A Numerical Study of the Dynamics of Vector-Born Viral Plant Disorders Using a Hybrid Artificial Neural Network Approach

Most plant viral infections are vector-borne. There is a latent period of disease inside the vector after obtaining the virus from the infected plant. Thus, after interacting with an infected vector, the plant demonstrates an incubation time before becoming diseased. This paper analyzes a mathematical model for persistent vector-borne viral plant disease dynamics. The backpropagated neural network based on the Levenberg—Marquardt algorithm (NN-BLMA) is used to study approximate solutions for fluctuations in natural plant mortality and vector mortality rates. A state-of-the-art numerical technique is utilized to generate reference data for obtaining surrogate solutions for multiple cases through NN-BLMA. Curve fitting, regression analysis, error histograms, and convergence analysis are used to assess accuracy of the calculated solutions. It is evident from our simulations that NN-BLMA is accurate and reliable.


Introduction
Plant disease epidemiology studies how diseases affect plant populations and how to combat plant diseases. Using spatial and temporal plant epidemiology models can provide useful statistical and mathematical data about disease transmission. In the mid-20th century, plant epidemiological models became prominent [1]. Examples of actual uses of this type of model include cassava mosaic disease [2], pine wilt disease [3], and potato late blight [4]. Later, new methods for studying nonlinear dynamics and numerical simulations helped solve complex ecological problems [5,6]. This accelerated the creation of more realistic and complicated plant disease models.
An essential part of the plant epidemiological system is modeling the interactions between infected and healthy plant populations, either directly or via a vector. Infected vectors feed on healthy plants, infecting them. Similarly, non-infected vectors become infected by diseased plants. The vector-borne plant disease is classified as persistent, semipersistent, or non-persistent based on the infectious agent's residence period in the vector [7,8]. The vector ingests viruses while feeding on infected plant sap in persistent transmission. The salivary glands then release the viruses into the plant tissue as they penetrate the digestive system. The persistent mode of transmission differs from the other two because it takes a long time for a vector to become infected with the virus and become infectious [7,9]. In the case of vectors, this time lag is referred to as the latent phase of infection.
given in Table 1 [2,35], which was previously analyzed by Basir et al. [35] for persistent vector-borne viral plant disease dynamics for the effect of both latent period and incubation delay of the dynamics of the deceased. This model is numerically analyzed using a gradient-based numerical technique. Numerous studies claimed that gradient-based techniques, such as RK-4, take up much more computer time than soft computing methods with comparable accuracy and that it is difficult to produce accurate global estimates of the truncation error [36,37]. For instance, at each step of the RK-4 method, the derivative must be evaluated n times. Here, 'n' is the order of accuracy of the RK-4 method, which is a significant drawback of gradient-based algorithms [38]. Moreover, RK-4 suffers from divergence for complex systems [39]. Failure in the case of singularity is another hurdle in using these gradient-based numerical techniques. Keeping these disadvantages in mind, the authors of this paper aimed to suggest an alternative gradient-free approach that can handle problems, such as model (1), with accuracy and reliability. The key features of this study are outlined as follows: • In this paper, we analyzed an established mathematical model (1)

Problem Formulation
This section develops a mathematical model for persistent vector-borne viral plant disease dynamics. The model considers plant and vector populations without explicitly including the mosaic virus. H(t) signifies healthy plants, while the infected plants are represented by I(t), Q(t) represents uninfected, and W(t) represents the infected whiteflies population.
Due to restricted plantation space and natural resources, logistic growth r and carrying capacity K are considered for healthy plants [2]. A healthy plant becomes infected when it comes into contact with an infected vector. When an infected vector and a susceptible plant are present, λ is the transmission rate, and λHW is the number of sensitive individuals moving from the susceptible compartment to the infected compartment.
An insect pest, such as a whitefly, shifts its host in response to changing biological and environmental conditions. They generally move between fields of crops [40,41]. They breed in the fields. The Holling type III survival curve describes their life course because of the high death rate they experience early on [41]. Whiteflies (adults and nymphs) can transmit illness.
Crops are typically planted and reaped at specific times of the year. Most crops are reaped a few months after they are produced. A few vectors travel from close or distant patches and reproduce in the vegetation. Vectors grow by migrating from another patch because of reproducing in the same patch or vegetative area. For the same reason, seasonal fluctuations in vector populations are ignored [35].
An open system is considered in this model. Assume Π is the rate of vector birth and migration into the system. No vertical virus transmission is allowed, and a vector cannot infect another vector. Viruses do not destroy or defend vectors. The vector retains the virus and does not recover. However, the infective insects do not get sick from the virus [31]. Let the mortality rate of plants and vectors be represented by µ and d, respectively. Infection-related plant death is expected to be higher than average plant mortality. m 1 is the infection-related mortality of infected plants. Thus, the overall plant mortality rate is m = µ + m 1 . Consider β to be the conversion between uninfected vectors (i.e., Q) and the infected plant (i.e., I). So, βQI signifies entering the number of uninfected vectors Q into the infected vectors W compartment.
In truth, both plant and vector infection takes time. Let τ 1 ∈ R + be the healthy plant's incubation time following successful infection. At time t, the disease transmission is given by the expression λe −mτ 1 H(t − τ 1 )W(t − τ 1 ), where the positive constants described previously are λ and µ. The term e −mτ 1 denotes the chance of a healthy plant surviving through the incubation time [t − τ 1 , t], i.e., the number of susceptible plants that came into touch with an infected vector at time t − τ 1 and lived up to time t to become infected plants.
Again the latent period in a vector is The number of uninfected vectors met an infected vector at time t − τ 2 and survived until time t to become infected [35]. Based on the given assumptions, the mathematical model is The initial biological conditions are The parameters used in the model (1) assigned some numerical values for solving the model numerically, and Table 1 shows its description and numerical values.

Design Methodology
This section examines artificial neural networks (ANN) using a novel approach to machine learning by focusing on the supervised neuronal learning mechanisms of these networks to utilize the study of the model for persistent vector-borne viral plant disease dynamics.

Artificial Neural Network (ANN)
An artificial neural network is a network of interconnected core components known as neurons that receives various inputs and generates only one output; each neuron represents a mapping. A neuron's output is a function of the total of its inputs produced by the activation function.

Activation Function
To introduce nonlinear properties, an activation function is used in an ANN. In a neural network, (Xi, Wi) stands for inputs, weights, and f (Xi), which is the input function that is sent to the network's output. This output function can then be used as an input for any additional layers or the final output [42][43][44].
The number of hidden units can be optimized using a multilayer perceptron (MLP). Both the weights and biases of the connections were enhanced as well. The construction of a standard MLP with one hidden layer is as follows: X i represents the inputs, where W ij and b j represent connection weights and biased vectors, respectively. Here, a log-sigmoid function is used as an activation function in the feed-forward neural network model, which is given below.
The MLP, also known as the feed-forward neural network (FNN), is a type of neural network with a hidden layer between the input and output layers. This layer is called the "hidden layer." The number below the hidden layer represents the number of neurons used inside the network. Figure 1 shows an artificial neural network controller. A backpropagated Levenberg-Marquardt method is used to train the feed-forward neural network. Local minima can be found using the LM algorithm, which is built-in in many applications.
Additionally, NN-BLMA is implemented in two phases. Figure 2 depicts the Algorithm's whole workflow, including all of its steps.

•
For collecting the initial reference data set, we solve the model (1) numerically by using a state-of-the-art technique. Here we use the RK − 4 method, which commonly gave better results, in Mathematica using the "NDSolve" package. The numerical technique generates 5001 in the range of [0, 50] with a 0.01 stepsize. • After that, the NN-BLMA is executed by using "nftool", a built-in MATLAB tool, to train, validate and test the targets (reference data set). The design technique uses 60% of the targets for training and 20% each for validation and testing. The maximum iteration is set to 1000 with a 60 number of neurons. Table 2 presents the parameters for the design scheme execution, and Algorithm 1 is the pseudo-code of the designed NN-BLMA.  The novel machine learning of NN-BLMA is easy to apply, handles nonlinear problems, and is also a gradient-free technique that converges faster than other machine learning technique [45][46][47][48].

Algorithm 1 Pseudocode of NN-BLMA:
Starting of NN-BLMA Construction: Construct inputs and reference data set using RK-4 method in Mathematica Data selection: Input and target data must be selected in non-linear format, i.e., matrices. Startup: Taking number of neurons and distributing the reference data set into training, testing and validation • 60 Hidden neurons • 60% data for training • 20% data for testing • 20% data for validation Architecture: Each input is given a weight, and the input to the transfer function is formed by adding the weights of all of the inputs together along with the bias. Stopping criteria: If all of the conditions listed below are met, the previous process will end automatically.
• Mu reach to its maximum value • Number of iteration reaches to maximum • Performance value reaches to minimum • Validation's performance became less then maximum fail • Gradient's performance dropped below minimum gradient The network is generalised using training data. If the outputs are good, proceed to Saving Output; otherwise, retrain the network.
Retraining: Change the startup parameters and train the network again Saving outputs: End the process by saving the results graphically as well as numerically Ending of NN-BLMA

Numerical Experimentation and Discussion
To study the design algorithms' performance and efficiency, we discuss various cases of the nonlinear model of vector-borne viral plant disease dynamics. The cases are based on variation in two parameters (i.e., plants' natural mortality rate, µ, and vector mortality rate, d). We set the same numerical value for both parameters in the first case. In case two, there is a slight decrease in the µ parameter and a slight increase in the parameter d, while in the third case, there is an increase in the parameter µ and a decrease in the d parameter compared with the first case. Figure 3 illustrates the mathematical model and the cases detail for vector-borne viral plant disease dynamics.

Cases
The cases of the model are based on variations in plants' natural mortality rate (i.e., μ) and vector mortality rate (i.e., d).  The design technique generates output data sets with probabilities of 60% of the sample data for testing, 20% for training, and 20% for validation. The performance graph of the design technique shows us its mean squared error (MSE). Figures 4-6 depict the best validation performance provided by the design technique because the error is minimized after some epochs of training but may increase on the validation data set as the network begins to overfit the training data. The training is halted after six consecutive rises in the validation error, and the best performance is picked from the epoch with the lowest validation error. The case 1 performance values are in the range of 2.9721 × 10 −9 , 7.1129 × 10 −9 , 3.0066 × 10 −8 and 2.8222 × 10 −5 . Similarly, the case 2 and case 3 performance values are in the range of 1.3057 × 10 −9 , 3.6923 × 10 −11 , 1.17878 × 10 −9 , 1.9703 × 10 −4 , and 9.9788 × 10 −11 , 3.1230 × 10 −9 , 2.4709 × 10 −8 , 2.7474 × 10 −4 , respectively.                The tables below provide the data information provided by the computing system. The tables show the best performance values in training, testing, validation, etc. Table 3 displays the best performance data for case 1, while the best performance data for case 2 and case 3 are displayed in Tables 4 and 5, respectively. These tables also show the hidden neuron count, iterations, and time spent. Table 3. Performance values of the design NN-BLMA, and time spent by the computing system to obtain solutions for case 1.  Table 4. Performance values of the design NN-BLMA, and time spent by the computing system to obtain solutions for case 2.  Table 5. Performance values of the design NN-BLMA, and time spent by the computing system to obtain solutions for case 3. The histogram of errors between targets and outputs after training a neural network is shown in Figures 13-15. Different color bars show the errors in the training, validation, and testing data. The error bars in which most of the points lie are very close to the zero error line, which means targets and the outputs are well matched and have the fewest errors, which shows the accuracy of our design technique. The error values for case 1 lie in the range 10 −3 to 10 −4 , 10 −4 to 10 −6 , 10 −4 to 10 −6 , and 10 −2 to 10 −3 . For case 2 and case 3 the error values lie in the range 10 −4 to 10 −5 , 10 −5 to 10 −7 , 10 −4 to 10 −5 , 10 −3 to 10 −4 , 10 −2 to 10 −3 , 10 −4 to 10 −6 , 10 −4 to 10 −6 , 10 −3 to 10 −5 , and 10 −2 to 10 −3 , respectively.  Further, Figure 16 compares the numerical solution of the model obtained by the "ND-Solve" package in Mathematica (targets) to the solution obtained by executing NN-BLMA (outputs). The solid lines show the solution obtained by solving the model numerically by the "NDSolve" package in Mathematica, while the circles show the solution by NN-BLMA. In the figure, we see that the solutions obtained from NN-BLMA come exactly on the targets' solutions lines, which shows how accurate our design technique is. These figures also indicate the model's variation due to some parameters in the model. It is obvious from the figures that healthy plants and uninfected whiteflies rise when there is an increase in plant mortality rate and a drop in vector mortality rate. In contrast, a drop in plant mortality rate and an increase in vector mortality rate leads to a rise in infected plants and whiteflies. The comparison of statistical data given by the 'NDSlove' package in Mathematica with the outputs of NN-BLMA is illustrated in the tables below. Table 6 illustrates the comparative analysis of both the solutions for case 1, while the comparison for case 2 and case 3 are illustrated in Tables 7 and 8

Conclusions
In this paper, we analyzed a mathematical model for persistent vector-borne viral plant disease dynamics. The model includes equations for healthy and infected plants and uninfected and infected whiteflies. The selected set of parameters for numerical simulation is for the cause of the mosaic disease in cassava. To see the impact of variation in the mortality parameters on the model, we made different cases in which we vary both plant and vector mortality parameters. The reference data (targets) for NN-BLMA were generated by solving the model numerically for all the cases in Mathematica. The designed technique uses the targets to train, test, and validate the ANN and to see the impact of variation in plants' natural and vectors' mortality rates. The key points concluded from the study are given below.

•
From the study, we see an increase in the mortality rate of plants, along with a decrease in the mortality rate of vectors, increases in healthy plants and uninfected whiteflies, and decreases in infected plants and infected whiteflies. In contrast, a drop in the mortality rate of plants and an increase in the mortality rate of vectors results in a decrease in healthy plants and uninfected whiteflies and an increase in the number of infected plants and infected whiteflies. • Further, the accuracy of the design technique is illustrated through extensive graphical and tabular data, which include the best performance in terms of the mean squared error, histogram, and regression analyses.