ,where Γ>0i=1,2,3,4. Then,(45)yiε−yi∗L2Q=εYiεL2Q.
Hence,yiε⟶yi∗ in L2Q, i = 1, 2, 3, 4. Denote(46)H=−M1∗−μv∗μ−μv∗−M3∗+μ−μv∗μ−μv∗+ωM1∗−μ−σM3∗00σ−μ−γ0μv∗μv∗μv∗+γ−μ−ω+μv∗,where M1∗=∂Fy1∗,y2∗,y3∗,y4∗/∂y1, M3∗=∂Fy1∗,y2∗,y3∗,y4∗/∂y3, and Y=Y1,Y2,Y3,Y4.
Hence, system (38)–(40) can be written in the form(47)∂Y∂t=AY+HY+Lv,t∈0,T,Y0=0,and its solution can be expressed as(48)Yt=∫0tSt−sHsYsds+∫0tSt−sLvsds.
By (43) and (48), we deduce that(49)Yεt−Yt=∫0tSt−sHεsYε−Y+YsHεs−Hsds.
Thus, all the coefficients of the matrixHɛ tend to the corresponding coefficients of the matrix H in L2Q. An application of Gronwall’s inequality yields to Yiε⟶Yi in L2Q as ɛ ⟶ 0, for i = 1, 2, 3, 4.
Letv∗ be an optimal control of (1)–(4), y∗=y1∗,y2∗,y3∗,y4∗ be the optimal state, Z be the matrix defined by Z=100001000000000−1, K=K1,K2,0,K3, Z∗ be the adjoint matrix associated to Z, H∗ be the adjoint matrix associated to H, and p=p1,p2,p3,,p4 be the adjoint variable; we can write the dual system associated to system (1)–(4):(50)−∂p∂t−Ap−H∗p=Z∗ZK,t∈0,T,pT,x=0.Lemma 1.
Under hypotheses of Theorem1, if y∗,v∗ is an optimal pair, then there exists a unique strong solution p∈W1,20,T;HΩ to system (50) with pi∈ℒT,Ω for i = 1, 2, 3, 4.Proof.
Similar to Theorem1, by making the change of variable s = T − t and the change of functions qis,x=piT−s,x=pit,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.
Letα > 0, v∗ be an optimal control of (1)–(4) and let y∗andp∈W1,20,T;HΩ with yi∗andpi∈ℒT,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∗. Then,(51)v∗=minvmax,max0,μy1∗+y2∗+y3∗+y4∗p1−p4α.Proof.
We supposev∗ is an optimal control and y∗=y1∗,y2∗,y3∗,y4∗=y1,y2,y3,y4v∗ are the corresponding state variables. Consider vε = v∗ + ɛh ∈ Uad and corresponding state solution yε=y1ε,y2ε,y3ε,y4ε=y1,y2,y3,y4vε; we have(52)J′v∗h=limε⟶01εJvε−Jv∗=limε⟶01ε∫0T∫ΩK1y1ε−y1∗t,xdxdt+∫0T∫ΩK2y3ε−y3∗t,xdxdt−∫0T∫ΩK3y4ε−y4∗t,xdxdt+α2∫0T∫Ωvε2−v∗2t,xdxdt=limε⟶0∫0T∫ΩK1y1ε−y1∗εt,xdxdt+∫0T∫ΩK2y3ε−y3∗εt,xdxdt−∫0T∫ΩK3y4ε−y4∗εt,xdxdt+α2∫0T∫Ωεh2+2hv∗t,xdxdt=∫0T∫ΩK1Y1t,xdxdt+∫0T∫ΩK2Y3t,xdxdt−∫0T∫ΩK3Y4t,xdxdt+α∫0T∫Ωhv∗t,xdxdt=∫0TZK,ZYHΩdt+∫0Tαv∗,hL2Ωdt.
We use (37) and (50), and we have(53)∫0TZK,ZYHΩdt=∫0TZ∗ZK,YHΩdt=∫0T−∂p∂t−Ap−H∗p,YHΩdt=∫0Tp,∂Y∂t−AY−HYHΩdt=∫0Tp,LhHΩdt=∫0TL∗p,hL2Ωdt.
SinceJ is Gateaux differentiable at v∗ and Uad 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 ∈ Uad.
We takeh = u − v∗and we use (52) and (53); then, J′v∗u−v∗=∫0TL∗p+αv∗,u−v∗L2Ωdt. We conclude that J′v∗u−v∗≥0 equivalent to ∫0TL∗p+αv∗,u−v∗L2Ωdt≥0 for all u ∈ Uad. By standard arguments varying u, we obtain(54)αv∗=−L∗p.
Then,(55)v∗=μy1∗+y2∗+y3∗+y4∗p1−μy1∗+y2∗+y3∗+y4∗p4α.
Asv∗ ∈ Uad, we have(56)v∗=minvmax,max0,μy1∗+y2∗+y3∗+y4∗p1−p4α.


## 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 vmax=1 [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.
Notations
Value
Description (units)
S
0
x
,
y
5
for
x
,
y
∈
Ω
i
i
=
1,2
10
for
x
,
y
∉
Ω
i
Initial susceptible population
people
/
km
2
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
I
0
x
,
y
3
for
x
,
y
∈
Ω
i
i
=
1,2
0
for
x
,
y
∉
Ω
i
Initial infected population
people
/
km
2
R
0
x
,
y
1
for
x
,
y
∈
Ω
Initial recovered population
people
/
km
2
σ
−1
1
/
0.09
Average duration of latent
μ
0.01
Rate of deaths
day
−
1
β
1
Transmission constant
γ
−1
1
/
0.04
Infective periods
ω
0.2
The rate of losing immunity
day
−
1
d
S
0.2
Diffusion rate for susceptible
d
E
0.3
Diffusion rate for exposed
d
I
0.5
Diffusion rate for infected
d
R
0.5
Diffusion rate for recovered
t
1,60
Time period
dayFigures1–4 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 Figure3, 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)/∉ Ωi, 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 Figures5 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.



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*Source: 4749365-2020-04-04.xml*