DISCRETE MATHEMATICAL MODELING AND OPTIMAL CONTROL FOR BAYOUD DISEASE OF DATE PALM

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this work, we propose a discrete mathematical model that describes the Bayoud, the disease caused by the Fusarium oxysporum fungi [24], which spreads by contact between palm roots and the Fusarium oxysporum soil fungi, the susceptible palm (S), the infected (I), the recovered palm (R), the Fusarium oxysporum fungi (C). We also focus on the importance of treatments to find the optimal strategies to minimize the number of infected palms and also the number of Fusarium oxysporum fungi population and maximize the number of the recovered palm under treatment. We propose three optimal control strategies for this disease in order to slow down the growth of Fusarium oxysporum, the agent of Bayoud’s disease. The result of this optimal system is solved numerically by Matlab. Consequently, the obtained results confirm the performance of our optimization strategy.


INTRODUCTION
The date palm (Phoenix dactylifera L.) is one of the oldest cultivated plant species, best adapted to the difficult climatic conditions of the Saharan and pre-Saharan regions. It produces fruits rich in nutrients, provides a multitude of secondary products, and is subsequently a key This vascular fusarium, commonly known as Bayoud, particularly affects the best dateproducing palm varieties; several million trees have been destroyed two−thirds of date palm in North Africa, especially in Morocco (more than 10 million trees) and in the western and central parts of Algeria in 20th centuries [17]. Fusarium oxysporum f.sp. was also reported as a potential danger to date production in California. It is reported that Fusarium spp., which negatively affected social and economic ecosystems in these regions.
One of the first typical external symptoms of a Bayoud attack is a unilateral drying and bleaching of one or more palms (leaflets and rachis) in the middle crown of the palm. This symptom is at the origin of the name of this disease, Bayoud derives from the Arabic word "abyed" which means white [23], and from the special form of Fusarium oxysporum which is responsible for it, albedinis, from the Latin albus (white).
The same symptoms then appear on neighboring palms, and the attack then spreads to the whole palm, which rapidly withers.
But the symptoms are not always so typical and it has already happened to confuse an attack of Bayoud with drying out due to water stress. A thorough examination of diseased plants is therefore necessary to locate and identify the parasite.
One of the first typical external symptoms of a Bayoud attack is a unilateral drying and bleaching of one or more palms, (leaflets and rachis) in the middle crown of the palm. This symptom is at the origin of the name of the disease, Bayoud derives from the Arabic word abyed which means white, and from the special form of Fusarium oxysporum which is responsible for it, albedinis, from the Latin albus (white). The same symptoms then appear on neighbouring palms, and the attack then spreads to the whole palm, which rapidly withers.
But the symptoms are not always so typical and it has already happened to confuse an attack of Bayoud with drying out due to water stress. A thorough examination of diseased plants is therefore necessary to locate and identify the parasite [ [16] in which ney considered a fixed population with only three compartments, asceptible (S), infected (I) and recovered (R).
Mathematically the spatial spread of palms infected with the fungus Fusarium oxysporum in a Date Palm Oasis can be first modeled using a system of SIRS −C with four compartments.
• The compartment S : The number of palm trees belonging to the area that are not sick but likely to become sick (e.g. old, infected, poorly watered, rocky soil, ...).
• The compartment I : The number of palm trees infected with bayoude.
• The compartment R : The number of palm trees that have already had the disease and are now immune to bayoud.
• The compartment C : The growth rate of the fungus Fusaruim oxyporum.
The following diagram shows the flow directions of the palms between the compartments.

Model equations.
The susceptibles palm S i becomes infected at rate γ when they come in contact with the Fusarium oxysporum . That is, the change in population is equal to −γS k C k .
In addition, individuals from the recovered group become susceptible again at a certain rate " η " to give ηR k , Thus, we have: The infected palm begins with adding what was just removed from the susceptible population, γS k C k and then a reduction in two ways i.e. palm can either recover or they are killed by the virus. They recovered from the virus at rate " θ " and are killed at rate " α ". Thus, we have: The recovered palm R i is increased by those that recovered from the virus and reduced by the number of ppalm that join the susceptible group at rate "λ ". This can be expressed as: The fact that the Fusarium oxysporum fungi population is increasing exponentially is not biologically satisfactory, because even if a population arrives in an environment containing all the necessary resources, which is the case for invasive species, a population cannot increase exponentially to infinity [12][19]. Self-regulation phenomena will therefore take place [9]. These phenomena are taken into account in Verhulst's model (1838), also known as the logistic growth model. This model, in discrete time, is written: with µ the growth rate and K the carrying capacity of the environment.
infection model by the following system of difference equations:

Numerical simulation without control.
In this section, we shall solve numerically the optimal control problem for our SIRS − C 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 four difference equations and boundary conditions. The optimality system can be solved by using an iterative method. Using an initial guess for the infected population increases too, and for the recovered it decreases. So, we can deduce that importance of controlling these populations.

THE OPTIMAL CONTROL PROBLEM APPROACH
In the absence of the available effective treatment for Bayoud's disease caused by the fungus Fusarium oxysporum, infected palms are not controlled, they can serve the disease, and the fungus can spread. According to the previous simulations, it can be seen that without the introduction of an external strategy, the entire palm population is doomed to disappear. So adopted the following mathematical model we introduce our controls into system (1) as the control measures to fight the spread of the fungus Fusarium oxysporum; we extend our system by including three kinds of controls v and w. The first control v is the selection of cultivars and clones resistant to Bayoud and of good date quality, and also the use of fungicides with systemic or endotherapeutic action. the second straegy is applying the control w on the Fusarium oxysporum fungi population for stop its propagation.
for exemple ε 1 = 1 and ε 2 = 0 it means that we apply only a single control,ε 1 = 1 and ε 2 = 1 it means that we apply two controls.
and we consider the objective functional : where the parameters a k > 0 , b k > 0 and c k > 0 are the cost coefficients; they are selected to weigh the relative importance of S k , I k , R t k ,C t k , v k , and w k at time k.N is the final time. In other words, we seek the optimal controls v k , and w k such that where U ad ad is the set of allowable controls defined by : The sufficient condition for the existence of optimal controls (v k , w k ) for problems ( 3.1 ) and (3.2 ) comes from the following theorem: There exists an optimal control v * k , w * k such that is finite, and there exists a sequence (v n k , w n k ) ∈ U ad such that and corresponding sequences of states S n → S, I n → I, R n → R, C n → C Since there is a finite number of uniformly bounded sequences, there exist (v * k , w * k ) ∈ U ad and (S * , I * , R * , C * ) ∈ R t end +1 such that, on a subsequence, Finally, due to the finite dimensional structure of system (3.1) and the objective function is an optimal control with corresponding states (S * , I * , R * , C * ). Therefore In order to derive the necessary condition for optimal control, the pontryagins maximum principle in discrete time. This principle converts into a problem of minimizing a Hamiltonian H at time step k defined by where f i,k+1 is the right side of the system of difference equations (3.1) of the i th state variable at time step k + 1 Theorem 3.2. Given an optimal control v * k , w * k and the solutions S * k , I * k , R * k and C * k of the corresponding state system (3.1), there exist adjoint functions λ 1,k , λ 2,k , λ 3,k , and λ 4,k satisfying With the transversability conditions at time N, λ 1,N = λ 5,N = 0, λ 2,N = B N , λ 3,N = A N , and λ 4,N = C N Furthermore, for k = 0, 1, 2 . . . N − 1 the optimal controls v * k , and w * k are given by Proof. The Hamiltonian at time step k is given by For k = 0, 1 . . . N − 1 the optimal controls u k , v k , w k can be solved from the optimality condition, we get :

NUMERICAL SIMULATION WITH CONTROL
In this part of the numerical simulation, we deal with three cases. In the first case we apply a single control, in the second case two controls. In the case of a single control, we start with the optimal control v that is applied for the infected population and its objective is to minimize this population. Next, we discuss the results obtained by applying control w that has as purpose the minimization of the Fusarium oxysporum fungi population. Then, we move on to show the importance of increasing the number of controls in the study.

Date palm resistant to bayoud and the genetic control. The first strategy to control
Bayoud is to select disease-resistant material of good date quality. Use the resistance of certain palm varieties to Bayoud [7]. The search for more resistant palms is also the general direction taken to control vascular fusarium of the different cultivated palms. This method has the disadvantage of being long to implement, not allowing the conservation of certain sensitive varieties that are highly appreciated for their yield and the quality of their fruit [27].
As another solution, the world has rallied around a genetic fight against the disease[2] [5]. A second strategy in the field has been to induce the date palm's defense reactions using salicylic acid (SA) [18] .
This led to a significant reduction in mortality rates of date palms inoculated with Foa. This result was obtained in correlation with a marked increase in the content of phenolic compounds, H2O2 and malondialdehyde on the one hand and phenylalanine ammonia-lyase and peroxidase activities on the other. In addition, we noted that activation of these components of date palm resistance by SA was greater after inoculation of the pathogen [20].  This method is ruled out, as the practical possibilities for the use of systematic fungicides for tracheomycoses are very limited [21]. In addition, these products are not very stable in the soil and may favor the selection of resistant strains [22]. If used repeatedly over many years, these chemicals can harm the environment [29].
In the figure 4 , we illustrate the results obtained after applying the control on the Fusarium oxysporum fungi population. Here, the alarming result is that even if we apply this control for minimizing the Fusarium oxysporum fungi population over time, this last one still increasing.
However, the number of susceptible is minimized, it decreases to zero. The number of infected becomes lower than that in the case without control, it does not exceed 200. The recovered increases and after a certain time it decreases so as not to exceed the number of recovered in the case without control.  We can deduce that the results obtained in this case are more convincing than those of the case of a single control w. Because, the control w alone leads to an increase in the number of the Fusarium oxysporum fungi population, however, when a second control is added, we can minimize the number of individuals of this population. And the same goes for the susceptible and the infected controls.

CONCLUSION
In this work, in spite of their inapplicability in areas infested with on a large scale, the results of our study show that combination of reproduction of species resistant to Bayoud's disease and genetic control remains the most effective way to fight the disease, and in case of introduction

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