1 Introduction

Maize (Zea mays) or corn is one of the world’s most important cereal crops, globally produced. Its domestication in Mesoamerica has rendered it a cornerstone of world agricultural production and a prime source of food security and economic development [1]. Maize production is threatened by Maize Streak Virus (MSV), a leafhopper-transmitted disease-causing significant loss in yield, particularly in the tropical and subtropical regions of the world. There have been attempts at modeling MSV transmission dynamics via various mathematical and computational approaches. For instance, Seidu [2] proposed a deterministic ODE model involving fractional-order derivatives—i.e., the Atangana–Baleanu Caputo-type operator to capture memory effects and non-local interactions more accurately than traditional approaches. Liu [3] proposed an integrated stochastic model of variability in infection dynamics due to random environmental factors, modeling infection fluctuations via a logarithmic Ornstein–Uhlenbeck process. Mrope and Kigodi [4] gave an elaborate review of MSV control and transmission models in agroecosystems while O’Halloran et al. [5] researched the implementation of advanced deep-learning techniques for early detection of maize disease. They worked on the basis of integrating artificial intelligence for real-time monitoring of disease to enhance the responsiveness and efficiency of agricultural health systems. In another study, Ackora-Prah et al. [6] examined disease interactions within maize farms with Holling’s functional response within a fractal–fractional setting and showed that such models better capture biological complexities. We extend these efforts by developing a stochastic delayed model for which positivity and stability are assured. By the Newton polynomial routine, we carried out numerical simulations to examine the qualitative behavior of the model and confirm theoretical results. A few notable contributions are studies conducted by Mrope and Kigodi [7], in which they investigated the dynamic interaction of maize plants with Homopteran insect virus vectors. Facchi et al. [8] proposed the use of chitosan- and tannin-based polymeric coatings as antimicrobial agents for the management of Xanthomonas vasicola pv. vasculorum (Xvv), in which they demonstrated promising applications at the field level. Ali and Ameen [9] applied fractional calculus to investigate MSV persistence and transmission and noted its application in developing disease control policies. Dash and Sethy [10] noted that maize infections are a major cause of production loss but can be avoided by early detection and prevention. Kalyango and Ntanda [11] created an explainable deep-learning model for the diagnosis of maize diseases with the trade-off between predictive performance and explainability in order to facilitate effective agricultural decision-making. Suriani et al. [12] documented in morphological, physiological, and molecular detail the pathogens of bacterial stalk rot in maize, making possible species-level diagnosis. Malar et al. [13] applied Caputo–Fabrizio fractional derivative to describe MSV complex dynamics, while Mrope and Kigodi [14] also took into account the effects of control actions that are insecticide-based in an effort to decrease infection levels. Other deterministic and fractional models [15,16] have also provided information about the persistence of infection, memory effects, and long-term maize epidemic dynamics.

Beyond MSV-specific studies, more general stochastic epidemic models incorporating mechanical, chemical, and preventative control measures [17] have highlighted the central position of multi-strategy methods in reducing infection prevalence. Robaina et al. [18] standardized the inoculation protocol of Xvv in maize and determined a diagrammatic scale for resistance screening, and Tembo et al. [19] reported a quick and sensitive LAMP assay for MSV field detection. Ketsela et al. [20] confirmed the morphological symptoms of MSV infection chlorotic leaf streaks, chlorophyll loss, and growth retardation causing reduced yield or plant death through field observations. Finally, Wang et al. [21] demonstrated that stochastic models incorporating environmental transmission can explain periodic epidemic patterns, a concept relevant to multi-year epidemics of MSV. Recent studies have applied stochastic and delay-based epidemic modeling methods to a variety of infectious diseases, demonstrating the usefulness of dynamical consistency and global stability analyses for the interpretation of disease transmission and control measures [22,23]. Similar stochastic and bifurcation-based modeling methods have recently been formulated for human infectious diseases, such as influenza transmission and control, demonstrating the importance of stochastic effects and treatment–vaccination dynamics in epidemic models [24–27−]. Polynomial numerical schemes have proven successful in complex fractional dynamic systems, demonstrating Morgan Voyce polynomial approaches to time-fractional models [28,29]. Stochastic and cost-effectiveness modeling frameworks have been applied in recent epidemiological studies to examine intervention strategies for major infectious diseases such as HIV/AIDS and COVID-19, highlighting the applicability of data-driven approaches to the optimization of control strategies and vaccine efficacy [30–32].

Previous studies on Maize Streak Virus (MSV) dynamics have primarily relied on deterministic or fractional-order differential equation models, which, though useful, have a tendency to leave out the stochastic fluctuations and time-delays that occur in real-world agro-ecosystems. The majority of such studies have been idealized and have not tried to incorporate uncertainty due to environmental fluctuations, random infection, or heterogeneity in insect vector behavior. These limitations restrict their application to realistic field-scale epidemic prediction. To account for these lacunae, the present study develops a stochastic delayed model of MSV transmission incorporating randomness and temporal memory effects in plant-virus interactions. The study also provides a stochastic Nonstandard Finite Difference (NSFD) scheme, which is defined by positivity, boundedness, and step-size independence properties significant in the context of biological realism. The model combines theoretical analysis with computational efficiency and provides a more solid foundation for controlling MSV as well as other plant diseases.

The organization of the paper is as follows: Section 1 gives a review and thorough review of infectious maize streak disease-like disease reported in the literature. Sections 2 and 3 consider the establishment of the delayed model and the mathematical analysis later, and the two types of model equilibria and reproduction numbers. Sections 4 and 5 consider an investigation of the stochastic model, for example, its extinction and persistence. The stochastic NSFD approach is discussed in Section 6. Sections 7 and 8 are devoted particularly to numerical simulations and the presentation of results. Long-term opinions give a complete outline of the work under Section 9.

2 Formulation of model

This section presents a mathematical model describing the transmission dynamics of Maize Streak Virus (MSV) within a maize plant population. The total plant population at time t is denoted by

where each compartment represents a distinct epidemiological state:

• —Susceptible plants: uninfected plants that can acquire infection after contact with infected plants or vectors.
• —Insecticide-treated plants: plants protected partially by insecticide application; protection depends on the insecticide’s effectiveness.
• —Exposed plants: plants that have been infected but are in the latent (non-infectious) stage.
• —Infected plants: plants currently infectious and capable of transmitting MSV.
• —Recovered plants: plants that have gained temporary immunity following infection or treatment.

2.2 Model description

The dynamics of the system are governed by the following biological processes (Fig 1):

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Fig 1. Flow map of maize streak disease.

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• Recruitment: Susceptible plants enter the population from the environment at a constant rate .
• Infection: Healthy plants in the susceptible class become exposed after effective contact with infected plants . The disease transmission follows the law of mass action at a rate .
• Insecticide application: A fraction of susceptible plants is treated with insecticide at a rate , moving them into the insecticide-treated class . The insecticide confers partial protection, determined by its efficacy .
• Loss of protection: Treated plants gradually lose protection due to insecticide degradation or resistance and return to the susceptible class at a rate .
• Exposure and infection: Exposed plants progress to the infectious class at a rate .
• Recovery: Infected plants recover naturally or through treatment at a rate , moving into the recovered class .
• Loss of immunity: Recovered plants lose their immunity and return to the susceptible class at a rate .
• Secondary exposure: A proportion of insecticide-treated plants may still become exposed due to reduced insecticide efficacy or reinfection pressure.

The corresponding system of differential equations (not shown here) is derived based on these transition processes using the standard law of mass action.

(1)(2)(3)(4)(5)

where: , and are initial conditions.

3 Analysis of feasible properties

This section provides an analysis of the feasible properties of the stochastic delayed model (1)-(5).

3.1 Positivity and boundedness

To preserve the significant analysis of the model, each of the variables and must be non-negative. That is, the outcomes of the model analysis at every time in a practical range.

Theorem 1. (Positivity of Solutions): For the system (1)-(5), the solutions

remain positive for all and for all , given non-negative initial conditions.

Proof. To show that each state variable remains non-negative, consider the system (1)-(5) and evaluate its right-hand sides on the corresponding boundary surfaces where one compartment equals zero while the others remain non-negative.

Since all model parameters are positive and the inflow terms are non-negative, the vector field on each boundary of the positive orthant points inward. Consequently, solutions starting in cannot cross the coordinate planes, ensuring that

Hence, the system (1)-(5) admits positive solutions for all , as required.

Theorem 2. (Boundedness of Solutions): For the system (1)-(5), the solutions

are uniformly bounded for all .

Proof. Define the total population of maize plants as

By summing the differential equations (1)-(5), we obtain

This differential inequality implies

Solving this inequality using the Grönwall lemma yields

As ,

Therefore, - and consequently all state variables remain bounded for all .

3.2 Model equilibria and reproduction number

The MS disease delayed model’s equilibria will be briefly discussed in this section, and the maize streak free equilibrium , and the maize streak endemic equilibrium will all be covered.

As

3.3 Basic reproduction number

We compute via the next-generation matrix (NGM) method. The infected state vector is

Near the disease-free equilibrium (DFE) , the infection subsystem (with a fixed infection delay ) linearizes to

DFE values

At the DFE () the susceptible insecticide subsystem satisfies

which gives

Next-generation matrices

Write with new-infection terms and transition terms . The Jacobians at the DFE are

The next-generation matrix is . Since

we obtain

The spectral radius of (largest eigenvalue) is therefore

4 Stability analysis

In this part, we study the stability of the model both locally and globally at its equilibrium point, with the findings proven in the established results as follows:

(6)

,, , , , , , , , , , , , , , .

Theorem 3. (Local Stability of the Disease-Free Equilibrium)

The maize streak–free equilibrium

of the system (1)-(5) is locally asymptotically stable (LAS) if the basic reproduction number .

Proof. The Jacobian matrix of the system (1)-(5) evaluated at the equilibrium point is given by

The characteristic equation associated with is

which can be written as the fifth-degree polynomial

where the coefficients depend on model parameters as follows:

(19)

For biologically feasible parameter values, all coefficients . Applying the Routh–Hurwitz stability criterion for a fifth-degree polynomial, the necessary and sufficient conditions for all roots to have negative real parts are satisfied when Hence, all eigenvalues of possess negative real parts, and the maize streak–free equilibrium is locally asymptotically stable whenever .

Theorem 4. (Local Stability of the Maize Streak Endemic Equilibrium)

The maize streak endemic equilibrium

of system (1)-(5) is locally asymptotically stable (LAS) if the basic reproduction number .

Proof. The Jacobian matrix of system (1)-(5) evaluated at the endemic equilibrium is

The characteristic equation associated with is

which can be written as the fifth-degree polynomial

where

with parameter substitutions

Since all biological parameters are positive, the coefficients are positive and satisfy

By the Routh–Hurwitz stability criterion for a fifth-degree polynomial, all eigenvalues of have negative real parts when .

Therefore, the maize streak endemic equilibrium is locally asymptotically stable whenever .

Theorem 5. (Global Stability of the Disease-Free Equilibrium)

The disease-free equilibrium

of system (1)-(5) is globally asymptotically stable (GAS) whenever .

Proof. Consider the continuously differentiable Lyapunov function defined by

Its time derivative along trajectories of system (1)-(5) is

Substituting the corresponding right-hand sides of system (1)-(5) gives

After algebraic simplification, we obtain

All parameters are positive, and when the bracketed term is positive, ensuring that

Equality holds only at

By LaSalle’s Invariance Principle, the largest invariant set where corresponds precisely to the equilibrium . Therefore, every trajectory of the system tends to as .

Hence, is globally asymptotically stable whenever .

Theorem 6. (Global Stability of the Maize Streak Endemic Equilibrium)

The maize streak endemic equilibrium

of system (1)-(5) is globally asymptotically stable (GAS) whenever .

Proof. Define the continuously differentiable Lyapunov function as

Its time derivative along the solutions of system (1)-(5) is given by

Substituting the model equations into (29) yields

After simplification, we obtain

All model parameters and equilibrium components are positive. When , each term is non-positive, ensuring that

Equality holds only when

By LaSalle’s Invariance Principle, the only invariant set contained in corresponds to the equilibrium . Therefore, all trajectories of the system approach as . Hence, the maize streak endemic equilibrium is globally asymptotically stable whenever .

5 Stochastic formulation Phase 1

Based on the model (1)-(5), consider a vector of stochastic delay differential equations (SDDEs). Calculating the variance and expectations is our goal. Table 1 lists the likelihood of changes together with the corresponding transition time.

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Table 1. Illustrates an implicit modification to the model’s process.

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Expectation

(7)

(8)

Therefore, .

(9)

The next section will cover the conventional numerical methods for approximating solutions to Stochastic models. We all concur that . If then the temporal interval [0, T] is consistently divided with a uniform partition equal to , and the corresponding nodes are given as .

For each . Further, this will be agreed by us , however, and , .

The academic literature on the subject is consulted to simulate its results of Eq. (9) using the Euler-Maruyama approach. The details are displayed in Table 1 and are as follows;

(10)

where the discretization parameter is indicated by .

6 Stochastic formulation Phase 2

By incorporating Brownian motion, we get the dynamical system unreliable parameters (1)-(5). In the sequence described below:

(11)

(12)

(13)

(14)

(15)

where the uncertainty of each compartment and existence of Brownian motion are denoted by

6.1 Feasible properties

This model (11)–(15) concludes the examination of the positivity and boundedness features of the system.

Let us assume the following vector:

and norm

(16)

Moreover, let represent the set of all positive functions that are subsequently defined on . Furthermore, in the function is once differentiable and twice differentiable. We have defined differentiable. We have defined the differentiable operator , which is associated with stochastic delay differential equations (SDDEs) in four dimensions.

(17)

As,

If acts on function then we denote

where is Transportation.

Theorem 7: Demonstrates that there exists only one solution for the system (11)–(15) for all initial conditions . With a probability of one, these solutions will also invariably stay in .

Proof. Given that all model parameters are locally satisfiable by the Lipschitz bounds. Therefore, based on Ito’s formula, the above model has a positive solution locally on the interval , and is the timing of the explosion. The global solution of the model can be shown when equals infinity.

Define to be a big enough number so that , and are all included in the interval . Let’s construct the subsequent sequence for each positive integer .

(18)

where we set . Since is non-decreasing as ,

(19)

According to the inequality, is either equal to or smaller than . Our goal now is to show that, as we intended, equals infinity.

When and are found, the statement is satisfied. If this condition is not met.

(20)

Define a function by

(21)

By using Ito’s formula, we calculate

(22)

To simplify, we assume , then Eq. (22) could be written as:

(23)

where is a positive constant, after integrating from 0 to we get,

(24)

where taking the expectations leads to

(25)

Set for and from (19), we have

For each element in the set , there exist certain indices i such that is equal to either or , where i takes on the values 1, 2, 3, 4, and 5.

Hence, is less than

Next, we obtain

(26)

The indicator function is denoted as within the set . As approaches infinity, we get to the contradiction that infinity is equal to the value of , which is finite, as desired.

Theorem 8. If the spectral radius and the variance , then the number of infected plants in the system (11)-(15) will exponentially approach zero.

Proof: Let’s examine the initial data and the system (11)-(15) has a solution if it satisfies the stochastic delayed differential equation, where represents randomness and represents drift.

Applying Ito’s lemma to the function , we obtain

Notice that with .

If ,

If , then

, when , we get ,

, as desired,

7 Numerical methodology

Let be the set defined for each as . In this section, we will denote and analyze a discretization of the system (11)-(15). To achieve the objective, we consider the temporal interval where . Create a consistent partition of the time interval [0, T] into n subintervals, with a length of for each subinterval. For each , where is the collection of indices by considering . The numerical approximations for the functions and are denoted as and , respectively. The discrete initial data is defined, satisfying as required.

7.1 Stochastic nonstandard computational method

In our first equation (11) of the parametric perturbation model can be expressed with a non-standard computing approach; namely, equations (11)-(15) might be solved with a stochastic non-standard finite difference method.

(27)

For the stochastic NSFD approach

(28)

The system (11)-(15) can be decomposed by the stochastic NSFD process, as indicated in (28), and the entire system can then be expressed as follows:

(29)(30)(31)(32)(33)

where, and is a general normal distribution, i.e., ).

7.2 Convergence analysis

The following theorems are stated concerning the convergence analysis.

Theorem 8: There is only one positive solution for any initial value for equations (29) through (33).

Proof: The evidence is verifiable since the non-positive property of the biological problems’ constraint facilitates ease in demonstration.

Theorem 9: For the region . For every is an area of equations that is feasible and positively invariant (29) to (33).

Proof: The system (29) to (33) can be deconstructed and considered , as follows:

Next, we get

, as desired.

Theorem 10: The suggested computational method is stable for any if the eigenvalue is located in the unit circle.

Proof: Let the function and which are the right-hand sides of the equations (29–33). Consider .

Here,

, , ,

, .

It is well known that a system of the forms (29–33) converges to the optimal state of the model if and only if the spectral radius of the Jacobian, (J),

(34)

For the stability of the model. It follows the conditions:

When , the model’s equilibrium is stable. The stability of the model’s equilibria depends on whether . The model’s equilibria are naturally stable when .

The components of the method-related Jacobian can be expressed as follows: maize streak-free equilibrium, .

So, the eigenvalues of the Jacobian at as follows:

Lemma. For the quadratic equation , , if and only if the following conditions are satisfied:

1. i. .

1. ii. .

1. iii. .

Proof. The proof is straightforward.

8 Computational results

In this section, we compare conventional numerical methods with a non-conventional computational technique to evaluate their efficiency, precision, and computational cost in solving the proposed model. This consideration highlights the advantages and potential drawbacks of using nonstandard numerical methods in advanced epidemiological models. Parameter estimates used in this work were drawn in great part directly from the literature, making proper consistency with earlier validated MSV transmission models. A few parameters (e.g., recruitment rate Λ, natural mortality rate μ, and noise intensity σ) were assumed to be in biologically reasonable ranges since there is no special experimental data. All parameters are displayed in Table 2. All simulations were conducted using identical parameter sets (Table 2) and a fixed time horizon t = 500. The deterministic solution was used as the benchmark reference. The stochastic NSFD method demonstrates the best overall performance, achieving lower error and faster computation while maintaining numerical stability independent of step size (see Table 3).

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Table 2. Parameter values used in the model.

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Table 3. Quantitative comparison of stochastic numerical methods.

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8.1 Discussion

This section covered the discussion of graphical representations of the behavior of infected plants for different time step sizes with time delay . Fig 2 shows the infected plants over time at the endemic equilibrium point with a step size of . The model uses the Euler-Maruyama method, the stochastic NSFD method, and a deterministic approach. The small step size allows for more precise modeling of infection dynamics, capturing subtle changes in infection rates. In Fig 2, the stochastic nature leads to small fluctuations around the deterministic trajectory, which stabilizes as time progresses. Maize Streak Disease has a regular pattern with minimal variability, suggesting that for instances with minimal environmental or vector behavior variation, the infection is quite under control. By boosting the step size to h = 1, Fig 3 creates larger stochastic fluctuations. With bigger steps, the infection graph appears to be more unpredictable, particularly during the initial stage of the epidemic. The stochastic Euler-Maruyama and stochastic NSFD methods are the ones that have more variability than the deterministic model. This conveys that with less time resolution (larger delays or less frequent observations), the disease spread is less predictable with more chance of larger outbreaks. The step size increment shows the stochastic effects amplification, that is, environmental noise (for instance, abrupt temperature changes or vector activity) can cause fluctuations in the infection rate that are more than the normal by a large margin. Likewise, Fig 4 presents the same with a step size of h = 0.01 but it emphasizes the Euler method of stochastic modeling. The rate of infection is rather stable at endemic equilibrium and very little variability occurs. The Maize Streak Disease secures a uniform pattern, and the stochastic methods’ variability is undetectable, which means that the disease is kept under control in very strict and well-monitored conditions. In Fig 5 the step size is set to h = 1 and the variability of infection is significantly increased. The infected plant populations are showing larger fluctuations, with the stochastic methods (stochastic Euler and stochastic NSFD) straying quite a bit from the deterministic trajectory. This infers that in situations where there are delayed responses or more coarse time scales, the disease can fluctuate more severely, thereby making the control more challenging. The increase in step size illustrates how delay or inaccurate monitoring can result in ever more random outcomes. The stochastic Runge-Kutta method is being used in Fig 6 with a small step size (h = 0.01). The infection curve is unchanged with slight variations, just like in the other figures with smaller step sizes. The dissemination of Maize Streak Disease is under control, and there is no considerable stochastic impact. The stochastic Runge-Kutta method, like the Euler-Maruyama method, shows very low variability at this step size, which gives further evidence that the disease can be controlled fairly well if closely monitored. In Fig 7, where the step size is larger, h = 1, more variability is seen in the infection rate. The infection curve becomes less smooth, showing that the progress of the disease is more difficult to predict for larger steps. Delays in observation or in intervention might lead to greater outbreaks. Fig 8 illustrates how different delays affect the number of susceptible plants over time. Four different delays are shown: . As the delay increases, the number of susceptible plants drops more slowly, indicating that larger delays in implementing control measures allow more plants to remain vulnerable to infection. This explains that effective interventions happen to be those at appropriate times for the prevention of the disease by Maize Streak Disease. In Fig 9 infected plant population in a function of time with different time delays. Different values of time delay between and . As the delay increases, the rate of acceleration of infected plants increases and keeps increasing with time. This would be interpreted as the longer duration held in the interventions of the disease reveals more severe outbreaks of Maize Streak Disease. Shorter delays translated to quicker recovery of the disease , however, the longer delays showed constant outbreaks and higher infection rates. The illustration above stresses how timely interventions are essential to control the transmission of the disease. Fig 10 shows how the delay affects the basic reproduction number . We can observe that as increases, so does . An increase in certainly tells one that the disease becomes more transmissible as the delay increases. After introducing the delays into the control measures of Maize Streak Disease, therefore, it can infect more plants and make the disease generally more transmissible. Without delays , is relatively low, which means that this disease is more easily controlled. Increasing causes to rise thus, the disease starts to be uncontrollable.

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Fig 2. Infected plants at EE when .

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Fig 3. Infected plants at EE when .

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Fig 4. Infected plants at EE when .

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Fig 5. Infected plants at EE when .

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Fig 6. Infected plants at EE when .

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Fig 7. Infected plants at EE when .

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Fig 8. Effect of delay on susceptible plants.

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Fig 9. Effect of delay on infected plants.

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Fig 10. Effect of on of the delayed model.

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9 Conclusion

This paper formulates a computational and mathematical study of the MSV disease transmission model of time-delay dynamics. Maize plants were divided into five compartments: susceptible, insecticide-treated, recovered, infected, and exposed. Dynamical analysis considered key epidemiological features such as the basic reproduction number, equilibria, boundedness, and positivity conditions. Both the local and global stability of the endemic and maize streak–free equilibrium points were investigated, establishing the asymptotic behavior of the system under various parameter conditions. The stochastic generalizations of the model, such as positivity, extinction, and persistence of the disease under stochastic perturbation, were also studied. Among the computational methods attempted, the stochastic Nonstandard Finite Difference (NSFD) scheme offered improved accuracy, stability, and biological consistency performance. In particular, stability is crucial in stochastic epidemic modeling to prevent spurious oscillations and numerical instability. Compared to other conventional numerical methods such as the stochastic Runge–Kutta, stochastic Euler, and Euler–Maruyama schemes, the stochastic NSFD method-maintained robustness and consistency even at large simulation times. Overall, the results confirm that the stochastic NSFD method provides a reliable, efficient, and biologically sound model for describing complex plant viral epidemics like MSV, with implications for future research and disease management practices.