OPTIMAL CONTROL STRATEGY WITH DELAY IN STATE AND CONTROL VARIABLES OF TRANSMISSION OF COVID-19 PANDEMIC VIRUS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this work, we are interested in the study of the dynamics of transmission of COVID-19 disease, described by SIR epidemic model, with time delay in state and control variables. Our goal is to characterize the optimal control pair, which minimizes the number of susceptible and infected individuals and maximizes the number of people recovered. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The numerical resolution of the optimal system, has shown the effectiveness of our adopted strategy.


INTRODUCTION
Coronaviruses are a large family of viruses that can be pathogenic in animals or humans. We know that several coronaviruses in humans can cause respiratory infections ranging from the common cold to more serious illnesses such as Middle East Respiratory Syndrome (MERS) and Severe Acute Respiratory Syndrome (SARS). The last coronavirus that was discovered is responsible for coronavirus disease 2019 . This new virus and disease was unknown before the outbreak in Wuhan (China) in December 2019. COVID-19 is mainly spread by respiratory droplets expelled by people who cough or have other symptoms, such as fever or fatigue. Many sufferers have only mild symptoms.Several mathematical models have been developed to simulate, analyze and understand the Coronavirus [1,2]. We can cite, Tahir and al [3] who proposed prevention strategies for the MERS-Coronavirus mathematical model with stability analysis and control optimal, Zhi-Qiang Xia et al. [4] modeled the transmission of the Middle East respiratory syndrome Coronavirus in the Republic of Korea. To describe the dynamics of a population infected with the COVID-19 virus, we consider a general SIR epidemic model with incubation time and a modified saturated incidence rate [5,6,7], the model used consists of three compartments SIR: susceptible, infected and recovered. We note their densities at time t by. S, I, R, respectively: The initial condition for the above system is S(η) = ψ 1 (η) , I(η) = ψ 2 (η) , and R(η) = To minimize the number of susceptible and infected individuals and to maximize the individuals recovered, We have adopted a control strategy consists of optimally combine, the raising people's awareness of the danger of covid 19 in order to take hygienic precautions necessary, and treatment of infected individuals, so that the cost of implementing the two interventions is minimized. The organization of this work is as follows. In Sect. 2, we formulate the optimal control problem and we use the principle of Pontryagin Maximum with delay given in Göllmann et al. [8] to characterize it. In Sect. 3, we give the numerical method and the simulation results.
Finally, we draw conclusions in Sect. 4.

OPTIMAL CONTROL PROBLEM
Our goal is to minimize the number of susceptible and infected individuals, and to maximize the recovered individuals. We include two controls (u, v), which represents the percentage of susceptible and infected individuals sensitized by the danger of the covid 19 and treated respectively per unit of time. In order to have a realistic model, we take in account that the transfer of infected individuals treated from the class of infected to the recovered class is subject to delay. Thus, the delay is introduced into the system as follows: at the time t only a percentage of infected individuals who have been treated. There is τ 2 unit of time,i.e. at times t − τ 2 , are deleted from the infected class and added to the recovered class. So the model (1) becomes: It is easy to show that there exists a unique solution (S(t), I(t), R(t)) of system (2).
For biological reasons, we assume that the initial data for system (2) satisfy: The problem is to minimize the objective functional given by: Where A, B, F, c 1 and c 2 are positive constants to keep a balance size of the terms. We seek the optimal control (u * , v * ) such that:

2.1
The existence of a Optimal Control. The existence of the optimal control pair can be obtained using a result by Fleming and Rishel in [9] and by Lukes in [10].
There exists an optimal control pair (u * , v * ) ∈ U ad such that Proof. To prove the existence of an optimal control pair it is easy to verify that: 1. The set of controls and corresponding state variables is nonempty.
2. The admissible set U ad is convex and closed.
3. The right hand side of the state system (2) Pontryagin's minimum Principle with delay given in [8] provides necessary conditions for an optimal control problem. This principle converts (2), (4), and (5) into a problem of minimizing an Hamiltonian, H, with where f i is the right side of the differential equation of the state variable, and λ i , i = 1, 2, 3 are the adjoint functions.
Furthermore, the optimal control pair (u * , v * ) is given by Proof. Using the Pontryagin's Maximum principle with delay in state and control variables, we obtain the adjoint equations and transversality conditions such that: The optimal control (u * , v * ) pair can be solved from the optimality condition:

NUMERICAL RESULTS AND DISCUSSIONS
In this section, we solve numerically the optimality [11] system (7) which is a tow-point boundary value problem, with separated boundary conditions at times t 0 = 0, and t f and we present the results found.
Let there exists a step size h > 0, and integers (n, m) ∈ IN 2 with τ = mh and t f − t 0 = nh .
For reasons of programming, we consider knots to left of t 0 and right of t f , and we obtain the following partition: Then, we have t i = ih(−m ≤ i ≤ n + m). Next we define the state and adjoint variables S(t), I(t), R(t), λ 1 (t), λ 2 (t), λ 3 (t) and u(t), v(t) in terms of nodal points S i , I i , R i , λ i 1 , λ i 2 , λ i 3 , u i and v i . Now using combination of forward and backward difference approximations, we obtain the Algorithm 1.

Algorithm 1
Step1 end f or f or i = n, ..., n + m, do        Finally, Figures 5 and 6 show the optimal controls u and v obtained to block the spread of the disease.

CONCLUSION
In this work,we propose a more realistic controlled model by including, on the one hand, the time delay which represents the time necessary for the transfer from the susceptible class to the infected class, and on the other hand, take into consideration the time delay in going from the infected class to recovered class after the proposed treatment. in this context, we applied the optimal control theory to prove the existence of the optimal control pair, and his explicit expression was obtained By using Pontryagin's maximum principle with delay .For the numerical simulation, we propose an algorithm based on the forward and backward difference approximation. The numerical results show that the our optimal strategy is effective in reducing the number of susceptible and infected individuals and maximizing the recovered individuals.