Modelling the Control of the Impact of Fall Armyworm (Spodoptera frugiperda) Infestations on Maize Production

School of Computational and Communication Science and Engineering, e Nelson Mandela African Institution of Science and Technology (NM-AIST), P.O. Box 447, Arusha, Tanzania Department of Mathematics, Humanities and Social Science (MHSS), National Institute of Transport (NIT), P.O. Box 705, Dar-es-Salaam, Tanzania Institute of Mathematical Science, Strathmore University, Nairobi, Kenya Department of Biometry and Mathematics, Botswana University of Agriculture and Natural Resources, Private Bag 0027, Gaborone, Botswana


Introduction
Food loss due to Fall Armyworm (FAW-Spodoptera frugiperda) is currently one of the biggest threats to food security, particularly in large parts of the developing world. As reported by the United Nations [1] and Shiferaw et al. [2], the world's population is expected to reach 9.3 billion by the end of 2050, with an approximate yearly increase of more than 80% of the global increase, and a quarter of this increase is expected to occur in developing countries.
is unprecedented global increase in the number of people poses a serious challenge for maize producers and policymakers, especially regarding the minimization of food losses due to the effect of interaction between maize crops and FAW. is issue has been a matter of active research for many years, with the main challenge lying in the unavoidable trade-off between the reduction of FAW, the financial costs involved, and the environmental impacts. e severity and extent of FAW outbreaks are enhanced following the onset of the wet season when the wind-borne immigrations of adult moths are attracted to lay eggs, which transform into caterpillars within 2 to 5 days [3]. e newly hatched caterpillars benefit from the flush of green maize vegetation resulting from the rain and develop rapidly over three weeks and outbreaks can have a very high density. e caterpillars can severely devastate maize plantations over several thousand square kilometres with a very high population density [4,5]. ese severe effects of FAW outbreaks particularly occur when rainstorms follow droughts [3]. e damage of whorl maize leaves by the caterpillars as they grow leaves semitransparent patches called windows, which interfere with the growth of maize plants.
ese outbreaks travelled over thousands of kilometres to East African countries such as Tanzania (January 2017), Kenya (April 2017), and Uganda (May 2017) [4]. Numerous studies have been conducted to understand these periodic outbreaks. Mathematical models based on weather patterns and estimated dates of arrival of the moths from rainfall forecasting and caterpillar counts from field inspections have also been formulated to predict possible outbreaks [10][11][12].
In a recent study, the mitigation of the damaging effects of FAW was performed using control measures such as treating the larvae with insecticides such as azadirachtin, synthetic pesticides such as DDT, and aqueous neem seed extracts [6]. e success was dose-dependent and produced a range of undesirable adverse side effects on sensitive wildlife and human health [13]. Paez [8] used biological control for the African Armyworm, which also can be applied to FAW, and Liang [13] used nuclear polyhedrosis virus (NPV), which kills armyworms and helps control outbreaks. Murua [14] proposed the control of the FAW using larvae parasites.
e system was very reliable, since 95% of the pest larvae were destroyed but the system was unable to reduce the pest eggs to a sufficiently low level to maintain control.
ere exists a rich set of mathematical models of FAW effects on maize production which were reported by several authors [6,8,15]. Models like stage-structured with and without time delay which often considered mature predators (natural enemy) have also been applied; however, most of these models, despite incorporating Holling types I, II, III, and IV and Beddington-DeAngelis functional responses as their baseline, did not target the interaction between maize and FAW [8,[16][17][18][19][20][21][22][23][24].
Considering the importance of maize to a majority of countries producing maize in Africa, the present work aims to utilize ordinary differential equation in exploring the implications of FAW infestation in a maize field planted with initial number of maize seeds at time t � 0 and obtaining maximum harvest at the end of the season. To come up with the intended results, we propose two subgeneric models, each with stage-structured in both of populations (maize and FAW) to determine the population dynamics in the presence and absence of immigration of the adult moth and estimate the yield when control measures such as pesticides and harvesting are deployed.

Model Formulation
e two submodels introduced herein consist of two populations: maize and FAW, both of which are stage-structured giving a total of five populations. We consider maize growth from emergence to maturity at any given time t > 0, growing through two time periods: Period I and Period II. Period I, which is assumed to take place over a time period [0, t 1 ], denotes the vegetative stage and comprises planting of maize seeds, seed emergence, development of whorl leaves, and tasselling, while period II, which takes place over a time period [t 1 , t 2 ], denotes the reproductive stage and consists of corncob, kernel development, and maturity.
On the other hand, the FAW population at any time t > 0 has been subdivided into egg population, caterpillar population, and the adult moth population. Although the FAW has six larval instar stages, we have considered this as single group called caterpillar in order to reduce complexity of the model. We assume that weather condition, environment condition, and planting system of maize seeds favor seed germination and their corresponding growth in both stages with no natural death rate before harvest.
In this regard, we let x 1 (t) represent the population density of maize in the vegetative stage and let x 2 (t) represent the population density of maize in the reproductive stage, while w(t), y(t), and z(t) represent the population density of eggs, caterpillars, and the adult moths, respectively. We also assume that caterpillar with a mortality rate μ y is the only threat to maize throughout its growth period and the adult moth takes over in the reproduction process. When food is limited, the older caterpillar of FAW exhibits a cannibalistic behavior on the smaller larvae [25,26]. e population density of eggs is replenished through the laying of eggs by the adult moth at a constant rate ρ per day and reduced through hatching into caterpillar and destruction (mortality) at rates c and μ w , respectively. On the other hand, the adult moth's population density is refilled through the caterpillar's development into an adult moth at a constant rate δ and reduced through mortality at a rate μ z . We further assume that, in a season (i.e., Period I and Period II), there is no maize seed replantation and the model in both stages has no maize recruitment. erefore, the classes x 1 (t) and x 2 (t) decline constantly due to caterpillar attack at the rates of α and η, respectively, with a destruction rate λ in each stage. e study assumes nonnegative values of the model parameters and variables in context with populations being considered.
e formulation of this model is also supported by the following assumptions: (i) Planting of maize seed is done at t � 0; therefore, the development rate of each maize plant from the vegetative stage to the reproductive stage is the same and continuous.  Tables 1 and 2, respectively.   2 International Journal of Differential Equations e model explanations above can be represented schematically as shown in Figure 1. e model described in Figure 1 is transformed into two interconnected or coupled model systems for the vegetative and reproductive stages. e two systems at the transitional period (i.e., t � t 1 ) are connected by the initial condition x 1 (t 1 ) � x 2 (t 1 ), where x 1 (t 1 ) represents the number of maize plants in the vegetative stage progressing into the reproductive stage, while x 2 (t 1 ) represents the number of maize plants that have progressed into the reproductive stage.
e model in vegetative stage (0 ≤ t ≤ t 1 ) is with initial conditions: (2) with initial conditions: x 2 (t) � 0, for t < t 1 , e parameters e 1 and e 2 are the conversion rates of maize biomass into caterpillar biomass in the vegetative and reproductive stages, respectively. e formulated systems (1) and (3) are interconnected such that the solutions at time t � t 1 of system (1) are the initial conditions for system (3). To prove this behavior, we first investigate the basic properties of the model systems as follows.

Basic Properties of the Models
e effect of interaction between maize and FAW is studied by analyzing systems (1) and (3) to determine if these models are mathematically and epidemiologically well posed. e study is carried out for maize growing in the vegetative and reproductive stages, which are model systems (1) and (3), respectively. We consider the basic properties of the model: positivity of solution and invariant region.

Variables
Description Population density of the maize plants growing in the vegetative stage at any time t x 2 (t) Population density of the maize plants growing in the reproductive stage at any time t y(t) Population density of the caterpillars at any time t z(t) Population density of adult moths at any time t w(t) Population density of the eggs laid at any time t  (1) and (3) to be epidemiologically meaningful and well posed, it is required to prove that all solutions of these systems with their respective initial conditions remain positive for all t > 0. is is established by the following theorem. (1) and (3); then, the solution sets (x 1 (t 1 ), y(t 1 ), z (t 1 ), w(t 1 )) and (x 2 (t 2 ), y(t 2 ), z(t 2 ), w(t 2 )) for systems (1) and (3), respectively, are positive for all t > 0 in the interval [0, t 2 ].
Proof. In a biological sensible way x 2 (t) � 0 in vegetative stage and x 1 (t) � 0 in reproductive stage, to prove the theorem, we consider two cases: models in the vegetative stage and the reproductive stage. Case 1. We consider system (1), the model in the vegetative stage (0 ≤ t ≤ t 1 ). Considering the first equation, dx 1 /dt � − αx 1 y − λx 1 . Grouping and arranging like terms, one gets Integrating over a time interval 0 to t leads to where x 0 � x 1 (0) and, for real values of y(t) ≥ 0 and (αy(s))ds ≥ 0, Similarly, from the second equation, dy/dt � e 1 αx 1 y + cw − δy − μ y y. Setting Continuing in the same manner, we can also show that z(t) ≥ 0 and w(t) ≥ 0, meaning that x 1 (t) ≥ 0, y(t) ≥ 0, z(t) ≥ 0, and w(t) ≥ 0 exist in Ω 1 and the population densities for species denoted by the state variables in model (1) are nonnegative. Case 2. For system (3), we consider the model in reproductive stage 0 ≤ t ≤ t 1 .
We can prove the first, second, third, and fourth equations of system (3) using the same techniques as in Case 1 and conclude that x 2 (t) ≥ 0, y(t) ≥ 0, z(t) ≥ 0, and w(t) ≥ 0 exist in Ω 2 , meaning that the population densities for species denoted by the state variables in system (3) (1) and (3) is bounded. We use the box invariant of Metzler matrix method as applied by Aloyce [27] to show the existence of invariant regions. We prove this as follows: systems (1) and (3) can be written as with X � (x 1 , y, z, w) T and the constant F � (0, 0, 0, 0) T for the model system (1) and with Y � (x 2 , y, z, w) T and the constant G � (0, 0, 0, 0) T for the model system (3). From equation (10), Maize growing in reproductive stage Maize growing in vegetative stage γw δy pz μ w w e 1 αx 1 y μ y y e 2 ηyx 2 μ 2 z

International Journal of Differential Equations
and, from equation (11), Referring to equations (12) and (13), it follows that A(X) and B(Y) are Metzler matrices for systems (1) and (3), respectively, ∀X, Y ∈ R 4 + in which all off-diagonal terms are nonnegative with F ≥ 0 and G ≥ 0. erefore, according to Aloyce [27], systems (10) and (11) are positive invariants in R 4 + . ese results lead us to the conclusion that systems (1) for 0 ≤ t ≤ t 1 and (3) for t 1 ≤ t ≤ t 2 with their respective initial conditions have bounded nonnegative solutions. Since the solution for systems (1) and (3) is well posed and bounded, the following section gives the numerical analysis to assess the impact of the interaction between maize plants, eggs, caterpillars, and the adult moths population throughout the time interval [0, t 2 ].

Numerical Simulation
Numerical simulations for systems (1) and (3) are carried out to illustrate the impact of FAW infestation on maize using a set of reasonable parameter values, where some of the data were obtained from literature and others were estimated based on the idea given by Li [28] and Tumwiine [29]. e parameters are chosen following realistic ecological observations. Using MATLAB software in simulating system (1) for time t ∈ (0, t 1 ), three cases are investigated with their respective maize yields. Case 1 concerns the effect of the interaction on maize when there are different numbers of the adult moths assumed at t � 0. Case 2 concerns the effects of the interaction when there is immigration of the adult moth at time t ≠ 0. Case 3 concerns the effect of the interaction when deploying controls. For both cases with x 1 (0) � 500, y(0) � 0, and w(0) � 0, the final solutions, that is, x 1 (t 1 ), y(t 1 ), z(t 1 ), and w(t 1 ), for each number of the adult moths assumed will be used as an initial value in simulating system (3) for time t ∈ (t 1 , t 2 ). erefore, before simulating these models, we first discuss the population dynamics of maize and FAW and then estimate and adopt some of the parameter values based on current literature.

Population Dynamics of Maize.
Maize plants grow through two main stages in a season: the vegetative stage (emergence to the tasselling stage) and the reproductive stage (tasselling to maturity). Maize planted at the time in an environment that may contain planted seeds germinates in 0-7 days. According to Kebede [7], maize in the vegetative and reproductive stages takes 63-97 days, respectively, to reach maturity from emergence. However, other maize seeds take about 90 or 151 days to reach maturity [27]. e infestation of FAW starts when an adult moth migrates into maize plants and lays eggs which then hatch into caterpillars that damage maize leaves and kernel development in the vegetative and reproductive stages, respectively. is infestation occurs continuously, where the pest is endemic throughout the year. Systems (1) and (3) represent the interaction between maize plants and FAW. After the simulation of the model, it will be possible to identify the output of the maize plants at t 1 � 63 days and t 2 � 160 days.

Population Dynamics of FAW.
e FAW is a holometabolous insect (undergoes complete metamorphosis including eggs, caterpillar, and adult moth) [7]. Only the adult moths that survive for 10 days (7 to 21 days) reproduce and migrate to another location [30]. When the adult moth immigrates into a maize field, it deposits most of her 100-200 eggs per mass. After a preoviposition period of 3 and 4 days of life, oviposition occurs for up to 3 weeks [7,31]. e total egg production per adult moth averages 1500 to a maximum of 2000 in its lifetime. e egg stage takes 2-3 days before hatching into a caterpillar in a season depending on the climatic conditions. e caterpillar that develops into the adult moth survives for about 14 and 30 days during the warm summer and cooler months, respectively. Maize plants in the vegetative and reproductive stages are susceptible to caterpillar attack, reducing their ability to manufacture food and in turn reducing yield. In Section 4.3, we estimate and adopt the parameters to be used in the simulation.

Parameters Estimation and Adoption.
Due to the unavailability of real data for all parameters related to systems (1) and (3), the suppositional values of the different parameters have been considered as follows: according to Tumwiine and Daudi [29,31], mortality/death rate is the reciprocal of the average of the life span of an organism, whereby the life span is the duration an organism survives before dying; that is, Since the duration of eggs, caterpillar, and adult moth is 2.5, 14, and 10 days, respectively, their natural mutation/ transformation rates are 0.400, 0.0071, and 0.161 per day, respectively. e development period for an egg into caterpillar and caterpillar into adult moth averages 17.5 and 14 days, respectively. is implies that the development rates of eggs into caterpillars and caterpillars into adult moths are 0.58 and 0.071 per day, respectively. Finally, the adult moth lays up to 2000 eggs in 10 days of its lifetime, and because a cycle may take up to 30 or 60 days depending on the weather conditions, the adult lays 0.017 eggs per day. According to Li [28], the survival/development rate is calculated as International Journal of Differential Equations with N 1 and N 2 representing the number of pests that progress into the next development stage and number of pests in the previous development stage, respectively. Table 3 summarizes the parameter values used in systems (1) and (3).
Considering the above data, we plot and analyze system (1) for assessing the impact of the interaction between maize and FAW from time t � 0 to time t 1 � 63 days and system (3) from time t � 0 to time t 2 � 160 days under the following cases.

Impact of Population Dynamics in the Absence of Immigration.
e simulation of the model systems (1) and (3) under this section is done by assuming that, for each number of the adult moths considered in the field such as z 1 (0) � 15, z 2 (0) � 30, z 3 (0) � 45, and z 4 (0) � 60; x 1 (0) � 500 is the initial number of maize seeds planted, while y(0) � 0 and w(0) � 0 define the initial numbers of caterpillars and eggs, respectively. In this regard, the simulation of these models is shown in Figure 2.
e simulation indicates that, for each number of adult moths assumed at t � 0, we observe an exponential decrease of the adult moth's population size before attaining its equilibrium value due to eggs-laying process and mortality rate (Figure 2(a)). Figures 2(b) and 2(c) demonstrate that the egg-laying process increases egg production to its maximum environment carrying capacity, a situation which also increases the density of caterpillars in the same manner due to hatching. e increase in caterpillar's population size for each number of adult moths assumed exponentially decreases the biomass of maize plants, an effect that reduces the efficiency of photosynthesis and in turn reduces the number of maize plants. e results for each number of the adult moths assumed at t � 0, egg production, caterpillar density, and their corresponding effects on maize plants are summarized in Table 4, where both data are taken at t 1 � 63 days. Now, we use the results for the model system (1) displayed in Table 4 taken at t 1 � 63 as an initial value in simulating model system (3) from t 1 � 63 to t 2 � 160 days, and the simulation results are shown in Figure 3.
We also observe in Figure 3 that the adult moths, eggs, and caterpillars are at equilibrium but maize continues decreasing to a certain endemic level at t 2 � 160 days. e number of adult moths, egg population, and their corresponding effects on maize as seen from Figures 3(a)-3(d) are in Table 5.

Impact of Population Dynamics in the Presence of
Immigration. In Section 4.4, the model systems (1) and (3) were simulated with an assumption about the number of maize seeds planted and a different number of adult moths prevailing at t � 0 with zero initial number of eggs and caterpillars population. is section assumes that there exist some exogenous factors (such as temperature, wind, and further human actions) that lead to the immigration of the adult moths in the maize field, whereby, at t > 0, x 1 (0) � 500, y(0) � 0, z(0) � 0, and w(0) � 0, the initial population densities of the adult immigrant moths lay eggs after entering the field. To incorporate the migratory behavior of the pest in models (1) and (3) at time t > 0, we assume that there is no emigration of adult moths in both vegetative and reproductive stages after immigration. erefore, we let σ be the immigration rate of the pest. Assuming no mortality rate during the immigration process, the number of adult moths in the field will abruptly change in a short period. e models of (1) and (3), after incorporation of immigration, appear as shown in model systems (16) and (18).
For t ≤ t 1 , subject to For t 1 ≤ t ≤ t 2 , subject to x 2 (t) � 0, for t < t 1 , To examine the effects of immigration of the moth on the population dynamics of maize plants, models (16) and (18) with different values of immigration rates in the absence of the control measure are simulated. e adult moth on average lays eggs in batches of 100-200 per mass which hatch into caterpillars within 2-5 days. e caterpillar is a 6 International Journal of Differential Equations     From Figure 4, the results indicate that immigration of the adult moths into maize field at different rates (i.e., σ 1 � 15, σ 2 � 30, σ 3 � 45, and σ 4 � 60) reproduces prolifically and reaches its environmental carrying capacity which also increases eggs production and density of the caterpillars to its environmental carrying within 63 days (see Figures 4(a)-4(c)).
is increase adversely affects maize growth and exponentially decreases the biomass of these plants. Figures 4(a)-4(d) also justify that there is a delay of the caterpillar in damaging maize from t � 0 to t � 10 days. During this period, the adult moth lays eggs and the eggs hatch into caterpillars. On the other hand, maize plants grow keeping up their carrying capacity (k � 500), where after t � 10 days the population size for maize declines abruptly for each immigration rate. Table 6 shows the different immigration rates and their corresponding increase in eggs production, density of caterpillar, and the impact on maize growth in 63 days.
However, when the results for the model system (16) displayed in Table 6 are used as initial values in simulating model system (18) from t 1 � 63 to t 2 � 160 days, we obtain the results shown in Figure 5.
e results show that the numbers of adult moths, egg production, and density of caterpillars are at equilibrium (Figures 5(a)-5(c)) but continue decreasing for each number of the adult moths assumed to immigrate due to the existence of the caterpillars in the population ( Figure 5(d)). e numbers of adult moths, egg production, and density of caterpillars at equilibrium as well as the corresponding effects on maize growth for each immigration rate assumed in 97 days are displayed in Table 7.
We generally conclude by comparing the results from  Figure 2(a), we observe that the number of moths declines very abruptly due to egg-laying process and mortality rate, while the numbers of eggs laid and caterpillars as we see in Table 4 and However, Figures 4(a)-4(c) and Table 6 indicate that the density of caterpillars, moths, and eggs production are progressively increasing to days before attaining their equilibrium values which appear to continue even after t �

Maize
Adult moths Caterpillars Eggs 265  2  2  6  6  47  8  415  243  4  4  11  11  94  185  408  223  6  6  17  17  141  22  401  205  8  8  22  22  185  401 In the table, International Journal of Differential Equations 9 is situation has a huge impact on maize biomass as we see in Figure 5(d) and Tables 6 and 7; the averages of maize plants remaining are 160 and 142 in the vegetative and reproductive stages, respectively, due to this effect. ese comparisons give us a way to focus and decide where to introduce our control to reduce the effect, where, according to Section 4.5, immigration of the moths at t > 0 seems to be more destructive compared to the moth existing at t � 0 as seen in Section 4.4. In this regard, we introduce our controls to the model with the   Table 3 with initial values x 1 (0) � 500, y(0) � 0, z(0) � 0, and w(0) � 0 from t � 10 to t � 63 and at different immigration rates: σ 1 � 15, σ 2 � 30, σ 3 � 45, and σ 4 � 60. (a) e adult moth immigrating into the maize field. (b) e increase in population size of the eggs laid after immigration. (c) e increase in the number of the caterpillars after the hatching process. (d) e decline in maize population size after immigration. Hence, results in each of the immigration rates at t � 63 days are the initial solutions for the model system (18).   Figure 5: e numerical simulation for the model system (18) from t � 63 to t � 160 using the final results in Table 6 obtained in t 1 � 63 days. Figures 5(a)-5(c) maintain the equilibrium values for the number of moths, egg production, and density of caterpillars from the early stages of maize growth, while (d) indicates the reduction of maize for each immigration rate. immigration of the moth to reduce the effects associated with this behavior.

Impact of Population Dynamics with Some Control Strategies
In the context of Integrated Pest Management (IPM), pest control strategies include a suitable combination of biological, cultural, and chemical control techniques, where most of these controls, for example, biological controls, were mathematically described by Paez [8] in terms of a smooth ordinary differential equation. Daudi [31] also identified the impossibilities of controlling the FAW using biological control. In this regard, our model shall bear a combination of controls, which includes harvesting effort (cultural methods) and spraying of pesticides. Harvesting is an important and effective cultural control method to prevent and control the explosive growth of the FAW in the maize field. Based on the Monitoring, Surveillance, and Scouting approach, harvesting is the removal of damaged cobs and stems from the field to reduce the number of caterpillars and thus infestation of the next crop [33]. ere are two types of harvesting: nonzero constant and linear harvesting [34]. Due to the unexpected invasion of the number of moths which finally give us an abundant number of caterpillars, which always increases depending on the moth, only a linear harvesting rate will be incorporated in this model. Assuming that only the caterpillar which is unable to fly will be harvested in the field, we let h represent the catchability coefficient of the caterpillar at a given time. We further assume that the harvesting rate in both stages is proportional to the size of caterpillar population. Alongside this control, we introduce another control method of applying a pesticide to precede the linear harvesting.
According to Liang [13], a pesticide is a chemical or biological agent that kills the pest in a density-dependent manner. When introduced in the model, according to Liang [13], it affects the FAW due to chemical toxicity but has no direct effects on maize plants. We assume that pesticide control is more effective on both caterpillars and moths. We use model equations (16) and (18) as our baseline model systems to introduce both controls (pesticide and harvesting). Since we are interested in killing the caterpillar and the adult moth, we let u denote the strength of the chemical pesticide sprayed and let d 1 and d 2 denote the damage coefficients due to sprayed pesticide on the caterpillar and the moth, respectively. We also assume that spraying of the pesticide is done continuously and in both stages of maize growth. Below is the model with control parameters. e model in the vegetative stage (0 < t < t 1 ) is subject to (2). e model in the reproductive stage (t 1 ≤ t < t 2 ) is subject to (4). Since our problem is not based on a case study, for the simulation purposes, we make use of the data from the literature as seen in Table 3 as well as d 1 � 3, d 2 � 11, u � 0.08, and h � 1.7861, which were used by Paez and Hui [8,35] to simulate the model systems (20) and (21) with controls being applied under the assumption that the moth is immigrating in the field at different rates per day such as σ � 15, 30, 45, 60 moths/day.
From the results shown in Figures 6-9, deploying pesticide and harvesting provides a satisfactory answer to our goal as the maximum control of the pest dynamics was achieved. Comparing the results in Tables 6 and 7, the caterpillar, eggs, and adult moths as we see in Figure 6(a) were unable to reach their environmental carrying capacity and were suppressed to 3, 16, and 17, respectively, with a resultant increase in maize plants to 492 within 63 days, while in Figure 6(b) the pest was suppressed to 3 caterpillars, 36 moths, and 31 eggs with a subsequent increase in maize plants to 446 in 97 days.
Furthermore, in Figure 7(a), the pest was reduced to 4 caterpillars, 36 moths, and 31 eggs and the maize plants increased to 491 in 63 days, while in Figure 7(b) the pest was decreased to 2 caterpillars, 37 moths, and 30 eggs and the maize plants peaked to 445 within 97 days. Moreover, Figure 8(a) shows that the caterpillars, adult moths, and eggs decreased to 5, 55, and 54, respectively, and the maize plants increased to 443, whereas in Figure 8(b) the pest was reduced to 4 caterpillars, 54 moths, and 47 eggs with a resultant increase in maize plants to 444. We finally observe in Table 7: Numbers of adult moths, eggs, caterpillars, and maize densities when there is no immigration of the moth.

Maize
Adult moths Caterpillars Eggs In the table, Figure 9(a) that the caterpillars, adult moths, and eggs were suppressed to 4, 72, and 62, respectively, and the maize plants increased to 449, while in Figure 9(b) the pest was reduced to 3 caterpillars, 54 moths, and 47 eggs with an increase in maize plants to 443.
e results in Figures 6-9 confirm that the suppression rate of the moths, eggs, and caterpillars in Figure 6 was more effective than that in Figures 7-9 due to the small number of the adult moths that immigrated into the field. It can also be seen from Figures 8 and 9 that the numbers of eggs and adult  Figure 7: Effects of deploying pesticide and harvesting when 30 adult moths immigrate into the field after 10 days. Subfigure 7(a) shows the simulation of the model system (20) from time t > 0 to t 1 � 63 days with initial values x 1 (t) � 500, y(0) � 0, z(t) � 0, and w(t) � 0. Subfigure 7(b) displays simulation of the model system (21) from time t � 63 to t 2 � 160 using results of subfigure 7(a) taken at t 1 � 63 as initial values. moths remaining in the field after applying control were large compared to the numbers shown in Figures 6 and 7 due to a large number of moths immigrating (σ � 45 and σ � 60) into the field at time t � 10 days. We therefore conclude that it is essential to use surveillance and monitoring systems for timely control of the pest; and it is better to apply control once a small population of all pest life cycle stages have been established.  Figure 9: Effects of deploying pesticide and harvesting when 60 adult moths immigrate into the field after 10 days. Subfigure 9(a) shows the simulation of the model system (20) from time t > 0 to t 1 � 63 days with initial values x 1 (t) � 500, y(0) � 0, z(t) � 0, and w(t) � 0. Subfigure 9(b) displays simulation of the model system (21) from time t � 63 to t 2 � 160 using results of subfigure 8(a) taken at t 1 � 63 as initial values.

Discussion and Concluding Remark
In this paper, a generic model with two submodels for assessing the impact of the FAW infestation on the maize production in the vegetative and reproductive stages was formulated. e two submodels interconnected by the initial condition x 1 (t 1 ) � x 2 (t 1 ) at the transitional period (i.e., t � t 1 ) were stage-structured in both populations of maize and FAW. e preliminary analysis of the two submodels revealed that the two systems had a unique and positively bounded solution for all time t ≥ 0. A numerical analysis of the submodels was also carried out under two different cases. Case 1 concerns different number of adult moths in the field which were assumed at t � 0 (with no immigration) and Case 2 concerns the existence of exogenous factors that lead to the immigration of the adult moths into the field at a time t.
e results indicated that the destruction of maize biomass which is accompanied by the decrease in maize plants to an average of 160 and 142 in the vegetative and reproductive stages, respectively, was observed to be higher in Case 2 than in Case 1 due to the subsequent increase in egg production and density of the caterpillars in a few (10) days after immigration. is severe effect of maize biomass caused by the unprecedented numbers of caterpillars, eggs, and adult moths in the field after adult moths' immigration gave us an idea of extending the two submodels to include controls such as pesticide and harvesting. e results indicated that the proposed approach significantly suppressed the numbers of caterpillars, eggs, and adult moths with a resultant increase in maize plants to averages of 467 and 443 in the vegetative and reproductive stages, respectively.
is work is not exhaustive as we expect in the future to deploy optimal control of the FAW on maize plants to minimize not only the cost of using pesticides and harvesting but also the side effects on human health and the environmental impact. is approach will also help to determine the maximum sustainable yield of maize crops and improve the quantity and quality of food to farmers.

Data Availability
All the data used are obtained from different studies, which have been cited in the manuscript.

Conflicts of Interest
e authors declare no conflicts of interest.

Authors' Contributions
S. Daudi contributed to formulation of two stage-structured submodels and analysis. L. Luboobi, M. Kgosimore, and D. Kuznetsov contributed to supervision, writing, and review of the paper.