#### Abstract

We established a mathematical model based on the sense of biological survey in the field of agriculture and introduced various control methods on how to prevent the crops from destructive pests. Basically, there are two main stages in the life cycle of natural enemies like insects: mature and immature. Here, we construct a food chain model of plant pest natural enemy. In natural enemies, there are two stages of construction. Also, we consider three classes of diseases in the pest population, namely, susceptible, exposed, and infectious in this proposed work. In order to categorize the considered models into the class of Impulsive Differential Equations (IDEs), in our study, we specifically consider two ecosystems, which define the impact of control mechanisms on the impulsive releasing of virus particle natural enemies and infectious pests at particular time. Additionally, the importance of spraying virus particles in pest control is discussed; then, we obtain two types of periodic solutions for the system, namely, plant pest extinction and pest extinction. By utilizing the small amplitude perturbation techniques and Floquet theory of the impulsive equation, we obtain the local stability of both periodic solutions. Moreover, the comparison technique of IDE shows the sufficient conditions for the global attractivity of a pest extinction periodic solution. With the assistance of the comparison results, we draw a numerical calculation for the addressed models. Finally, we extend the study of the two models for pest management models: with and without the existence of virus particle.

#### 1. Introduction

Of late, it has been of immense interest to inspect the dynamical properties of impulsive perturbations on population models. In the field of agriculture, the main problem faced by farmers is to find an effective pest control method. Numerous methods like physical, chemical control or biological methods can be used to control pests. It is widely acknowledged that pest is a destructive insect and its surge affects economic as well as ecological problems critically [1, 2]. Evidence demonstrates that annually the pests induce shrinkage in rice, in pulses, in wheat, in sugar cane, in oil seeds, and in the case of cotton [3]. One of the most important population models is the predator-prey system, which has been discussed by various authors. Acknowledging the predatory-prey model (stocking or harvesting) is essential, as it comprises human actions. Human action invariably happens instantaneously or in a short time. Then, we reintegrate an impulsive perturbation after detaching the action of humans and models. These human models are based on short-term perturbations which are generally in the structure of IDE in the modeling. Therefore, IDEs give instinctive descriptions of similar systems [4]. Various illustrations are provided by Bainov [5]: relating impulsive vaccination [6], impulsive birth [7, 8], population ecology [9, 10], and chemotherapeutic treatment of disease [11–13]. While considering plant preservation, we affix considerable emphasis to managing insect pest systems and diseases biologically, depending on the respective food chain, and systematically discharge the natural enemies of pests in order to accomplish or eliminate pests’ purpose. As an illustration, to effectively control the ectoparasitoid, *Scleroderma guani* is discharged periodically to execute the dissemination medium of *Bursaphelenchus xylophilus* Nickel and *Monochamus alternatus* Hope. There are ample measures of literature which can be used in the control of microbial disease to conceal pests and making use of mathematical model to discuss the dynamics of it [14–20]. And for forest insect pests, only a few applications are there. To manage forest insect pests, discharging of the natural enemies can be considered an effective method.

In 1760, Daniel Bernoulli, a pioneer in the field, presented a solution for his mathematical model in small pox. Mathematical methods are widely used for finding the mechanisms behind the spreading of infectious diseases; in particular, the epidemic outbreak among animals has gained a lot of attention. Epidemic models found an important category of mathematical ecology. Andreson and May [21, 22] studied different types of SIR epidemic models. The disease’s incubation time is negligible in SIR models, and as a result, each vulnerable individual becomes infected and then recovers with either temporary or permanent immunity. In many recent works, researchers divided the diseased population into 2 or 3 components as in SI and SEI models, using microbial diseases as a control input. Researchers like Jiao, Wu, Shi, and Song [23–26] consider SEI models as they give a more realistic explanation of biological problems where the susceptible population moves to exposed pests. Xiang [27] made a study on a relevant pest management SEI model and hence proposed a model given as follows:

Here, time , , and denote densities of susceptible, exposed, and infectious pests, respectively. In the lack of , the growth of exponentially with carrying capacity and is the intrinsic birth rate constant, the individual susceptible population and infectious population get in contact with them until the contact rate is given by , and represents the inverse of the latent period. , and are positive constants, and the death rate of the infectious and exposed pests is denoted by the parameter . If is the period of the impulse effect, then the release amount of the infected pests is denoted by at .

The above model is modified by adding natural enemies, and the corresponding model is given in (2) and (3). Many authors [24, 28–31] studied the predator-prey system, which is a relevant population model. Also, we assume that the ability to attack prey is almost the same for each individual predator in the classical predator-prey model [32].

The individuals of the natural enemy are classified into either mature or immature in this paper; also, we assume that immature natural enemy does not attack the prey. This can be considered reasonable as in the case of many mammals in which the immature natural enemy is brought up by their parents. In these cases, the reproductive rate and the attacking rate are negligible. Inspired by these models, we analyze the stability of the eco-epidemiological plant pest natural enemy model with different impulsive strategies, but it lacks the ability to attack the infected pest. In our approach, pests and natural enemies are treated as prey and predators, respectively. Severe assumptions are made for the mathematical simplicity of this model. Some details are given in the coming section. The key results of this paper are summarized as follows:(i)Based on the survey, only a few studies have been done on plant pest natural enemy paradigm in eco-epidemiology. In particular, we are considering the different life stages of natural enemies and also diseases in pest populations with three classes.(ii)This manuscript deals with different impulsive strategies; particularly, models with viruses and without viruses are discussed. The numerical investigation concludes with a comparative study of these two models.(iii)In our model, we are using multiple impulsive strategies. So, in case of any shortage in any of the impulsive control inputs, we can stabilize our system by altering other control inputs. That’s why this model is practically more useful compared to other models.(iv)This model is very effective and the period of releasing these impulsive controls can be lengthened compared to other integrated models.

And the remaining work is constructed as follows. This paper is structured into 5 sections. In S, we construct an eco-epidemiological model with stage structure and formulate two different models such as with virus particle and without virus particle. Section 3 deals with periodic solutions and main lemmas, followed by global attractivity and local stability of periodic solutions which are investigated in Section 4. Comparative study and discussions are given in Section 5. In the final section, future works and conclusion are given.

#### 2. Formation of Mathematical Modelling

The following assumptions are established in order to create a mathematical model that discusses the entire behavior of a plant, pest, virus, and natural enemy.(i) Logistically, the plant population is increasing. The density of pest population captures plant represented by with rate : plant predation rate by susceptible pests. Thus, the evolution equation is Diseases in pest populations can be divided into three categories: susceptible, exposed, and infectious. Varied contexts necessitate different functional reactions, according to Holling [33] in 1965. As a result, the typical Lotka-Volterra systems were more practical than they had ever been. Holling II response function means nonlinear saturated incidence rate, , where is the contact number of susceptible pest and infected pest per unit time, so gives the force of infection and the effect of inhibition caused by behavioral changes in sensitive individuals owing to their increased numbers or crowding effect is determined by . denotes the inverse of the latent period. Natural enemies’ mature and immature life phases are represented by and , respectively. As the density of pests increases, the natural enemies with a predation rate can only consume a limited quantity of pests. Susceptible pests are consumed only by the mature natural enemy, while the exposed and infective pests are not affected by them. is the death rate of susceptible pests. Also, the mortality rates of exposed and infected pests are represented by . Thus, the equations are depends on and the natural enemy’s death rate and maturity rate are and , accordingly. Then, the corresponding equation is At a rate of , when the immature natural enemy population grows, the mature natural enemy population grows as well. is the mortality rate of a mature enemy population. Thus, evolution model is Releasing the number of infected pests, the immature and mature natural enemies are , and , respectively, which are released periodically at a particular time , where is the impulsive period and .

By using the above-stated hypotheses, we will propose 2 mathematical models: the first model: without a virus particle and the second model: with a virus particle.

##### 2.1. Model without Virus Particle

The use of pesticides can even affect nontarget species badly. To prevent this, we can use some biological control like viruses. Take, for example, the case of North America where certain forest areas to a large extent were affected by defoliation due to the presence of larvae of gypsy moths. Here, *Lymantria dispar* multicapsid nuclear virus was sprayed extensively to control the larvae. Those larvae that consumed the virus perished and the carcasses that remained on foliage further facilitated the presence of the virus to infect the rest of the larvae still present on the foliage as well. Another prime example for the use of viruses as a biological control measure was seen in Australia where the mammalian virus and rabbit hemorrhagic disease virus were used to control the invasive European rabbit population. However, the control measure turned counterproductive since an extensive amount of rabbit population in the country got eliminated when some rabbits which were under quarantine managed to get away [34]. To control specific insect pests, baculoviruses are well-known substitutes for chemical pesticides. These viruses cause infection only if they get exposed to the host [35]. But certain viruses can live on nonliving organisms for a time duration, and when the host gets in control of these viruses, it will get infected. As we know, even COVID-19 can live for about 3 hr in air, 4 hr in copper, 24 hr in cardboard, days in stainless steel, and 3 days in polypropylene plastic. The most commonly used virus or baculovirus will hereafter refer to nucleopolyhedroviruses. And these viruses are characterized for their species-specific, narrow spectrum insecticidal applications. Also, they are characterized for not having any negative impacts on any other living species like plants, birds, mammals, etc. From these, we can see the importance of using virus particles in pest control. Now, we are going to modify the above mathematical model by introducing the virus particle as one more control input.

##### 2.2. Model with Virus Particle

The virus is thought to spread mainly from pests to pests. This can happen between pests that are in close contact with one another. But the virus can also spread from contact with infected surfaces or objects. For example, if we spread a virus particle, a pest can be infected by touching a surface or object that has the virus on it. By using this concept, we modify the above model.

Further assumptions can be added to the above assumptions for modification of system (8) and (9). Let be the virus particle; they attack susceptible pests and make them infected. Infected pests when die release the virus. is the production rate of virus from infected pests and represents the death rates of virus particles.

If the virus is released periodically, with releasing amount , when . Then, (8) and (9) becomes

The parametric description of the above-mentioned models is given in Table 1.

#### 3. Preliminaries

Consider the solution of system (8) & (9) and is a piecewise continuous function , therefore, is continuous in , is a natural number, and exists. Here, we recall some preliminaries and establish results for the following sections.

Lemma 1. *(see [4]) : The left continuous functionat , satisfies the inequalitieswhere and are constants; thus,*

*If we reverse the directions of*(12)*, we will also reverse the directions of inequality in*(13)*.*Lemma 2. *Consider**, with**, and* *for all solutions**for large**in system* (8) and (9)*.*

*Proof. *Consider is a solution of (8) and (9).

Assume and . Let . We getLet and . From Lemma 1 for , we getAs a result, is uniformly bounded; there is a constantsuch that , and , for all *t* large enough.

Lemma 3. *Consider , with , and for all solutions for large in (10) and (11).*

*Proof. *We can easily prove this lemma by similar techniques used in Lemma 2.

Lemma 4. *Let**be a positive periodic solution of the system**and for every solution**of* (17)*, we obtain**as**, for**,**When pests become extinct, we have**For (19) and (20), from Lemma 4,is a positive solution of the system (19) and (20), which is globally asymptotically stable.**Using on (19) and (20),**, ,**We get the following stroboscopic map of (22) by following the periodic discharge of impulses:**, (24) has a fixed point which is unique and positivewhich satisfy if and if . By [36], we obtained that is globally asymptotically stable. Then, the periodic solution of (22) iswith initial value is globally asymptotically stable.**In the case of system (10) and (11), when pests are extinct, we obtain the (19) and (20) together with the following equations:**Substituting into (29), then we obtain**, ,**We get the following stroboscopic map of (30) by following the periodic discharge of impulses:**, (32) has a unique positive fixed pointwhich satisfy if and if . By [36], we obtained that is globally asymptotically stable. Then,with initial valuewhich is globally asymptotically stable.**After that, we will take a look at the subsystem of (8) and (9),**, there exist a stable equilibrium which is globally asymptotic and , unstable equilibrium. Periodic solutions are as follows:*(1)*: plant pest extinction periodic solution*(2)*: pest extinction periodic solution*

#### 4. Stability Analysis

By Floquet's theory of the linear T-periodic impulsive equation, we are deriving the stability of pest eradication periodic solution and plant pest eradication periodic solution of models with and without virus particles. And also, we give a comparative result in this section, which shows the effectiveness of the model with virus particles.

Theorem 1. *Let**be any solution* (8)* and (9); the plant pest eradication periodic solution** is unstable.*

*Proof. *Considering the local stability of the periodic solution , we havewhere , and 7 are small-amplitude perturbations of the solution. The linearized form of (8) and (9) isLet be the fundamental matrix of (38) and (39):whereThen, linearization of impulsive conditions of (8) and (9) yieldsThe corresponding monodromy matrix of (8) and (9) isWe obtain , where is the identity matrix (38) and (39). The fundamental solution matrix is as follows:It is not necessary to compute the exact value of in the following research. This is a list of the eigenvalues of the monodromy matrix :Since , by Floquet’s theory of IDE, we found that the (8) and (9)’s plant pest extinction periodic solution is unstable.

Theorem 2. *is any solution* (8) and (9); *the pest eradication periodic solution**is locally asymptotically stable iff **, where*

*Proof. *As in the previous case, we may establish the local stability of the periodic solution . Letwhere , and 7 are small-amplitude perturbations of the solution. That is, the linearized form of (8) and (9) isIf is the fundamental matrix of (48) and (49), then holds:whereThe linearization of impulsive conditions of (8) and (9) givesThe monodromy matrix that corresponds to (8) and (9) isThen, the eigenvalues areThe periodic solution of the system (8) and (9) for plant pest extinction is locally asymptotically stable iff i.e., . Hence, the proof.

Next, we are going to consider the subsystem of (10) and (11). The two periodic solutions are as follows:(1): pest extinction periodic solution(2): plant pest extinction periodic solution

Theorem 3. *Let be any solution of the system (10) and (11); then*(i)* The pest eradication periodic solution is locally asymptotically stable iff, where*(ii)

*The plant pest eradication periodic solution**is unstable.**Proof. *(i)It is possible to determine the local stability of periodic solution in a similar way to the earlier study. Consider Then, the linearized form of (10) and (11) is If is the fundamental matrix of (56) and (57), then holds: The linearization of impulsive conditions of (10) and (11) becomes The corresponding monodromy matrix of (10) and (11) is(i)And the eigenvalues are The periodic solution of the system (10) and (11) for plant pest extinction is locally asymptotically stable iff , that is, .(ii)It is possible to determine the periodic solution is unstable in a similar way to the earlier study.Next, we are going to establish the global attractivity of the pest eradication periodic solution of (8) and (9).

Theorem 4. * is any solution (8) and (9); the pest eradication periodic solution is globally attractive provided ,*

*Proof. *Consider is any solution of (8) and (9). The first equation of system (8) and (9) can be rewritten aswhich yields