OPTIMAL CONTROL STRATEGY WITH MULTI-DELAY IN STATE AND CONTROL VARIABLES OF A DISCRETE MATHEMATICAL MODELING FOR THE DYNAMICS OF DIABETIC POPULATION

1Laboratory of Analysis, Modeling, and Simulation (LAMS), Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, BP 7955, Sidi Otheman, Casablanca, Morocco 2Laboratory of Dynamical Systems,Mathematical Engineering Team (INMA), Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco 3URAC04, LaMSD, Department of Mathematics, Faculty of Sciences Mohammed Premier University, Oujda,


INTRODUCTION
According to the World Health Organization (WHO) [1]. Diabetes is a chronic disease that occurs either when the pancreas does not produce enough insulin or when the body cannot use effective insulin or the pancreas does not produce insulin completely. What is Insulin: Insulin is a hormone that regulates blood sugar, which plays a key role in maintaining the normal rate of blood sugar. When a defect in the hormone secretion results in high blood sugar, it is one of the common effects of uncontrolled diabetes.
In 2012, the rise of glucose in blood led to a rise in the number of deaths, which amounted to about 2.2 million people. In 2014, 8.5% of adults aged 18 years and older had diabetes. In 2016, diabetes was a direct cause of death for 1.6 million people, and this number was increase to 4.2 million in 2019.
Nowadays, all the world suffers from the high number of people with diabetes. According to the 9th edition of the International Diabetes Federation (IDF) [2], Diabetes is the fourth leading cause of death, with more than 463 million diabetics, especially those aged over 45, as well as children and adolescents under 20 years of age. It is expected that the number of people with diabetes rises on the horizon of the year 2045 to about 629 million people, Therefore, effective solutions must be sought to reduce the number of diabetics.
According to IDF, diabetes is a costly chronic disease, especially when it is not diagnosed at the time or not treated. It has serious psychological, ethical and behavioral consequences, which can be life-threatening and lead to premature death, in addition to the costs of medical care and treatment expenses, which overburden the concerned community and the family as a whole.
Mathematical modeling of diabetes is not new. Differential equations play an important role in modeling different types of problems. There are different models devoted to mathematical modelling. In a related research work, Boutayeb et al. [3] and Derouich et al [4] proposed a mathematical model for the dynamics of the population of diabetes patients with or without complications using a system of ordinary differential equations, Kouidere et al [5] proposed a discrete mathematical model with highlighting the impact of living environment Also, many researches have focused on this topic and other related topics ( [6] , [7] , [13] , [19 − 20] ...), But they did not take into account the measuring the extent of interaction with the means of treatment or awareness campaigns. That may cause delays response and this is the objective of this work.
To make the modelling of this phenomena more realistic, we consider an optimal control problem governed by a system of difference equations with time to delay. we study an optimal control problem with time to delay in the state and control variable are described in a descrite PDEC model of kouidere et al [5] and a time delays representing the measuring the extent of interaction with the means of treatment or awareness campaigns. Then we derive first-order necessary conditions for existence of the optimal control and develop a numerical method to solve them.
We note, as mentioned above, that most researchers about diabetes and its complications focus on continuous and discrete time models and describe differential equations. Recently, more and more attention has been paid to the study of optimal control with delay in the state or state and control (see [8; 9; 15 − 18, · · · ] and references cited There).
In this paper, in section 2, we represent our discrete PDEC mathematical model of kouidere et al [5] , that describes the dynamic of a population of diabetes with highlighting the impact of living environment. In section 3, we present a optimal control problem with delay in our proposed model where we give some results concerning the existence of the optimal control and we caracterize the optimal controls using the Pontryagin's maximum principle in discrete time. Numerical simulations through MATLAB are given in section 4. Finally we conclude the paper in section 5.

A MATHEMATICAL MODEL
We consider a discrete mathematical model PDEC of kouidere et al [5], that describes the dynamics of a population of diabetics . We divide the population denoted by N into four compartments : People who are likely to have diabetes through genetics P, the individual diabetics without complications D, people who are likely to have diabetes through the effect of living environment or psychological problems E and the individual diabetics with complications C.
We present the diabetic model by the following system of difference equations: And P 0 ≥ 0, D 0 ≥ 0.E 0 ≥ 0 and C 0 ≥ 0 are the given initial states.
Where -Λ 1 : denote the incidence rate of pre-diabetes -Λ 2 : denote the incidence rate of environment effect

THE OPTIMAL CONTROL PROBLEM
The strategy of control, we adopt consists of an awareness program in order to minimize the negative the effect of living environment in diabetic people without complication, Our main is to minimize the number of people evolving from the stage of pre-diabetes to the stages of diabetes with and without complications . In this model, we include three control u = (u 0 , u 1 , · · · , u T −1 ) , v = (v 0 , v 1 , · · · , v T −1 ) and w = (w 0 , w 1 , · · · , w T −1 ), that represent consecutively the awareness program through media and education, treatment, and psychological support with follow-up at time k. in order to have realistic and logic model, we need to take in consideration that the movement of controlled individuals from the compartment of diabetics without complications (DWC) to diabetics with complications (DC), and transition by the contact between DWC and the effect of the environment E to DC, and the transition of the ordinary people (E) to DWC because of the effect of the environment is subject to a delay. Thus, the time delay is introduced into the system as follows: at the moment,Thus, the delay is introduced into the system as follows: in time, only a percentage of individuals (DWC, DC, E) that have been treated and controlled τ i time unit ago, that is to say that at the time k − τ i with i ∈ {1, 2, 3}, are removed to other compartments. So the mathematical system with time delay in state and control system of variables is given by the nonlinear retarded system of difference equations : In addition, for biological reasons, we assume, for ϕ ∈ {−τ, ..., 0}, that P ϕ , D ϕ, E ϕ and C ϕ are nonnegative continuous functions and u ϕ = 0, v ϕ = 0 and w ϕ = 0.
Our main gaol is to minimize the number of diabetics with complications and maximizethe number of diabetics without complications during the time step k = 0 to T , and also minimize the cost spent in this, strategy of control.
Then, the problem is to minimize the objective functional Where A k , B k and G k are the cost coefficients. They are selected to weigh the relative importance of u k , v k and w k at time k, T is the final time.
In other words, we seek the optimal controls u * ,v * and w * such that Where U is the set of admissible controls defined by 3.1. The optimal control: Existence.
We first show the existence of solutions of the system, after that we will prove the existence of optimal control. Theorem 1. Consider the control problem with the system. There are three optimal controls corresponding sequences of states P i , D i , E i and C i . Since there is a finite number of uniformly bounded sequences, there exist (u * , v * , w * ) ∈ U 3 and P * , D * , E * and C * ∈ R T +1 such that, on a subsequence, (u i , v i , w i ) → (u * , v * , w * ), P i → P * , D i → D * , E i → E * , and C i → C * . Finally, due to the finite dimensional structure of system (2) and the objective function L(u, v, w) and also (u * , v * , w * ) is an optimal control with corresponding states P * , D * , E * and C * . Therefore

Characterization of the Optimal Control.
In order to derive the necessary condition for optimal control, the pontryagins maximum principle, in discrete time, given in was used. This principle converts into a problem of minimizing a Hamiltonian, H k at time step k defined by where f i,k+1 is the right side of the difference equation of the i th state variable at time step k + 1.
Theorem 2. Given the optimal controls (u * , v * , w * ) and the solutions P * , D * , E * and C * of the corresponding state system (2), there exists adjoint variables λ 1,k , λ 2,k ,λ 3,k and λ 4,k satisfying: Furthermore, for k = 0, 1, 2...T , the optimal controls u * , v * and w * are given by Proof. In order to derive the necessary condition for optimal control, the pontryagins maximum principle in discrete time given in [10 − 12, 13, 14] was used. This principle converts into a problem of minimizing a Hamiltonian H k at time step k defined by For, k = 0, 1...T − 1 the optimal controls u k , v k , w k can be solved from the optimality condition, By the bounds in U of the controls, it is easy to obtain u * k , v * k and w * k in the form of system.

NUMERICAL SIMULATION
In this section, we shall solve numerically the optimal control problem for our PDEC model.
Here, we obtain the optimality system from the state and adjoint equations.
The proposed optimal control strategy is obtained by solving the optimal system which consists of six difference equations and boundary conditions. The optimality system can be solved by using an iterative method. Using an initial guess for the control variables, u k , v k and w k , That is, we use the value parameter in table 1,the optimality system is a two-point boundary value problem with separated boundary conditions at time steps k = 0 and k = T . We solve the optimality system by an iterative method with forward solving of the state system followed by backward solving of the adjoint system. We start with an initial guess for the controls at the first iteration and then before the next iteration we update the controls by using the characterization.
We continue until convergence of successive iterates is achieved.
We proposed control strategy in this work helps to achieve several objectives.

4.
3. Strategy C: Prevention and protection E from diabetes. We note from the figure 3, this decrease can be explained by the fact that the longer the delay in the positive response to the awareness and awareness campaigns carried out by the government and specialists as well as civil society associations. lead to a negative reaction as mentioned above, we will not get the desired results, and the number of diabetic patients will increase due to negative convergence with the effect of the living medium, for example, when τ = 2 the value was 1.42 × 10 7 , And when τ = 7 the value decreased to 1.38 × 10 7 , and , the devaluation τ = 10 increased to 1.31 × 10 7 after 120 months.