Mathematical Modelling of Host-Pest Interaction in the Presence of Insecticides and Resistance: A Case of Fall Armyworm

Gatwiri, Moreen Brenda; Ronoh, Marilyn; Ngari, Cyrus Gitonga; Kitavi, Dominic Makaa

doi:https://doi.org/10.1155/2024/2886786

Journal of Mathematics

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Abstract Introduction Numerical Simulation Discussion and Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 2886786 | https://doi.org/10.1155/2024/2886786

Mathematical Modelling of Host-Pest Interaction in the Presence of Insecticides and Resistance: A Case of Fall Armyworm

Moreen Brenda Gatwiri,¹Marilyn Ronoh,¹Cyrus Gitonga Ngari,²and Dominic Makaa Kitavi¹

Academic Editor: Firdous A. Shah

Received03 Apr 2023

Revised15 Oct 2023

Accepted10 Nov 2023

Published04 Jan 2024

Abstract

Several pest management programs have been developed to control rising agricultural pest populations. However, the challenge of rapid evolution and pest resistance towards control measures continues to cause high production losses to maize farmers in Africa. Few models have attempted to address the issue of fall armyworm (FAW) but have barely incorporated the effect of insecticide resistance. Models with resistance would help predict the dynamics of the FAW population, thus mitigating losses. The main objectives of this work were to develop, analyze, and numerically simulate a susceptible-infected deterministic mathematical model expressing the FAW-maize interaction and population dynamics under insecticidal sprays and resistance FAW larvae. Three model steady states are established. Their local stability is conducted using either the eigenvalue or the Routh–Hurwitz stability criteria, and their global stability is analyzed using either the Castillo–Chavez, Perron eigenvector, or the Lyapunov methods. An expression for the basic reproduction number , together with the sensitivity analysis of its parameter values, is provided. Numerical analysis is conducted on various model parameter values. The results established all the model steady states to be locally and globally asymptotically stable at . Also, resistance increased the infection rates by increasing the FAW larvae survival rate and reducing the insecticidal efficacy and . This work informs the agriculturists and policymakers on pest control with the best ways to use insecticides to minimize pest resistance and enhance efficacy in production. Pest control measures should be modified to lower the FAW survival rate and all model parameters contributing to resistance formation by FAW larvae to minimize FAW-host interaction, thus reducing crop damage.

1. Introduction

Fall armyworm (FAW), scientifically known as Spodoptera frugiperda, is an agricultural pest species of the order Lepidoptera, a larval stage for the fall armyworm moth [1]. It is a polyphagous, sporadic pest that has continuously caused crop destruction and yield losses to both organic and inorganic farmers globally [2]. Maize (corn) is considered a staple food and a source of food security in most African countries; however, farmers continue to face the threat of significant production losses due to climate change, pests, and diseases [3]. Recent research studies show that maize is the most preferred host plant of FAW [4]. Various integrated control mechanisms for both organic and inorganic maize farming have been put in place to control the pest-host (FAW-maize) interactions, but natural selection and mutation have caused FAW resistance towards set control mechanisms [5].

Recent studies show that improperly managed FAW invasion in a maize plantation can cause a significant reduction in the quality and quantity of the harvest [6]. In Africa, the conducive weather conditions and the availability of FAW-preferred host favor rapid FAW reproduction. This makes FAW the most dominant and endemic pest in Africa and thus a great threat to production and food security [7]. Synthetic insecticides are the main control methods adopted globally against FAW pest invasions, especially in Africa where governments are spending huge funds buying and distributing insecticides to their farmers [8]. However, continuous use of the insecticides increases the chances of pest resistance against the control method and thus high production cost [9, 10].

With maize being a staple food in Kenya and also FAW’s most preferred host plant [6], the negative impacts of FAW on maize production significantly affect Kenya’s Big Four Agenda of achieving 100% national nutrition and food security and the entire African economic developments [11]. The poor-quality maize yields negatively affect the country’s GDP due to poor market access [12]. Managing FAW populations is also expensive for most undeveloped African countries. Generally, FAW’s invasion of the agricultural sector in Africa poses a great threat to the achievement of the CAADP Malabo Declaration of halving poverty by 2025 and the achievements of the 2030 Sustainable Development Goals (SDGs) of improving food security, eradicating poverty, and achieving sustainable production and consumption plans [7, 12].

Predicting the dynamics of a pest population and evaluating the existing pest control measures could significantly reduce the number and the cost of pest management, thus improving crop production, food security, and sustainability [13]. There is an increased need to study FAW-maize interaction and the application of insecticides with the evolution of resistance to develop better control methods that are more efficient, effective, and economical [12, 14]. Mathematical modeling offers an avenue to explore such important factors in agricultural production [15]. For instance, crop growth models issue physiological approaches for the simulations of pest destructions and crop interactions [16].

The disease infection rate in the maize population could be decreased through control intervention measures such as chemical insecticides aimed at reducing susceptible-infected maize contact rates [17]. In the present work, we will consider chemical insecticides as the major control method against FAW. However, resistance alleles and migration rates significantly influence FAW population dynamics [18]. Ordinary differential equations (ODEs) have been used in developing and analyzing stage-structured FAW-maize interaction models [13, 17–19]. However, limited attention has been paid to host-pest interaction models, particularly in insect pest management measures [20]. Thus, we apply ODEs and the concept of host-pest interaction models to study FAW larvae population dynamics in maize populations.

From the literature, various deterministic mathematical models describing the dynamics of agricultural pest populations under various pest control measures have been developed [17–19, 21–24]. Moreover, mathematical models evaluating the effects of pests and insecticides on crop production have been developed [13, 25]. However, despite many pest control models in agricultural production, comprehensive parametric research is limited. Also, limited host-pest interaction models address various pest management measures. From the previous studies of models on agricultural pests, we did not find a mathematical model on the host-pest (FAW-maize) interaction assessing the interaction and population dynamics under an insecticidal control strategy. Through natural selection and mutation, a proportion of FAW pests are considered to express resistance traits against insecticides, thus increasing the host-pest interactions and reducing the insecticide’s efficacy [5].

To address this gap, this study develops a susceptible-infected (SI) compartmental model for two interacting populations, FAW-maize population, assessing the effects of insecticide sprays and resistance factors on the interaction patterns and population dynamics. SI models are used to model the rate and transmission dynamics of infectious human or plant diseases [26, 27]. This study assumes that the FAW larvae depend largely on the maize population for food and survival and considers insecticides which are the most commonly used control methods against FAW. This research study will increase the understanding of the fall armyworm-maize interaction patterns and the best control measures to employ when the FAW species is in its larval stage while minimizing larvae resistance formation. This will significantly improve on crop production enhancing food security and economic development [28].

A study by Daudi 2021 proposed a stage-structured model for the control of FAW impacts on maize production. Two interacting maize-FAW submodels were developed with the maize population divided into two stages: vegetative and reproductive stages, while the FAW population was divided into three stages, that is, the egg , caterpillar , and adult stages. In this study, we introduce the knowledge of epidemiological disease models to model the population dynamics of maize-FAW interaction. We develop a susceptible-infected (SI) compartmental model for two interacting populations: the maize population and the FAW population as shown in (Figure 1). In SI models, infection occurs when a susceptible individual comes into contact with an infected individual, thus contracting the disease [29]. The FAW larvae infect the susceptible maize population after making contact. We introduce two factors: FAW control through insecticides and resistance, as important factors affecting the interaction patterns, the spread of the disease, and the model population dynamics. The model is analyzed, and numerical simulation is conducted using the MATLAB ODE solver with the data obtained from previously published articles cited accordingly in this work.

2. Model Description and Formulation

In formulating the maize-FAW SI model shown in Figure 1, we consider two maize sections: organic (O) and insecticidal (I). The organic section (O) is without any FAW control methods, while the insecticidal section (I) is under insecticidal spraying. The FAW larvae infect the maize population through contact. The FAW larvae population has been divided into two compartmental classes, that is, the normal larval and the resistant larval populations. A proportion of normal larvae progress into resistance larvae at a constant rate after contact with insecticides. The two maize sections interact naturally with both the normal larvae and the resistant larvae . The maize population transits from the susceptible into the infected compartmental class after making contact with either the resistant or the normal larvae that transmit the disease. The maize population in the two sections then progresses from susceptible into the infected compartments through two forces of infection and . Susceptible and infected maize populations at any time t in the organic section (O) are denoted as and , respectively, while the susceptible and infected maize populations at any time t in the insecticidal section (I) are denoted as and , respectively.

The natural recruitment rates of the maize population occur at a constant rate . The natural recruitment rate contributing to the FAW larvae population is a constant rate through a survival rate . The FAW population slowly kills the host maize population by residing in it, infecting and feeding on the maize biomass [13]. This contributes to an increased recruitment rate of FAW at a rate and a reduction in the maize population over time. The natural harvesting rate of the maize population is a constant rate denoted by π. The FAW larvae population declines at a constant rate which is either by the natural death rate or progression into the pupal FAW life cycle at a constant rate . The exposure to insecticides causes death of the FAW larvae at constants rates δ, with denoting the normal larvae insecticidal-induced death rate and denoting the resistance larvae insecticidal-induced death rate, with .

The total population in the model at time t will be, where .

.

The infection force at any time t in the organic section (O) is denoted as and that in the insecticidal section (I) is denoted as :where and , , and infection rates .

In this study, the following assumptions were made during model formulation:(1)To reduce model complexity, the model only considers one larvae stage of the FAW life cycle. The other FAW stages are represented by the FAW recruitment rate and progression rate .(2)FAW larvae are the only pest interacting with the maize population at time t.(3)Insecticidal sprays are the only control methods adopted against the FAW population.(4)The term normal larvae denote the larvae not expressing the resistance traits.(5)The immigration and emigration rates of the adult FAW larvae population are considered to be negligible.(6)The model assumes homogenous mixing of the FAW and maize population at any time t.

The following state variables in Table 1 and model parameters in Table 2 are discussed as used in model formulation.

Considering the state variables, parameters, and assumptions discussed above, we developed the model flowchart diagram given in Figure 1.

The system of ordinary differential equations governing the interaction of Fall Armyworm larvae with maize is given by the following equations:

3. Model Analysis

3.1. Positivity of Solutions

Positivity ensures that the model is well-posed and the equations lie on the feasible region of the system and thus realistic in representing pest-host interaction with positive values [30]. Since the model system describes a living population of the FAW larvae-maize interaction, then the state variables and the model parameters are positive at any time . We prove the positivity of solutions of our model system shown in Figure 1 by stating and proving Theorem 1 below [31].

Theorem 1. Let the initial dataset be . Then, the solution set is positive for all .

Proof. Let the variables be solutions to the system of nonnegative initial conditions:Starting with equation (1),Clearly, by the inspection method, on the assumption of nonnegative model variables and parameters.
We need to show thatBy separation of variables, we obtainUpon integration with respect to time , where C₁ is a constant of integration at .
That is, .
Thus,Hence, the first equation is positive.
By applying the same procedure, we obtainHence, the solution set for the model system is proved to be positive at all . Hence, the model equations lie in the feasible region.

3.2. Boundedness/Invariant Region

To prove for boundedness of the solution of our model system, we state and prove the following theorem as applied by [32].

Theorem 2. The solution to the study model equations in Section 2 is uniformly bounded in a proper subset such that .

With and , then all the solutions to the model system move into and remain in

Proof. To get the boundedness of the solution for the maize population at any time t, we take the time derivative of our total maize population along its solution to getwhere and
Next, we need to show that every solution originating from Ω stays in Ω and is bounded by × . Consider Lemma 3 [33] stated below.

Lemma 3. The linear differential equation , where are constants, has infinitely many solutions labeled by as

Proof. is also denoted as
Introducing an integrating factor μ, we get .
Let , and thus,
The left-hand side can be expressed as a total derivative of a product of two functions:Replacing the value in the equation above, we obtainUpon integration,, and thus, as , we have
Similarly, for the FAW larvae population,where and
Upon integration,It then follows that the solution to the model equations exists in the region defined bySuch that This proves the boundedness of the solution inside Ω which means that all the solutions to the model equations (2)–(7) starts and stays in Ω for all time . Generally, the solution to the initial value problems defined in Ω exists and is unique in the given interval. They remain bounded in the positively invariant and bounded region Ω; hence, the system is biologically and ecoepidemiologically well-posed, and the dynamics of the model can be sufficiently studied in Ω.

3.3. Equilibrium Point Analysis

All systems of nonlinear differential equations may have none, one, many, or even infinite steady states [34]. In this study, we discuss the disease/larvae-free equilibrium points (), insecticidal/control-free equilibrium points (), and disease endemic equilibrium points ().

3.3.1. Disease/Larvae-Free Equilibrium Points ()

The value of is obtained by setting all the infectious classes to zero, , with and to get

3.3.2. Insecticidal/Control-Free Equilibrium Point ()

Let denote the control-free equilibrium points. We solve the control equilibrium points by expressing it in terms of the force of infection evaluated at the control-free equilibrium point. We set , , , and to get

We evaluated for the value of as . We let , , and . Where , and

3.3.3. Disease Endemic Equilibrium Point ()

An endemic equilibrium point is a state in the model system where the disease in the population approaches a constant [22]. Let denote the disease endemic equilibrium point.

Solving for the system of equations (2)–(7) in terms of the force of infections and ,

Now, substituting the values of in and and introducing the relation which implies , and then replacing into the values of the endemic equilibrium points to obtain the value of .

For the model system to lie in the positively invariant region, we let the value of

if and only if .

3.4. Basic Reproduction Number

is used to estimate the number of secondary infections that could arise if one infectious individual is introduced into a completely susceptible population [35]. We determine the value of using the next-generation matrix as discussed in [17]. We evaluate the Jacobian matrix of the model system of equations at DFEP and determine the spectral radius using Wolfram Mathematica software:

The vector for the infected and infectious classes is denoted as , and the vector for the uninfected classes is denoted as

Using the notation f to denote the matrix for new infection and to denote the matrix of the transfer of infections in the system,

Evaluating the Jacobian matrices of f and at DFEP to get , , and as the disease-free equilibrium point,with and

Given that and were previously described as disease-free equilibrium points and their expression given in equation (23).

We let , , and

The dominant eigenvalue in is the value of It is the spectral radius of matrix G:

The eigenvalues of the next-generation matrix G are determined using Wolfram Mathematica as

Hence, the value of is

3.4.1. Biological Interpretation of

As shown in studies [17, 36], is defined as the expected number of secondary infections generated from a single primary infection in a completely susceptible population. In our model, the interactions present are organic maize-normal larvae interaction, organic maize-resistant larvae interaction, insecticidal sprayed maize-normal larvae interaction, and insecticidal sprayed maize-resistant larvae interaction. The basic reproduction number is consequently calculated as the sum of these interactions as follows:where and

With , , and ,

represents the number of new infections on the insecticidal maize population arising from both the normal larvae and the resistant larvae :

represents the number of new infections on the organic maize population arising from both the normal larvae and the resistant larvae :

3.5. Stability of Equilibrium Points

In this study, we determined the global asymptotic stability of the three model steady states using Castillo–Chavez, the Perron eigenvector, and the Lyapunov methods. The steady states are locally asymptotically stable if all the eigenvalues of the Jacobian matrix evaluated at each (E) have negative real parts. The equilibrium point is asymptotically unstable if at least one of the eigenvalues has a positive real part. This is the linearity stability analysis theorem.

Let the model system of equations be denoted in vector form as

We then write in matrix form as

The matrix is used to evaluate the Jacobian matrix for the model system and thus the stability analysis for the various model stationary points.

3.5.1. Local Stability Analysis of the Disease-Free Equilibrium Point ()

Theorem 4. The disease-free equilibrium points for the FAW larvae-maize interaction model described by equations (1)–(6) are locally asymptotically stable if the below conditions are satisfied:(1)(2)(3)(4)

Otherwise, is unstable.

Proof. Evaluating the Jacobian matrix at DFEP (shown in equation (23)), we getThe first eigenvalue of the above Jacobian matrix is which is less than zero and hence stable. Applying the Routh–Hurwitz stability criterion as shown by [37], the roots to characteristic equation (43) below have negative real parts if the coefficients are nonnegative and matrices for :Using Mathematica software, the characteristic polynomial of the above matrix is evaluated aswhere are the eigenvalues.
By the Routh–Hurwitz criterion for stability analysis, the model system is proved to be locally asymptotically stable at disease-free steady state if and only if , , , and shown in Table 3. Given that all the eigenvalues have negative real parts, it implies that the disease-free equilibrium point is locally asymptotically stable.

3.5.2. Global Stability Analysis of the Disease-Free Equilibrium Point

To prove the global stability of the DFE points, we use a matrix-theoritic method defined by [38] to construct a Lyapunov function L involving the Perron eigenvector. Let and be defined as

Also, let us define the value with the values of as defined previously in Section 3.2 in equations (30) and (31), respectively. is defined as the perron eigenvector or the left eigenvector corresponding to the eigenvalue .

Theorem 5. Consider F, V, and f(x) as defined in equations (30), (31), and (45). If f(x) ≥ 0 in , the model system F ≥ 0, ≥ 0 and ≤ 1, then the function is a Lyapunov function for the model systems (2)–(7) in Ω.

Proof. We start by getting the derivative of along the solutions to model equations (1)–(6):As shown above, since we have that and in , then the last term is negative. If , then in , and thus, is taken to be the Lyapunov function for the model system.
We determineTherefore,as generated in Section 3.2.Thus,Equation (50) becomes the Lyapunov function for the model system shown by equations (1)–(6).
By Perron–Frobenius, we letWithto be irreducible and positive in , it follows then that Hence, by LaSalle’s invariant principle [39], shows that and
Thus, our disease-free equilibrium point ( is globally asymptotically stable.
We also prove that the disease-free equilibrium point is globally asymptotically stable using the Castillo–Chavez method as shown below.
We apply the method established by [40]. We start first by rewriting the system of model equations in the form:With representing the uninfected classes and representing the infected and the infectious classes, denotes the disease-free equilibrium point of the system
According to the Castillo–Chavez stability theorem if the following conditions are satisfied in the points given above, then the global asymptotic stability of is guaranteed. The conditions include is globally asymptotically stable

Theorem 6. The disease-free equilibrium point is globally asymptotically stable.

Proof. We start by dividing the model into subsystems:We then generate two vector-valued functions:Evaluating for the reduced system from condition (1), to get,We note that the system has dynamics of an asymptomatic system independent of initial conditions in Ω.
We then compute and prove that .
Let be a Jacobian matrix of taken in and evaluated at . It is also defined as an M matrix since all the nondiagonal elements are nonnegative:Evaluating the value of AZ,With the equation , then the value ofSince , then DFE is proved to be globally asymptotically stable.

3.5.3. Local Stability Analysis of the Control-Free Equilibrium Point

Theorem 7. The control-free equilibrium point is locally asymptotically stable.

Proof. The local stability analysis of the control-free steady state is achieved by first generating the Jacobian matrix evaluated at the control-free steady state evaluated in Section 3.3.2:where , , , , , and as shown in Section 3.3.2.
Using Wolfram Mathematica, the eigenvalues are found to beThrough back substitution, all the Eigen values are negative, and thus, the control-free equilibrium point is evaluated to be stable.

3.5.4. Global Stability Analysis of the Control-Free Equilibrium Point

If a Lyapunov function to a linearized nonlinear system is obtained and exists, then it shows that the model system is asymptotically stable [41].

Theorem 8. The control-free equilibrium point is globally asymptotically stable (i) if the insecticidal-free equilibrium is feasible and (ii) if the equilibrium point is a locally asymptotically stable solution.

Proof. We consider a Lyapunov method for stability analysis, an approach adopted by [42]. We start by constructing a Lyapunov function: represents a constant selected such that , represents the compartments classes, and represents the control-free equilibrium point of the compartmental classes.
Expanding the Lyapunov function and substituting the compartments,We evaluate the derivative of equation (69) with respect to time to getwhere .
Substituting the values of with the model values, we getAs shown in Section 3.1, the model equations are positively invariant, and hence,and also,It can be seen that the only invariant set in Ω where = 0 is the set . Since f(x) = 0 by LaSalle’s invariance principle [39], is globally asymptotically stable on Ω iff . Otherwise, it is unstable.

3.5.5. Global Stability Analysis of the Endemic Equilibrium Point

Theorem 9. For the endemic equilibrium point to be globally stable, (i) the endemic equilibrium point must be feasible and (ii) the endemic equilibrium point must be locally stable.

Proof. We assume that the endemic equilibrium point is locally asymptotically stable since the linearization method evaluated at the equilibrium point proves to be mathematically complicated.
Applying the Lyapunov method as used by [42] and constructing the appropriate Lyapunov function, we getwhere represents a constant selected such that , represents the compartments classes, represents the disease endemic equilibrium point of the compartment classes.
Expanding the Lyapunov function and substituting the compartments,Differentiating equation (75) with respect to time, we getIt is equivalent towhere .
As in Section 3.1, the model equations are positively invariant, and hence,It can be seen that the largest invariant set in Ω where = 0 is the set . Since f(x) = 0 by LaSalle’s invariance principle [39], is globally asymptotically stable on Ω iff . Otherwise it is unstable.

3.6. Sensitivity Analysis of

Sensitivity analysis of helps identify the key parameters that significantly affect the FAW larvae-maize interaction model. This helps determine the key parameters to consider in the control strategies against FAW larvae infestation in the maize population by managing the value of the basic reproduction number and the infection. We follow a sensitivity method as conducted by [17, 18].

Definition 10. The normalized forward sensitivity index of a variable , depending differentially on parameter , is defined by an equation:where represents the basic reproduction number and represents all the main parameters.
In our study, we have the value of given asThe sensitivity index of to isA similar procedure is used to calculate the sensitivity indices for the other parameters around the basic reproduction number; the results are shown in the Table 4.

3.6.1. Interpreting the Sensitivity Indices

Sensitivity indices for the basic reproduction number presented in the table above show positive and negative values for the parameter values. A positive sensitivity index denotes a direct linkage between the parameter and the basic reproduction number, while a negative sensitivity index denotes an inverse linkage between the parameter and the basic reproduction number. From the table, ( have positive sensitivity indices. This means that the parameter values have a great impact on spreading the disease among the maize population upon an increase in their parameter values. This is because the value of will tend to increase by increasing the parameter values, further increasing the number of secondary infections in the susceptible maize population.

The parameters have negative sensitivity indices. This means that the value of decreases when their values are increased. This results in a decrease in the rate of infection, lowering the secondary infections in the susceptible maize population. The value of being equal to 1 means a unit increase in results in a unit increase in the value of and vice versa.

4. Numerical Simulation

4.1. Parameter Estimation

Numerical analysis of the model is conducted using a MATLAB inbuilt solver based on the Runge–Kutta order of 5 with parameter values cited in Table 5 as obtained from published studies together with a few estimated values. The initial value states are used as as shown by similar studies on the fall armyworm-maize interaction [17, 19, 43]. The simulations are conducted at a time range of between 0 and 60 days which is the vegetative stage of the maize population and the most interactive phase with the FAW population. The resulting simulation graphs are presented in Figures 2–11.

4.2. Simulation Results

In Figures 2 and 3, the susceptible maize populations {} increase exponentially with time until they reach the endemic equilibrium point. The curve for the organic infected maize in Figure 2 is observed to be higher than the curve for the infected insecticidal maize population in Figure 3. This is the effect of uncontrolled FAW larvae-organic maize interactions, which subsequently leads to a high force of infection resulting in a higher infected organic maize population.

In Figure 4, when is decreased by 50%, the infected organic maize population decreases exponentially with time. This is due to the direct linkage between the force of infection and the basic reproduction number. At a higher value of , the infected organic maize population increases exponentially. Similar observations are made in Figure 5; however, the infected insecticidal maize population occurs in lower rates as compared to the organic section. This is due to the lower values of as compared to .

According to the model flowchart (Figure 1), represents the survival rate for the FAW larvae. From Figures 6 and 7, reducing the value of to lower values than the baseline value subsequently reduces the population of the normal and the resistant larvae. This has a direct effect on the force of infection in the maize population.

denote the insecticidal-induced death rate on the resistant and normal larvae, respectively. In Figures 8 and 9, increasing the values of and reduces the population of the resistant and the normal larvae. This in turn reduces the infection rate leading to a lower basic reproduction number. We also observe that the resistant larvae are large in numbers as compared to normal larvae which pose a greater risk in the insecticide control measures.

From Figures 10 and 11, represents the rate at which the normal larvae progress into resistant larvae after contact with insecticidal sprays. At higher values of , more normal larvae, which are easy to eradicate from the model population by insecticides, progress into resistant larvae. This subsequently reduces the normal larvae population and increases the resistant larvae population.

5. Discussion and Conclusion

This study developed and analyzed a deterministic ecoepidemiological model on the maize-FAW interaction in the presence of insecticides and resistance factors. The model was proved to be uniformly bounded and positively invariant. Three equilibrium points, that is, the disease/larvae-free, control-free, and endemic equilibrium points, were established and evaluated to be locally and globally asymptotically stable at . Furthermore, an expression for the basic reproduction number and its sensitivity analysis were conducted. The results showed that an increase in and a decrease in greatly increased the FAW population dynamics, increased the maize-FAW larvae interaction, and hence the spread of the disease to the susceptible maize population. Through numerical simulation, graphical results of the FAW-maize interaction and population dynamics are presented by applying parameter values obtained from literature and cited studies accordingly.

The analysis of in the host-pest (maize-FAW) interaction model helps determine how effective the insecticide control measures against the FAW larvae are and how to effectively use the control measures to reduce the value of to a value of less than one [13]. Increasing the forces of infection increased the number of infected maize populations, while reducing the infection forces resulted in a decrease in the number of infected maize populations in both the organic and insecticidal sections. This was attributed to the direct impact of the force of infection on the basic reproduction number. The insecticide control measures used to control FAW-maize interactions in the insecticidal maize section ensured a reduced contact rate, and thus, the number of resulting secondary infections at any time t was lower than in the organic uncontrolled FAW-maize interactions.

is a parameter value used in the model to represent a constant rate at which the normal larvae progress into resistant larvae after insecticide spraying. When insecticides are used to control the FAW larvae population, a few mutants in the population tolerate chemical insecticides better, while the normal population succumbs to them [44]. Resistance increased the infection rates by increasing the FAW larvae survival rate and reducing the insecticidal efficacy by lowering the larvae insecticidal-induced death rates and . However, while resistance affects both the organic and the insecticidal controlled maize sections, the FAW-maize interactions and thus the infection rates are lower in the controlled insecticidal sections than in the organic uncontrolled section.

Generally, the results from the model analysis showed that the FAW survival rate , resistance formation , and the insecticidal-induced death rates and are essential in controlling both the normal and the resistant FAW larvae. Control intervention aimed at reducing the infection rate in organic and insecticidal maize populations should aim at reducing these parameter factors. This is by using high-efficacy insecticides, resulting in higher FAW larvae death rates , thus reducing the FAW survival rate The sensitivity analysis of showed that the FAW survival rate significantly affects the FAW-maize interactions. This informs both the organic and the inorganic farmers on the importance of using chemical control methods that are highly effective in reducing the FAW survival rate

Various integrated FAW-maize management approaches should be adopted where several pest control methods are used together since no control method has been reported to work best in isolation. This will help reduce the high rate of resistance formation observed in FAW larvae, thus reducing their population dynamics. African countries should conduct proper civic education on pest control methods, FAW-maize interaction patterns, and resistance formation in insecticides used to ensure environmental conservation and minimize pest resistance formation. This is due to FAW’s unique characteristics of high migration, mutation, and reproduction, which makes its control a bit expensive and difficult. However, the findings from this study are not exhaustive. In future studies, we will consider developing an optimal larvae survival control theory with resistance factors to achieve a profitable FAW larvae control strategy.

Data Availability

The research data used for the model analysis were obtained from previously published articles duly cited in this paper. Table 5 and other pertinent locations in this work provide references to the published articles.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author would like to thank the Board of Postgraduate Studies and the Department of Mathematics and Statistics at the University of Embu for their support to join the Master’s program and continue in research work.

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