OPTIMAL CONTROL STRATEGY FOR A DISCRETE TIME EPIDEMIC MODELS OF MYCOBACTERIUM TUBERCULOSIS INFECTIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, our aim is to study the optimal control strategy of a mathematical model of the tuberculosis transmission in the discrete case, and to investigate, in discrete time, optimal control strategy in which the controls are: vaccination and treatment and sensibilisation. The studied population is divided into five compartments SL1IL2R. Our objective is to find the best strategy to reduce the number of S, L1, I and L2. So, the Pontryagin’s maximum principle, in discrete time, is used to characterize the optimal control. The numerical simulation is carried out using MATLAB. The obtained results confirm the performance of the optimization strategy.


INTRODUCTION
"Just sleep and eat nutritious foods" was the advice given to patients in the 1800s infected with tuberculosis, or formerly known for a long time as consumption [12]. Tuberculosis (TB) has never stopped making victims through the times and in all known human civilizations. Even today it is considered as the most infectious disease that has led to the most deaths in the history of humanity. TB remains one of the leading causes of illness and death in the world, estimated one third of the world's population is infected with TB. Such a human reservoir triggers about 8 million new TB cases and 2 million deaths each year according to WHO [26]. The identification of Mycobacterium tuberculosis "MTB" (or Bacillus of Koch "BK") on March 24, 1882, by biologist Robert Koch, followed by the invention of The BCG vaccine starts with Albert Calmette and Camille Guerin, thy were two French scientists who from 1905 had been working on developing a vaccine against TB. BCG is abbreviation of Bacillus Calmette-Guerin, meaning the bacilli of Calmette and Guerin, then the discovery of streptomycin in 1943 by Selman Waksman, have eventually allowed to revolutionize the vital and functional prognosis of patients with TB [12]. Tuberculosis is a contagious disease, secondary to infection with "bacillus of Koch" (Mycobacterium tuberculosis). This bacterial agent is transmitted by air via the droplets contaminated by the bacterium, which are suspended in the exhaled air by patients, especially during coughing. Inhaling a small number of contaminated droplets is enough to infect an individual. The displacement of populations (travelers, refugees) has largely contributed recently in the spread of the disease in the world, People who are more likely to acquire TB infection are the following: (1) People recently exposed to someone who has symptomatic TB disease; (2) People who live in congregate settings with high risk persons; (3) People who live or have lived in countries where TB is common; (4) People who are health care workers who are in contact with TB patients when proper infection control procedures are not followed.
Many people who acquire TB infection do not have symptoms and may never develop TB disease. These people have latent TB infections (LTBI) [24]. After exposure to the bacillus of tuberculosis, some people develop a primary infection, the "primary infection", which is controlled by the immune system in 90% of cases: tuberculosis is labeled "latent". The bacillus remains in the body, but the immune system prevents its multiplication.
In 10% of infected people, the bacillus is not sufficiently controlled by the immune system and these people develop a form of so-called "active" tuberculosis, which will cause illness and complications. The organs most often affected by tuberculosis infection are the lungs (more than two-thirds of cases): it is "pulmonary tuberculosis", which is also the contagious form of the disease, Tuberculosis can also infect lymph nodes ("lymphadenopathy"), skin, kidneys, brain ("meningitis"), bones, intestines: it is "extra-pulmonary tuberculosis", which is the noncontagious form . After contact with the bacillus of Koch occurs an incubation phase where the bacteria fight against the immune defenses of the infected person in order to develop. It lasts from one to three months and usually goes unnoticed, but Koch's bacillus, which can remain dormant in the body for years, can also wake up to develop the infection because of the secondary weakening of the immune system of the person affected (HIV, chemotherapy, immunosuppressive treatments) [3]. With early antibiotic treatment and well followed, "tuberculosis disease" usually heals without leaving sequelae (treatment combining 4 antibiotics = quadrotherapy). On the other hand, if the treatment is not treated correctly, the cure will not be obtained and the bacillus will become resistant to the usual antibiotics obliging to resort to heavier and more complicated treatments [8].
In Morocco, tuberculosis remains a major public health problem,. . It is estimated that more than 87% of incident TB cases are detected and treated. Furthermore, the treatment success rate is more than 88% among TB patients who are put on treatment. These high rates in detection and treatment success are likely to contribute to significantly decreasing TB-related deaths. The analysis of the data generated by the NTP information system suggests that the transmission of TB is likely declining in general population. Even though there is a steady annual decrease in TB incidence, this decrease is low in general population. At this decrease pace, the decline in TB burden is likely to remain significant for many of the coming years.
The goal of this national strategic plan (NSP), which covers the timeframe from 2018 to 2021, is to fit within the sustainable development goals and to reduce the number of TB deaths by 40% in 2021 compared to 2015. Indeed, this NSP aims at increasing more TB case detection and treatment success rate, especially in highly urbanized regions, through the improvement and strengthening of the existing NTP services, the involvement of all care providers and the reinforcement of TB services for high-risk groups and vulnerable populations.
Moreover, in order to reduce the number of TB deaths, this NSP also aims at improving and strengthening TB/HIV joint activities and the programmatic management of drug-resistant TB.
To develop and implement these interventions, highlighted above, it is clear that the managerial capacities of the NTP need to be improved and reinforced at all levels [27] and [16]. [2].
The reasons for adopting discrete modeling are as follows: Firstly, the statistical data are collected at discrete moments (day, week, month, or year). So, it is more direct and more accurate and timely to describe the disease using discrete time models than continuous time models.
Secondly, the use of discrete time models can avoid some mathematical complexities such as choosing a function space and regularity of the solution. Thirdly, the numerical simulations of continuous time models are obtained by the way of discretization.
Based on the aforementioned reasons, we will develop in this paper a discrete time model studying the dynamics of Koch bacillus spread and introduce a mortality rate due to active MTB infection. In addition, in order to find the best strategy to reduce the number of susceptible, infected who have active MTB or recently and persistent infected latent, we will use four control strategies, namely vaccination and treatment programs, tests to detect the disease and take the TB drugs regularly and to complete them. In this paper, we construct a discrete SL 1 IL 2 R Mathematical TB Model. In Section 2, the mathematical model is proposed. In Section 3, we investigate the optimal control problem for the proposed discrete mathematical model. Section 4 consists of numerical simulation through MATLAB.The conclusion is given in Section 5.

FORMULATION OF THE MATHEMATICAL MODEL
In the present paper, following, we consider a TB mathematical model taken from [7],where reinfection and post-exposure interventions, consisting of a system of non-linear ordinary differential equations representing population dynamics. in The model without controls, the population total is divided into five categories: (S) : susceptible, who have never encountered the Mycobacterium; (L 1 ): early latent, that is, individuals recently infected (less than two years) but not infectious; (I): infected, that is, individuals who have active tuberculosis and are infectious; (L 2 ): persistent latent, that is, individuals who were infected and remain latent; (R): recovered, that is, individuals who were previously infected and treated.
Individuals in the early latent compartment L 1 can progress either to active disease (I) with rate φ δ or to a persistent latent infection (L 2 ) with rate (1 − φ )δ , following the approach in [28]. Parameter φ reflects that only 5% of infected individuals will ever develop active TB [21] , [20]. We choose δ such that the progression rate from early infections to active disease is φ δ = 0.6yr −1 , which roughly approximates the data by [22] , describing the proportions of disease development after conversion. For the rates of reactivation we adopt ω = 0.0002yr −1 for untreated latent infections [23] , [25] and ω R = 0.00002yr −1 . for those who have undergone a therapeutic intervention. As in Gomes et al. (2004a) [9], the partial susceptibility factor affecting the rate of exogenous reinfection of untreated individuals, σ , is fixed at 0.25, in accordance to the highest estimates of protection conferred by BCG vaccination see [1] .In treated patients this factor becomes σ R , for which several exploratory values are adopted. Treatment of different infection stages is implemented at specific rates: τ 0 applies to active TB and represents the rate of recovery (typically as a result of treatment, though here it also accounts for the infrequent natural recovery); τ 1 and τ 2 apply, respectively, to the latent classes L 1 and L 2 as the rates at which chemotherapy or a post-exposure vaccine is applied. The rate τ 0 is fixed at 2yr −1 , corresponding to an average duration of infectiousness of 6 months, while τ 1 and τ 2 are considered at different exploratory values. see [10].
The proportions of the population in each category change, as represented by the diagram in Fig. 1; According to Silva and Torres [See [19]] , the Tuberculosis modelled is described by the nonlinear time-varying state equations: where Λ is the recruitment rate, µ is the natural per-capita mortality rate, d is the per-capita TB induced mortality rate, β is the transmission rate, with The initial conditions for system (1) are: The values of the model parameters presented in the control system (1) are given in Table 1. The values of the rates β , δ , µ, σ , σ R , ω, ω R , φ , τ 0 , τ 1 , and τ 2 are taken from [10] and the references cited therein.

THE OPTIMAL CONTROL OF A TUBERCULOSIS MODEL
The model includes control variables representing vaccination or prevention and treatment measures, which are continuously implemented during a considered period of disease treatment: We now consider the TB model (1) and introduce four control functions u 1 (.), u 2 (.), u 3 (.) and u 4 (.), and four real positive model constants ε 1 , ε 2 , ε 3 and ε 4 , The resulting model is given by the following system of non-linear differential equations:    Then, the objective functional is presented as follows: (3) where the parametrs B t > 0,C t > 0, D t > 0, E t > 0 and A i,t > 0, for i = 1, 2, 3, 4 are the cost coefficients. They are selected to weigh the relative importance of S t , L 1,t , I t , L 2,t and u 1,t , u 2,t , u 3,t , u 4,t at time t.
T is the final time. We are minimizing the number of susceptible individuals, early latent individuals, infected individuals and persistent latent individuals during the time steps t = 0 to T − 1, and at the final time and also minimizing the cost of administering the control.
In other words, we seek the optimal control u * = (u * 1 , u * 2 , u * 3 , u * 4 ) such that : Where U ad is the set of admissible controls defined by: The sufficient condition for the existence of optimal controls (u 1 , u 2 , u 3 , u 4 ) for problem (2) and (3) comes from the following theorem: Theorem 1. There exists an optimal control (u * 1 , u * 2 , u * 3 , u * 4 ) such that: subject to the control system (2)  and corresponding sequences of states S j , L j 1 , I j , L j 2 , and R j . Since there is a finite number of uniformly bounded sequences, there exist (u * 1 , u * 2 , u * 3 , u * 4 ) ∈ U ad and S * , L * 1 , I * , L * 2 , and R * ∈ IR T +1 such that, on a subsequence, (u Finally, due to the finite dimensional structure of system (2) and the objective function J(u 1 , is an optimal control with corresponding states S * , L * 1 , I * , L * 2 , and R * . Therefore inf is achieved. see [14] In order to derive the necessary conditions for optimal control, the Pontryagin's maximum principle, in discrete time, given in [17] was used. This principle converts into a problem of minimizing a Hamiltonian, H t at time step t defined by: where f j,t+1 is the right side of the system of difference equations (2) of the j th state variable at time step t+1 .
By the bounds in U ad of the controls, it is easy to obtain u * 1,t , u * 2,t , u * 3,t , and u * 4,t in the forme of (9).

NUMERICAL SIMULATIONS
4.1. Algorithm: In this section, we present the result obtained by solving numerically the optimality system.
Step 2: for i=0;1;....;T-1,do: Step 3: for i=0;1;....;T write:    In this strategy,we will use four controls: the BCG vaccination u 1 , the effort on early detection and treatment of recently infected u 2 , the effort that prevents the failure of treatment in active TB infectious individuals u 3 and the effort to identified the persistent latent individuals and put them under treatments u 4 to optimize the objective function J(u).
In Fig. 5, we observe that there is a significant decrease in the number of susceptible individuals, early latent individuals, infected, and persistent latent individuals, controled compared with those not controled, and an increase the number of recovered individuals.

CONCLUSION
In this paper, we introduced a discrete modeling of TB tuberculosis in order to minimize the number of susceptible individuals, early latent individuals, infectious individuals, and persistent latent individuals, we also introduced four controls which, respectively, represent BCG vaccination, and the effort on early detection and treatment of recently infected, and the effort that prevents the failure of treatment in active TB infectious individuals, and the effort to identifie the persistent latent individuals and put them under treatment. We applied the results of the control theory and we managed to obtain the characterizations of the optimal controls. The

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.