A Fractional-Order Mathematical Model of Banana Xanthomonas Wilt Disease Using Caputo Derivatives

: This article investigates a fractional-order mathematical model of Banana Xanthomonas Wilt disease while considering control measures using Caputo derivatives. The proposed model is numerically solved using the L1-based predictor-corrector method to explore the model’s dynamics in a particular time range. Stability and error analyses are performed to justify the efficiency of the scheme. The non-local nature of the Caputo fractional derivative, which includes memory effects in the system, is the main motivation for incorporating this derivative in the model. We obtain varieties in the model dynamics while checking various fractional order values.


Introduction
Banana Xanthomonas Wilt (BXW), a destructive bacterial disease caused by a bacterium called Xanthomonas campestris pv.musacearum (Xcm), has been identified as the major disease that threatens banana farming in East Africa [1].The vectors, like bats, birds, and flying insects (e.g., bees), spread the Xcm bacteria from an infected banana plant to a susceptible banana plant.The long-distance spread of Xcm is mainly caused by birds and bats [2].The common symptoms of BXW are yellowing and withering of leaves, untimely ripening and rotting of the fruit, shriveling and blackening of male gusset bloom, yellow drip presented on the cross-cut of the banana plant pseudo-trunk, and lastly, plant death [3,4].The authors in [5] observed community mobilization as a key to controlling BXW disease management.In [6], the authors analyzed the possibilities of removing the infected plant and leaving the uninfected plant to grow.The authors in [7] also investigated that time-to-time removing the infected plants from the mat is the best control compared to removing the complete mat, which is costly, time-consuming, and requires more labor.In [8], the authors explored BXW control techniques in Rwanda.
Several mathematical models have been derived by researchers to understand the transmission dynamics of BXW disease and provide possible control techniques.In [9], the authors derived a model to understand the transmission structure of the BXW epidemic by vectors with control measures.In [10], the authors proposed a non-linear model to analyze the role of contaminated measures in the reiteration of BXW.In [11], the researchers proposed a model considering roguing and debudding controls for the BXW transmission.Nakakawa et al. [12] considered the vertical and vector mode transmissions in the BXW dynamical model.Kweyunga et al. [13] developed a model of BXW considering both horizontal and vertical modes of transmission.In [14], the authors derived a non-linear model to analyze the role of neglected control measures in the BXW transmission.
Nowadays, fractional calculus [15][16][17] is being applied to solve various real-world problems in terms of mathematical modeling.Different types of fractional derivatives [18,19] have been successfully used to model various problems.More specifically, several deadly epidemics have been modeled by using mathematical models in a fractional-order sense.It is a well-known fact that fractional-order operators are non-local in nature and may be more effective for modeling history-dependent systems.Moreover, a fractional order can be fixed as any positive real number that better fits the real data.So, by using such an operator, an accurate adjustment can be made in a model to fit with real data for better predicting the outbreaks of an epidemic.Recently, several applications of fractional derivatives have been recorded in epidemiology.In [20][21][22][23][24][25], the authors have studied the dynamics of the COVID-19 disease by using fractional-order models.In [26], the authors proposed the mathematical modeling of typhoid fever in terms of fractional-order operators.In [27], a fractional-order model of the Chlamydia disease is proposed.In [28], the dynamics of the Chagas-HIV epidemic model using various fractional operators are explored.In [29], the authors derived a novel non-linear model for the dynamics of tooth cavities in the human population.In [30], the authors performed an analysis of the stability and bifurcation of a delay-type fractional-order model of HIV-1.In [31], the authors solved a fractionalorder HIV-1 infection of CD4+ T-cells considering the impact of antiviral drug treatment.In [32], the authors defined a fractal-fractional model of the AH1N1/09 virus.In [33], the authors studied the dynamics of a fractional-order hostparasitoid population model describing insect species.In [34], the authors used a wavelet-based numerical method for a fractional-order model of measles using Genocchi polynomials.In [35], some theoretical analyses of the Caputo-Fabrizio fractional-order model for hearing loss due to the mumps virus with optimal controls were proposed.
Several numerical methods have been proposed by researchers to solve fractional-order problems.In [36], the authors derived a new generalized form of the predictor-corrector (PC) scheme to investigate fractional initial value problems (IVPs).Kumar et al. [37] introduced a new method to simulate fractional-order systems with various examples.In [38], the PC method was derived to simulate delayed fractional differential equations.A modified form of the PC scheme in terms of the generalized Caputo derivative to solve delay-type systems has been introduced in [39].Odibat et al. [40] have derived the generalized differential transform method for solving fractional impulsive differential equations.The authors in [41] introduced a novel finite-difference predictor-corrector (L1-PC) scheme to solve fractional-order systems in the sense of the Caputo derivative.In [42], the authors proposed a new form of L1-PC scheme to solve multiple delay-type fractional-order systems.In [43], a novel numerical scheme to solve fractional differential equations in terms of Caputo-Fabrizio derivatives was proposed.In [44], the authors derived a difference scheme for the time fractional diffusion equations.In [45], a second-order scheme for the fast evaluation of the Caputotype fractional diffusion equations has been derived.In [46], the authors defined a fractional clique collocation method for numerically solving the fractional Brusselator chemical model.In [47], the researchers derived efficient matrix techniques for solving the fractional Lotka-Volterra population model.
To date, the aforementioned studies of mathematical modeling of the BXW disease [9][10][11][12][13][14] have yet to be analyzed using fractional derivatives.In this paper, we generalize the non-linear control-based model of BXW [14] by using Caputo fractional derivatives.The motivation behind this generalization is that fractional derivatives are non-local and may be more effective to include memory effects in the model.
The rest of this paper is designed as follows: In Section 2, some preliminaries are recalled.The model description in the Caputo sense is given in Section 3. The numerical analysis containing the solution algorithm, error estimation, and stability are given in Section 4. The graphical simulations are performed in Section 5. Concluding remarks are given in Section 6.

Preliminaries
The preliminaries are as follows: Definition 1.A function (real) f(s), s > 0 belongs to the space (a) , [16] The Riemann-Liouville (R-L) fractional integral of ( ) ∈ ≥ − is defined as follows: is given by ( ) where m = [ω] + 1 and [ω] are the integer-part of ω.Definition 4. [16] The Caputo fractional derivative of ) Remark 1.The most common difference between the R-L and Caputo fractional derivatives is that the R-L derivative problems contain fractional initial conditions, whereas Caputo's definition uses classical conditions.Also, the derivative of a constant function is zero by the Caputo derivative but not by the R-L definition.

Model description
Here, we define the Caputo-type fractional-order generalization of a BXW disease model, including some control measures, which was given in [14].We know that fractional derivatives are non-local differential operators that allow memory effects in the system, which is a very important feature for studying disease outbreaks more accurately.The model contains two population sizes: the banana population (N p ) and the insect vector population (N v ).The population of banana plants involves three different classes: susceptible plants (S p ), asymptomatic infectious plants (A p ), and symptomatic infected plants (I p ).The population of vectors involves two classes: susceptible vectors (S v ) and vectors contaminated with Xcm bacteria (I v ).An environment contaminated with Xcm bacteria is defined by E b .The model is given as follows: (1 ) , ( ) with the initial conditions (0) 0, (0) 0, (0) 0, (0) 0, (0) 0, (0) 0, where C D ω is the Caputo fractional derivative operator of order ω.For setting the same dimensions t −ω at both sides of the fractional-order model, we applied the power ω on the parameters, those are in time unit t −1 .
The compartmental diagram of the model is given in Figure 1.(1 ) ( ) , .
The model parameters with numerical values are defined in Table 1.The positivity and boundedness of the model solution can be explored by considering the invariant region of the model, derived as follows: From the system (4), we have , ( ) Therefore, the invariant region for N p is given by Again, from the system (4), we have ( ) Therefore, the invariant region for N v is given by Furthermore, Considering the aforementioned non-negative initial conditions, the proposed model ( 2) is positive invariant and solutions remain positive and bounded in the region The disease-free equilibrium  0 of the model ( 2) is defined by , , , , ) ,0,0,0, ,0 .
By using ( 14), we have

Solution existence
Here, we check the existence and uniqueness of the solution with the application of some well-known mathematical results.In this regard, let us consider the above given IVP Consider the Volterra integral equation of the given IVP in equation ( 17) Using the iterative scheme methodology on the non-linear kernel Φ, we define the expression Therefore, we get the following expression Theorem 1.The given IVP in equation ( 17) has a unique solution under the contraction for Φ.Proof.From equation (18), we have  1) Then, the successive iterations give [ ] Applying the norm, we get If n → ∞, the right-hand side of equation ( 19) converges to zero.Then,

∫ which gives the existence of the solution ζ(t).
Now, for the uniqueness, consider two different solutions ζ(t) and ζ 1 (t).Then, Hence, there exists a unique solution for the proposed IVP (17).Therefore, we conclude that the proposed fractional-order model (2) has a unique solution.
, where 0 and ( ) is a positive, non-decreasing, and continuous function.Then, there exists a solution ζ * of equation (17), such that ( ) where ( ) and δ is a positive constant.
Proof.We define the metric d on space  by ( ) Define an operator : , It is easy to say 0 0 ( , ) The operator  is a strictly contractive operator, which can be seen by the following expression: Since ρ is non-decreasing, we have By using Theorem 2, there is a solution ζ * of IVP (17), such that ( , ) .( 1) Hence, the solution of the proposed model is stable.

Numerical analysis on the model
In this section, we perform the necessary numerical simulations (solution derivation, error estimation, and stability) to derive the solution of the proposed fractional-order model ( 2) by using the L1-PC scheme [41].

Derivation of the solution
According to the L1-PC method, the Caputo fractional derivative is numerically defined by where 1 1 ( 1) .( 2) We approximate C D ω ζ(t) by the formula (33), and put it into (32) to get where ζ k defines the approximate value of the solution of (32) at t = t k and 2) (34) can be rewritten as: After rewriting the terms (35), we get the following from

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Contemporary Mathematics ( 2) Define Remark that k a s ′ has the following characteristics: .
In view of equations ( 38) and (37), take the following form: We can see that equation ( 39) is of the form ( ), .
Hence, using the scheme of Daffatardar-Gejji-Jafari method gives an approximate value of ζ n given by 1 ,0 1 0 Therefore, this approximated solution of the DGJ scheme gives the following predictor-corrector algorithm called the L1-PC method.), Using the above given methodology, the approximation equations of the proposed model ( 2) in terms of L1-PC method are derived as follows: 1 where  ),

Error analysis
The brief analysis on the error estimation of L1-PC scheme has been given in the studies [41,49,50] and now investigated below.The error estimate is given by here C is a positive constant depends on ω and ζ.Derive r n by [ ] In view of (44), .
To derive the error estimation, we will use the lemmas given below.Lemma 1. [51] For 0 < ω < 1 and k a s ′ (as given in equation ( 38)), we have ζ be the approximate solution calculated from the algorithm (40).Then, for 0 < ω < 1, we have where ' k a s are given in equation (38).Lemma 3. [41] Using Lemma 3 in equation ( 48), we get .
where C 1 is a constant defined above.

Stability analysis
Here, h is the step size given in equation (32).
Further, observe that

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Contemporary Mathematics Using discrete form of Gronwall's inequality and equation (38), we obtain where c is a constant and Using ( 50) and ( 51) in ( 49), we get where C is a constant.

Graphical simulations
In this section, we perform the graphical simulations to understand the behavior of the proposed model in a time range t  [0,20].The initial conditions are used as follows: S p (0) = 4,000, A p (0) = 500, I p (0) = 200, E b (0) = 500, S v (0) = 3,500, and I v (0) = 500.The parameter values are taken from Table 1 along with the control measures; the participatory community education programs (ξ = 0.7), vertical transmission control (δ = 0.6), and the clearance of Xcm bacteria in the soil (ψ = 0.5).
In Figure 2, the variations in the susceptible plants Sp and susceptible vectors Sv are plotted at fractional-order values ω = 0.9 and ω = 0.8, along with the integer-order case ω = 1.Here, we notice that as the fractional order decreases, the susceptible plant and vector population also decreases.
In Figure 3, the changes in the population of asymptomatic infected plants Ap and infected plants Ip are plotted at the same orders: ω = 1,0.9and ω = 0.8.Here, we notice the variations at given fractional orders after the time range [0, 5].Between the time range [5,20] months, when the fractional order decreases, the infection slightly increases.
In Figure 4, the variations in the Xcm bacteria in the soil Eb and infected vectors Iv are plotted at the given fractional-order values.From Figure 4(b), we notice that, reaching the end point of the time t = 20, all fractional-order outputs nearly converge.
In  From the given graphical simulations, we notice that the fractional-order values result in variations in the behavior of the model dynamics.Such effects cannot be captured by using integer-order derivatives, which justifies the advantage of fractional derivatives.The graphs are plotted using MATLAB-2021a.

Conclusion
A fractional-order mathematical model of the BXW disease using Caputo derivatives has been considered in this study.The proposed model has been numerically solved using an L1-based predictor-corrector scheme.The analysis of the stability and error approximation of the proposed method has been established to justify the efficiency of the scheme.The graphical simulations justified the fact that fractional-order values result in variations in the model dynamics and that variations cannot be captured in the case of the integer-order model.In the future, some other fractional-order operators can be incorporated to analyze the proposed model's dynamics.Moreover, some other fractional-order models can be proposed to forecast the outbreaks of BXW.

Figure 1 .
Figure 1.Compartmental diagram of the model ,..., ) ( then solution ζ(t) exists and continuous.where Λ n (t) is the amount of error with Λ n (t) → 0 when n → ∞.Then, Now, is absolutely continuous and satisfies n

z
are the predictors and c n ζ is the corrector.
of the proposed IVP and p k positive quantities k and h'

Theorem 5 .
Suppose ( , )t ζ Φfollows the Lipschitz property with respect to the variable ζ with a constant L and ( established from the scheme(40), then the scheme (40) is stable.Proof.We have to prove that 0 0

Figure 2 .Figure 3 .Figure 4 .Figure 5 .
Figure 2. Variations in the susceptible population Sp and Sv at fractional order values ω

Table 1 .
Parameters with numerical values [14] Rate of recruitment of susceptible suckers Birth rate of susceptible vectors Rate of harvesting of old plants Vertical transmission rate from an infected plant Removal rate of infected plant Death rate caused by BXW Rate of infection caused by contaminated farming measures from asymptomatic infected plants Rate of infection caused by contaminated farming measures from symptomatic infected plants Contact rate between vectors and banana plants Probability of Xcm bacteria transmission from an infected vector to a susceptible plant when in contact Probability of Xcm bacteria transmission from contaminated soil to a susceptible plant Probability of Xcm bacteria transmission from an infected plant to a susceptible vector Death rate of the vectors Rate of recovery of infected vectors Transmission rate of asymptomatic infectious class to symptomatic infectious plants class Spreading rate of Xcm bacteria from symptomatic infectious plant to the soil Half saturation constant of Xcm bacteria in the environment Rate of natural clearance of bacteria in the environment bv αpAp (αp + d +r)Ip (1 ₋ ξ) βaSp qAp (1 ₋ δ) θIp ϕIp μvIv μvSv μvIv αγ3Sv (ψ+ μb)Eb , Consider ( )