# Optimal Control Strategy for SEIR with Latent Period and a Saturated Incidence Rate

Authors: Abta Abdelhadi; Laarabi Hassan
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

We propose an SEIR epidemic model with latent
period and a modified saturated incidence rate. This work investigates the fundamental role of the vaccination strategies to reduce the
number of susceptible, exposed, and infected individuals and increase
the number of recovered individuals. The existence of the optimal
control of the nonlinear model is also proved. The optimality system
is derived and then solved numerically using a competitive Gauss-Seidel-like implicit difference method.


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## Body

## 1. Introduction

Epidemiological models with latent or incubation period have been studied by many authors because many diseases have a latent or incubation period, during which the individual is said to be infected but not infectious. This period can be modeled by introducing an exposed class [1]. Therefore, it is an important subject to determine the optimal vaccination strategies for the models which take into account the incubation period.In this paper, our aim is to set up an optimal control problem related to the SEIR epidemic model. The dynamics of this model are governed by the following equations [2, 3]:
(1)dSdt=A-μS(t)-βS(t)I(t)1+α1S(t)+α2I(t),dEdt=βS(t)I(t)1+α1S(t)+α2I(t)-(σ+μ)E(t),dIdt=σE(t)-(μ+α+γ)I(t),dRdt=γI(t)-μR(t),
where S is the number of the susceptible individuals, E is the number of exposed individuals, I is the number of infected individuals, R is the number of the recovered individuals, A is the recruitment rate of the population, μ is the natural death of the population, α is the death rate due to disease, β is the transmission rate, α1 and α2 are the parameter that measure the inhibitory effect, γ is the recovery rate of the infective individuals, and σ is the rate at which exposed individuals become infectious. Thus 1/σ is the mean latent period.Now we introduce one controlu(t) which represents the percentage of susceptible individuals being vaccinated per unit of time. Hence, (1) becomes
(2)dSdt=A-μS(t)-βS(t)I(t)1+α1S(t)+α2I(t)-u(t)S(t),dEdt=βS(t)I(t)1+α1S(t)+α2I(t)-(σ+μ)E(t),dIdt=σE(t)-(μ+α+γ)I(t),dRdt=γI(t)-μR(t)+u(t)S(t).
In addition, for biological reasons, we assume that the initial data for system (2) satisfy
(3)S(0)=S0≥0,E(0)=E0≥0,I(0)=I0≥0,R(0)=R0≥0.
The rest of the paper is organized as follows. In Section 2, we use Pontryagin's maximum principle to investigate analysis of control strategies and to determine the necessary conditions for the optimal control of the disease. Mathematical results are illustrated by numerical simulations in Section 3. Finally, we summarize our work and propose the future focuses.


## 2. The Optimal Control Problem

The optimal control problem is to minimize the objective (cost) functional given by(4)J(u)=∫0tend‍[A1S(t)+A2E(t)+A3I(t)+12τu2(t)]dt
subject to the differential equations (2), where the first tree terms in the functional objective represent benefit of S(t), E(t), and I(t) populations that we wish to reduce, and the parameters A1, A2, and A3 are positive constants to keep a balance in the size of S(t), E(t), and I(t), respectively. We use in the second term in the functional objective (as it is customary) the quadratic term (1/2)τu2, where τ is a positive weight parameter which is associated with the control u(t), and the square of the control variable reflects the severity of the side effects of the vaccination.Our target is to minimize the objective functional defined in (4) by decreasing the number of infected, exposed, and susceptible individuals and increasing the number of recovered individuals by using possible minimal control variables u(t). In other words, the control variable u(t)∈Uad represents the percentage of susceptible individuals being vaccinated per unit of time and Uad is the control set defined by(5)Uad={u∣u(t)ismeasurable,0≤u(t)≤umax<∞,t∈[0,tend]}.


### 2.1. Existence of an Optimal Control

For the existence of an optimal control we use the result in Lukes [4], and we obtain the following theorem.Theorem 1.
There exists a control functionu*(t) so that
(6)J(u*(t))=minu∈UJ(u(t)).Proof.
To prove the existence of an optimal control it is easy to verify that(1)
the set of controls and corresponding state variables is nonempty,(2)
the admissible setUad is convex and closed,(3)
the right hand side of the state system (2) is bounded by a linear function in the state and control variables,(4)
the integrand of the objective functional is convex onUad,(5)
there exist constantsω1>0 and ω2>0, and ρ>1 such that the integrand L(S,E,I,u) of the objective functional satisfies ρ>1 and positive numbers ω1 and ω2 such that L(S,E,I,u)≥ω2+ω1(|u|2)ρ/2. The result follows directly from [5].


### 2.2. Characterization of the Optimal Control

Before characterizing the optimal control, we first define the Lagrangian for the optimal control problem (2) and (4) by
(7)L(S,E,I,u)=A1S(t)+A2E(t)+A3I(t)+12τu2(t)
and the Hamiltonian H for the control problem by
(8)H(S,E,I,R,u,λ1,λ2,λ3,λ4,t)=L(S,E,I,u)+λ1dS(t)dt+λ2dE(t)dt+λ3dI(t)dt+λ4dR(t)dt,
where λ1, λ2, λ3, and λ4 are the adjoint functions to be determined suitably. Next, by applying Pontryagin’s maximum principle [6] to the Hamiltonian, we obtain the following theorem.Theorem 2.
Given an optimal controlu*(t) and solutions S*(t), E*(t), I*(t), and R*(t) of the corresponding state system (2) and (4), there exists adjoint variables λ1, λ2, λ3, and λ4 that satisfy
(9)dλ1(t)dt=-A1+λ1(t)(μ+u*(t)+Λ1)-λ2(t)Λ1-λ3(t)u*(t),dλ2(t)dt=-A2+λ2(t)(μ+σ)-λ3(t)σ,dλ3(t)dt=-A3+λ1(t)Λ2-λ2(t)Λ2+λ3(t)(μ+α+γ)-λ4(t)γ,dλ4(t)dt=λ3(t)μ,
where Λ1=(βI*(1+α2I*)/(1+α1S*+α2I*)2) and Λ2=(βS*(1+α1S*)/(1+α1S*+α2I*)2) with transversality conditions
(10)λi(tend)=0,i=1,2,3,4.
Furthermore, the optimal control u*(t) is given by
(11)u*(t)=max(min((λ1(t)-λ4(t))S*(t)τ,umax),0).Proof.
Using the Pontryagin’s maximum principle we obtain the adjoint equations and transversality conditions such that(12)dλ1(t)dt=-∂H∂S=-A1+λ1(t)(μ+u*(t)+Λ1)-λ2(t)Λ1-λ3(t)u*(t),dλ2(t)dt=-∂H∂E=-A2+λ2(t)(μ+σ)-λ3(t)σ,dλ3(t)dt=-∂H∂I=-A3+λ1(t)Λ2-λ2(t)Λ2+λ3(t)(μ+α+γ)-λ4(t)γ,dλ4(t)dt=-∂H∂R=λ4(t)μ,
and by using the optimality conditions we find
(13)∂H∂u=τu*(t)-λ1(t)S*+λ4(t)S*=0,atu=u*(t)whichgivesu*(t)=(λ1(t)-λ4(t))S*(t)τ.
Using the property of the control space, we obtain
(14)u*(t)=0if(λ1(t)-λ4(t))S*(t)τ≤0,u*(t)=(λ1(t)-λ4(t))S*(t)τif0<(λ1(t)-λ4(t))S*(t)τ<umax,u*(t)=umaxif(λ1(t)-λ4(t))S*(t)τ≥umax.
So the optimal control is characterized as
(15)u*(t)=max(min((λ1(t)-λ4(t))S*(t)τ,umax),0).
Therefore, using the characterization of the optimal control, we have the following optimality system:
(16)S˙*=A-(μ+u*)S*-βS*I*1+α1S*+α2I*,E˙*=βS*I*1+α1S*+α2I*+(σ+μ)E*,I˙*=σE*-(μ+α+γ)I*,R˙*=γI*-μR*+u*S*,λ˙1=-A1+λ1(μ+u*+βI*(1+α2I*)(1+α1S*+α2I)2)-λ2(βI*(1+α2I*)(1+α1S*+α2I*)2)-λ3u*,λ˙2=-A2+λ2(σ+μ)-λ3(t)γ,λ˙3=-A3+λ1(βS*(1+α1S*)(1+α1S*+α2I*)2)-λ2(βS*(1+α1S*)(1+α1S*+α2I*)2)+λ3(t)(μ+α+γ)-λ4γ,λ˙4=λ4μ
with λ1(tend)=0, λ2(tend)=0, λ3(tend)=0, λ4(tend)=0, S(0)=S0, E(0)=E0, I(0)=I0, and R(0)=R0.


## 2.1. Existence of an Optimal Control

For the existence of an optimal control we use the result in Lukes [4], and we obtain the following theorem.Theorem 1.
There exists a control functionu*(t) so that
(6)J(u*(t))=minu∈UJ(u(t)).Proof.
To prove the existence of an optimal control it is easy to verify that(1)
the set of controls and corresponding state variables is nonempty,(2)
the admissible setUad is convex and closed,(3)
the right hand side of the state system (2) is bounded by a linear function in the state and control variables,(4)
the integrand of the objective functional is convex onUad,(5)
there exist constantsω1>0 and ω2>0, and ρ>1 such that the integrand L(S,E,I,u) of the objective functional satisfies ρ>1 and positive numbers ω1 and ω2 such that L(S,E,I,u)≥ω2+ω1(|u|2)ρ/2. The result follows directly from [5].


## 2.2. Characterization of the Optimal Control

Before characterizing the optimal control, we first define the Lagrangian for the optimal control problem (2) and (4) by
(7)L(S,E,I,u)=A1S(t)+A2E(t)+A3I(t)+12τu2(t)
and the Hamiltonian H for the control problem by
(8)H(S,E,I,R,u,λ1,λ2,λ3,λ4,t)=L(S,E,I,u)+λ1dS(t)dt+λ2dE(t)dt+λ3dI(t)dt+λ4dR(t)dt,
where λ1, λ2, λ3, and λ4 are the adjoint functions to be determined suitably. Next, by applying Pontryagin’s maximum principle [6] to the Hamiltonian, we obtain the following theorem.Theorem 2.
Given an optimal controlu*(t) and solutions S*(t), E*(t), I*(t), and R*(t) of the corresponding state system (2) and (4), there exists adjoint variables λ1, λ2, λ3, and λ4 that satisfy
(9)dλ1(t)dt=-A1+λ1(t)(μ+u*(t)+Λ1)-λ2(t)Λ1-λ3(t)u*(t),dλ2(t)dt=-A2+λ2(t)(μ+σ)-λ3(t)σ,dλ3(t)dt=-A3+λ1(t)Λ2-λ2(t)Λ2+λ3(t)(μ+α+γ)-λ4(t)γ,dλ4(t)dt=λ3(t)μ,
where Λ1=(βI*(1+α2I*)/(1+α1S*+α2I*)2) and Λ2=(βS*(1+α1S*)/(1+α1S*+α2I*)2) with transversality conditions
(10)λi(tend)=0,i=1,2,3,4.
Furthermore, the optimal control u*(t) is given by
(11)u*(t)=max(min((λ1(t)-λ4(t))S*(t)τ,umax),0).Proof.
Using the Pontryagin’s maximum principle we obtain the adjoint equations and transversality conditions such that(12)dλ1(t)dt=-∂H∂S=-A1+λ1(t)(μ+u*(t)+Λ1)-λ2(t)Λ1-λ3(t)u*(t),dλ2(t)dt=-∂H∂E=-A2+λ2(t)(μ+σ)-λ3(t)σ,dλ3(t)dt=-∂H∂I=-A3+λ1(t)Λ2-λ2(t)Λ2+λ3(t)(μ+α+γ)-λ4(t)γ,dλ4(t)dt=-∂H∂R=λ4(t)μ,
and by using the optimality conditions we find
(13)∂H∂u=τu*(t)-λ1(t)S*+λ4(t)S*=0,atu=u*(t)whichgivesu*(t)=(λ1(t)-λ4(t))S*(t)τ.
Using the property of the control space, we obtain
(14)u*(t)=0if(λ1(t)-λ4(t))S*(t)τ≤0,u*(t)=(λ1(t)-λ4(t))S*(t)τif0<(λ1(t)-λ4(t))S*(t)τ<umax,u*(t)=umaxif(λ1(t)-λ4(t))S*(t)τ≥umax.
So the optimal control is characterized as
(15)u*(t)=max(min((λ1(t)-λ4(t))S*(t)τ,umax),0).
Therefore, using the characterization of the optimal control, we have the following optimality system:
(16)S˙*=A-(μ+u*)S*-βS*I*1+α1S*+α2I*,E˙*=βS*I*1+α1S*+α2I*+(σ+μ)E*,I˙*=σE*-(μ+α+γ)I*,R˙*=γI*-μR*+u*S*,λ˙1=-A1+λ1(μ+u*+βI*(1+α2I*)(1+α1S*+α2I)2)-λ2(βI*(1+α2I*)(1+α1S*+α2I*)2)-λ3u*,λ˙2=-A2+λ2(σ+μ)-λ3(t)γ,λ˙3=-A3+λ1(βS*(1+α1S*)(1+α1S*+α2I*)2)-λ2(βS*(1+α1S*)(1+α1S*+α2I*)2)+λ3(t)(μ+α+γ)-λ4γ,λ˙4=λ4μ
with λ1(tend)=0, λ2(tend)=0, λ3(tend)=0, λ4(tend)=0, S(0)=S0, E(0)=E0, I(0)=I0, and R(0)=R0.


## 3. Numerical Simulations

In this section, we solve numerically the optimality system (16) using the Gauss-Seidel-like implicit finite-difference method developed by Gumel et al. [7], and we use in this simulation the parameter values given in Table 1.Table 1
Values of the parameters.
Value
Description
S
0
=
80
Initial susceptible population
E
0
=
20
Initial exposed population
I
0
=
3
Initial infected population
R
0
=
20
Initial recovered population
μ
=
0.4
Natural death of the population
α
=
0.99
Death rate due to disease
α
1
=
0.1
Parameter that measures the inhibitory effect
α
2
=
0.5
Parameter that measures the inhibitory effect
β
=
0.6
Transmission rate
γ
=
0.9
Recovery rate
σ
=
0.5
The rate at which exposed individuals become infectious
A
=
100
Recruitment rate
A
1
=
100
Weight parameter
A
2
=
100
Weight parameter
A
3
=
100
Weight parameter
τ
=
1
0
7
Weight parameterFigure1 shows that a significant difference in the number of susceptible, exposed, infected, and recovered individuals with and without control from the twenty days of vaccination, after that it begins to go to the stable state.Evolution of different classes of individuals with or without control.
(a)
(b)
(c)
(d)
(e)


## 4. Conclusions and Future Research

We have presented in this paper the SEIR model with latent period and a modified saturated incidence rate. Our aim is to outline the steps in setting up an optimal control problem, so we presented an efficient numerical method based on optimal control to identify the best vaccination strategy of SEIR model. Our numerical results show that the optimal vaccination strategies for the diseases have a latent period to reduce the number of susceptible, exposed, and infected individuals and increase the number of recovered after twenty days of vaccination.In future research, we determine the optimal control strategies for the delayed SIR model and compare it with that presented in this work. It is an important subject to study these two types of modeling the incubation period.



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