A Comparative Analysis of Different Fractional Optimal Control Strategies to Eradicate Bayoud Disease in Date Palm Trees

Abstract

Bayoud disease, caused by Fusarium oxysporum f. sp. albedinis, is a major threat to date palm trees. It leads to lower crop yields, financial losses, and decreased biodiversity. The complexity of the disease presents challenges to effective disease management. This study introduces a mathematical model comprising six compartments for palm trees: susceptible trees, resistant varieties, exposed trees, infected trees, isolated trees under treatment, and recovered trees, along with a contaminant water compartment. The model emphasizes the role of resistant varieties, contamination of irrigation water, and the treatment of infected trees in disease control. Theoretical analyses guarantee positivity, boundedness, and the existence of a unique solution. The existence of equilibrium points (disease-free and endemic) and the reproduction number (${\mathcal{R}}_0$) of the model are calculated analytically and validated through numerical simulations. Stability analysis at disease-free and endemic equilibrium points is conducted in terms of ${\mathcal{R}}_0$. Sensitivity analysis identifies key parameters influencing disease dynamics and is helpful to identify the potential control parameters. An optimal control problem is formulated to minimize infection spread and associated costs via preventive isolation and treatments, irrigation water treatment, and the promotion of resistant varieties. Numerical simulations demonstrate the efficacy of these strategies, highlighting the potential of resistant varieties and treatment measures in reducing infection rates and enhancing tree health. This research offers valuable insights into sustainable Bayoud disease management, underscoring the importance of mathematical modeling in addressing agricultural challenges.

1. Introduction

The fungus Fusarium oxysporum induces Fusarium wilt disease that badly affects date palms. This disease mainly attacks the tree’s vascular system and is known as Bayoud disease [1]. Since its discovery in the early 20th century, Bayoud has significantly reduced agricultural productivity and caused major financial losses. The fungus spreads through tiny structures in the soil that can survive for a long time, so it is difficult to eliminate the disease. Over time, Bayoud has emerged as a significant global menace to palm trees, resulting in extensive damage and plant fatalities. There are different types of fungus, and some of them may only infect certain kinds of date trees [2]. Effective disease management includes maintaining hygiene by removing and disposing of infected trees and taking preventive measures to reduce their spread. Bayoud disease continues to be a significant problem for agriculture and biodiversity despite efforts to reduce its impact. Therefore, date palm growers and farming experts must be able to recognize the symptoms of Bayoud disease early for quick detection and management. The disease usually begins with browning and wilting of adult leaves, which spreads from the lower fronds upwards. Affected trees may exhibit delayed growth, decreased vigor, and overall poor health. As the disease spreads, more fronds dry off, beginning with the lower leaves and gradually progressing up. In severe situations, the palm’s crown may weaken or fall as a result of vascular injury. Infected trees typically perish within a few weeks to several months. The symptoms may vary depending on the palm species and Fusarium strain. Early detection and therapy of the disease are critical for preventing its spread and minimizing its impact. Without intervention, the disease can cause palm trees to die within a few months to a year after symptoms first appear.

Date palms are prone to various viruses that harm the plants and cause major financial losses. Many of these diseases result from fungal infections [3]. One such fungus is Fusarium, a soil-borne pathogen with elongated spores that thrive in soil and organic matter. Because Fusarium can adapt to different environments, it is found worldwide. While most strains of Fusarium oxysporum are harmless to plants, some can cause dieback in date palms, leading to reduced yields [4,5,6]. A key early sign of the disease is the drying and discoloration of fronds, usually in the middle crown of the palm. This symptom gives the disease its name. The infection can spread to nearby palms, causing similar damage. In some cases, the symptoms resemble those of water stress, so careful inspection is needed to confirm the presence of the fungus. Currently, there is no direct treatment for Bayoud illness. However, various preventive measures can be taken to protect healthy date palm groves. These include planting resistant palm varieties, using chemical treatments like fumigation and fungicides, and removing, burning, or isolating infected trees. Naturally resistant or specially bred date palm varieties have been effective in controlling disease spread and reducing financial losses [7,8]. Chemical treatments, while useful, can harm beneficial soil microbes, accumulate in the food chain, and pose risks to human health and the environment [9]. They may also encourage the development of resistant fungal strains [10]. Despite these concerns, chemical control remains a practical option if the infection source is identified early and appropriate fungicide levels are used. Preventive measures, such as avoiding the transfer of infected plant material between orchards, are also recommended [11]. Research shows that early detection plays a crucial role in controlling and preventing the spread of the disease [12,13,14]. A study in [15] proposed a mathematical model to understand how fungal diseases spread in plants. This model helps determine the right fungicide dosage for effective plant protection.

Mathematical modeling is a valuable tool for researchers and decision-makers to study and predict how infectious diseases spread in populations. These models provide important insights that help develop strategies for disease prevention, control, and resource management [16,17,18]. One common method used in disease modeling is compartmental modeling [19,20,21]. Many researchers have applied mathematical modeling and optimal control theory to reduce the spread of epidemic diseases and manage the costs of control strategies [16,20,22]. For example, a study on pine wilt disease, which beetles cause, found that the most effective control strategies included deforestation, tree injections, and insecticide spraying [23]. Another study [24] developed a delay mathematical model to examine how time delays affect disease control and suggested ways to prevent pine wilt. Similarly, research in [25] identified advertising and pesticide use as the best methods for controlling pests. In their study [26], the authors introduced a new database containing images of date palm fronds, including both healthy and Bayoud-affected fronds. They used various augmentation techniques to increase the database size. They then developed an automated system for detecting Bayoud disease using a deep learning algorithm. The trained model achieved precision and recall rates of up to 100%, with a very small loss value of $4\times {10}^{-10}$. Their deep learning-based model successfully identified and classified disease-affected fronds. This demonstrates that the proposed database accurately represents Bayoud disease and helps extract important features, making disease recognition easier. In [27], the author developed and investigated a compartmental model that represents the interactions of palm trees, Rhynchophorus ferrugineus (red palm weevil), and entomopathogenic nematodes as part of an integrated pest control strategy. They determined the system’s equilibrium points and performed a stability study to better understand its behavior. In addition, they developed a linear quadratic regulator (LQR) to regulate the spread of the red palm weevil in a locally linearized system. The feedback control law is simple and easy to implement, eliminating the need for sophisticated cost computations and simplifying the optimal control problem. In their research [28], the authors presented a high-quality, chromosome-scale genome assembly of the virulent Fusarium oxysporum isolate Foa 44, providing valuable insights into the genetic basis of Bayoud disease. The genome assembly consists of 11 chromosomes and 40 unplaced contigs, totaling 65,971,825 base pairs. It has a GC content of 47.77% and contains 17.30% transposable elements (TEs). Through gene prediction and annotation, they identified 20,416 protein-coding genes. By analyzing gene and repeat densities and aligning them with the genome sequence of Fusarium oxysporum f. sp. lycopersici (FOL) 4287, they identified core and lineage-specific regions in Foa 44, shedding light on its genome structure. Additionally, a phylogenomic analysis based on 3292 core genes from the BUSCO dataset revealed a distinct clade of FOA isolates within the Fusarium oxysporum species complex (FOSC). In [29], the authors proposed a basic discrete susceptible-infected-recovered (SIR) model, incorporating a compartment for the fungus Fusarium oxysporum. They then analyzed the model using optimal control theory. Their findings indicate that increasing the reproduction of resistant palm tree species and implementing genetic control strategies are the most effective ways to prevent the spread of Bayoud disease.

This research contributes to the scientific literature on plant disease modeling by addressing a critical agricultural challenge and providing valuable insights for the effective and sustainable management of disease. We proposed a novel mathematical model that illustrates the progression of Bayoud disease in date palms and provides guidelines for efficient disease control. The model includes six tree compartments: susceptible trees, resistant varieties, exposed trees, infected trees, trees undergoing treatment and isolation, recovered trees, and a contaminant water compartment. This framework provides an opportunity to explore key factors influencing disease dynamics, emphasizing the potential benefits of resistant date palm varieties and the effectiveness of protective measures. A crucial aspect of the model is the inclusion of a resistant variety compartment, which is essential for analyzing genetic resistance to Bayoud disease. Resistant date palm varieties, whether naturally occurring or bred, can help mitigate disease spread and reduce economic losses. The trees in the isolated treatment compartment are assumed to have no root contact with other trees and are supplied with a separate irrigation system. Additionally, the soil around the roots of isolated trees is assumed to be replaced with fresh, fungi-free soil. A thorough mathematical analysis of the proposed model will be conducted to examine key properties, including positivity, boundedness, and the existence and uniqueness of solutions. Stability will be assessed at both disease-free and endemic equilibrium points to identify conditions under which the disease may be eradicated or persist. An optimal control framework will also be developed to identify cost-effective disease management strategies, integrating treatment, protection, and the promotion of resistant varieties [30,31]. Our manuscript is designed as follows: the formulation of a new mathematical model that describes the dynamics of Bayoud disease affecting date palm trees, and we conduct a theoretical analysis covering the existence of a unique solution, its boundedness and positivity, the identification of equilibrium states, and the calculation of the reproduction number (Section 2). Conditions for the local and global asymptotic stability analysis at the disease-free and endemic equilibrium points are determined theoretically and validated through numerical simulations by varying both the initial conditions and the range of the model’s parameters (Section 3). We analyze the impact of infection rate using the sensitivity analysis tool and identify the parameters that have more influence on the dynamics of disease (Section 4). The formulation of a fractional-order optimal control problem is discussed in detail to recommend effective disease control strategies (Section 5). A detailed discussion on the numerical results, graphical insights, and discussions (Section 6), along with concluding remarks on the directions for future research in disease management for date palm trees, are also provided (Section 7).

2. Formulation of the Mathematical Model

A study in [32] introduced a model for pest control using biological insecticides. In [33], researchers developed a mathematical model to understand the spread of Bayoud disease and find ways to reduce infections in date palm trees. They suggested the best fungicide application rates and optimal quarantine measures to control the disease. However, there is still little research on mathematical models for Bayoud disease and effective ways to manage its spread. Fractional calculus extends traditional calculus by introducing derivatives and integrals of non-integer orders, providing a powerful framework for analyzing complex systems like epidemics [34,35,36]. Fractional epidemic models are more detailed and effective than standard integer-order models because they naturally incorporate memory and hereditary effects, allowing past states to influence present dynamics [37,38,39]. Various fractional operators, such as Riemann-Liouville (RL), Caputo, and Caputo-Fabrizio (CF), are used in epidemic modeling [40,41]. Among them, the ABC fractional operator has gained recognition for its advanced modeling capabilities [42,43,44]. It overcomes the limitations of other fractional operators by combining non-local and non-singular properties, offering a more realistic representation of memory effects. In one study [45], researchers analyzed the dynamics of a two-stage plant disease (TSPD) using two fractional operators: the Caputo fractional derivative (CFD) and the Caputo-Fabrizio fractional derivative (CFFD), both with arbitrary orders in the range $\omega \in (0,1]$. They examined the impact of curative and preventive treatments on plant disease transmission. The study confirmed that the model produces non-negative and stable solutions, which is essential in population dynamics. In this study, we propose a $S{S}_{R}EIQR$ compartmental model, using the ABC fractional-order model, to capture the dynamics of Bayoud disease. The following assumptions are made for the proposed model.

• The total date palm trees ($N\left(t\right)$) are divided into six time-dependent compartments/classes, i.e., susceptible ($S\left(t\right)$), resistant varieties ($S_R\left(t\right)$) exposed/latent ($E\left(t\right)$), infectious ($I\left(t\right)$), quarantined ($Q\left(t\right)$), and recovered/removed ($R\left(t\right)$).
• A class for contaminant water $\left(W\right(t\left)\right)$ is considered for transmission of Bayoud through water, i.e., rain and water irrigations.
• It is assumed that Bayoud disease is transmitted through direct contact with the roots of infected and susceptible trees as well as through contaminated water.
• It is assumed that the quarantined class $\left(Q\right(t\left)\right)$ is isolated from the other classes and is under treatment. Therefore, there is no interaction of $Q\left(t\right)$ with other trees or contaminated water.
• The death or removal of date palm trees other than Bayoud is considered a natural death ($\nu$) and assumed to be the same in all compartments. Death from Bayoud disease is considered to occur only in $I\left(t\right)$ and $Q\left(t\right)$.
• The recovery of the trees after infections is considered in $R\left(t\right).$
• All the parameters and state variables are positive.
• All the state variables are continuously differentiable on the given domain, i.e., $t\in [0,\infty ).$

Thus, the total population at time t is described by the following equation.

$$N\left(t\right)=S\left(t\right)+S_R\left(t\right)+E\left(t\right)+I\left(t\right)+Q\left(t\right)+R\left(t\right).$$

(1)

Traditional integer-order models frequently fail to capture memory effects inherent in biological systems. To solve this problem, the traditional integer-order derivative ${\displaystyle \frac{d}{dt}}$ is replaced by the fractional ABC operator ${}_{0 }{}^{ABC}D_t^{\varsigma }$. This change allows us to better capture and study the memory effects on the dynamics of Bayoud illness. Before continuing, it is necessary to review the ABC derivative’s primary features. We first recall some fundamental notions related to Atangana–Baleanu fractional derivatives [46].

Definition 1 

([46]). If the differentiable function $\mathrm{\Phi }:[a,b]\to \mathbb{N}$ is defined on $[a,b]$ such that $\mathrm{\Phi }\in H^1(a,b)$, $b>a$ and $\varsigma \in (0,1]$, then the Atangana–Baleanu derivative of Φ in Caputo sense is defined as:

$$\begin{array}{c}\hfill {}_{a }{}^{ABC}D_t^{\varsigma }\mathrm{\Phi }\left(t\right)={\displaystyle \frac{F\left(\varsigma \right)}{1-\varsigma }}{\int }_a^t\dot{\mathrm{\Phi }}\left(\mathrm{\Omega }\right)E_{\varsigma }\left[-\varsigma {\displaystyle \frac{{(t-\mathrm{\Omega })}^{\varsigma }}{1-\varsigma }}\right]d\mathrm{\Omega },\end{array}$$

(2)

where $E_{\varsigma }$ is a well-known one parameter Mittag–Leffler function and $F\left(\varsigma \right)$ is a normalizing function such that $F\left(0\right)=1,F\left(1\right)=1$.

Definition 2 

([46,47]). An ABC fractional integral with a non-local kernel is defined by

$$\begin{array}{c}\hfill {}_{a }{}^{ABC}I_t^{\varsigma }\mathrm{\Phi }\left(t\right)={\displaystyle \frac{1-\varsigma }{F\left(\varsigma \right)}}\mathrm{\Phi }\left(t\right)+{\displaystyle \frac{\varsigma }{F\left(\varsigma \right)\mathrm{\Gamma }\left(\varsigma \right)}}{\int }_a^t\mathrm{\Phi }\left(\mathrm{\Omega }\right){(t-\mathrm{\Omega })}^{\varsigma -1}d\mathrm{\Omega }.\end{array}$$

(3)

The tree population, denoted as $S\left(t\right)$, is healthy yet vulnerable to disease, with a recruitment rate of $\mathrm{\Pi }$. The susceptible trees transition to the exposed compartment $E\left(t\right)$ following exposure to an infectious tree $I\left(t\right)$ and contaminant water $W\left(t\right)$ at the rates of ${\alpha }_2$ and ${\alpha }_3$, respectively. Trees that exhibit susceptibility but subsequently develop resistance to the virus, whether through natural resistance, genetic modification, or fungicide application, are categorized into the resistant variety compartment $S_R\left(t\right)$ and recruited at the rate of ${\alpha }_1$ from the $S\left(t\right)$. Trees exhibiting weak resistance or immunity from $S_R\left(t\right)$ may become infected following effective contact with $I\left(t\right)$ and $W\left(t\right)$, subsequently transferring to the exposed compartment $E\left(t\right)$. Let ${\beta }_2$ and ${\beta }_3$ represent the contact rate of weakly resistant varieties with infectious trees and contaminant water. Trees with high immunity or resistance are classified as recovered trees and subsequently transferred to the recovered/removal compartment $R\left(t\right)$ at a rate of ${\beta }_1$. Exposed trees become infectious at a rate of $\gamma$, transitioning to compartment $I\left(t\right)$. The infected trees are quarantined/isolated at rate $\tau$ for protective treatment. The trees in treatment compartment $Q\left(t\right)$ are expected to have no root contact and to utilize independent irrigation systems. Additionally, we posit that the soil infested around the roots of these trees has been replaced with fresh, fungi-free soil. Fungicide spray and other protective measures are employed to treat these trees. It is assumed that the trees experience natural mortality at a rate of $\nu$. It is assumed that ${\delta }_I$ and ${\delta }_Q$ represent the rates at which trees in compartments $I\left(t\right)$ and $Q\left(t\right)$ perish due to disease and are removed to prevent disease transmission. The Figure 1 depicts the progression of Bayoud disease across various compartments.

The governing equations describing the dynamics of the Bayoud disease through compartments are given as follows.

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }S\left(t\right)=\hfill & \mathrm{\Pi }-({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )S,\hfill \end{array}$$

(4)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }{S}_R\left(t\right)=\hfill & {\alpha }_{1}S-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R,\hfill \end{array}$$

(5)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }E\left(t\right)=\hfill & ({\alpha }_{2}I+{\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R-(\gamma +\nu )E,\hfill \end{array}$$

(6)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }I\left(t\right)=\hfill & \gamma E-(\tau +{\delta }_I+\nu )I,\hfill \end{array}$$

(7)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }Q\left(t\right)=\hfill & \tau I-(\theta +{\delta }_Q+\nu )Q,\hfill \end{array}$$

(8)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }R\left(t\right)\hfill & =\theta Q+{\beta }_1{S}_R-\nu R,\hfill \end{array}$$

(9)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }W\left(t\right)= \hfill & {\psi }_{1}I-{\psi }_{2}W,\hfill \end{array}$$

(10)

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & S\left(0\right)=S_0>0,S_R\left(0\right)={S_R}_0\ge 0,E\left(0\right)=E_0\ge 0,I\left(0\right)=I_0\ge 0,\hfill \\ \hfill & Q\left(0\right)=Q_0\ge 0,R\left(0\right)=R_0\ge 0,W\left(0\right)=W_0\ge 0.\hfill \end{array}\end{array}$$

(11)

The model states that $S\left(t\right),S_R\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right),R\left(t\right),\mathrm{and}W\left(t\right)$ real functions are considered continuously differentiable over the domain $[0,\infty )$. The following is the description of the model parameters.

• $\mathrm{\Pi }$: The rate at which susceptible date palm trees are recruited.
• $\nu$: Natural death rate of trees.
• ${\alpha }_1$ Recruitment rate of resistant varieties.
• ${\alpha }_2$: Interaction rate between susceptible and infectious trees through direct contact (i.e., roots).
• ${\alpha }_3$: Interaction rate between susceptible and contaminant water (i.e., rain and water reservoirs).
• ${\beta }_1$ Recovery/removal rate of the resistant varieties.
• ${\beta }_2$: Interaction rate between resistant varieties and infectious trees through direct contact (i.e., roots).
• ${\beta }_3$: Force of infection due to contaminant water in resistant varieties (i.e., rain and water reservoirs).
• $\gamma$: Recruitment rate of infectious trees.
• $\tau$: Quarantine and treatment rate of infectious trees.
• $\theta$: Rate of recovery/removal of quarantine and undertreatment trees.
• ${\delta }_Q$: Mortality rate of under isolation and treatment plants caused by disease.
• ${\delta }_I$: Mortality rate of infectious trees caused by disease.
• ${\psi }_1$: Inward flow of the disease-containing water.
• ${\psi }_2$ Outward flow of water leaving the containment water compartment.

For further analysis, we put the proposed model (Equations (4)–(11)) into the following compact form.

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathrm{\Phi }& =\mathcal{G}\left(\mathrm{\Phi }\left(t\right)\right),0<t<T_f<\infty \hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \mathrm{\Phi }\left(0\right)& ={\mathrm{\Phi }}_0,\hfill \end{array}$$

(13)

where functions $\mathrm{\Phi }\left(t\right):[0,T_f]\to R_{+}^7$ and $\mathcal{G}:R_{+}^7\to R_{+}^7$ are described as:

$$\mathrm{\Phi }\left(t\right)={\left(S\left(t\right),S_R\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right),R\left(t\right),W\left(t\right)\right)}^T,$$

with

$${\mathrm{\Phi }}_0={\left(S\left(0\right),S_R\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right),R\left(0\right),W\left(0\right))\right)}^T,$$

and

$$\mathcal{G}\left(\mathrm{\Phi }\left(t\right)\right)=\left(\begin{array}{c}{\mathcal{G}}_1\\ {\mathcal{G}}_2\\ {\mathcal{G}}_3\\ {\mathcal{G}}_4\\ {\mathcal{G}}_5\\ {\mathcal{G}}_6\\ {\mathcal{G}}_7\end{array}\right)=\left(\begin{array}{c}\mathrm{\Pi }-({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )S\\ {\alpha }_{1}S-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R\\ ({\alpha }_{2}I+{\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R-(\gamma +\nu )E\\ \gamma E-(\tau +{\delta }_I+\nu )I\\ \tau I-(\theta +{\delta }_Q+\nu )Q\\ \theta Q+{\beta }_1{S}_R-\nu R\\ {\psi }_{1}I-{\psi }_{2}W\end{array}\right).$$

2.1. Qualitative Properties of Bayoud Disease Model

This section provides a theoretical examination of the Bayoud disease model (Equations (4)–(11)). It examines important model aspects and demonstrates its suitability for numerical approximation. We ignore the contaminant water compartment/class (W) in this section because it is simply a flow of Bayoud disease-containing water.

Theorem 1. 

The solution $y\left(t\right)=(S\left(t\right),S_R\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right),R\left(t\right))$ of the Bayoud model (Equations (4)–(11)) is bounded.

Proof. 

Applying ${}_{0 }{}^{ABC}D_t^{\varsigma }$ to (Equation (1)), we obtain

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }N\left(t\right)=& {}_{0 }{}^{ABC}D_t^{\varsigma }S\left(t\right)+{}_{0 }{}^{ABC}D_t^{\varsigma }{S}_R\left(t\right)+{}_{0 }{}^{ABC}D_t^{\varsigma }E\left(t\right)+{}_{0 }{}^{ABC}D_t^{\varsigma }I\left(t\right)\hfill \\ & +{}_{0 }{}^{ABC}D_t^{\varsigma }Q\left(t\right)+{}_{0 }{}^{ABC}D_t^{\varsigma }R\left(t\right),\hfill \\ \hfill =& \mathrm{\Pi }-{\delta }_{I}I-{\delta }_{T}Q-\nu N.\hfill \end{array}$$

Since ${\delta }_{I}I+{\delta }_{T}Q\ge 0$, so

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }N\left(t\right)\le \mathrm{\Lambda }-\nu N\left(t\right).\end{array}$$

We apply the Laplace transform on both sides of the above inequality to obtain:

$$\begin{array}{c}\hfill \mathcal{L}\left\{{}_{0 }{}^{ABC}D_t^{\varsigma }N\left(t\right)\right\}\left(s\right)\le {\displaystyle \frac{\mathrm{\Pi }}{s}}-\nu \mathcal{L}\left\{N\left(t\right)\right\}\left(s\right),\end{array}$$

or

$$\begin{array}{c}\hfill {\displaystyle \frac{H\left(\varsigma \right)s^{\varsigma }}{\varsigma +(1-\varsigma )s^{\varsigma }}}N\left(s\right)+\nu N\left(s\right)\le \mathrm{\Pi }{s}^{-1}+{\displaystyle \frac{H\left(\varsigma \right)N\left(0\right)s^{\varsigma -1}}{\varsigma +(1-\varsigma )s^{\varsigma }}},\end{array}$$

(14)

Inequality (14) is solved to obtain:

$$\begin{array}{c}\hfill N\left(s\right)\le {\displaystyle \frac{\mathrm{\Pi }\mathrm{\Lambda }}{\nu }}{\displaystyle \frac{s^{\varsigma -(\varsigma +1)}}{s^{\varsigma }+\mathrm{\Lambda }}}+{\displaystyle \frac{\mathrm{\Lambda }}{\nu \varsigma }}\left[\mathrm{\Pi }(1-\varsigma )+H\left(\varsigma \right)N\left(0\right)\right]{\displaystyle \frac{s^{\varsigma -1}}{s^{\varsigma }+\mathrm{\Lambda }}},\end{array}$$

(15)

where $\mathrm{\Lambda }={\displaystyle \frac{\varsigma \nu }{H\left(\varsigma \right)+(1-\varsigma )\nu }}$.

The application of the inverse Laplace transform to both sides of (15) yields:

$$\begin{array}{c}\hfill N\left(t\right)\le {\displaystyle \frac{\mathrm{\Pi }\mathrm{\Lambda }}{\nu }}{E}_{\varsigma ,\varsigma +1}(-\mathrm{\Lambda }{t}^{\varsigma })+{\displaystyle \frac{\mathrm{\Lambda }}{\nu \varsigma }}\left[\mathrm{\Pi }(1-\varsigma )+H\left(\varsigma \right)N\left(0\right)\right]E_{\varsigma ,1}(-\mathrm{\Lambda }{t}^{\varsigma }).\end{array}$$

(16)

Since the Mittag–Leffler function

$$\begin{array}{c}\hfill E_{\varsigma ,\varsigma +1}(-\mathrm{\Lambda }{t}^{\varsigma })={\displaystyle \frac{1}{\mathrm{\Lambda }{t}^{\varsigma }}}\left[1-E_{\varsigma ,1}(-\mathrm{\Lambda }{t}^{\varsigma })\right],\end{array}$$

is bounded for all $t>0$, thus exhibiting asymptotic behavior. From (16), it is evident that $N\left(t\right)\le {\displaystyle \frac{\mathrm{\Pi }}{\nu }}$ as $t\to \infty$. Therefore, $N\left(t\right)$ and all other variables in the model (4)–(11) are constrained within specific limits.    □

Theorem 2. 

Given the non-negative initial conditions, the solution space $y\left(t\right)$ of the Bayoud model (Equations (4)–(11)) remains positive for all time $t>0$.

Proof. 

Let Equation (4) of the Bayoud model (Equations (4)–(11)).

$${}_{0 }{}^{ABC}D_t^{\varsigma }S=\mathrm{\Pi }-({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )S.$$

It is already proved that the solutions of the Bayoud model (Equations (4)–(11)) are bounded, thus, we can define a constant c as:

$$c=max({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu ).$$

Then,

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }S\left(t\right)\ge -cS\left(t\right).\end{array}$$

(17)

Application of the Laplace transform yields us:

$$\begin{array}{c}\hfill {\displaystyle \frac{H\left(\varsigma \right)s^{\varsigma }}{\varsigma +(1-\varsigma )s^{\varsigma }}}\left[S\left(s\right)\right]-{\displaystyle \frac{H\left(\varsigma \right)s^{\varsigma -1}}{\varsigma +(1-\varsigma )s^{\varsigma }}}S\left(0\right)\ge -c\left[S\left(s\right)\right].\end{array}$$

This can be solved to obtain

$$\begin{array}{c}\hfill \left[S\left(s\right)\right]\ge {\displaystyle \frac{H\left(\varsigma \right)S\left(0\right)\mathrm{\Delta }}{c\varsigma }}{\displaystyle \frac{s^{\varsigma -1}}{s^{\varsigma }+\mathrm{\Delta }}},\end{array}$$

where $\mathrm{\Delta }={\displaystyle \frac{c\varsigma }{H\left(\varsigma \right)+c(1-\varsigma )}}$.

Now, the application of the inverse Laplace transform along with the property of the Mittag–Leffler function gives:

$$\begin{array}{c}\hfill S\left(t\right)\ge {\displaystyle \frac{H\left(\varsigma \right)S\left(0\right)\mathrm{\Delta }}{c\varsigma }}{E}_{\varsigma ,1}\left(-\mathrm{\Delta }{t}^{\varsigma }\right)>0.\end{array}$$

(18)

Therefore, the solution variable $S\left(t\right)>0$ for all $t\ge 0$. Similarly, other state variables corresponding to any non-negative beginning data can be demonstrated to be positive for every $t\ge 0$. Thus, the solutions in ${\mathbb{R}}_{+}^6$ remain positive indefinitely.    □

On the basis of the above results, the feasible invariant region is defined as

$$\begin{array}{c}\hfill \mathrm{\Omega }=\left\{(S\left(t\right),S_R\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right),R\left(t\right))\in {\mathbb{R}}_{+}^6:0<N\left(t\right)\le {\displaystyle \frac{\mathrm{\Pi }}{\nu }}\right\},\end{array}$$

having non-negative initial conditions in ${\mathbb{R}}_{+}^6$.

2.2. Equilibrium Points

To find equilibrium points, we consider the rate of change in all the state variables equal to zero, i.e., ${}_{0 }{}^{ABC}D_t^{\varsigma }S\left(t\right)={}_{0 }{}^{ABC}D_t^{\varsigma }{S}_R\left(t\right)={}_{0 }{}^{ABC}D_t^{\varsigma }E\left(t\right)={}_{0 }{}^{ABC}D_t^{\varsigma }I\left(t\right)={}_{0 }{}^{ABC}D_t^{\varsigma }Q\left(t\right)={}_{0 }{}^{ABC}D_t^{\varsigma }R\left(t\right)={}_{0 }{}^{ABC}D_t^{\varsigma }W\left(t\right)=0,$ and to find the disease-free equilibrium point of an epidemic model, we assume that there is no infection present, i.e., $E=I=0$. Therefore, the resulting DFE point of the proposed model (Equations (4)–(11)) is given as follows:

$$\begin{array}{c}\hfill X^0=(S^0,S_R^0,E^0,I^0,Q^0,R^0,W^0)=\left({\displaystyle \frac{\mathrm{\Pi }}{{\beta }_1+\nu }},{\displaystyle \frac{{\alpha }_1\mathrm{\Pi }}{({\beta }_1+\nu )({\alpha }_1+\nu )}},0,0,0,{\displaystyle \frac{{\beta }_1{\alpha }_1\mathrm{\Pi }}{\nu ({\alpha }_1+\nu )({\beta }_1+\nu ),0}}\right),\end{array}$$

(19)

where it is interesting to note that the sum of all state variables equals to ${\displaystyle \frac{\mathrm{\Pi }}{\nu }}$:

In the endemic equilibrium (EE) condition, where $I\ne 0$, the fungus remains within the trees. The EE state is calculated by resolving the stationary equations of the model (4)–(11) with the constraint $I\ne 0$. The calculated EE point is defined as follows:

$$\begin{array}{c}\hfill X^{\ast }=(S^{\ast },S_R^{\ast },E^{\ast },I^{\ast },Q^{\ast },R^{\ast },W^{\ast }),\end{array}$$

(20)

From the model (Equations (4)–(11)):

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }W& ={\psi }_{1}I-{\psi }_{2}W=0⟹W^{\ast }={\displaystyle \frac{{\psi }_1}{{\psi }_2}}{I}^{\ast },\hfill \\ \hfill {}_{0 }{}^{ABC}D_t^{\varsigma }Q& =\tau I-(\theta +{\delta }_Q+\nu )Q=0⟹Q^{\ast }={\displaystyle \frac{\tau }{\theta +{\delta }_Q+\nu }}{I}^{\ast },\hfill \\ \hfill {}_{0 }{}^{ABC}D_t^{\varsigma }S& =\mathrm{\Pi }-({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )S=0,⟹S^{\ast }={\displaystyle \frac{\mathrm{\Pi }}{{\alpha }_1+\left({\alpha }_2+{\alpha }_3\frac{{\psi }_1}{{\psi }_2}\right)I^{\ast }+\nu }},\hfill \\ \hfill {}_{0 }{}^{ABC}D_t^{\varsigma }{S}_R& ={\alpha }_{1}S-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R=0,⟹S_R^{\ast }={\displaystyle \frac{{\alpha }_1{S}^{\ast }}{{\beta }_1+\left({\beta }_2+{\beta }_3\frac{{\psi }_1}{{\psi }_2}\right)I^{\ast }+\nu }},\hfill \\ \hfill {}_{0 }{}^{ABC}D_t^{\varsigma }R& =\theta Q+{\beta }_1{S}_R-\nu R=0,⟹R^{\ast }={\displaystyle \frac{\theta Q^{\ast }+{\beta }_1{S}_R^{\ast }}{\nu }}.\hfill \end{array}$$

From Equations (6) and (7),

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }E& =({\alpha }_{2}I+{\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R-(\gamma +\nu )E=0,\hfill \\ ⟹& E^{\ast }={\displaystyle \frac{\left({\alpha }_2+{\alpha }_3\frac{{\psi }_1}{{\psi }_2}\right)I^{\ast }{S}^{\ast }+\left({\beta }_2+{\beta }_3\frac{{\psi }_1}{{\psi }_2}\right)I^{\ast }{S}_R^{\ast }}{\gamma +\nu }},\end{array}$$

$$\begin{array}{cc}\hfill & E^{\ast }={\displaystyle \frac{\mathrm{\Pi }{I}^{\ast }\left[\left({\alpha }_2+{\alpha }_3\frac{{\psi }_1}{{\psi }_2}\right)\left({\beta }_1+\left({\beta }_2+{\beta }_3\frac{{\psi }_1}{{\psi }_2}\right)I^{\ast }+\nu \right)+{\alpha }_1\left({\beta }_2+{\beta }_3\frac{{\psi }_1}{{\psi }_2}\right)\right]}{\left({\alpha }_1+\left({\alpha }_2+{\alpha }_3\frac{{\psi }_1}{{\psi }_2}\right)I^{\ast }+\nu \right)\left(\gamma +\nu \right)\left({\beta }_1+\left({\beta }_2+{\beta }_3\frac{{\psi }_1}{{\psi }_2}\right)I^{\ast }+\nu \right)}},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }I& =\gamma E-(\tau +{\delta }_I+\nu )I=0,⟹E^{\ast }={\displaystyle \frac{(\tau +{\delta }_I+\nu )I^{\ast }}{\gamma }}.\hfill \end{array}$$

(22)

Comparing Equations (21) and (22), we obtain:

$$\begin{array}{cc}\hfill & (\tau +{\delta }_I+\nu )(\gamma +\nu )\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)I^{\ast 2}\hfill \\ \hfill & +\{(\tau +{\delta }_I+\nu )(\gamma +\nu )\left[({\alpha }_1+\nu )\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)+({\beta }_1+\nu )\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\right]\hfill \\ \hfill & -\gamma \mathrm{\Pi }\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\}{I}^{\ast }\hfill \\ \hfill +\left\{\right(\tau & +{\delta }_I+\nu )(\gamma +\nu )({\alpha }_1+\nu )({\beta }_1+\nu )-\gamma \mathrm{\Pi }\left[\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)({\beta }_1+\nu )+{\alpha }_1\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\right]\}=0.\hfill \end{array}$$

Or,

$$\begin{array}{c}\hfill A{I}^{\ast 2}-B{I}^{\ast }+C=0,\end{array}$$

(23)

where

$$\begin{array}{cc}\hfill A=& (\tau +{\delta }_I+\nu )(\gamma +\nu )\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right),\hfill \\ \hfill B=& (\tau +{\delta }_I+\nu )(\gamma +\nu )\left(({\alpha }_1+\nu )\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)+({\beta }_1+\nu )\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\right)\hfill \\ & -\gamma \mathrm{\Pi }\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right),\hfill \\ \hfill C=& (\tau +{\delta }_I+\nu )(\gamma +\nu )({\alpha }_1+\nu )({\beta }_1+\nu )-\gamma \mathrm{\Pi }\left[\left({\alpha }_2+{\alpha }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)({\beta }_1+\nu )+{\alpha }_1\left({\beta }_2+{\beta }_3{\displaystyle \frac{{\psi }_1}{{\psi }_2}}\right)\right].\hfill \end{array}$$

Thus, Equation (23) provides two solutions for $I^{\ast }$ based on the values of the model’s parameters: one is positive, and the other is negative, i.e.,

$$\begin{array}{cc}\hfill I^{\ast }=& {\displaystyle \frac{-B+\sqrt{B^2-4AC}}{2A}}\left(\mathrm{Positive}\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill I^{\ast }=& {\displaystyle \frac{-B-\sqrt{B^2-4AC}}{2A}}\left(\mathrm{Negative}\right).\hfill \end{array}$$

(25)

Since the tree population cannot be negative, we will only consider positive values of $I^{\ast }$ from this point onward. To better understand this, we numerically solve the equations for $I^{\ast }$ in relation to the model parameters, which helps validate the theoretical results (Figure 2).

Reproduction Number

We now determine the reproduction number, ${\mathcal{R}}_0$, for the Bayoud model (Equations (4)–(11)). This key parameter plays a significant role in determining whether an epidemic will propagate within a population or ultimately diminish. The value of ${\mathcal{R}}_0$ is obtained through the next-generation matrix method [48,49]. It is specifically described as the largest eigenvalue of the matrix $\mathbb{F}{\mathbb{G}}^{-1}$, where $\mathbb{F}$ is the Jacobian matrix that shows new infections and $\mathbb{G}$ is the Jacobian matrix that shows the rest of the transitional terms of the equations of the infectious compartments. The matrices corresponding to new infections and various transitions are specified as follows:

$$\mathcal{F}=\left(\begin{array}{c}({\alpha }_{2}I+{\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R\\ \\ 0\\ \\ 0\end{array}\right),\mathcal{H}=\left(\begin{array}{c}(\gamma +\nu )E\\ \\ -\gamma E+(\tau +{\delta }_I+\nu )I\\ \\ -{\psi }_{1}I+{\psi }_{2}W\end{array}\right).$$

The Jacobian of the matrices $\mathcal{F}$ and $\mathcal{H}$ evaluated at DFE point $R^0$ are

$$\mathbb{F}=\left(\begin{array}{ccccc}0& & {\displaystyle \frac{{\alpha }_2\mathrm{\Pi }}{{\beta }_1+\nu }}+{\displaystyle \frac{{\alpha }_1{\beta }_2\mathrm{\Pi }}{({\beta }_1+\nu )({\alpha }_1+\nu )\nu }}& & {\displaystyle \frac{{\alpha }_2\mathrm{\Pi }}{{\beta }_1+\nu }}+{\displaystyle \frac{{\alpha }_1{\beta }_2\mathrm{\Pi }}{({\beta }_1+\nu )({\alpha }_1+\nu )\nu }}\\ \\ 0& & 0& & 0\\ \\ 0& & 0& & 0\\ \end{array}\right),\mathbb{G}=\left(\begin{array}{ccc}\gamma +\nu & 0& 0\\ \\ -\gamma & \tau +{\delta }_I+\nu & 0\\ \\ 0& -{\psi }_1& {\psi }_2\end{array}\right).$$

Thus, the basic reproduction number ${\mathcal{R}}_0$ of the model (Equations (4)–(11)) is given as:

$${\mathcal{R}}_0={\displaystyle \frac{\mathrm{\Pi }\gamma ({\psi }_2({\alpha }_1{\alpha }_2\nu +{\alpha }_2{\nu }^2+{\alpha }_1{\beta }_2)+{\psi }_1({\alpha }_1{\alpha }_3\nu +{\alpha }_3{\nu }^2+{\alpha }_1{\beta }_3))}{(\gamma +\nu )(\tau +{\delta }_I+\nu )({\alpha }_1+\nu )({\beta }_1+\nu )\nu {\psi }_2}}.$$

(26)

3. Local and Global Stability of Equilibrium Points

In this section, we analyze the stability of the proposed model at both of the equilibrium points.

3.1. LAS at DFE

Since we are dealing with an epidemic model, hence, as a first case, we analyzed the stability at the DFE point $X^0$ in terms of ${\mathcal{R}}_0$. Therefore, we state and prove the following theorems:

Theorem 3. 

The proposed Bayoud model (Equations (4)–(11)) is locally asymptotically stable (LAS) at the disease-free equilibrium (DFE) point $X^0$ if ${\mathcal{R}}_0<1$ and unstable for ${\mathcal{R}}_0>1$.

Proof. 

This theorem can be proved using the Jacobian matrix approach. For this purpose, we need to find the Jacobian matrix around a disease-free equilibrium point, i.e.,

$$\begin{array}{c}\hfill J^{\ast }=\left(\begin{array}{ccccccc}-({\alpha }_1+\nu )& 0& 0& -{\alpha }_2{S}^0& 0& 0& -{\alpha }_3{S}^0\\ \\ {\alpha }_1& -({\beta }_1+\nu )& 0& -{\beta }_2{S}_R^0& 0& 0& -{\beta }_3{S}_R^0\\ \\ 0& 0& -(\gamma +\nu )& {\alpha }_2{S}^0+{\beta }_2{S}_R^0& 0& 0& {\alpha }_3{S}^0+{\beta }_3{S}_R^0\\ \\ 0& 0& \gamma & -(\tau +{\delta }_I+\nu )& 0& 0& 0\\ \\ 0& 0& 0& \tau & -(\theta +{\delta }_Q+\nu )& 0& 0\\ \\ 0& {\beta }_1& 0& 0& \theta & -\nu & 0\\ \\ 0& 0& 0& {\psi }_1& 0& 0& -{\psi }_2\end{array}\right).\end{array}$$

(27)

As we know, a system is said to be locally asymptotically stable at equilibrium if all the eigenvalues are negative around this equilibrium point and unstable otherwise. We obtain the following eigenvalues using Maple (version 7) software.

$$\begin{array}{cc}\hfill {\lambda }_1=& -\nu ,\hfill \\ \hfill {\lambda }_2=& -({\alpha }_1+\nu ),\hfill \\ \hfill {\lambda }_3=& -({\beta }_1+\nu ),\hfill \\ \hfill {\lambda }_4=& -(\gamma +\nu ),\hfill \\ \hfill {\lambda }_5=& -(\theta +{\delta }_Q+\nu ),\hfill \\ \hfill {\lambda }_6=& -(\tau +{\delta }_Q+\nu )\left[1-{\mathcal{R}}_0+{\displaystyle \frac{\gamma {\psi }_1({\beta }_3{S}_R^0+{\alpha }_3{S}^0)}{\left({\gamma }_{\nu }\right)(\tau +{\delta }_I+\nu )}}\right],\hfill \\ \hfill {\lambda }_7=& -(\tau +{\delta }_I+\nu )\left[{\displaystyle \frac{{\mathcal{R}}_0-1}{{\lambda }_6}}\right].\hfill \end{array}$$

(28)

All the eigenvalues are negative when ${\mathcal{R}}_0<1$ and at least Equation (28) is positive otherwise. Thus, the proposed Bayoud model (Equations (4)–(11)) is locally asymptotically stable at the DFE point $X^0$.    □

Global Stability Analysis

The Castillo–Chavez method [50] plays a crucial role in determining the global stability of the disease-free stationary point ${\mathcal{X}}^0$. To implement this method, it is required to present the model (4)–(11) in the specific format described below.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{V}\left(t\right)=& \mathrm{\Lambda }(\mathbb{V},\mathbb{I}),\hfill \\ \hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{I}\left(t\right)=& \mathrm{\Psi }(\mathbb{V},\mathbb{I}),\mathrm{\Psi }(\mathbb{V},0)=0.\hfill \end{array}\end{array}$$

(29)

In (29), $\mathbb{V}={(S,S_R,R)}^T$ represents the uninfected tree compartments and $\mathbb{I}={(E,I,Q,W)}^T$ denotes the compartments with infected trees. The point $X^0=({\mathbb{V}}^0,0)$ is a disease-free point of the given disease model.

According to Castillo–Chavez, the following conditions must hold for global asymptotic stability of the system (29).

$$\begin{array}{cc}\hfill \mathbf{Cond}\mathbf{1}:{}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{V}\left(t\right)& =\mathrm{\Lambda }(\mathbb{V},0),{\mathbb{V}}^0\mathrm{is}\mathrm{asymptotically}\mathrm{stable}\mathrm{on}\mathrm{a}\mathrm{global}\mathrm{scale},\hfill \\ \hfill \mathbf{Cond}\mathbf{2}:{}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{I}\left(t\right)& =\mathrm{\Psi }(\mathbb{V},\mathbb{I})=\mathcal{AI}-\widehat{\mathrm{\Psi }}(\mathbb{V},\mathbb{I}),\widehat{\mathrm{\Psi }}(\mathbb{V},\mathbb{I})\ge 0,\forall (\mathbb{V},\mathbb{I})\in \mathrm{\Omega },\hfill \end{array}$$

where $\mathcal{A}={\mathcal{D}}_{\mathbb{I}}\mathrm{\Psi }({\mathbb{V}}^0,0)$ is an $M-matrix$, and $\mathrm{\Omega }$ is the feasible region.

Theorem 4. 

The disease-free state $X^0$ of the proposed Bayoud disease model (Equations (4)–(11)) is globally asymptotically stable provided ${\mathcal{R}}_0<1$.

Proof. 

We must show that system (29) satisfies the conditions $\mathbf{Cond}\mathbf{1}$ and $\mathbf{Cond}\mathbf{2}$.

For $\mathbb{I}=0$, Equation (29) takes the form:

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{V}\left(t\right)=\left(\begin{array}{c}\mathrm{\Pi }-({\alpha }_1+\nu )S\\ {\alpha }_{1}S-({\beta }_1+\nu )S_R\\ {\beta }_1{S}_R-\nu R\end{array}\right).\end{array}$$

Equivalently,

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{V}\left(t\right)=\left(\begin{array}{c}-({\alpha }_1+\nu )(S-S^0)\\ -({\beta }_1+\nu )(S_R-{S_R}^0)\\ -\nu (P-P^0)\end{array}\right),\end{array}$$

or

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{V}\left(t\right)=\mathcal{L}(\mathbb{V}-{\mathbb{V}}^0),\end{array}$$

(30)

where the matrix

$$\mathcal{L}=\left[\begin{array}{ccc}-({\alpha }_1+\nu )& 0& 0\\ 0& -({\beta }_1+\nu )& 0\\ 0& 0& -\nu \end{array}\right],$$

has negative eigenvalues, i.e., ${\lambda }_1^{\ast }=-({\alpha }_1+\nu )<0,{\lambda }_2^{\ast }=-({\beta }_1+\nu )<0,{\lambda }_3^{\ast }=-\nu <0$.

The general solution of Equation (30) is given as:

$$\begin{array}{c}\hfill \mathbb{V}\left(t\right)-{\mathbb{V}}^0=c{e}^{\mathcal{L}t},\end{array}$$

(31)

where

$$e^{\mathcal{L}t}=\left[\begin{array}{ccc}{e}^{{\lambda }_1^{\ast }t}& 0& 0\\ 0& e^{{\lambda }_2^{\ast }t}& 0\\ 0& 0& e^{{\lambda }_3^{\ast }t}\end{array}\right]$$

Since $e^{{\lambda }_i^{\ast }t}\to 0$ as $t\to \infty \forall i=1,2,3$, so $\mathbb{V}\left(t\right)\to {\mathbb{V}}^0$ as $t\to \infty$. Thus, for ${}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{V}\left(t\right)=\mathrm{\Lambda }(\mathbb{V},0)$, ${\mathbb{V}}^0$ is globally asymptotically stable.

The second expression of the system (29) can be translated to the following form.

$$\begin{gathered} {}_{0 }{}^{ABC}D_t^{\varsigma }\left[\begin{array}{c}E\\ \\ I\\ \\ Q\\ \\ W\\ \end{array}\right]=\left[\begin{array}{cccc}-(\gamma +\nu )& {\alpha }_2{S}^0+{\beta }_2{S}_R^0& 0& {\alpha }_3{S}^0+{\beta }_3{S}_R^0\\ \\ \gamma & -(\tau +{\delta }_I+\nu )& 0& 0\\ \\ 0& \tau & -(\theta +{\delta }_Q+\nu )& 0\\ \\ 0& {\psi }_1& 0& -{\psi }_2\\ \end{array}\right]\left[\begin{array}{c}E\\ \\ I\\ \\ Q\\ \\ W\\ \end{array}\right] \\ -\left[\begin{array}{c}({\alpha }_{2}I+{\alpha }_{3}W)(S^0-S)+({\beta }_{2}I+{\beta }_{3}W)(S_R^0-S_R)\\ \\ 0\\ \\ 0\\ \\ 0\end{array}\right], \end{gathered}$$

or

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\mathbb{I}}{dt}}=& \mathbb{AI}-\widehat{\mathrm{\Psi }}(\mathbb{V},\mathbb{I}),\hfill \end{array}$$

(32)

where

$$\begin{array}{c}\hfill \mathbb{A}={\mathcal{D}}_{\mathbb{I}}\mathrm{\Psi }({\mathbb{V}}^0,0)=\left[\begin{array}{cccc}-(\gamma +\nu )& {\alpha }_2{S}^0+{\beta }_2{S}_R^0& 0& {\alpha }_3{S}^0+{\beta }_3{S}_R^0\\ \\ \gamma & -(\tau +{\delta }_I+\nu )& 0& 0\\ \\ 0& \tau & -(\theta +{\delta }_Q+\nu )& 0\\ \\ 0& {\psi }_1& 0& -{\psi }_2\\ \end{array}\right],\end{array}$$

is an $M-matrix$ and

$$\begin{array}{c}\hfill \widehat{\mathrm{\Psi }}(\mathbb{V},\mathbb{I})=\left[\begin{array}{c}({\alpha }_{2}I+{\alpha }_{3}W)(S^0-S)+({\beta }_{2}I+{\beta }_{3}W)(S_R^0-S_R)\\ \\ 0\\ \\ 0\\ \\ 0\\ \end{array}\right].\end{array}$$

Since $S^0\ge S$ and $S_R^0\ge S_R$, the column matrix $\widehat{\mathrm{\Psi }}(\mathbb{V},\mathbb{I})\ge 0,\forall (\mathbb{V},\mathbb{I})\in \mathrm{\Omega }.$ Hence, the disease-free point $X^0$ is globally asymptotically stable.    □

Global stability in the disease-free state indicates that irrespective of any perturbation, the disease will not reemerge in the tree population. This characteristic emphasizes the tree population’s remarkable resilience, enabling rapid recovery from any outbreaks. Moreover, continuous monitoring and intervention measures will be necessary to sustain this stability and guarantee that the situation remains regulated.

3.2. LAS at EE

Theorem 5. 

The proposed Bayoud model (Equations (4)–(11)) is locally asymptotically stable at the endemic equilibrium (EE) point $X^{\ast }$ if ${\mathcal{R}}_0>1$ and unstable for ${\mathcal{R}}_0<1$.

Proof. 

We apply the Jacobian matrix approach to prove the stability of the $X^{\ast }$:

$$\begin{array}{c}\hfill J^{\ast }=\left(\begin{array}{ccccccc}-({\alpha }_1+\nu +\mathcal{A})& 0& 0& -{\alpha }_2{S}^{\ast }& 0& 0& -{\alpha }_3{S}^{\ast }\\ \\ {\alpha }_1& -({\beta }_1+\nu +\mathcal{B})& 0& -{\beta }_2{S}_R^{\ast }& 0& 0& -{\beta }_3{S}_R^{\ast }\\ \\ \mathcal{A}& \mathcal{B}& -(\gamma +\nu )& {\alpha }_2{S}^{\ast }+{\beta }_2{S}_R^{\ast }& 0& 0& {\alpha }_3{S}^{\ast }+{\beta }_3{S}_R^{\ast }\\ \\ 0& 0& \gamma & -(\tau +{\delta }_I+\nu )& 0& 0& 0\\ \\ 0& 0& 0& \tau & -(\theta +{\delta }_Q+\nu )& 0& 0\\ \\ 0& {\beta }_1& 0& 0& \theta & -\nu & 0\\ \\ 0& 0& 0& {\psi }_1& 0& 0& -{\psi }_2\end{array}\right),\end{array}$$

(33)

where $\mathcal{A}={\alpha }_2{I}^{\ast }+{\alpha }_3{W}^{\ast }=({\alpha }_2+{\displaystyle \frac{{\alpha }_3{\psi }_1}{{\psi }_2}})I^{\ast }$ and $\mathcal{B}={\beta }_2{I}^{\ast }+{\beta }_3{W}^{\ast }=({\beta }_2+{\displaystyle \frac{{\beta }_3{\psi }_1}{{\psi }_2}})I^{\ast }.$

Since it is challenging to solve the Jacobian matrix analytically at the endemic equilibrium point, we solve the Jacobian matrix numerically at the endemic equilibrium point ($X^{\ast }$) and all the eigenvalues are negative for those values of the parameters when the proposed Bayoud model (Equations (4)–(11)) is at EE state (Figure 3).    □

Theorem 6. 

The endemic equilibrium state $X^{\ast }$ of the proposed model (Equations (4)–(11)) is globally asymptotically stable when ${\mathcal{R}}_0>1$.

Proof. 

We employ a Voltera-type Lyapunov function $\mathbb{L}(S,S_R,E,I,Q,R)$ defined as follows:

$$\begin{array}{cc}\hfill \mathbb{L}=& (S-S^{\ast }-S^{\ast }log{\displaystyle \frac{S}{S^{\ast }}})+(S_R-S_R^{\ast }-S_R^{\ast }log{\displaystyle \frac{S_R}{S_R^{\ast }}})+(E-E^{\ast }-E^{\ast }log{\displaystyle \frac{E}{E^{\ast }}})\hfill \\ & +(I-I^{\ast }-I^{\ast }log{\displaystyle \frac{I}{I^{\ast }}})+(Q-Q^{\ast }-Q^{\ast }log{\displaystyle \frac{Q}{Q^{\ast }}})+(R-R^{\ast }-R^{\ast }log{\displaystyle \frac{R}{R^{\ast }}})\hfill \\ \hfill & +(W-W^{\ast }-W^{\ast }log{\displaystyle \frac{W}{W^{\ast }}}),\hfill \end{array}$$

The function $\mathbb{L}$ is differentiated to obtain the following expression.

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{L}\left(t\right)& =\left(1-{\displaystyle \frac{S^{\ast }}{S}}\right) {}_{0 }{}^{ABC}D_t^{\varsigma }S\left(t\right)+\left(1-{\displaystyle \frac{S_R^{\ast }}{S_R}}\right) {}_{0 }{}^{ABC}D_t^{\varsigma }{S}_R\left(t\right)+\left(1-{\displaystyle \frac{E^{\ast }}{E}}\right) {}_{0 }{}^{ABC}D_t^{\varsigma }E\left(t\right)\hfill \\ & +\left(1-{\displaystyle \frac{I^{\ast }}{I}}\right) {}_{0 }{}^{ABC}D_t^{\varsigma }I\left(t\right)+\left(1-{\displaystyle \frac{Q^{\ast }}{Q}}\right) {}_{0 }{}^{ABC}D_t^{\varsigma }Q\left(t\right)+\left(1-{\displaystyle \frac{R^{\ast }}{R}}\right) {}_{0 }{}^{ABC}D_t^{\varsigma }R\left(t\right)\hfill \\ & +\left(1-{\displaystyle \frac{W^{\ast }}{W}}\right) {}_{0 }{}^{ABC}D_t^{\varsigma }W\left(t\right).\hfill \end{array}$$

Using differential of Equations (4)–(11), we obtain:

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{L}\left(t\right)& =\left(1-{\displaystyle \frac{S^{\ast }}{S}}\right)\left(\mathrm{\Pi }-({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )S\right)+\left(1-{\displaystyle \frac{S_R^{\ast }}{S_R}}\right)\left({\alpha }_{1}S-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R\right)\hfill \\ & +\left(1-{\displaystyle \frac{E^{\ast }}{E}}\right)\left(({\alpha }_{2}I+{\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R-(\gamma +\nu )E\right)+\left(1-{\displaystyle \frac{I^{\ast }}{I}}\right)\left(\gamma E-(\tau +{\delta }_I+\nu )I\right)\hfill \\ & +\left(1-{\displaystyle \frac{Q^{\ast }}{Q}}\right)\left(\tau I-(\theta +{\delta }_Q+\nu )Q\right)+\left(1-{\displaystyle \frac{R^{\ast }}{R}}\right)\left(\theta Q+{\beta }_1{S}_R-\nu R\right)+\left(1-{\displaystyle \frac{W^{\ast }}{W}}\right)\left({\psi }_{1}I-{\psi }_{2}W\right).\hfill \end{array}$$

We can rewrite the preceding expression in the following form.

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{L}\left(t\right)& =\left(1-{\displaystyle \frac{S^{\ast }}{S}}\right)(\mathrm{\Pi }-({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )S^{\ast }-({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )(S-S^{\ast }))\hfill \\ & +\left(1-{\displaystyle \frac{S_R^{\ast }}{S_R}}\right)({\alpha }_{1}S-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R^{\ast }-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )(S_R-S_R^{\ast }))\hfill \\ & +\left(1-{\displaystyle \frac{E^{\ast }}{E}}\right)\left(({\alpha }_{2}I+{\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R-(\gamma +\nu )E^{\ast }-(\gamma +\nu )(E-E^{\ast })\right)\hfill \\ & +\left(1-{\displaystyle \frac{I^{\ast }}{I}}\right)(\gamma E-(\tau +{\delta }_I+\nu )I^{\ast }-(\tau +{\delta }_I+\nu )(I-I^{\ast }))\hfill \\ & +\left(1-{\displaystyle \frac{Q^{\ast }}{Q}}\right)(\tau I-(\theta +{\delta }_Q+\nu )Q^{\ast }-(\theta +{\delta }_Q+\nu )(Q-Q^{\ast }))\hfill \\ & +\left(1-{\displaystyle \frac{R^{\ast }}{R}}\right)({\beta }_1{S}_R+\theta Q-\nu R^{\ast }-\nu (R-R^{\ast }))\hfill \\ & +\left(1-{\displaystyle \frac{W^{\ast }}{W}}\right)({\psi }_{1}I-{\psi }_2{W}^{\ast }-{\psi }_2(W-W^{\ast })).\hfill \end{array}$$

A simple arrangement of the terms of the above equation leads us to the following expression.

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{L}\left(t\right)={\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_2,\end{array}$$

where

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_1& =\mathrm{\Pi }+({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu ){\displaystyle \frac{{\left(S^{\ast }\right)}^2}{S}}+({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu ){\displaystyle \frac{{\left(S_R^{\ast }\right)}^2}{S_R}}+{\alpha }_{1}S+({\alpha }_{2}I+{\alpha }_{3}W)S\hfill \\ & +({\beta }_{2}I+{\beta }_{3}W)S_R+(\gamma +\nu ){\displaystyle \frac{{\left(E^{\ast }\right)}^2}{E}}+\gamma E+(\tau +{\delta }_I+\nu ){\displaystyle \frac{{\left(I^{\ast }\right)}^2}{I}}+\tau I+(\theta +{\delta }_Q+\nu ){\displaystyle \frac{{\left(Q^{\ast }\right)}^2}{Q}}\hfill \\ & +\theta Q+{\beta }_1{S}_R+\nu {\displaystyle \frac{{\left(R^{\ast }\right)}^2}{R}}+{\psi }_{1}I+{\psi }_2{\displaystyle \frac{{\left(W^{\ast }\right)}^2}{W}},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_2& =({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu )S^{\ast }+\mathrm{\Pi }{\displaystyle \frac{S^{\ast }}{S}}+({\alpha }_1+{\alpha }_{2}I+{\alpha }_{3}W+\nu ){\displaystyle \frac{{(S-S^{\ast })}^2}{S}}\hfill \\ \hfill & +{\alpha }_{1}S{\displaystyle \frac{S_R^{\ast }}{S_R}}+({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R^{\ast }+({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu ){\displaystyle \frac{{(S_R-S_R^{\ast })}^2}{S_R}}\hfill \\ \hfill & +\left(({\alpha }_{2}I+{\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R\right){\displaystyle \frac{E^{\ast }}{E}}+(\gamma +\nu )E^{\ast }+(\gamma +\nu ){\displaystyle \frac{{(E-E^{\ast })}^2}{E}}\hfill \\ \hfill & +\gamma E{\displaystyle \frac{I^{\ast }}{I}}+(\tau +{\delta }_I+\nu )I^{\ast }+(\tau +{\delta }_I+\nu ){\displaystyle \frac{{(I-I^{\ast })}^2}{I}}\hfill \\ \hfill & +\left(\tau I\right){\displaystyle \frac{Q^{\ast }}{Q}}+(\theta +{\delta }_Q+\nu )Q^{\ast }+(\theta +{\delta }_Q+\nu ){\displaystyle \frac{{(Q-Q^{\ast })}^2}{Q}}\hfill \\ \hfill & +(\theta Q+{\beta }_1{S}_R){\displaystyle \frac{R^{\ast }}{R}}+\nu R^{\ast }+\nu {\displaystyle \frac{{(R-R^{\ast })}^2}{R}}+\left({\psi }_{1}I\right){\displaystyle \frac{R^{\ast }}{R}}+{\psi }_2{W}^{\ast }+{\psi }_2{\displaystyle \frac{{(W-W^{\ast })}^2}{W}}.\hfill \end{array}$$

Every parameter in the Bayoud disease model is non-negative; therefore, ${}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{L}\left(t\right)\le 0$ provided ${\mathrm{\Gamma }}_1\le {\mathrm{\Gamma }}_2$. Moreover,

$${}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{L}\left(t\right)=0\iff {\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2,$$

Likewise,

$${}_{0 }{}^{ABC}D_t^{\varsigma }\mathbb{L}\left(t\right)=0\iff S=S^{\ast },S_R=S_R^{\ast },E=E^{\ast },I=I^{\ast },Q=Q^{\ast },R=R^{\ast },W=W^{\ast }.$$

The LaSalle invariance principle [51] demonstrates that the endemic equilibrium point $X^{\ast }$ is asymptotically stable. As a result, any solution trajectory starting from an initial condition within the feasible region $\mathrm{\Omega }$ will approach $X^{\ast }$ as time t approaches infinity.    □

4. Sensitivity Analysis

In this section, we explored the influence of each parameter on the disease’s reproduction number, also known as sensitivity analysis. Sensitivity analysis is a critical technique for understanding the variables (parameters) involved in the development of an infectious disease in a community and identifying the most sensitive parameters allows us to build appropriate control tactics. We use the normalized sensitivity technique [52,53,54] to acquire the sensitivity index of a parameter (i.e., $\alpha$) in our proposed model. The formula is given as follows:

$$\begin{array}{cc}\hfill S_{\alpha }& ={\displaystyle \frac{\alpha }{{\mathcal{R}}_0}}{\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial \alpha }}.\hfill \end{array}$$

There are two main parameters involved in the reproduction number of an epidemic model: parameters that can be controlled, i.e., infectious interaction, vaccination, isolation, or treatment rate, and the others are uncontrollable, i.e., birth and death rates. Therefore, the sensitivity index is important for those parameters that can be controlled to minimize the infection. Our calculation of the (local and global) sensitivity analysis indicates that ${\alpha }_2$ (force of infection due to infected trees), ${\beta }_1$ (rate of recovery of the resistance varieties), $\gamma$ (recruitment rate of the infected compartment), and ${\alpha }_3$ (force of infection due to containment water) and their combinations are the most sensitive to the ${\mathcal{R}}_0$ of the proposed model (Equations (4)–(11)). Similarly, $\mathrm{\Pi }$ (the birth rate) and $\nu$ (the death rate of Bayoud trees) are the most sensitive trivial (uncontrollable) parameters. Lastly, some parameters always do not affect the reproduction number and have a sensitivity index equal to zero, e.g., $\theta$, and ${\delta }_Q$ (

Table 1. Sensitivity index for reproduction number ${\mathcal{R}}_0$ for all model parameters. For instance, halting the growth of palm trees or removing them is not a viable solution. Similarly, the sensitivity index shows that the permanent protection/recovery rate (${\beta }_1$) of the resistant variants of palm trees is one of the best ways to minimize the spread of the Bayoud disease, but not an ideal strategy. On the other hand, the rate of becoming infectious ${\beta }_2$ is the most sensitive parameter of the model and can increase (decrease) in 1% in the value of ${\alpha }_2$ will increase (decrease) $45\%$ in the ${\mathcal{R}}_0$. The second and third sensitive parameters are ${\alpha }_2$ and, $\gamma$, respectively, and will change in ${\mathcal{R}}_0$ almost $41\%$ and $23\%$, respectively.

Parameter	Values	Sensitivity Index	Relationship
$\nu$	0.01721	−0.8420566793	Inverse
$\mathrm{\Pi }$	0.40	1	Direct
${\alpha }_1$	0.1361	0.05111985811	Direct
${\alpha }_2$	0.4312	0.4139418206	Direct
${\alpha }_3$	0.232	0.1306726238	Direct
${\beta }_1$	0.2093	−0.9240210148	Inverse
${\beta }_2$	0.00912	00.4516085578	Direct
${\beta }_3$	0.00013	0.003776997955	Direct
$\gamma$	0.0591	0.2255274544	Direct
$\tau$	0.1041	−0.2059306444	Indirect
${\delta }_I$	0.3842	−0.7600245299	Inverse
${\psi }_1$	0.389	0.1344496217	Direct
${\psi }_2$	0.663	−0.1344496220	Inverse
$\theta$	0.1165	0	None
${\delta }_Q$	0.0369	0	None

, Figure 4 and Figure 5).

Global sensitivity analysis enables the simultaneous variation of all parameters across their entire range [55]. This approach helps assess the relative influence of each input parameter and identify interactions that affect the model’s output. By analyzing how input parameters vary within a specified range, we can determine which parameters and interactions have the most significant impact on the overall behavior of the model. For global sensitivity analysis, there are a number of methods that can be used. These include the weighted average of local sensitivity analysis, the partial rank correlation coefficient (PRCC), multiparametric sensitivity analysis, the Fourier amplitude sensitivity test (FAST), and Sobol’s method. Systems pharmacology models can utilize all these methods. Among these, we widely use the Latin Hypercube Sampling (LHS) method for global sensitivity analysis. The LHS Monte Carlo method with PRCC is used for parameter sensitivity analysis in this study to find the pharmacokinetic parameters that have the most impact on the model dynamics. We analyzed the effect of all parameters on the peaks of infected and exposed palm trees. It can be seen that ${\alpha }_1,{\alpha }_2,\gamma$, and ${\delta }_I$ are the most globally sensitive parameters (Figure 6) as observed in the local sensitivity analysis (Table 1).

5. Fractional Control Problem and Optimization

The following section seeks to provide an optimal control problem that will help us determine the most effective disease management and control measures [16,56]. To achieve this, we first adjust the disease model (4)–(11) by including properly chosen time-dependent control variables. After that, we create the objective functional, which will serve as the basis for formulating an optimal control issue [57,58].

5.1. Bayoud Model Adjusted with Controls

We update the Bayoud disease model (4)–(11) by incorporating four time-varying control variables: $u_1\left(t\right),u_2\left(t\right),u_3\left(t\right),$ and $u_4\left(t\right)$, each representing a unique intervention strategy. Specifically, $u_1\left(t\right)$ corresponds to an awareness campaign designed to reduce interactions between susceptible trees and infected roots, $u_2\left(t\right)$ involves the treatment of irrigation water to reduce fungal contamination, $u_3\left(t\right)$ focuses on the quarantine and direct treatment of infected trees, and $u_4\left(t\right)$ aims to improve access to water treatment services. These control measures create a more dynamic and realistic approach to managing the disease’s spread over time. The choice of these controls is based on their practical significance in the management of diseases.

In particular, $u_1\left(t\right)$ and $u_2\left(t\right)$ are responsible for efforts aimed at limiting disease transmission due to root contact and contaminated water, while $u_3\left(t\right)$ governs the isolation and treatment of infected trees, minimizing their interaction with healthy ones. The control variable $u_4\left(t\right)$ expands the access to water purification services, further reducing the risk of spreading the pathogen. By integrating these controls, the model becomes more adaptable to develop an optimal control framework to reduce the prevalence of diseases while maximizing resource use. The treated trees are assumed to be isolated from contact with the roots of other trees and are irrigated independently. In addition, the soil surrounding the roots of the treated trees is replaced with fungus-free soil to help prevent reinfection. The disease model with the addition of controls is given below.

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }S\left(t\right)=& \mathrm{\Pi }-({\alpha }_1+(1-u_1){\alpha }_{2}I+(1-u_2){\alpha }_{3}W+\nu )S,\hfill \end{array}$$

(34)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }{S}_R\left(t\right)=\hfill & {\alpha }_{1}S-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R,\hfill \end{array}$$

(35)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }E\left(t\right)=\hfill & ((1-u_1){\alpha }_{2}I+(1-u_2){\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R-(\gamma +\nu )E,\hfill \end{array}$$

(36)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }I\left(t\right)=\hfill & \gamma E-(u_3+{\delta }_I+\nu )I,\hfill \end{array}$$

(37)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }Q\left(t\right)=\hfill & u_{3}I-(\theta +{\delta }_Q+\nu )Q,\hfill \end{array}$$

(38)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }R\left(t\right)=\hfill & \theta Q+{\beta }_1{S}_R-\nu R,\hfill \end{array}$$

(39)

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }W\left(t\right)=\hfill & (1-u_4){\psi }_{1}I-{\psi }_{2}W,\hfill \end{array}$$

(40)

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & S\left(0\right)=S_0>0,S_R\left(0\right)={S_R}_0\ge 0,E\left(0\right)=E_0\ge 0,I\left(0\right)=I_0\ge 0,\hfill \\ & Q\left(0\right)=Q_0\ge 0,R\left(0\right)=R_0\ge 0,W\left(0\right)=W_0\ge 0.\hfill \end{array}\end{array}$$

(41)

5.2. Optimality Conditions for the Control Problem

The objective of the optimal control problem is to minimize the number of exposed and infected trees while fostering the growth of resistant species. To achieve this, we use Pontryagin’s maximum principle [16] and define a cost functional. This functional helps measure the trade-offs between the costs of implementing control measures and the benefits of reducing the disease impact.

$$J(E,I,u)={\int }_0^{T_f}\left({\omega }_{1}E\left(t\right)+{\omega }_{2}I\left(t\right)+{\displaystyle \frac{1}{2}}{c}_1{u}_1^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_2{u}_2^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_3{u}_3^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_4{u}_4^2\left(t\right)\right)dt.$$

(42)

Here, $T_f$ represents the final time, the constants ${\omega }_i>0,i=1,2,c_j>0,j=1,\dots ,4$ are, respectively, the costs associated with the states $E\left(t\right)$, $I\left(t\right)$ and the controllers $u_1\left(t\right),u_2\left(t\right),u_3\left(t\right),u_4\left(t\right)$.

The problem is to determine an optimizer $u^{\ast }\left(t\right)\in \mathcal{U}$ that minimizes the functional (42), i.e.,

$$\mathrm{Find}\mathrm{minimizer}{u}^{\ast }\in \mathcal{U}\mathrm{for}\mathrm{the}\mathrm{functional}J(E,I,u)\mathrm{subject}\mathrm{to}\left(34\right)–\left(41\right).$$

(43)

The set of controls $\mathcal{U}$ is defined as follows:

$$\mathcal{U}=\left\{u\left(t\right)|0\le u\left(t\right)\le \tilde{u},0\le t\le T_f\right\}.$$

We consider the following Hamiltonian function to derive the optimality conditions necessary for the control problem (43).

$$H(t,\vartheta ,u,{\chi }_j)={\omega }_{1}E\left(t\right)+{\omega }_{2}I\left(t\right)+{\displaystyle \frac{1}{2}}{c}_1{u}_1^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_2{u}_2^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_3{u}_3^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_4{u}_4^2\left(t\right)+\sum _{j=1}^6{\chi }_j{\varphi }_j(t,\vartheta ,u),$$

where the state variables are represented by $\vartheta =(S,S_R,E,I,Q,R,W)$, the co-state variables are referred by ${\chi }_j,j=1,\dots ,7$, and ${\varphi }_j(t,\vartheta ,u),j=1,\dots ,7$ are the right parts of the state Equations (34)–(41). The Hamiltonian function can be written as follows.

$$\begin{array}{cc}\hfill H(t,\vartheta ,u,{\chi }_i)=& {\omega }_{1}E\left(t\right)+{\omega }_{2}I\left(t\right)+{\displaystyle \frac{1}{2}}{c}_1{u}_1^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_2{u}_2^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_3{u}_3^2\left(t\right)+{\displaystyle \frac{1}{2}}{c}_4{u}_4^2\left(t\right)\hfill \\ \hfill & +{\chi }_1\left(\mathrm{\Pi }-({\alpha }_1+(1-u_1){\alpha }_{2}I+(1-u_2){\alpha }_{3}W+\nu )S\right)\hfill \\ \hfill & +{\chi }_2\left({\alpha }_{1}S-({\beta }_1+{\beta }_{2}I+{\beta }_{3}W+\nu )S_R\right)\hfill \\ \hfill & +{\chi }_3(((1-u_1){\alpha }_{2}I+(1-u_2){\alpha }_{3}W)S+({\beta }_{2}I+{\beta }_{3}W)S_R-(\gamma +\nu )E)\hfill \\ \hfill & +{\chi }_4(\gamma E-(u_3+{\delta }_I+\nu )I)\hfill \\ \hfill & +{\chi }_5(u_{3}I-(\theta +{\delta }_Q+\nu )Q)\hfill \\ \hfill & +{\chi }_6\left(\theta Q+{\beta }_1{S}_R-\nu R\right)\hfill \\ \hfill & +{\chi }_7\left((1-u_4){\psi }_{1}I-{\psi }_{2}W\right).\hfill \end{array}$$

(44)

Theorem 7. 

Let $\tilde{S},{\tilde{S}}_R,\tilde{E},\tilde{I},\tilde{Q},\tilde{R},\tilde{W}$ represent the optimal solutions of the model (34)–(41) corresponding to the controllers $u_1^{\ast },u_2^{\ast },u_3^{\ast },u_4^{\ast }$ for the problem (43) that optimizes $J(E,I,u_1,u_2,u_3,u_4)$ over the control set $\mathcal{U}$. Then, there exists the following adjoint system of equations.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_1\left(t\right)=-{\displaystyle \frac{\partial H}{\partial S}},{}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_2\left(t\right)=-{\displaystyle \frac{\partial H}{\partial S_R}},{}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_3\left(t\right)=-{\displaystyle \frac{\partial H}{\partial E}},{}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_4\left(t\right)=-{\displaystyle \frac{\partial H}{\partial I}},\hfill \\ \hfill & {}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_5\left(t\right)=-{\displaystyle \frac{\partial H}{\partial Q}},{}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_6\left(t\right)=-{\displaystyle \frac{\partial H}{\partial R}},{}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_7\left(t\right)=-{\displaystyle \frac{\partial H}{\partial W}},\hfill \end{array}\end{array}$$

(45)

with the final conditions:

$$\begin{array}{c}\hfill {\chi }_i\left(T_f\right)=0,i=1,\dots ,7.\end{array}$$

The control set $\{u_1^{\ast },u_2^{\ast },u_3^{\ast },u_4^{\ast }\}$ is characterized by:

$$\begin{array}{cc}\hfill u_1^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{({\chi }_3\left(t\right)-{\chi }_1\left(t\right)){\alpha }_{2}SI}{c_1}},0\right\}\right\},\hfill \\ \hfill u_2^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{({\chi }_3\left(t\right)-{\chi }_1\left(t\right)){\alpha }_{3}SW}{c_2}},0\right\}\right\},\hfill \\ \hfill u_3^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{({\chi }_4\left(t\right)-{\chi }_5\left(t\right))I}{c_3}},0\right\}\right\},\hfill \\ \hfill u_4^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{{\psi }_{1}I{\chi }_7\left(t\right)}{c_4}},0\right\}\right\}.\hfill \end{array}$$

We obtain the following system of linear adjoint equations from (45).

$$\begin{array}{cc}{}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_1\left(t\right)=\hfill & {\alpha }_1({\chi }_1-{\chi }_2)+\left[(1-u_1){\alpha }_{2}I+(1-u_2){\alpha }_{3}W\right]({\chi }_1-{\chi }_3)+\nu {\chi }_1,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_2\left(t\right)=& ({\beta }_1+\nu ){\chi }_2+({\beta }_{2}I+{\beta }_{3}W)({\chi }_2-{\chi }_3)-{\beta }_1{\chi }_6,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_3\left(t\right)=& -{\omega }_1+\gamma ({\chi }_3-{\chi }_4)+\nu {\chi }_3,\hfill \end{array}$$

(48)

$$\begin{array}{c}{}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_4\left(t\right)=-{\omega }_2+{\alpha }_2(1-u_1)({\chi }_1-{\chi }_3)S+{\beta }_2({\chi }_2-{\chi }_3)S_R\\ +({\delta }_I+\nu ){\chi }_4+u_3({\chi }_4-{\chi }_5)-(1-u_4){\psi }_1{\chi }_7,\end{array}$$

(49)

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_5\left(t\right)=& \theta ({\chi }_5-{\chi }_6)+({\delta }_Q+\nu ){\chi }_5,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_6\left(t\right)=& \nu {\xi }_6,\hfill \end{array}$$

(51)

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }{\chi }_7\left(t\right)={\alpha }_3(1-u_2)({\chi }_1-{\chi }_3)S+{\beta }_3({\chi }_2-{\chi }_3)S_R+{\psi }_2{\chi }_7.\end{array}$$

(52)

These equations are supported by the conditions:

$$\begin{array}{c}\hfill {\chi }_j\left(T_f\right)=0,j=1,\dots ,7.\end{array}$$

(53)

The expressions for the controls are determined based on the evaluation of the optimality conditions ${\displaystyle \frac{\partial H}{\partial u_i}}=0,i=1,\dots ,4$.

$$\begin{array}{cc}\hfill u_1\left(t\right)=& {\displaystyle \frac{({\chi }_3\left(t\right)-{\chi }_1\left(t\right)){\alpha }_{2}SI}{c_1}},\hfill \\ \hfill u_2\left(t\right)=& {\displaystyle \frac{({\chi }_3\left(t\right)-{\chi }_1\left(t\right)){\alpha }_{3}SW}{c_2}},\hfill \\ \hfill u_3\left(t\right)=& {\displaystyle \frac{({\chi }_4\left(t\right)-{\chi }_5\left(t\right))I}{c_3}},\hfill \\ \hfill u_4\left(t\right)=& {\displaystyle \frac{{\psi }_{1}I{\chi }_7\left(t\right)}{c_4}}.\hfill \end{array}$$

The optimal control characterization with bounds is written as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_1^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{({\chi }_3\left(t\right)-{\chi }_1\left(t\right)){\alpha }_{2}SI}{c_1}},0\right\}\right\},\hfill \\ \hfill u_2^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{({\chi }_3\left(t\right)-{\chi }_1\left(t\right)){\alpha }_{3}SW}{c_2}},0\right\}\right\},\hfill \\ \hfill u_3^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{({\chi }_4\left(t\right)-{\chi }_5\left(t\right))I}{c_3}},0\right\}\right\},\hfill \\ \hfill u_4^{\ast }\left(t\right)& =min\left\{\tilde{u},max\left\{{\displaystyle \frac{{\psi }_{1}I{\chi }_7\left(t\right)}{c_4}},0\right\}\right\}.\hfill \end{array}\end{array}$$

(54)

The condition

$$\begin{array}{c}\hfill {}_{0 }{}^{ABC}D_t^{\varsigma }{\vartheta }_j\left(t\right)={\displaystyle \frac{\partial H}{\partial {\chi }_j}},j=1,\dots ,7,\end{array}$$

(55)

will eventually give us to the Bayoud disease model (34)–(41).

Equations (46)–(55) establish the essential optimality conditions to solve the optimal control problem (43). These conditions include the state equations, which describe the system’s dynamics. They also incorporate adjoint equations, which provide valuable insights for optimization. Together, these elements create a strong framework for identifying optimal controls and their associated state trajectories.

To determine an optimizer $u^{\ast }\left(t\right)$ for problem (43), we follow the steps described in the algorithm provided below (Algorithm 1). We numerically approximate the necessary optimality conditions (46)–(55) using the Toufik-Atangana-type forward and backward in time methods, implemented in MATLAB (version-24).

Algorithm 1 Procedure for finding the minimizer of the control problem (43).

With $\ell =0$, take $u_{\ell }={({\left(u_1\right)}_{\ell },{\left(u_2\right)}_{\ell },{\left(u_3\right)}_{\ell })}^T\in \mathcal{U}$. Approximate the state Equations (34)–(41) and the co-state Equations (46)–(53) using the control $u_{\ell }$. Find $u={(u_1,u_2,u_3,u_4)}^T$ using expressions (54). Update $u_{\ell }$ by refining it as $u_{\ell }=(u+u_{\ell })/2$. If $\left|\right|{\varpi }_{\ell }-{\varpi }_{\ell -1}\left|\right|<tol\left|\right|{\varpi }_{\ell }\left|\right|$, for $\ell >0$, then STOP the iterative procedure, otherwise $\ell \to \ell +1$ and go back to Step 2.

Here, $\varpi$ denotes each of the model variables ${\vartheta }_j,j=1,\dots ,7$, co-state variables ${\chi }_j,j=1,\dots ,7$ and the control variable $u={(u_1,u_2,u_3,u_4)}^T$.

6. Results and Discussion

This section presents a comprehensive analysis of the results of solving the necessary optimality conditions. To numerically simulate the state and adjoint equations, we discretize the time interval $[0,T_f]$ into n segments, each of uniform size $h=T_f/n$. We approximate solutions of these equations at discrete time points $t_i=ih$ for $i=0,1,\dots ,n$ using the Toufik–Atangana type numerical scheme [59]. The detailed derivation of the scheme for a fractional epidemic model is presented in [22]. The cost functional (43) is approximated using Simpson’s 1/3 rule.

This research focuses on creating an optimal control strategy to mitigate the spread of Bayoud disease affecting date palm trees by efficiently managing exposed and infected trees while minimizing associated costs. The proposed model features four key control measures: an awareness campaign aimed at isolating infected trees $\left(u_1\right)$, irrigation water treatment $\left(u_2\right)$, quarantine and treatment (direct application of fungicides or related treatment measures to infected trees) control $\left(u_3\right)$, and service coverage for water treatment $\left(u_4\right)$. The objective is to identify the best way to implement these controls to decrease the number of exposed and infected trees while minimizing the cost functional $J(E,I,u)$, which accounts for both disease prevalence and the costs of intervention. Through the analysis of the following control scenarios, this study aims to identify the most effective and economically feasible strategies for restricting the disease and fostering a sustainable management approach.

We examine the issue presented in problem (43) by exploring four distinct scenarios. Each case will be analyzed to assess the specific effects of different regulations on the dynamics of Bayoud disease transmission. The purpose is to gain a deeper understanding of how varying measures influence the spread of the virus and to identify which strategies might be most effective in restricting the spread of the disease.

Case-1: Control of disease through awareness alone: To examine the effect of awareness control $u_1$ on the dynamics of the disease, we consider the control problem (43) with $c_2=0,c_3=0,$ and $c_4=0$, where $u_2=0,u_4=0$ and $u_3$ is replaced by $\tau$, representing a constant rate of quarantine and treatment. Figure 7, Figure 8 and Figure 9 demonstrate the graphical illustration of Case-1.

The control variable $u_1$ (related to awareness campaigns for isolating and treating infected trees) changes over time depending on different values of $\varsigma$. Higher values of $\varsigma$ indicate a stronger memory effect, meaning previous states of the system have a significant influence on the present. This leads to a gradual but more sustainable adjustment in the control variable. The increasing trend of $u_1$ followed by stabilization indicates that a successful awareness campaign necessitates a strong initial effort but can later be maintained at a consistent level to control the disease spread efficiently. Analysis of the cost functional graph shows a decrease in overall costs as the optimization process progresses. The rate at which optimal costs are achieved varies with different fractional orders. While higher fractional orders may lead to increased costs initially, they ultimately support better long-term disease management due to the cumulative effects of previous interventions.

The susceptible tree population $S\left(t\right)$ increases after optimization, illustrating that the implementation of $u_1$ effectively shields the trees from infections. Likewise, the resistant tree population $S_R\left(t\right)$ also rises, suggesting that long-term approaches could foster a gradual accumulation of resistance, whether through natural selection or proactive protective measures. The exposed tree population $E\left(t\right)$ experiences a significant decline after optimization, with a stronger effect observed for higher $\varsigma$. This trend is also reflected in the infected tree population $I\left(t\right)$, where reductions are more marked with increasing $\varsigma$. The rise in quarantined trees $Q\left(t\right)$ during treatment indicates that isolation strategies are maintained over a longer duration, particularly with higher fractional orders. Furthermore, the number of recovered trees $R\left(t\right)$ is also increasing, which underscores the effectiveness of awareness campaigns to curb new infections. Water contamination $W\left(t\right)$ decreases due to optimized control, highlighting an indirect benefit of $u_1$. Since water contamination is linked to infected trees, reducing infections through awareness and isolation helps limit the spread of the pathogen via irrigation. The memory effect in fractional-order systems indicates that previous contamination levels continue to influence the current state; however, effective control measures help mitigate long-term threats.

In conclusion, the results show that optimizing $u_1$ leads to a significant decrease in exposed and infected trees while increasing susceptible and protected populations.

Case-2: Control of disease through irrigation water treatment alone: To explore the effect of irrigation water treatment control $u_2$ on the dynamics of the disease, we consider the control problem (43) with $c_1=0,c_3=0,$ and $c_4=0$, where $u_1=0,u_4=0,$ and $u_3$ is replaced by $\tau$, representing a constant rate of quarantine and treatment. Figure 10, Figure 11 and Figure 12 illustrate Case-2 graphically. The findings from the optimal control problem regarding irrigation water treatment control $u_2$ offer valuable insights into its effectiveness in managing Bayoud disease in date palm trees. By comparing these results with the previous scenario, where the main control was awareness-based isolation $u_1$, we can assess the relative impact of these strategies on disease suppression and cost efficiency.

The control variable $u_2$ varies over time for different values of the fractional-order $\varsigma$. The control levels are relatively lower than those of $u_1$, indicating that irrigation water treatment might not require as aggressive an intervention as awareness-based isolation to effectively reduce the disease. The gradual rise and stabilization of $u_2$ indicate that maintaining a steady water treatment strategy is successful in minimizing pathogen transmission through irrigation. The trends in the cost functional reveal the differences in economic efficiency between the two control methods. Although the cost function declines with optimization iterations, its absolute values are considerably higher than those observed in the $u_1$ scenario. This indicates that water treatment tends to be more costly for disease management compared to awareness-based isolation.

The evaluation of the susceptible tree population $S\left(t\right)$ reveals an increase following the optimization, indicating that water treatment is effective in mitigating the spread of disease through contaminated irrigation. However, in comparison to $u_1$, the rise in $S\left(t\right)$ is less significant, implying that while water treatment is helpful, it may not be as impactful as direct isolation strategies for maintaining healthy trees. A similar pattern is seen in the resistant tree population $S_R\left(t\right)$, where the increase post-optimization is relatively modest. This suggests that while controlling waterborne transmission lowers infection rates, it does not play a major role in fostering disease-resistant tree populations. The number of exposed trees $E\left(t\right)$ declines under control $u_2$, but this reduction is notably less effective than in the $u_1$ scenario, indicating that while contaminated water is one transmission route, the spread through direct contact and proximity (addressed by $u_1$) may have a more significant influence on disease transmission. The trend in the infected tree population $I\left(t\right)$ reflects a similar pattern, suggesting that although water treatment is valuable, it might need to be complemented with other strategies for effective disease management. The number of recovered trees $R\left(t\right)$ also rises with $u_2$ but at a slower rate than in the $u_1$ scenario. Additionally, $W\left(t\right)$ experiences a slight decline in the $u_2$ control case compared to $u_1$. The $u_1$ strategy had a more indirect influence on $W\left(t\right)$, focusing mainly on tree isolation rather than addressing environmental contamination.

Case-3: Control of disease through quarantine and treatment alone: To examine the effect of quarantine and treatment control $u_3$ on the dynamics of the disease, we consider the control problem (43) with $c_1=0,c_2=0,$ and $c_4=0$, where we take $u_1,u_2,u_4$ equal to zero. Figure 13, Figure 14 and Figure 15 demonstrate the graphical illustration of Case-3.

The results of the optimal control problem with $u_3$ (quarantine and treatment) highlight the importance of isolating and treating infected trees to restrict Bayoud disease. The control variable $u_3$ increases initially before stabilizing, suggesting that a strong initial effort is needed to quarantine and treat infected trees. Furthermore, higher fractional-order values $\varsigma$ show a prolonged influence of past disease states, emphasizing the long-term effects of these strategies. The functional cost J decreases with time, but its values are still higher than those of the $u_1$ (awareness-based isolation) and $u_2$ (irrigation water treatment) cases. This means that quarantine and treatment require significant resources, making this strategy quite expensive.

The populations of susceptible trees, $S\left(t\right)$, and resistant trees, $S_R\left(t\right)$, both increase after optimization. However, this increase is less pronounced compared to the $u_1$ case. The number of exposed trees, $E\left(t\right)$, and the number of infected trees, $I\left(t\right)$, decrease following optimization, highlighting the effectiveness of quarantine in preventing exposed trees from progressing to infection. Additionally, the treatment of infected trees reduces disease prevalence. However, compared to $u_1$, the reduction is slightly less, suggesting that early isolation, as seen in the awareness campaign, is more effective in preventing exposure in the first place. This suggests that while quarantine and treatment measures are beneficial, they are not as effective in preventing initial exposure as awareness-based isolation. The number of quarantined trees, $Q\left(t\right)$, and the recovered trees $R\left(t\right)$ increase after optimization. The increase in the quarantined trees is more pronounced compared to previous cases, confirming that this strategy focuses on containing and treating infected trees rather than preventing infections outright. Water contamination $W\left(t\right)$ decreases, though not as significantly as in the $u_2$ (irrigation water treatment) case.

Case-4: Control of disease through water treatment service coverage alone: To examine the effect of water treatment service coverage, $u_4$, on the dynamics of the disease, we consider the control problem (43) with $c_1=0,c_2=0,$ and $c_3=0$, where we take $u_1=0,u_2=0$ and $u_3$ is replaced by $\tau$. Figure 16, Figure 17 and Figure 18 demonstrate the graphical illustration of Case-4.

The results of the optimal control problem with $u_4$ (service coverage of water treatment control) provide insights into the effectiveness of expanding access to water treatment in mitigating Bayoud disease. The control variable $u_4$ follows a gradual increase before stabilizing, similar to previous control measures, indicating that sustained service coverage is necessary to maintain effective disease control. The control levels are lower compared to quarantine and treatment $u_3$, suggesting that expanding water treatment coverage requires less intensive intervention than direct quarantine efforts. The cost functional decreases over-optimization iterations, but its values are relatively high, comparable to those in the $u_2$ (irrigation water treatment) case. However, the steady decline in cost function values over iterations indicates that optimizing service coverage does improve cost-effectiveness in the long run.

The tree populations that are susceptible, $S\left(t\right)$, resistant, $S_R\left(t\right)$, quarantined, $Q\left(t\right)$, and recovered, $R\left(t\right)$, show an increase following optimization. However, this growth is not as significant as that observed in the $u_1$ (awareness-based isolation) scenario. This suggests that while expanding service coverage is advantageous, direct awareness initiatives are still more effective in preventing initial exposure. Additionally, the number of exposed trees, $E\left(t\right)$, and the infected tree population, $I\left(t\right)$, also decrease after optimization, but the reduction is less compared to the $u_1$ situation. Notably, there is a decrease in water contamination, $W\left(t\right)$, after optimization, which aligns with the results seen in the $u_2$ scenario. This indicates that enhancing service coverage for water treatment effectively minimizes the transmission of environmental pathogens.

Among the four control strategies analyzed for managing Bayoud disease, awareness-based isolation ($u_1$) stands out as the most effective in reducing the number of exposed ($E\left(t\right)$) and infected ($I\left(t\right)$) trees while maintaining the lowest cost. This strategy effectively prevents disease transmission by promptly isolating infected trees, thereby limiting the spread of the disease and reducing the need for costly interventions in the future.

7. Conclusions

In this study, we developed and analyzed a compartmental mathematical model for Bayoud disease in date palm trees. The model includes compartments for susceptible, resistant, exposed, infected, quarantined and under treatment, and recovered trees. By including memory effects and long-range interactions, the fractional derivative in the Bayoud disease model gives a more accurate picture of how the disease changes over time. We established the model’s positivity and boundedness, ensuring that the solutions are biologically feasible. Additionally, we proved the existence and uniqueness of solutions, confirming that the model is well-posed. Our stability analysis revealed that the disease-free equilibrium is stable when the basic reproduction number (${\mathcal{R}}_0$) is less than one. In contrast, the endemic equilibrium becomes stable when ${\mathcal{R}}_0$ is greater than one, indicating that the disease can persist in the absence of effective control measures.

The control of Bayoud disease in date palm trees is analyzed using four different approaches: awareness campaigns, irrigation water treatment, quarantine and treatment, and service coverage for water treatment. Among these approaches, awareness-based isolation ($u_1$) emerges as the most effective and cost-efficient option. Optimizing $u_1$ leads to a notable reduction in both exposed and infected trees while increasing the susceptible and resistant populations. The initially high level of intervention effort stabilizes over time, making it a sustainable approach. In comparison, irrigation water treatment ($u_2$) and service coverage for water treatment ($u_4$) demonstrate moderate effectiveness but come with higher costs. Quarantine and treatment ($u_3$) successfully reduced infection rates but at a significant economic burden. The analysis concludes that a combined strategy, focusing on awareness campaigns along with targeted quarantine measures, strikes the best balance between cost-effectiveness and disease control, promoting long-term management and sustainability.

Future work could investigate the effect of climate change on Bayoud disease dynamics by incorporating temperature and humidity-dependent parameters. Another direction is the consideration of host-vector interactions that may allow us to enhance our understanding of disease transmission and its implications.