OPTIMAL CONTROL OF A SEIR MODEL WITH CONFINEMENT OF COVID’19

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this work, we proposed a mathematical model with an isolation rate and a control strategy, such as confinement, to control the spread of the COVID’19 epidemic in society. We formulated an optimal control problem and solved it. Also, a characterization of the optimal control is given using Pontryagin’s Maximum Principle.


INTRODUCTION
Corona-virus disease (COVID'19) is an epidemic of viral-looking pneumonia that is different from Sars-Cov viruses. It was officially announced by Chinese health authorities and the World Health Organization (WHO). Unfortunately, this virus was able to spread very quickly leading to an increase in the number of deaths every day. Therefore, it is imperative to control the spread of this disease using non-pharmaceutical strategies such as the confinement.
Since the announcement of the first cases of COVID'19 in Morocco [9], the country has moved up a gear in terms of measures to keep the Corona-virus under control, which consist of closing schools, restaurants and all gathering places, and announcing a complete quarantine throughout the country. Figure (1) shows the cumulative number of confirmed cases of Corona-virus in Morocco between 2 March and 28 December 2020, showing that the quarantine and isolation measures announced in mid-March contributed to reducing the contact rate. Therefore, the incidence curve did not accelerate due to the slow and structured increase. Whereas after announcing the lifting of the quarantine on 25 June, Morocco recorded a significant increase in the number of confirmed cases.
Mathematical modelling methods are used to understand the dynamics of infectious diseases and how to control their spread using epidemiological models that are considered the most effective for formulating disease and epidemic control strategies. From this standpoint, many mathematical models have been applied to Corona-virus. Some authors have proposed models to estimate the risks of transmission and effects of the virus [1], while others have suggested epidemic models to predict the potential local and global spread of the epidemic Corona-virus [2].
Although most researchers have focused on calculating the baseline reproductive R 0 number and predicting the number of infected and deaths [3,4,5,6], some authors have been interested in formulating models to investigate certain control strategies, such as social distance and health education [7,8].
In this paper, we applied the optimal control theory to propose the optimal strategy in order to mitigate the spread of the COVID'19 epidemic by minimizing the susceptible individuals and the infected individuals, where confinement was used as a control.
The model inspired by the COVID'19 pandemic associated with confinement is presented in the following section. In section 3, we provide necessary condition for the existence of an optimal control and its characterization using Pontryagin's Maximum Principle.

COVID'19 MODEL WITH CONFINEMENT
To study the transmission dynamics of the novel Corona-virus epidemic, we use in our work an extension of a SEIR model. This choice is motivated by the fact that this model takes into account the latency period of the disease. That is to say that during this period, the individual is infected without any transmission of the disease to other individuals. In this model, we consider a time-dependent control u(t) representing the confinement, that is bounded between 0 and U max , with U max < 1.
The population is made up of five classes of individuals (compartments), susceptible S, exposed E, infected without protection I and we introduce the compartment R 1 for infectious individuals, but confined. In other words, individuals whose movement is restricted and completely removed from individuals likely to be infected, and their infection is confirmed by tests in laboratories approved by the state. Whereas, the compartment R 2 contains contagious and isolated cases, but they are not counted, that is to say that she has not undergone any test, either because some of them are suffering from symptoms of low severity, either because some live in areas far from laboratories and where live in retirement homes and some are afraid to test and prefer to isolate themselves and adopt a health protocol to deal with the virus. It should be noted that the two compartments R 1 and R 2 also include both cases which are no longer contagious, either because they have cured or because they have died.
The parameter α is defined as the effective contact rate, the parameter β is the incubation rate at which people infected in the latent phase become infectious, η is the isolation rate of infectious people and therefore removed from the chain of transmission, and δ is the fraction of infectious individuals who are counted among the confirmed cases in the time of isolation, it is assumed that this fraction is constant, although it may change over time.
The total population is given by Thus, we can write the model mathematically as:

OPTIMAL CONTROL PROBLEM
The aim of our work is to analyse the problem of optimal control to suggest the most effective mitigation strategy for the COVID'19 epidemic by minimizing the level of susceptible and infected individuals and therefore maximizing the number of recovered individuals.
In our model, we have included the control u which represents the confinement, so our goal is to determine an optimal control u * to stop the spread of the epidemic by minimizing, over a finite time horizon [0,t f ], the objective functional.
We define the objective functional J as follows: where constants C 1 , C 2 and ρ are positive. The weight parameter ρ is associated with the control u. C 1 describes the cost related to the susceptible class while C 2 indicates the cost incurred due to the symptomatic compartment. In addition, the objective functional J corresponds to the total cost due to the Corona-virus epidemic and its control strategy. Further, the integrand of the objective functional is given by L (S, E, I, R 1 , R 2 , u) = C 1 S(t) + C 2 I(t) + 1 2 ρu 2 (t). More precisely, the optimal control problem can be defined as follows: where is the set of admissible controls.

Existence of optimal control.
In order to solve the optimal control problem, it is first necessary to show the existence of the solution of system (1). Consider the state variables S(t), E(t), I(t), R 1 (t), R 2 (t) and the control u(t) with non negative initial conditions as given (2), then the system can be written as X t is derivative of X with respect to time t. It is clearly that the system (6) is a non linear system with a bounded coefficient.
We pose Then, The second term on the right hand side of the equation (7) satisfies where M 1 > 0 and M 2 > 0 are independent of the variables S(t) and I(t) ≤ N respectively. Therefore, Also, we get Thus, it follows that the function H satisfies the Lipschitz condition, uniformly with respect to non negative state variables. Therefore, there exists a solution of the system (1). Now, we present a result that will show the existence of an optimal control that minimizes the objective functional J in a finite interval, subjected to the system (1).
Proof. To prove the existence of an optimal control, we use the conditions of theorem (III.4.1) and its corresponding corollary in Fleming and Rischel [10].
Note that the set of solution to the system (1)-(2) with control variables in Ω is nonempty. Let u 1 , u 2 ∈ Ω such that 0 ≤ u 1 ≤ u max and 0 ≤ u 2 ≤ u max . Then, for any ε ∈ [0, 1], one has which implies that Ω is convex and close. Also the state system is linear in the control variable u with coefficients depending on time and state variables. In addition, it is easy to verify that all ε ∈ [0, 1] and all u 1 , u 2 ∈ Ω, where X = {S, E, I, R 1 , R 2 }. Thus, the integrand of the objective functional is convex on Ω.

Characterization of optimal control.
Theorem 1 assures us the existence of the solution of the problem (4) before attempting to calculate the optimal control. Thereafter, we characterize this control by applying Pontryagin's Maximum Principle to the Hamiltonian.

CONCLUSION
Among the most important measures taken to limit the propagation of Corona-virus is quarantined and isolation of infected individuals, and it is the control option we supported in this work. We have demonstrated the existence of optimal control over our model that minimizes the number of susceptible individuals and infected individuals. The results discussed in this article will be valuable for our future work because we aim to use other optimal control strategies in the form of vaccination, health awareness and intensive prevention.

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