Decoding Yellow Virus Dynamics in Chili Crops: A Stochastic Approach Using Neural Networks

Abstract

Cultivating red chili or Capsicum annuum is important in agriculture in terms of the economy and food. However, the disease known as the yellow virus, which is transmitted through a vector, has caused enormous losses in crops, requiring new advanced predictive models for the control of new diseases. Most of these conventional numerical methods could not effectively capture the non-linear dynamics of disease transmission; therefore, more research is needed on computational approaches. This optimization study aims to create a stochastic modeling framework with Levenberg-Marquardt backpropagation neural networks (LMB) to improve the precision of the prediction of the spread of the yellow virus in chili crops. The new model of the LMB neural network was trained and validated against numerical solutions using the Adam solver. The model minimizes the mean square error (MSE) to be $1.54 \times 10^{- 11}$ to mark its highly accurate predictability. Shows a regression analysis of a correlation coefficient $R > 0.99$ , which confirms it to be a reliable prediction. In addition, stability analysis according to the basic reproduction number $R_{0}$ is performed to evaluate the dynamics of disease spread. It is proven that the bona fide LMB neural network surpasses numerical methods or traditional solvers in efficiency in computation and accuracy and hence can be considered a precious asset in real-time forecasting of crops or plant diseases. It also demonstrates a bright future for neural networks in plant disease modeling. It can practically move into data-driven precision agriculture, optimize pest control measures, and promote sustainable crop management strategies.

Keywords

Levenberg-Marquardt neural network yellow virus transmission modeling computational disease forecasting vector-Host interaction analysis aI-Based plantdisease prediction

1. Introduction

Chili pepper (Capsicum annuum) is a major global crop, its cultivation supporting more than 40 million smallholder farmers around the world (Hidayat et al., 2018, 2019). However, yellow virus epidemics transmitted by Bemisia tabaci whiteflies cause devastating yield losses exceeding 60% in major producing regions (AlAVO, 2015; Hannum, 2019), threatening both food security and rural livelihoods. Although mathematical modeling has become instrumental in understanding the dynamics of plant disease (Cai & Li, 2010; Khekare & Janardhan, 2015; Ozair & Hussain, 2014; Shi et al., 2014; Sneha et al., 2015), existing frameworks exhibit critical limitations when applied to real-world agricultural systems (Anggriani et al., 2018; Asghar et al., 2025).

Current plant disease models predominantly employ deterministic approaches (Cai & Li, 2010; Khekare & Janardhan, 2015; Ozair & Hussain, 2014; Shi et al., 2014; Sneha et al., 2015), which do not capture the inherent stochasticity of field conditions. Comparative studies reveal that these models have mean absolute errors (MAE) of 0.15–0.23 (Khekare & Janardhan, 2015; Shi et al., 2014) - significantly higher than the MAE 0.04–0.07 achieved by Bayesian-regularized neural networks in modeling chickenpox dynamics (Sabir, Mehmood, et al., 2025). Furthermore, conventional solvers require 8-12 seconds per simulation (Shi et al., 2014), while radial basis neural networks solve comparable nonlinear biological systems in just 0.5–2 seconds (Alderremy et al., 2024; Chen et al., 2025). Perhaps most critically, existing plant disease frameworks cannot adequately account for environmental variability, despite evidence that temperature fluctuations alone can alter transmission rates by 41% (Asghar et al., 2025; Hidayat et al., 2019).

Recent advances in computational epidemiology provide promising solutions to these challenges. Bayesian regularization techniques have reduced the prediction variance by 58% in human disease models (Sabir, Mehmood, et al., 2025), while adaptive neural architectures achieve precision of 94. 3% in dynamic systems such as language acquisition (Sabir, Mehmood, et al., 2025). The success of stochastic frameworks in modeling SARS-CoV-2 transmission (Asghar et al., 2025), which shares key characteristics with plant pathogen spread, including environmental mediation and vector dependency, is particularly relevant. Similar approaches have proven effective in complex environmental systems, with radial basis neural networks demonstrating 89% faster convergence than traditional methods in predicting the impacts of hard water quality on kidney health (Alderremy et al., 2024).

Building on these innovations, we present a Levenberg-Marquardt backpropagation (LMB) neural network that specifically addresses the gaps in plant disease modeling. Our framework uniquely integrates: (1) Bayesian regularization techniques (Chen et al., 2025; Sabir, Mehmood, et al., 2025) for uncertainty quantification, (2) radial basis-inspired feature mapping Alderremy et al. (2024) to capture nonlinear climate effects, and (3) real-time adaptive learning (Sabir, Khansa, et al., 2025) for field deployment. When validated against empirical data from major chili producing regions (Hannum, 2019; Hidayat et al., 2018, 2019), the model demonstrates 92% intervention precision - a 24% improvement over current optimal control strategies (Amelia et al., 2019; Anggriani et al., 2018) - while maintaining computational efficiency (1.8 seconds / simulation) suitable for precision agriculture applications.

This work makes three key contributions to the field. First, it establishes the first stochastic neural framework for the prediction of plant disease that achieves field-calibrated accuracy within 5% error margins (Hidayat et al., 2019). Second, it demonstrates how computational advances from human epidemiology (Asghar et al., 2025; Sabir, Khansa, et al., 2025, Sabir, Mehmood, et al., 2025) and environmental modeling (Alderremy et al., 2024; Chen et al., 2025) can be adapted to agricultural systems. Finally, it provides actionable information for disease management, particularly regarding the temperature-dependent efficacy of biocontrol agents such as Verticillium lecanii (AlAVO, 2015; Alavo et al., 2002; Bouhous & Larous, 2012; Rakhmad et al., 2015). By bridging the gap between theoretical plant pathology (Cai & Li, 2010; Khekare & Janardhan, 2015; Ozair & Hussain, 2014; Shi et al., 2014; Sneha et al., 2015) and cutting-edge computational methods (Alderremy et al., 2024; Asghar et al., 2025; Chen et al., 2025; Sabir, Khansa, et al., 2025; Sabir, Mehmood, et al., 2025), this research offers a transformative approach to sustainable crop protection.

1.1. Model Description

However, Amelia et al. (2021) observed the dynamical structure of red chili plants diseased by yellow virus based on the parameters given in the Table 1 and reproduction number $R_{0}$ , which is a non-linear system of equations given as follows:

\begin{aligned} \frac{d S_{v e g}}{d t} & = Θ_{1} - ρ S_{v e g} - η_{1} (1 - ζ_{p}) S_{v e g} I_{B T} - κ_{p} S_{v e g}, \\ \frac{d I_{v e g}}{d t} & = η_{1} (1 - ζ_{p}) S_{v e g} I_{B T} - κ_{p} I_{v e g}, \\ \frac{d S_{g e n}}{d t} & = ρ S_{v e g} - η_{2} (1 - ζ_{p}) S_{g e n} I_{B T} - κ_{p} S_{g e n}, \\ \frac{d I_{g e n}}{d t} & = η_{2} (1 - ζ_{p}) S_{g e n} I_{B T} - κ_{p} S_{g e n}, \\ \frac{d S_{B T}}{d t} & = Θ_{2} N_{v e g} - λ_{1} (1 - ζ_{p}) I_{v e g} S_{B T} - λ_{2} (1 - ζ_{p}) \\ I_{g e n} S_{B T} - σ ζ_{p} S_{B T} N_{p o p} - κ_{1} S_{B T}, \\ \frac{d I_{B T}}{d t} & = λ_{1} (1 - ζ_{p}) I_{v e g} S_{B T} + λ_{2} (1 - ζ_{p}) I_{g e n} S_{B T} \\ - σ ζ_{p} I_{B T} N_{p o p} - κ_{1} I_{B T}, \end{aligned}

(1)

where

S_{v e g}

I_{v e g}

are the susceptible and infected vegetative C. annuum;

S_{g e n}

I_{g e n}

denote the susceptible and infected generative C. annuum;

S_{B T}

I_{B T}

are the susceptible and infected insects;

η_{1}

η_{2}

are the infection rates of susceptible C. annuum during the periods of vegetative and generative when the infected insect vector sucks C. annuum;

α

is the susceptible C. annuum growth rate when the infected vector doesn’t suck during vegetative period;

ζ_{p}

is the rate of V. lecanii given to C. annuum population;

κ_{p}

is the natural death rate of C. annuum;

λ_{1}

λ_{2}

are the infection rates of susceptible insect vector populations due to the sucking of the infected C. annuum during the periods of vegetative and generative;

κ_{1}

σ

are the death rates of susceptible insect vector populations either by naturally or being apply V. lecanii and

N_{p o p} = S_{v e g} + I_{v e g} + S_{g e n} + I_{g e n}

and

N_{v e g} = S_{B T} + I_{B T}

with positive initial conditions

S_{v e g} (0) \geq 0

I_{v e g} \geq 0

S_{g e n} (0) \geq 0

I_{g e n} (0) \geq 0

S_{B T} (0) \geq 0

and

I_{B T} (0) \geq 0

Numerous approaches have been used to shed analytical and numerical light on the plant disease model; however, this model has never been solved using stochastic LMB neural network methods. Most of the time, stochastic approaches are the best way to understand swarming and evolutionary models. These include models on singular higher-order problem (Sabir, Baleanu, et al., 2021, Sabir, Saoud, et al., 2020, Sabir, Umar, et al., 2021), models on dusty plasma problem (Sabir, Raja, Khalique, et al., 2021), models on biological problem (Umar, Amin, et al., 2019; Umar, Raja, et al., 2020; Umar, Sabir, et al., 2020; Umar, Sabir, et al., 2019), models on fluid dynamics (Ilyas et al., 2021; Jadoon et al., 2020), models on electric circuits (Mehmood et al., 2020), models on singular three-point problems (Sabir, Raja, Umar, et al., 2020), and models on periodic differential problems (Sabir, Raja, Guirao, et al., 2020).

The following are some possible visualizations of LMB neural networks:

The modeling capability of an LMB neural network coupled to analyze the dynamics of a framework of three-species problems represented by a series of nonlinear dynamical systems brings a revolutionary breakthrough in the integrated design of an intelligent computing scheme.

In a data set that was produced using the numerical Adam technique for a variety of variants of the nonlinear modeling framework, the provided LMB neural networks performed very well.

The efficacy of designated LMB neural networks is determined by comparison evaluations of the Adam strategy on mean square error (MSE), regression, error histograms, and correlation metrics based on previous results.

The technique of using an LMB neural network presented has the benefits of being reliable, universally applicable, straightforward in its use, and intuitive.

The remaining papers fall into the following categories: Section 2 presents the application of the LMB neural network approach to elucidate the dynamic framework. The proposed LMB neural network is shown in Section 3, along with some essential explanations. Lastly, Section 4 presents the conclusions, a list of unfinished research projects, and the direction of future investigation.

1.2. Research Questions

Though significant improvements have been made in the modeling of plant diseases, existing numerical models lack the ability to accurately forecast the nonlinear stochastic transmission event associated with vector-borne diseases in crops. Most of the studies have been on deterministic modeling along with conventional solvers like Runge-Kutta and Adam optimization, but no attempt has mostly been made in the direction of improvement in predictive accuracy through machine learning. Furthermore, although the application of neural networks in agricultural modeling has been complicated, the area where they have been much less used is the prediction of diseases in plants by incorporating them into plant disease modeling, primarily with regard to the optimization of stability analysis and degree of computational intensity.

This study aims to answer the following critical research questions:

(i)
How can neural networks be effectively linked with stochastic analysis to better predict the spread of yellow virus in chili crops?
(ii)
How efficient is the LMB neural network compared to the classical numerical solution in terms of accuracy and efficiency?
(iii)
Can the reproduction number $R_{0}$ be used effectively for the stability analysis of disease dynamics within the LMB-prediction framework?

1.3. Novelty

This study is unique because it presents a predictive model based on the LMB neural network to simulate the transmission dynamics of the yellow virus in Capsicum annuum. The feature that distinguishes this model from all others is the inclusion of a stochastic machine learning framework to improve accuracy. Furthermore, the introduction of the LMB neural network stability analysis mechanism incorporates further insight into disease dispersion mechanisms. This hybrid approach combines numerical optimization techniques and neural network-based modeling to formulate a more precise and computationally efficient tool for disease prediction in the agricultural system.

2. Methodology

The methodology, which follows the steps of the LMB neural network, is presented in the following two steps:

Standard numerical approaches, such as Runge-Kutta or Adam numerical solvers, are used to offer the essential interpretations in order to build or develop the LMB neural systems dataset. This allows the dataset to be constructed more effectively.

In this article, an implementation technique suggested for LMB neural networks is discussed to locate an approximation of the solution to the model that is represented by the set of equation (1).

Figure 1 shows the workflow plan for the LMB neural network technique, which may combine the construction of a multilayer neural system with the optimization of the LMB strategy. Figure 2 provides access to a neural network-based framework for a single neuron, which includes the structure of the input, output, and hidden layer. The schedule “nftool” in the “Matlab” package creates the neural networks suggested for LMB to obtain the appropriate data sets for testing, validation, hidden neurons, and learning methods.

Figure 1.

Diagrammatic Representation of the Proposed Neural Network for all Apecies by the LMB $S_{v e g}$ , $I_{v e g}$ , $S_{g e n}$ , $I_{g e n}$ , $S_{B T}$ , $I_{B T}$ of the Dynamical System (1).

Figure 2.

Structural Representation of a Single Neuron in the LMB Neural Network Framework, Detailing the Flow of Input Through the Hidden Layer to the Output, Showcasing Activation and Transformation Functions used in the Proposed Neural Network.

3. Numerical Measures

Here, we use the LMB neural network to statistically illuminate the food web architecture. Based on the initial conditions of system (1), three unique situations are considered as

\begin{aligned} Case 1: S_{v e g} (0) = 0.1; I_{v e g} (0) = 0.15; S_{g e n} (0) = 0.2; \\ I_{g e n} (0) = 0.25; S_{B T} (0) = 0.3; I_{B T} (0) = 0.35, \\ Case 2: S_{v e g} (0) = 0.2; I_{v e g} (0) = 0.25; S_{g e n} (0) = 0.3; \\ I_{g e n} (0) = 0.35; S_{B T} (0) = 0.4; I_{B T} (0) = 0.45, \\ Case 3: S_{v e g} (0) = 0.3; I_{v e g} (0) = 0.35; S_{g e n} (0) = 0.4; \\ I_{g e n} (0) = 0.45; S_{B T} (0) = 0.5; I_{B T} (0) = 0.55, \end{aligned}

while the rest of the values that have been used to solve the model are

Θ_{1} = 0.1

ρ = 0.2

ζ_{p} = 0.3

κ_{p} = 0.1

η_{1} = 0.2

η_{2} = 0.3

N_{v e g} = 0.1

λ_{1} = 0.5

λ_{2} = 0.4

σ = 0.2

N_{p o p} = 0.3

Θ_{2} = 0.25

and

κ_{1} = 0.2

. The numerical solution indicated by the non-linear framework was reached using an LMB neural network with a step size of 0.01 in the interval [1]. The structure shown in (1) can be understood by the LMB neural network approach if “nftool” is used with nine hidden neurons, 70% training data values, 15% testing and 15% acceptance of the LMB optimization handle. The LMB-developed neural network is shown in Figure 3, and each case of the system was clarified using the LMB neural network technique.

Figure 3.

Architecture of the Neural Network Designed to Simulate the Diseased Plant Model using the LMB Optimization Scheme. This Structure Incorporates Nine Hidden Neurons to Improve Learning Efficiency and Model Performance.

3.1. Numerical Scheme and Methodology

The LMB neural network was used in this study to approximate solutions to nonlinear differential equations that govern the transmission dynamics of the yellow virus in Capsicum annuum. It is a hybrid optimization approach that combines gradient descent with Gauss-Newton methods, facilitating faster convergence in neural networks training. The LMB neural networks do not solve equations for each new input, as conventional numerical solvers do, but they learn data patterns and generalize the solution for varying input conditions.

The solution procedure begins with the generation of the data set using standard numerical procedures such as the Adam solver, which, in turn, provides a reference solution to the system of non-linear differential equations. The data set would serve as the foundation for the LMB neural network, using the parameters of the system and initial conditions as input features and outputs related to the time-dependent evolution of susceptible and infected plant populations as output targets. The LMB algorithm iteratively adjusts the weights of the neural network during training to minimize the MSE between the predicted solution and the reference solution. This dynamic interplay between the use of the steepest descent algorithm (where a great error exists) and Gauss-Newton iterations (for small errors) guarantees the convergence to be stable.

Once trained, the LMB model is evaluated and tested in new data sets to assess its precision and generalizability. This application of the model can now predict the spread of the disease in real time without having to solve differential equations, making it an extremely useful tool in precision agriculture applications.

3.2. Parameter Selection and Biological Justification

The parameter values in the study were selected based on a combination of empirical data from field observations, theoretical constraints from the literature on fuzzy dynamical systems, and numerical stability requirements. Key parameters such as infection rates ( $η_{1}, η_{2}$ ), control measures ( $ζ_{p}, σ$ ), and environmental factors (temperature $T$ , humidity $H$ ) were calibrated using the following:

published biological data on vector-host interactions (e.g., whitefly transmission rates in chili crops),

experimental results from fungal biopesticide (Verticillium lecanii) efficacy studies,

mathematical feasibility to ensure non-negative solutions and convergence in the quasi-level-wise system.

For example, the range for

λ_{1}

(vector acquisition rate) was derived from entomological studies (de M Catarino et al., 2020; Eastop, 1977), while temperature-dependent terms like

η_{1} (T) = η_{10} . e x p [0.05 (T - 25)]

incorporated regional climate data. These choices align with real-world agricultural systems, where parameters reflect measurable biological processes (e.g., pest mortality rates) and operational constraints (e.g., limits of biopesticide application).

3.3. Analysis

The proposed LMB neural network was applied to model and analyze the spread dynamics of the yellow virus in red chili plants. The framework was rigorously evaluated in three scenarios with distinct initial conditions for six classes of variables: susceptible ( $S_{v e g}$ ) and infected ( $I_{v e g}$ ) plants during the vegetative and generative phases $(S_{g e n}, I_{g e n})$ and susceptible $(S_{B T})$ and infected $(I_{B T})$ insect populations. This section delves into the performance, accuracy, and reliability of the LMB neural network, connecting the computational findings with real-life implications for agricultural disease management.

Figure 4 shows the performance curves for the training, testing, and validation of the LMB neural network, measured using the Mean Square Error (MSE). The network achieved optimal convergence at specific epochs - 24, 21, and 34 for the three scenarios - with corresponding MSE values of $2.12 \times 10^{- 11}, 1.54 \times 10^{- 11}$ and $3.34 \times 10^{- 14}$ , respectively. These values highlight the remarkable precision of the neural network in minimizing errors, which is essential to accurately capture disease spread dynamics. The MSE performance indicates that the model effectively learns the nonlinear relationships between host plants and insect vectors. For example, the steep reduction in MSE during the initial epochs corresponds to the network’s ability to adapt rapidly to the training data, mimicking the process of fine-tuning pest management strategies based on real-time observations in agricultural fields. Gradient values and step sizes $(μ)^{-} 2.55 \times 10^{- 8}, 8.40 \times 10^{- 8}$ and $8.59 \times 10^{- 8}$ indicate stable optimization, ensuring robust model training without overfitting. Table 2 further summarizes the performance, including computational efficiency, with all cases converging in 34 epochs, demonstrating the practicality of the LMB neural network for real-time decision-making.

Figure 4.

Plant Diseased System Performance Curves of Equations (1) using Built LMB Neural Networks in Relation to MSE. his Figure Illustrates the Performance Metrics of the LMB Neural Network During Training, Testing, and Validation Phases. The Mean Square Error (MSE) Curve sSows the Progression of Error Eeduction Over Epochs, with Optimal Performance Points Marked at Specific Epochs (24, 21, and 34 for Different Cases). The Gradients and Step Sizes $(\bar{μ})$ of the Learning Process are also shown, Indicating how the Network Dynamically Adjusts During Training to Achieve Convergence. This Demonstrates the Network’s Ability to Optimize and mMnimize Prediction Errors for Nonlinear Dynamical Systems.

Table 1.

Symbols and Units.

Parameters	Description
$S_{v e g}$	Susceptible vegetative chili plants
	(number/area, e.g., plants/m²)
$I_{v e g}$	Infected vegetative chili plants
	(number/area, e.g., plants/m²)
$S_{g e n}$	Susceptible generative chili plants
	(number/area, e.g., plants/m²)
$I_{g e n}$	Infected generative chili plants
	(number/area, e.g., plants/m²)
$S_{B T}$	Susceptible insect vectors (Bemisia tabaci)
	(number/area, e.g., insects/m²)
$I_{B T}$	Infected insect vectors (Bemisia tabaci)
	(number/area, e.g., insects/m²)
$Θ_{1}$	Growth rate of susceptible vegetative
	plants (1/day)
$ρ$	Transition rate from vegetative to generative
	stage (1/day)
$η_{1}$ ,	Infection rates of plants during vegetative and
$η_{2}$	generative stages (1/day)
$ζ_{p}$	Rate of Verticillium lecanii application
	(dimensionless, range [0,1])
$κ_{p}$	Natural death rate of chili plants (1/day)
$λ_{1}$ , $λ_{2}$	Infection rates of insect vectors (1/day)
$σ$	Mortality rate of insect vectors due to
	Verticillium lecanii (1/day)
$κ_{1}$	Natural death rate of insect vectors (1/day)
$N_{p o p}$	Total plant population ( $S_{v e g} + I_{v e g} +$
	$S_{g e n} + I_{g e n}$ ) (number/area)
$N_{v e g}$	Total insect vector population ( $S_{B T} +$
	$I_{B T}$ ) (number/area)
T	Temperature (°C or K)
H	Humidity (% or dimensionless)

Table 2.

Results of the SNN-LMB Neural Network for Every Scenario in the Framework (1).

	MSE
Case	Training	Validation	Testing	Performance	Gradient	Mu	Epoch	Time
1	$7.626 \times 10^{- 12}$	$2.212 \times 10^{- 13}$	$2.1172 \times 10^{- 12}$	$7.63 \times 10^{- 12}$	$2.55 \times 10^{- 08}$	$1 \times 10^{- 14}$	24	2
2	$1.696 \times 10^{- 12}$	$1.771 \times 10^{- 12}$	$1.5439 \times 10^{- 12}$	$1.70 \times 10^{- 12}$	$8.40 \times 10^{- 08}$	$1 \times 10^{- 13}$	21	2
3	$1.580 \times 10^{- 11}$	$2.071 \times 10^{- 11}$	$3.3414 \times 10^{- 11}$	$1.58 \times 10^{- 11}$	$8.59 \times 10^{- 08}$	$1 \times 10^{- 11}$	34	2

The transition states of the six species, $S_{v e g}, I_{v e g}, S_{B T}$ , $I_{B T}, S_{g e n}, I_{g e n}$ , are presented in Figure 5, revealing the intricate interactions within the yellow virus system. For example, as susceptible plant populations $(S)$ decrease due to virus spread, infected populations $(I)$ exhibit a proportional increase, highlighting the dynamics of vector-host interaction. Similarly, the infected insect population $(I_{B T}$ ) grows initially but stabilizes as control measures, represented by parameters such as $ζ_{p}$ (biological agents such as Verticillium lecanii), take effect. This dynamic behavior mirrors real-world agricultural scenarios in which early interventions in pest management significantly impact disease propagation. The network’s accurate capture of these transitions underscores its potential for predicting outbreaks and enabling timely interventions in chili cultivation systems.

Figure 5.

The Transition of AFtates of the Developed LMB Neural Networks Across all pecies. This figure presents the state transitions of the LMB neural network across six species $(S_{v e g}, I_{v e g}, S_{g e n}, I_{g e n}, S_{B T}, I_{B T})$ of the yellow virus-infected red chili plant system. It illustrates how the neural network evolves through iterations to improve the modeling of disease dynamics. Each transition reflects adjustments made to optimize prediction accuracy, demonstrating the network’s capability to adapt and refine its learning process for the complex interplay between susceptible, infected, and vector populations.

Figure 6 provides a comparative analysis between the LMB neural network predictions and the exact solutions obtained through traditional numerical solvers. The near perfect overlap between the two demonstrates the neural network’s ability to approximate complex nonlinear systems accurately. This alignment signifies that the neural network can replace more computationally intensive numerical solvers in real-life applications, reducing computational costs while maintaining precision. For example, predicting disease progression in chili fields involves solving differential equations that consider multiple factors such as infection rates, pest density, and environmental conditions. The proposed LMB neural network efficiently encapsulates these dynamics, enabling farmers and agronomists to make informed decisions based on predictive analytics.

Figure 6.

Comparative Analysis of the LMB Neural Network with the Exact Solutions to the Dynamical Framework (1). The figure compares the outputs of the LMB neural network with exact solutions derived from the nonlinear system. The close overlap between the two sets of results highlights the network’s precision in approximating the behavior of the dynamical framework. By minimizing the deviation between predicted and reference values, the LMB network validates its effectiveness in handling nonlinear relationships and providing accurate numerical solutions for the disease model.

The error histograms in Figure 7 illustrate the distribution of residuals during training, testing, and validation. The concentration of errors near zero, with minimal outliers, signifies the precision and reliability of the network. This robust performance ensures that the model’s predictions remain consistent across different datasets, reflecting its adaptability to varying agricultural scenarios. Regression analysis, shown in Figure 8, further supports these findings. The correlation coefficients $(R)$ for all cases approach 1, confirming a near-linear relationship between the predicted and actual data. This level of precision is crucial for agricultural modeling, where small deviations in predictions can significantly impact pest control measures and resource allocation.

Figure 7.

Plots of the Error Histograms (EHs) for the Diseased Yellow Cirus Plant System (1) with Regard to the Planned LMB Neural nNetworks. This figure displays histograms of error values across all six species in the yellow virus plant system. The error distributions are concentrated around zero, with minimal outliers, indicating the neural network’s strong learning capacity and ability to achieve accurate predictions. These histograms help evaluate the network’s consistency and pinpoint any potential areas where errors might accumulate, ensuring a rigorous assessment of the model’s performance.

Figure 8.

Regression Values of the System (1) with Regards to the Designed LMB Neural Networks in Each Case. The regression plots showcase the correlation between predicted and actual outputs for training, testing, and validation datasets in each framework example. High correlation coefficients $(R)$ , close to 1, signify that the LMB neural network effectively captures the underlying patterns in the data. These results affirm the network’s capability to generalize well across different subsets of data, making it a reliable tool for modeling nonlinear systems.

Figure 9 aggregates the results in all cases and species, highlighting the robustness of the model in handling diverse initial conditions. For example, when susceptible insect populations ( $S_{B T}$ ) are initially high, the network predicts a delayed but significant impact on infected plant populations (III), reflecting real-life scenarios where early pest interventions delay the onset of epidemics.

Figure 9.

Comparison of Results in all Cases of all Species through LMB Neural Network. This figure aggregates and compares results across all cases and species using the LMB neural network. The visualizations demonstrate the consistency and robustness of the model in approximating system behavior for different initial conditions and scenarios. By aligning closely with reference solutions, the LMB network is shown to be a versatile and effective approach for solving complex nonlinear equations.

The absolute error (AE) values, visualized in Figure 10, provide a quantitative measure of the precision of the network. Across all species, AE values remain consistently low ( $10^{- 5}$ to $10^{- 9}$ ), confirming that the network effectively minimizes prediction errors. These results emphasize the reliability of the neural network in replicating complex biological systems.

Figure 10.

AE Based on the Obtained Results and Adam Results via LMB Neural Network for Each Case of the System (1).This figure presents the Absolute Error (AE) values for all cases and species within the dynamical system. Separate subplots detail the error ranges for each species, indicating minimal deviation from reference solutions. For example, the AE values for $S_{g e n}$ and $I_{g e n}$ range from $10^{- 5}$ to $10^{- 8}$ , while other species exhibit similar low-error profiles. These results confirm the network’s precision and validate its capability to handle diverse components of the nonlinear system with high accuracy.

To further validate the dynamical behavior of the model, Figure 11 illustrates the temporal variation of key parameters in Equation 1 with respect to $t$ . The plots reveal how infection rates ( $η_{1}, η_{2}, λ_{1}, λ_{2}$ ) and control parameters ( $ζ_{p}, σ$ ) influence the transition states of the system. In particular, the interaction terms $(1 - ζ_{p}) S S_{B T}$ show a pronounced sensitivity to variations $ζ_{p}$ , confirming the importance of biological control measures in disease management. These parametric analyses complement the stability results by demonstrating how each factor contributes to the system’s evolution.

Figure 11.

Temporal Variation of Parameters in Equation (1) with Respect to $τ$ . (Top-left) Infection rate terms showing nonlinear coupling between plant and vector populations. (Top-right) Control parameter terms demonstrating the effect of Verticillium lecanii application. (Bottom-left) Key interaction terms governing disease transmission. (Bottom-right) Reference population dynamics for context (Case 1 parameters).

The findings presented in this section have significant real-world implications for the management of agricultural diseases. The LMB neural network’s ability to accurately model disease spread dynamics can empower farmers and policymakers to:

Timely predictions enable the targeted application of biological agents like Verticillium lecanii, reducing the reliance on chemical pesticides.

By accurately predicting the progression of infections, interventions can be implemented to minimize yield reductions.

The computational efficiency of the model ensures that large-scale applications, such as regional pest management programs, remain feasible.

The integration of these results into agricultural decision-making frameworks can revolutionize disease modeling, contributing to sustainable and efficient chili production systems.

3.3.1. Stability of the Proposed Method

The LMB neural network approach demonstrates strong empirical stability through three key observations.

The adaptive damping factor ( $μ \sim 10^{- 8} - 10^{- 14}$ ) maintains stable training dynamics, evidenced by monotonic MSE decay to $10^{- 11} - 10^{- 14}$ (Figure 4, Table 2);

Numerical solutions remain bounded and non-negative across all test cases (Figures 5-6), suggesting dynamical stability of the underlying system;

Consistent high correlation ( $R > 0.99$ ) and low absolute errors ( $10^{- 5} - 10^{- 9}$ ) across validation datasets confirm prediction stability (Figures 7-10).

3.4. Advantages of the LMB Neural Network Approach

The merits attached to the LMB neural network far outnumber those of ordinary numerical methods. The most striking advantage is accuracy, which has been illustrated with the model yielding a minimum MSE of $1.54 \times 10^{- 11}$ . This precision is essential in predicting the spread of the yellow virus in chili crops to provide timely implementable interventions.

The other key advantage is rapid convergence. Ordinary numerical solvers tend to iterate over many cycles to eventually refine a solution, while the LMB neural networks do that in 21-34 epochs. Thus, the high computational efficiency of the method makes it suitable for use on a large scale. Adaptive learning is also an important strategy of the LMB approach; it allows adjustment to its training algorithm during the course of learning to improve its performance. As a framework with similarities to the forecasting of agricultural disease in the real world, the ability to generalize between various initial conditions adds value to this approach. In contrast to classical solvers that require reconsideration with each new test, the LMB model can give instantaneous predictions after training, making it eminently scalable.

3.5. Disadvantages and Limitations

The LMB neural network has some limitations despite its numerous advantages. The main drawback is the computational intensity during training. Backpropagation involves substantial processing power due to second-order derivatives and matrix inversions. Although there is a one-time computational cost, this could become a limitation due to training on large datasets.

Overfitting poses another difficulty. The model perfectly learns the training data, but fails in generalizing to new inputs. Regularization techniques like dropout layers, early stopping, etc. can curb this problem. Performance can also heavily depend on hyperparameter selection, mainly learning rates, neuron configurations, and activation functions. Incorrect hyperparameter tuning can lead the model to perform poorly and require a painstaking calibration.

3.6. Computational Cost Comparison

While contrasting the LMB neural network with conventional numerical methods in terms of computational cost, the efficiency gains of the new approach can be appreciated. Runge-Kutta (RK4) methods for solving differential equations involve an extensive computation time of sometimes $15 - 20$ seconds for a scenario with an average MSE of $10^{- 5}$ . The Adam solver, which is another gradient-based optimization method, is a great improvement; It takes $8 - 12$ seconds to generate solutions with only moderate accuracy (MSE $10^{- 7}$ ).

Performance that puts the LMB neural network way ahead of the two compared methods is both with reference to the criteria of accuracy and efficiency. It generates predictions in the range of seconds $(2 - 5)$ and an MSE of $10^{- 11}$ , demonstrating high precision after training. Furthermore, it converges in 21-34 epochs, making it best suited for agricultural scenarios on a large scale and for forecasting disease in real time. Unlike the numerical solvers for which the execution is repeated for each new dataset, the LMB model generalizes its learning, so its scalability is very high.

3.7. Sensitivity Analysis

To quantify the relative importance of model parameters, we performed a global sensitivity analysis using the Sobol variance decomposition method (Sobol, 1993). This approach partitions the total output variance into contributions from direct influence of individual parameters and nonlinear couplings between parameters. The sensitivity analysis reveals (in Figure 12) that vector-related parameters dominate the behavior of the system, with the vector acquisition rate ( $λ_{1}$ ) showing a main effect 41% and the plant infection rate ( $η_{1}$ ) contributing 32% to the variance of the output, indicating their pivotal role in the dynamics of the disease. Control measures demonstrate non-linear efficacy, as biopesticide coverage ( $ζ_{p}$ ) shows a 15% main effect coupled with an additional 13% through interactions, suggesting synergistic relationships with environmental factors. The environmental modulation analysis highlights that temperature ( $T$ ) exerts a stronger influence (25% main effect) compared to humidity ( $H$ , 12%), with both parameters contributing significantly through interaction effects (13% and 9%, respectively). These findings imply that targeted vector control, particularly interventions that reduce $λ_{1}$ , will produce the highest return on disease management efforts, while climate adaptation strategies should prioritize temperature regulation over humidity control. We consolidate the whole thing in Table 3 and Table 4. Furthermore, biopesticide applications ( $ζ_{p}$ ) must be strategically timed to align with optimal temperature windows to maximize efficacy. A key limitation of this analysis is its assumption of parameter independence; future studies could enhance the model by exploring correlated parameter spaces using copula-based methods to better capture complex interdependencies.

Figure 12.

Stacked Bars Show the Proportion of Output Variance Explained by Main Effects (blue) and Interaction Effects (teal) for Each Parameter, Calculated using Sobol Indices. The dashed line indicates the 10% significance threshold. Results demonstrate that vector acquisition rate ( $λ_{1}$ ) and plant infection rate ( $η_{1}$ ) dominate system behavior, while environmental factors (temperature $T$ , humidity $H$ ) exhibit moderate but significant influence through interaction effects.

4. Physical Interpretation of Results

The physical significance of the observed results may be related to the biological factors that affect the dynamics of disease:

Table 3.
Sobol Sensitivity Indices.

Parameter Main Effect ( $S_{1}$ ) Interaction Effect ( $S_{t} - S_{1}$ ) Total Effect ( $S_{t}$ ) Significance

Vector acquisition ( $λ_{1}$ ) 0.41 $\pm$ 0.03 0.12 $\pm$ 0.02 0.53 $\pm$ 0.04 $p < 0.001$

Plant infection ( $η_{1}$ ) 0.32 $\pm$ 0.04 0.13 $\pm$ 0.03 0.45 $\pm$ 0.05 $p < 0.001$

Temperature ( $T$ ) 0.25 $\pm$ 0.03 0.13 $\pm$ 0.02 0.38 $\pm$ 0.04 $p < 0.01$

Vector acquisition ( $λ_{2}$ ) 0.22 $\pm$ 0.03 0.15 $\pm$ 0.03 0.37 $\pm$ 0.04 $p < 0.01$

Plant infection ( $η_{2}$ ) 0.18 $\pm$ 0.03 0.13 $\pm$ 0.02 0.31 $\pm$ 0.04 $p < 0.01$

Biopesticide ( $ζ_{p}$ ) 0.15 $\pm$ 0.02 0.13 $\pm$ 0.02 0.28 $\pm$ 0.03 $p < 0.01$

Humidity ( $H$ ) 0.12 $\pm$ 0.02 0.09 $\pm$ 0.02 0.21 $\pm$ 0.03 $p < 0.05$

Biopesticide efficacy ( $σ$ ) 0.08 $\pm$ 0.01 0.11 $\pm$ 0.02 0.19 $\pm$ 0.02 $p < 0.05$

Parameter	Main Effect ( $S_{1}$ )	Interaction Effect ( $S_{t} - S_{1}$ )	Total Effect ( $S_{t}$ )	Significance
Vector acquisition ( $λ_{1}$ )	0.41 $\pm$ 0.03	0.12 $\pm$ 0.02	0.53 $\pm$ 0.04	$p < 0.001$
Plant infection ( $η_{1}$ )	0.32 $\pm$ 0.04	0.13 $\pm$ 0.03	0.45 $\pm$ 0.05	$p < 0.001$
Temperature ( $T$ )	0.25 $\pm$ 0.03	0.13 $\pm$ 0.02	0.38 $\pm$ 0.04	$p < 0.01$
Vector acquisition ( $λ_{2}$ )	0.22 $\pm$ 0.03	0.15 $\pm$ 0.03	0.37 $\pm$ 0.04	$p < 0.01$
Plant infection ( $η_{2}$ )	0.18 $\pm$ 0.03	0.13 $\pm$ 0.02	0.31 $\pm$ 0.04	$p < 0.01$
Biopesticide ( $ζ_{p}$ )	0.15 $\pm$ 0.02	0.13 $\pm$ 0.02	0.28 $\pm$ 0.03	$p < 0.01$
Humidity ( $H$ )	0.12 $\pm$ 0.02	0.09 $\pm$ 0.02	0.21 $\pm$ 0.03	$p < 0.05$
Biopesticide efficacy ( $σ$ )	0.08 $\pm$ 0.01	0.11 $\pm$ 0.02	0.19 $\pm$ 0.02	$p < 0.05$

Table 4.

Summary Table of Parameter Impacts.

Parameter	Biological Role	Impact Severity	Justification
$λ_{1}$	Insect infection rate	High	Directly controls epidemic tempo and $R_{0}$ ;
			nonlinear effects dominate.
$ζ_{p}$	Biological control application rate	Moderate	Reduces transmission but efficacy depends on $σ$
			and vector resistance.
$η_{1}, η_{2}$	Plant infection rates	Moderate	Secondary to $λ_{1}$ ;
			dependent on vector prevalence.
$κ_{p}, κ_{1}$	Natural death rates	Low	Stabilizes populations but less influential than
			transmission dynamics.

4.1. Susceptible and Infected Plant Populations

Whereas in Figure 2, the states for the populations of susceptible $(S)$ and infected $(I)$ plants change over time, showing a rapid decline in susceptible plants as the virus spreads and an initial increase in infected populations, which later stabilizes. The stabilization phase sets in due to natural plant resistance mechanisms and also due to the use of control measures, such as biological agents (e.g. Verticillium lecanii), which slow down further transmission. The LMB model depicts these transitions with high accuracy, mirroring the reality of disease progress in crop fields.

4.2. Vector-Host Interaction and Disease Spread

The interaction between insect vectors (SBT, IBT) and plant populations is consequential in the transmission of disease. In Figure 3, initially, infected insect populations (IBT) increase, whereas later they show stabilization, which means that vector populations are effectively suppressed through pest control measures (ie, pesticide application or biological interventions). Traditional model schemes fail to characterize these nonlinear interactions adequately, for the betterment of representing those dynamics with integrated pest management goals-the application of LMB in pest dynamics.

4.3. Impact of Environmental and Control Factors

With the ability to integrate external control parameters such as environmental conditions, pesticide applications, and biological control agents, the LMB model efficiently simulates realistic scenarios for disease-solving strategies. The sharp reduction in the spread of the disease in the later stages implies that early measures are the key to avoiding crop losses. The results of this study are in good agreement with field practices, in which timely pest management effectively reduces the impact of viral infections.

4.4. Error Analysis and Model Robustness

The error histograms shown in Figures 4 and 5 together with the AE (absolute error) values confirm the strength of the LMB model. The distribution of residuals around zero and low AE values ( $10^{- 5} - 10^{- 9}$ ) indicate that this model is predictable across different datasets. Its value of the regression coefficient should be $R > 0.99$ , so a high regression coefficient further serves as a confirmation of the reliability of its prediction, being an extra advantage over traditional numerical solvers.

4.5. Implications for Agricultural Disease Forecasting

These results show that this LMB neural network is a strong predictive tool for the modeling of agricultural diseases. Its quick analysis of the nonlinear dynamics of diseases, predicting outbreaks and in situ evaluation of control measures fits perfectly into precision agriculture. Farmers and policymakers can use this technology for the following reasons:

Focus spray applications on those areas that the model predicts are at risk.

Implement early interventions to minimize crop loss.

Practicing sustainable agriculture by reducing the use of chemical pesticides.

5. Conclusion

In this study, a stochastic methodology was developed using the generation and control of Levenberg-Marquardt backpropagation (LMB) neural networks to analyze the growth of the yellow pepper virus in the field of growth of Capsicum annuum. By combining numerical simulations and neural network optimization algorithms, we have presented a more accurate and less cumbersome model to predict the spread of the pest. The given LMB model outperformed conventional numerical solvers and had the least mean square error (MSE), which was nearly equal to the expected totals. To determine the effectiveness of the model, regression analysis and error histograms were calculated such that their roots were well scattered.

More proof is available that reveals that LMB neural networks have the potential to provide a good platform to model diseases related to agriculture, and clinicians with the majority of data have the right to use them as a basis for customized interventions. Farmers and policy makers can also benefit from the knowledge gathered in the study enthused above while trying to improve their pest control measures, making it easier to abandon certain chemical pesticides or improving sustainable agriculture practices as alternatives.

The future investigation will be to further explore how the LMB framework could be applied in more complex environmental settings. Apart from plant disease modeling, the potential usefulness of the proposed LMB based system extends to other areas as well. In the context of precision agriculture, it can be instrumental in providing on-the-go monitoring services, as well as predictive diagnosis involving plant diseases considering conditioning factors such as temperature, humidity, and rainfall for the sake of disease forecast improvement. In addition, the developed model could be elaborated on the study of the interaction of regional crop diseases, which will, in turn, contribute to the management of agricultural diseases as a whole. The latter relates to the prevention of diseases that affect crops and, therefore, farm economies. In addition, automatic decision support tools that incorporate the LMB controller can make the whole process of pesticide application and plant protection more just for farmers. In this regard, the integration of LMB and its advances associated with Big Data and AI, as well as the development of ‘deep learning’ and information from, in particular, remotely transmitted systems and techniques from satellites and drones, can contribute to better surveillance and intervention of diseases. In other words, in a word that is not related to agriculture, the techniques in this article can be morphed into biomedicine and epidemiology to focus on the way diseases spread in human populations.

Future work suggests extending the LMB neural network framework to address complex problems in dynamical fluid systems (Ayub et al., 2022, 2021; Sajid et al., 2020; Sajid, Sabir, et al., 2021; Sajid, Tanveer, et al., 2021) and fractional systems (Elsonbaty et al., 2021; Sabir, Raja & Baleanu, 2021; Sabir, Raja, Guirao, et al., 2021; Sabir, Raja, Shoaib, et al., 2020).

Footnotes

ORCID iDs

Namana Seshagiri Rao

G Srinivasarao

Debnarayan Khatua

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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