Research Article
An Optimal Control for a Two-Dimensional Spatiotemporal SEIR
Epidemic Model
Khalid Adnaoui
1
and Adil El Alami Laaroussi
2
1
Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science,
Faculty of Sciences Ben M’Sik, Hassan II University Casablanca, BP 7955, Sidi Othman, Casablanca, Morocco
2
Laboratory of Applied Sciences and Didactics, Higher Normal School Tetouan, Abdelmalek Essaadi University,
Tetouan, Morocco
Correspondence should be addressed to Khalid Adnaoui; khalid.adnaoui@gmail.com
Received 15 September 2019; Revised 15 February 2020; Accepted 18 February 2020; Published 4 April 2020
Academic Editor: Jingxue Yin
Copyright © 2020 Khalid Adnaoui and Adil El Alami Laaroussi. is is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
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.
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.
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.
2. The Basic Mathematical Model
2.1.eModel. In this paper, we consider the following SEIR
epidemic model (susceptible (S), exposed (E), infected (I),
and recovered (R)):
Hindawi
International Journal of Differential Equations
Volume 2020, Article ID 4749365, 15 pages
https://doi.org/10.1155/2020/4749365