# An Integrated Eco-Epidemiological Plant Pest Natural Enemy Differential Equation Model with Various Impulsive Strategies

Authors: Sayooj Aby Jose; R. Raja; Quanxin Zhu; J. Alzabut; M. Niezabitowski; Valentina E. Balas
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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 25% shrinkage in rice, 30% in pulses, 5−10% in wheat, 20% in sugar cane, 35% in oil seeds, and 50% 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:(1)dStdt=rSt1−StK−ϑ1StIt1+σ^StdEtdt=ϑ1StIt1+σ^St−ω+δ1EtdItdt=ωEt−δ1Itt≠nT,St+=StEt+=EtIt+=It+η^1t=nT.Here, timet, St,Et, and It denote densities of susceptible, exposed, and infectious pests, respectively. In the lack of It, the growth of St exponentially with carrying capacity K and r is the intrinsic birth rate constant, the individual susceptible population St and infectious population It get in contact with them until the contact rate is given by ϑ1StIt/1+σ^St, and ω represents the inverse of the latent period. K,r,ϑ1,σ^,δ1, and η^1 are positive constants, and the death rate of the infectious and exposed pests is denoted by the parameter δ1. If T is the period of the impulse effect, then the release amount of the infected pests is denoted by η^1 at t=nT,n∈ℕ.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].(2)dStdt=rSt1−StK−ϑ1StIt1+σ^St−ϑ2StNtdEtdt=ϑ1StIt1+σ^St−ω+δ1EtdItdt=ωEt−δ1ItdNtdt=δϑ2StNt1+hϑ2St−δ2Ntt≠nT,(3)St+=StEt+=EtIt+=It+η^1Nt+=Nt+ν^2t=nT.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. InS, 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)
ℋ1 Logistically, the plant population Pt is increasing. The density of pest population St captures plant Pt represented by PtSt with rate p1: plant predation rate by susceptible pests. Thus, the evolution equation is(4)dPtdt=Pt1−Pt−p1PtSt.ℋ2 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, ϑ1StIt/1+σ^St, where ϑ1 is the contact number of susceptible pest and infected pest per unit time, so ϑ1ItSt 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 1/1+σ^St. ω denotes the inverse of the latent period. Natural enemies’ mature and immature life phases are represented by N1 and N2, respectively. As the density of pests increases, the natural enemies with a predation rate ϑ2 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. δ is the death rate of susceptible pests. Also, the mortality rates of exposed and infected pests are represented by δ1. Thus, the equations are(5)dStdt=p1PtSt−ϑ1StIt1+σ^St−ϑ2StN2t−δSt,dEtdt=ϑ1StIt1+σ^St−ω+δ1Et,dItdt=ωEt−δ1It.ℋ3N1t depends on N2t and the natural enemy’s death rate and maturity rate are δ3 and κ, accordingly. Then, the corresponding equation is(6)dN1tdt=ϑ2StN2t−δ3+κN1t.ℋ4 At a rate of κ, when the immature natural enemy population N1t grows, the mature natural enemy population N2t grows as well. δ4 is the mortality rate of a mature enemy population. Thus, evolution model is(7)dN2tdt=κN1t−δ4N2t.ℋ5 Releasing the number of infected pests, the immature and mature natural enemies are η^1,ν1^, and ν2^, respectively, which are released periodically at a particular time t=nT, where T is the impulsive period and n∈ℤ+.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

(8)dPtdt=Pt1−Pt−p1PtStdStdt=p1PtSt−ϑ1StIt1+σ^St−ϑ2StN2t−δStdEtdt=ϑ1StIt1+σ^St−ω+δ1EtdItdt=ωEt−δ1ItdN1tdt=ϑ2StN2t−δ3+κN1tdN2tdt=κN1t−δ4N2tt≠nT,(9)Pt+=PtSt+=StEt+=EtIt+=It+η^1N1t+=N1t+ν^1N2t+=N2t+ν^2t=nT.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, 2−3 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 assumptionsℋ1−ℋ5 for modification of system (8) and (9). Let Vt be the virus particle; they attack susceptible pests and make them infected. Infected pests when die release the virus. ϑ3 is the production rate of virus from infected pests and δ2 represents the death rates of virus particles.If the virus is released periodically, with releasing amountη2^, when t=nT,n∈ℤ+. Then, (8) and (9) becomes(10)dPtdt=Pt1−Pt−p1PtStdStdt=p1PtSt−ϑ1StIt1+σ^St−ϑ2StN2t−ϑ3StVt−δStdEtdt=ϑ1StIt1+σ^St−ω+δ1Et+ϑ3StVtdItdt=ωEt−δ1ItdVtdt=ϑ3δ1It−δ2VtdN1tdt=ϑ2StN2t−δ3+κN1tdN2tdt=κN1t−δ4N2tt≠nT,(11)Pt+=PtSt+=StEt+=EtIt+=It+η^1Vt+=Vt+η2^N1t+=N1t+ν^1N2t+=N2t+ν^2t=nT.The parametric description of the above-mentioned models is given in Table1.Table 1
Parametric values for the models.
ParameterDescriptionϑ1Contact number of susceptible pest per unit time for infected pestР1Predation rate of plantϑ3Production rate of virus from infected pestkConversion rate of immature to mature natural enemyδ1Death rate of exposed and infected pest populationδ3Death rate of immature natural enemiesδ4Death rate of mature natural enemiesδDeath rate of susceptible pest populationδ2Death rate of a virus particleσ⌢Half saturation constantv⌢1Impulsive releasing amount of immature natural enemiesη⌢1Impulsive releasing amount of infected pestϑ⌢2Impulsive releasing amount of mature natural enemiesη⌢2Impulsive releasing amount of virus particleωInverse of a latent periodv2Predation rate of mature natural enemy


## 2.1. Model without Virus Particle

(8)dPtdt=Pt1−Pt−p1PtStdStdt=p1PtSt−ϑ1StIt1+σ^St−ϑ2StN2t−δStdEtdt=ϑ1StIt1+σ^St−ω+δ1EtdItdt=ωEt−δ1ItdN1tdt=ϑ2StN2t−δ3+κN1tdN2tdt=κN1t−δ4N2tt≠nT,(9)Pt+=PtSt+=StEt+=EtIt+=It+η^1N1t+=N1t+ν^1N2t+=N2t+ν^2t=nT.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, 2−3 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 assumptionsℋ1−ℋ5 for modification of system (8) and (9). Let Vt be the virus particle; they attack susceptible pests and make them infected. Infected pests when die release the virus. ϑ3 is the production rate of virus from infected pests and δ2 represents the death rates of virus particles.If the virus is released periodically, with releasing amountη2^, when t=nT,n∈ℤ+. Then, (8) and (9) becomes(10)dPtdt=Pt1−Pt−p1PtStdStdt=p1PtSt−ϑ1StIt1+σ^St−ϑ2StN2t−ϑ3StVt−δStdEtdt=ϑ1StIt1+σ^St−ω+δ1Et+ϑ3StVtdItdt=ωEt−δ1ItdVtdt=ϑ3δ1It−δ2VtdN1tdt=ϑ2StN2t−δ3+κN1tdN2tdt=κN1t−δ4N2tt≠nT,(11)Pt+=PtSt+=StEt+=EtIt+=It+η^1Vt+=Vt+η2^N1t+=N1t+ν^1N2t+=N2t+ν^2t=nT.The parametric description of the above-mentioned models is given in Table1.Table 1
Parametric values for the models.
ParameterDescriptionϑ1Contact number of susceptible pest per unit time for infected pestР1Predation rate of plantϑ3Production rate of virus from infected pestkConversion rate of immature to mature natural enemyδ1Death rate of exposed and infected pest populationδ3Death rate of immature natural enemiesδ4Death rate of mature natural enemiesδDeath rate of susceptible pest populationδ2Death rate of a virus particleσ⌢Half saturation constantv⌢1Impulsive releasing amount of immature natural enemiesη⌢1Impulsive releasing amount of infected pestϑ⌢2Impulsive releasing amount of mature natural enemiesη⌢2Impulsive releasing amount of virus particleωInverse of a latent periodv2Predation rate of mature natural enemy


## 3. Preliminaries

Consider the solutionXt=Pt,St,Et,It,N1t,N2t′ of system (8) & (9) and is a piecewise continuous function X:R+⟶R+6, therefore, Xt is continuous in nT,n+1T, n is a natural number, and XnT+ exists. Here, we recall some preliminaries and establish results for the following sections.Lemma 1.
(see [4]): The left continuous functionv∈PC′R+,Rat tk,k∈ℕ, satisfies the inequalities(12)v′t≤gtvt+ht,t≥t0,t≠tk,vtk+≤δkvtk+lk,t=tk,k∈ℕ,where g,h∈PCR+,R and δk≥0,lk are constants; thus,(13)vt≤vt0∏t0<tk<tδkexp∫t0tgudu+∑t0<tk<t∏tk<tj<tδjexp∫tktgudulk+∫t0t∏u<tk<tδkexp∫utgxdxhudu,t≥t0.
If we reverse the directions of (12), we will also reverse the directions of inequality in (13).Lemma 2.
ConsiderC>0, withPt≤C,St≤C,Et≤C,N1t≤C, and N2t≤Cfor all solutionsXt=Pt,St,Et,It,N1t,N2tfor largetin system (8) and (9).Proof.
ConsiderPt,St,Et,It,N1t,N2t is a solution of (8) and (9).
AssumeV^t=Pt,St,Et,It,N1t,N2t and δ¯=minδ,δ1,δ3,δ4. Let t≠nT. We get(14)D+V^t+δ¯V^t≤1+δ¯Pt−P2t≤C0,C0=1+δ¯24.
Lett=nT and V^t+=V^t+η^1+ν^1+ν^2. From Lemma 1 for t∈nT,n+1T, we get(15)V^t≤V^0exp−δ¯t+∫0tC0exp−δ¯t−sds+∑0<nT<tη^1+ν^1+ν^2exp−δ¯t−nT=V^0exp−δ¯t+C0δ¯1−expδ¯t+η^1+ν^1+ν^2exp−δ¯t−T−exp−δ¯t−n+1T1−exp−δ¯T<V^0exp−δ¯t+C0δ¯1−expδ¯t+η^1+ν^1+ν^2exp−δ¯t−T1−exp−δ¯T+η^1+ν^1+ν^2expδ¯Texpδ¯T−1⟶C0δ¯+η^1+ν1^+ν2^expδ¯Texpδ¯T−1,whentis large enough.
As a result,V^ is uniformly bounded; there is a constant(16)C:=C0δ¯+η^1+ν^1+ν^2expδ¯Texpδ¯T−1>0,
such thatPt≤C,St≤C,Et≤C,N1t≤C, and N2t≤C, for all t large enough.Lemma 3.
ConsiderC>0, with Pt≤C,St≤C,Et≤C,Vt≤C,N1t≤C, and N2t≤C for all solutions Xt=Pt,St,Et,It,Vt,N1t,N2t for large t in (10) and (11).Proof.
We can easily prove this lemma by similar techniques used in Lemma2.Lemma 4.
Lety^tbe a positive periodic solution of the system(17)y′t=k−νyt,t≠nTyt+=yt+η,t=nT,n∈ℕ,
and for every solutionytof (17), we obtainyt−y^t⟶0ast⟶∞, fort∈nT,n+1T,(18)y^t=kν+ηexp−νt−nT1−exp−νTwithy^0+=kν+η1−exp−νT.
When pests become extinct, we have(19)dItdt=−δ1It,dN1tdt=−δ3+κN1t,dN2tdt=κN1t−δ4N2t,t≠nT,(20)It+=It+η^1,N1t+=N1t+ν1^,N2t+=N2t+ν2^t=nT.
For (19) and (20), from Lemma 4,(21)I^t=η^1exp−δ1t−nT1−exp−δ1TandI^0+=η^11−exp−δ2T,N^1t=ν^1exp−δ3+κt−nT1−exp−δ3+κTandN^10+=ν^11−exp−δ3+κT,is a positive solution of the system (19) and (20), which is globally asymptotically stable.
UsingN^1t on (19) and (20),(22)dN2tdt=κN^1t−δ4N2tt≠nT,N2t+=N2t+ν2^t=nT.
Then, t∈nT,n+1T,(23)N2t=κν1^exp−δ3+κt−nT−exp−δ4t−nTδ4−δ3−κ1−exp−δ3+κT+N2nT+exp−δ4t−nT.
We get the following stroboscopic map of (22) by following the periodic discharge of impulses:(24)N2n+1T+=κν1^exp−δ3+κt−nT−exp−δ4t−nTδ4−δ3−κ1−exp−δ3+κT+N2nT+exp−δ4t−nT+ν2^,(25)=fN2nT+.
Then, (24) has a fixed point which is unique and positive(26)N2∗=κν1^exp−δ3+κt−nT−exp−δ4Tδ4−δ3−κ1−exp−δ3+κT1−exp−δ4T+ν2^1−exp−δ4T,which satisfy N2<fN2<N2∗ if 0<N2<N2∗ and N2∗<fN2<N2 if N2>N2∗. By [36], we obtained that N2∗ is globally asymptotically stable. Then, the periodic solution of (22) is(27)N^2t=κν^1δ4−δ3−κexp−δ3+κt−nT1−exp−δ3+κT+−κν^1/δ4−δ3−κ+ν^2exp−δ4t−nT1−exp−δ4T,with initial value(28)N^20+=κν^1δ4−δ3−κ11−exp−δ3+κT+−κν^1/δ4−δ3−κ+ν^21−exp−δ4T.That 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:(29)dVtdt=ϑ3δ1It−δ2Vtt≠nT,Vt+=Vt+η^2t=nT.
SubstitutingI^t into (29), then we obtain(30)dVtdt=ϑ3δ1It−δ2Vtt≠nT,Vt+=Vt+η^2t=nT.
Then, t∈nT,n+1T,(31)Vt=ϑ3δ1η^1exp−δ1t−nT−exp−δ2t−nTδ2−δ11−exp−δ1T+VnT+exp−δ2t−nT.
We get the following stroboscopic map of (30) by following the periodic discharge of impulses:(32)Vn+1T+=ϑ3δ1η^1exp−δ1t−nT−exp−δ2t−nTδ2−δ11−exp−δ1T+VnT+exp−δ2t−nT+η^2=fVnT+.
Then, (32) has a unique positive fixed point(33)V∗=ϑ3δ1η^1δ2−δ1exp−δ1T−exp−δ2T1−exp−δ1T1−exp−δ2T+η^21−exp−δ2T,which satisfy V<fV<V∗ if 0<V<V∗ and V∗<fV<V if V>V∗. By [36], we obtained that V∗ is globally asymptotically stable. Then,(34)V^t=ϑ3δ1η^1δ2−δ1exp−δ1t−nT1−exp−δ1T+η^2−ϑ3δ1η^1δ2−δ1exp−δ2t−nT1−exp−δ2T,with initial value(35)V^0+=ϑ3δ1η^1δ2−δ111−exp−δ1T+η^2−ϑ3δ1η^1δ2−δ111−exp−δ2T.which is globally asymptotically stable.
After that, we will take a look at the subsystem of (8) and (9),(36)dPtdt=Pt1−Pt.
Then, there exist a P=1 stable equilibrium which is globally asymptotic and P=0, unstable equilibrium. Periodic solutions are as follows:(1)
0,0,0,I^,N^1,N^2: plant pest extinction periodic solution(2)
1,0,0,I^,N^1,N^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.
LetPt,St,Et,It,N1t,N2tbe any solution (8)and (9); the plant pest eradication periodic solution0,0,0,I^,N^1,N^2is unstable.Proof.
Considering the local stability of the periodic solution0,0,0,I^,N^1,N^2, we have(37)Pt=τ1t,St=τ2t,Et=τ3t,It=I^t+τ4t,N1t=N^1t+τ6t,N2t=N^2t+τ7t,where τit,i=1,2,3,4,6, and 7 are small-amplitude perturbations of the solution. The linearized form of (8) and (9) is(38)dτ1tdt=τ1tdτ2tdt=−ϑ1I^t+ϑ2N^2t+δτ2tdτ3tdt=ϑ1I^tτ2t−ω+δ1τ3tdτ4tdt=ωτ3t−δ1τ4tdτ6tdt=ϑ2N^2tτ2t−δ3+κτ6tdτ7tdt=κτ6t−δ4τ7tt≠nT,(39)τ1t+=τ1tτ2t+=τ2tτ3t+=τ3tτ4t+=τ4tτ6t+=τ6tτ7t+=τ7tt=nT.
Letτt be the fundamental matrix of (38) and (39):(40)dτtdt=Aτt,where(41)A=1000000−ϑ1tI^+ϑ2N^2t+δ00000ϑ1I^t−ω+δ100000ω−δ1000ϑ2N^2t00−δ3+κ00000κ−δ4.
Then, linearization of impulsive conditions of (8) and (9) yields(42)τ1t+τ2t+τ3t+τ4t+τ6t+τ7t+=100000010000001000000100000010000001τ1tτ2tτ3tτ4tτ6tτ7t.
The corresponding monodromy matrix of (8) and (9) is(43)M=100000010000001000000100000010000001τT.
We obtainτT=τ0e∫0TAdt, where τ0 is the identity matrix (38) and (39). The fundamental solution matrix is as follows:(44)τT=eT000000e−∫0Tϑ1I^t+ϑ2N^2t+δdt00000⋆e−ω+δ1T00000⋆e−δ1T000⋆00e−δ3+κT00000⋆e−δ4T.
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 M:(45)λ1=eT>1,λ2=e−∫0Tϑ1I^t+ϑ2N^2t+δdt<1,λ3=e−ω+δ1T<1,λ4=e−δ1T<1,λ6=e−δ3+κT<1,λ7=e−δ4T<1.
Sinceλ1>1, by Floquet’s theory of IDE, we found that the (8) and (9)’s plant pest extinction periodic solution is unstable.Theorem 2.
Pt,St,Et,It,N1t,N2tis any solution (8) and (9); the pest eradication periodic solution1,0,0,I^,V^,N^1,N^2is locally asymptotically stable iffT≤Tmax, where(46)Tmax=ϑ1p1−δη^1δ1+ϑ2p1−δκν^1δ4−δ3−κ1δ3+κ+−κν^1δ4−δ3−κ+ν^21δ4.Proof.
As in the previous case, we may establish the local stability of the periodic solution1,0,0,0,N^1,N^2. Let(47)Pt=1+τ1t,St=τ2t,Et=τ3t,It=τ4t,N1t=N^1t+τ6t,N2t=N^2t+τ7t,where τit,i=1,2,3,4,6, and 7 are small-amplitude perturbations of the solution. That is, the linearized form of (8) and (9) is(48)dτ1tdt==−τ1t−p1τ2tdτ2tdt==−−p1+ϑ1I^t+ϑ2N^2t+δτ2tdτ3tdt=ϑ1I^tτ2t−ω+δ1τ3tdτ4tdt=ωτ3t−δ1τ4tdτ6tdt==ϑ2τ2tN^2t−δ3+κτ6tdτ7tdt=κτ6t−δ4τ7tt≠nT,(49)τ1t+=τ1tτ2t+=τ2tτ3t+=τ3tτ4t+=τ4tτ6t+=τ6tτ7t+=τ7tt=nT.
Ifτt is the fundamental matrix of (48) and (49), then τt holds:(50)dτtdt=Aτt,where(51)A=−1−p100000−−p1+ϑ1I^t+ϑ2N^2t+δ00000ϑ1I^t−ω+δ100000ω−δ1000ϑ2N^2t00−δ3+κ00000κ−δ4.
The linearization of impulsive conditions of (8) and (9) gives(52)τ1t+τ2t+τ3t+τ4t+τ6t+τ7t+=100000010000001000000100000010000001τ1tτ2tτ3tτ4tτ6tτ7t.
The monodromy matrix that corresponds to (8) and (9) is(53)M=100000010000001000000100000010000001τT.
Then, the eigenvalues are(54)λ1=e−T<1,λ2=e∫0Tp1−ϑ1I^t−ϑ2N^2t−δdt,λ3=e−ω+δ1T<1,λ4=e−δ1T<1,λ6=e−δ3+κT<1,λ7=e−δ4T<1.
The periodic solution of the system (8) and (9) for plant pest extinction is locally asymptotically stable iff λ2≤1 i.e., T≤Tmax. Hence, the proof.
Next, we are going to consider the subsystem of (10) and (11). The two periodic solutions are as follows:(1)
1,0,0,I^,V^,N^1,N^2: pest extinction periodic solution(2)
1,0,0,I^,V^,N^1,N^2: plant pest extinction periodic solutionTheorem 3.
LetPt,St,Et,It,Vt,N1t,N2t be any solution of the system (10) and (11); then(i)
The pest eradication periodic solution1,0,0,I^,V^,N^1,N^2 is locally asymptotically stable iffT≤T^max, where(55)T^max=ϑ1p1−δη^1δ1+ϑ2p1−δκν^1δ4−δ3−κ1δ3+κ+−κν^1δ4−δ3−κ+ν^21δ4+ϑ3p1−δϑ3δ1η^1δ2−δ11δ1+η^2−ϑ3δ1η^1δ2−δ11δ2.(ii)
The plant pest eradication periodic solution 0,0,0,I^,V^,N^1,N^2 is unstable.Proof.
(i)
It is possible to determine the local stability of periodic solution1,0,0,0,V^,N^1,N^2 in a similar way to the earlier study.ConsiderPt=1+τ1t,St=τ2t,Et=τ3t,It=τ4t,Vt=V^t+τ5t,N1t=N^1t+τ6t,N2t=N^2t+τ7t.Then, the linearized form of (10) and (11) is(56)dτ1tdt=−τ1t−p1τ2tdτ2tdt=−−p1+ϑ1I^t+ϑ2N^2t+ϑ3V^t+δτ2tdτ3tdt=ϑ1I^tτ2t−ω+δ1τ3t+ϑ3V^τ2tdτ4tdt=ωτ3t−δ1τ4tdτ5tdt=ϑ3δ1τ4t−δ2τ5tdτ6tdt=ϑ2N^2tτ2t−δ3+κτ6tdτ7tdt=κτ6t−δ4τ7tt≠nT,(57)τ1t+=τ1tτ2t+=τ2tτ3t+=τ3tτ4t+=τ4tτ5t+=τ5tτ6t+=τ6tτ7t+=τ7tt=nT.Ifτt is the fundamental matrix of (56) and (57), then τt holds:(58)τtdt=Aτt,A=−1−p1000000−−p1+ϑ1I^t+ϑ2N^2t+ϑ3V^t+δ000000ϑ1I^t+ϑ3V^t−ω+δ1000000ω−δ1000000ϑ3δ1−δ2000ϑ2N^2t000−δ3+κ000000κ−δ4.The linearization of impulsive conditions of (10) and (11) becomes(59)τ1t+τ2t+τ3t+τ4t+τ5t+τ6t+τ7t+=1000000010000000100000001000000010000000100000001τ1tτ2tτ3tτ4tτ5tτ6tτ7t.The corresponding monodromy matrix of (10) and (11) is(60)M=1000000010000000100000001000000010000000100000001τT.(i)
And the eigenvalues are(61)λ1=e−T<1,λ2=e∫0Tp1−ϑ1I^t−ϑ2N^2t−ϑ3V^t−δdt,λ3=e−ω+δ1T<1,λ4=e−δ1T<1,λ5=e−δ2T<1,λ6=e−δ3+κT<1,λ7=e−δ4T<1.The periodic solution of the system (10) and (11) for plant pest extinction is locally asymptotically stable iff λ2≤1, that is, T≤T^max.(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.
Pt,St,Et,It,N1t,N2t is any solution (8) and (9); the pest eradication periodic solution 1,0,0,I^,V^,N^1,N^2 is globally attractive provided T<Tmax,Proof.
ConsiderPt,St,Et,It,N1t,N2t is any solution of (8) and (9). The first equation of system (8) and (9) can be rewritten as(62)dPtdt≤Pt1−Pt,which yields limsupt⟶∞Pt=1; therefore, there is an integer m1>0; if t>m1, then Pt<1+ε0. From the fourth and tenth equations of (8) and (9),(63)dItdt≥−δ1It,t≠nT,It+=It+η^1,t=nT.
Let us have a look at the comparison system(64)dw1tdt=−δ1w1t,t≠nT,w1t+=w1t+η^1,t=nT.
Lemma4 shows that the system (64) has a periodic solution:(65)w^1t=η^1exp−δ1t−nT1−exp−δ1TnT,n+1T,n∈ℤ+,
i.e,w^1t is globally asymptotically stable. When t⟶∞, by Lemma 4 and comparison theorem of the IDE, It≥w1t and w1t⟶w^1t. Thus, there exist m2>m1,t>m2 with(66)It≥w1t>I^t−ε0,nT,n+1T,n∈ℤ+,n>m2.
Using the fifth and twelfth equations of (8) and (9),(67)dN1tdt≥−δ3+κN1t,t≠nT,N1t+=N1t+ν^1,t=nT.
Comparison system is(68)dw2tdt=−δ3+κw2t,t≠nT,w2t+=w2t+ν^1,t=nT.
From Lemma4, we get (68) has a periodic solution(69)w^2t=ν1^exp−δ3+κt−nT1−exp−δ3+κTnT,n+1T,n∈ℤ+,
i.e,w^2t is globally asymptotically stable. When t⟶∞, by Lemma 4 and comparison theorem of the IDE, N1t≥w2t and w2t⟶w^2t. Thus, there exist m3>m2,t>m3 with(70)N1t≥w2t>N^1t−ε0,nT,n+1T,n∈ℤ+,t>m3
By using (70), the sixth and thirteenth of (8) and (9), we get the following subsystem:(71)dN2tdt≥κν1^exp−δ3+κt−nT1−exp−δ3+κT−ε0−δ4N2t,t≠nT,N2t+=N2t+ν^2,t=nT.
Consider the following comparison system:(72)dw3tdt=κν1^exp−δ3+κt−nT1−exp−δ3+κT−ε0−δ4w3t,t≠nT,w3t+=w3t+ν^,t=nT.
From the above manner, we get the system (72) has a periodic solution(73)w^3t=κν^1δ4−δ3−κexp−δ3+κt−nT1−exp−δ3+κT+−κν^1/δ4−δ3−κ+ν^2exp−δ4t−nT1−exp−δ4T−ϑ2κε0δ4nT,n+1T,n∈ℤ+,
i.e,w^3t is globally asymptotically stable. When t⟶∞, by Lemma 4 and comparison theorem of the IDE, N2t≥w3t and w3t⟶w^3t. Then, there exist m4>m3,t>m4 with(74)N2t≥w3t>N^2t−ε0,nT,n+1T,n∈ℤ+,t>m4.
By system (8) and (9) we obtain(75)dStdt≤Stp11+ε0−ϑ1I^t−ε01+σ^K−ϑ2N^2t−ε0−δt≠nT,St+=Stt=nT.
Integrating the preceding equation over the pulses,(76)St≤SnT+exp∫nTn+1Tp11+ε0−ϑ1I^t−ε01+σ^K−ϑ2N^2t−ε0−δdt.
We get the stroboscopic map after the periodic discharge of impulses:(77)Sn+1T+≤SnT+exp∫nTn+1Tp11+ε0−ϑ1I^t−ε01+σ^K−ϑ2N^2t−ε0−δdt=SnT+Γ,where Γ=exp∫nTn+1Tp11+ε0−ϑ1I^t−ε0/1+σ^K−ϑ2N^2t−ε0−δdt<1, as T<Tmax.
Thus,SnT+≤S0+Γn and so SnT+⟶0 as n⟶∞. Hence, St⟶0 as n⟶∞. Then, there exists an ε1>0, sufficiently small, so that 0<St<ε1 for all t>m5. As a result, we can obtain an integer m6>0 such that ϑ1ItSt/1+σ^K<ε2 for all t≥m6 .
Again, in the system (8) and (9), we have dPt/dt≥P1−p1ε1−P,
That impliesliminft⟶∞Pt=1, i.e., Pt⟶1 as t⟶∞.
By the third equation of (8) and (9),(78)dEtdt≤ε2−ω+δ1Et,which implies(79)Et≤E0+e−ω+δ1t+ε2ω+δ1−ε2ω+δ1e−ω+δ1t⟶0ast⟶∞.
Thus, there exists an integerm7>m6 such that Et<ε3 for all t≥m7.
By the fourth and tenth equations of (8) and (9),(80)dItdt≤ωε3−δ1It,t≠nT.
Let us consider(81)dw4∗tdt=ωε3−δ1w4∗t,t≠nT,w∗t=w4∗t+η^1,t=nT.
By Lemma4, the system (81) has a periodic solution(82)w^4∗t=ωε3δ1+η^1exp−δ1t−nT1−exp−δ1TnT,n+1T,n∈ℤ+,
i.e,w^4∗t is globally asymptotically stable. When t⟶∞, by Lemma 4 and comparison theorem of the IDE, It≤w∗t and w∗t⟶w^∗t; there is an integer m8, so,(83)It≤w4∗t<w^4∗t+ε0,t≥m8.
Using the fifth and eleventh equations of (8) and (9) we get(84)dN1tdt≤ϑ2ε2M−δ3+κN1t,t≠nT,N1t+=N1t+ν^1,t=nT.
By Lemma4 and the comparison theorem of the IDE, there is ε4>0 such that(85)N1t<ϑ2ε1Mδ3+κ+ν1^exp−δ3+κt−nT1−exp−δ3+κT+ε4nT,n+1T,n∈ℤ+,
By (8) and (9),(86)dN2tdt≤κϑ2ε1Mδ3+κ+ν1^exp−δ3+κt−nT1−exp−δ3+κT−ε4−δ4N2t,t≠nT,N2t+=N2t+ν^2,t=nT.
Obtaining the previous manner, then there existsε5>0 such that(87)N2t<κν^1δ4−δ3−κexp−δ3+κt−nT1−exp−δ3+κT+−κν^1/δ4+δ3−κ+ν^2exp−δ4t−nT1−exp−δ4T−κδ4ϑ2ε1Mδ3+κ+ε4+ε5,nT,n+1T,n∈ℤ+.
Ast⟶+∞, we get It⟶I^t,Nt⟶N^1t, and N2t⟶N^2t, accounting that εi is small enough. Therefore, the pest eradication periodic solution 1,0,0,I^,N^1,N^2 is globally attractive.Theorem 5.
Pt,St,Et,Vt,It,N1t,N2tis any solution (10) and (11); the pest eradication periodic solution 1,0,0,I^,V^,N^1,N^2 is globally attractive provided T<T^max.Proof.
Same as that of Theorem4.Remark 1.
If the1,0,0,I^,V^,N^1,N^2 of system (10) and (11) and the 1,0,0,I^,N^1,N^2 of system (8) and (9) are locally asymptotically stable, then(88)T^max>Tmax.


## 5. Numerical Simulations and Discussion

This research discusses an integrated eco-epidemiological plant pest natural enemy model with several impulsive methods. The impulsive influence of infected pests, virus particles, and natural enemies on pest depopulation has already been studied. Next, we will calculate the impact of releasing a certain number of infected pests, virus particles, and mature and immature natural enemies on the systems (8) and (9) and (10) and (11). The impact of a variety of impulsive techniques is also explored. The matching parametric values are supplied in Table 2 and Table 3.Table 2
Parametric values for numerical.
ParameterDescriptionValue per weekϑ1Contact number of susceptible pest per unit time for infected pest2.6kConversion rate of immature to mature natural enemy0.4σ⌢Half saturation constant0.1δDeath rate of susceptible pest0.2δ3Death rate of immature natural enemies0.2δ1Death rate of exposed and infected pest population0.5δ4Death rate of mature natural enemies0.4v⌢1Impulsive releasing amount of immature natural enemies2η⌢1Impulsive releasing amount of infected pest0.7v⌢2Impulsive releasing amount of mature natural enemies4ωInverse of the latent period0.3p1Predation rate of plant1ϑ2Predation rate of mature natural enemy0.3Table 3
Parametric values for numerical.
ParameterDescriptionValue per weekδ2Death rate of the virus particle0.2η^2Impulsive releasing amount of virus particle2ϑ3Production rate of virus from infected pest0.5Initially, an infected pest and natural enemies are used as control inputs in a pest control model. They are released on the spur of the moment. The Threshold Limit for the Impulsive Period (TLIP) is provided byTmax=9.55 in this model (8) and (9). If T=9.55, the pest extinction periodic solution also is locally and globally stable, according to Theorem 2 and Theorem 4.We have included an extra impulsive control termed virus particle in our model. The TLIP in this model (10) and (11) is T^max=16.893. If T<16.893, the pest extinction periodic solution is both locally and globally stable, according to Theorem 3 and Theorem 4.There may be a lack of natural enemies, infected pests, or viral particles in some real-life scenarios. This will have an impact on the impulsive period’s threshold limit. As can be seen, Table4 provides a clear image of the impulsive period’s threshold limit in various conditions. As a result, the statistics in Table 4 are quite useful in dealing with these crises. It is clear that the model with virus particles is very effective and the period of releasing these impulsive control can be lengthened. Again, we can also conclude that if ν^1=ν^2=0, then the pests stay alive and natural enemies become extinct, so we need to release more amount of natural enemies and infected pests. For pest extinction, the impulsive discharging amount of mature natural enemy ν^1>0 should be increased when there is no impulsive discharging amount of immature natural enemy ν^1=0. Related outcomes are also obtained when ν^2=0. If η^1=0 and the rest of the parameters are the same, then we need to release natural enemies to make the susceptible and exposed pests extinct. In other words, the virus particle has a noticeable effect on pest population size and influences the size of the pest population to a great extent. The natural enemy can properly control pests only when Tmax is large.Table 4
Comparison of our models: TLIP.
ParametersTmaxT^maxη^1>0,ν^1>0,ν^2>09.5516.893η^1=0,ν^1>0,ν^2>0512.5η^1=0,ν^1=0,ν^2>03.7510η^1=0,ν^1>0,ν^2=01.257.5η^1>0,ν^1=0,ν^2>08.315.643η^1>0,ν^1=0,ν^2=04.5511.893η^1>0,ν^1>0,ν^2=05.813.143From Table4, it is evident that the TLIP of the model with virus particle (10) and (11) is always greater than the TLIP of the model without virus particle (8) and (9); that is, T^max>Tmax.In this study, we focused mainly on TLIP. From the detailed study, it is noticed that changing the parameter creates a greatimpact on the threshold limit. This is well picturized in the following 3D contour plot in Figures1–9. In Figures 1–4, the dynamics of Tmax in terms of p1,ϑ1,ν^1,η^1, and ν^2 is shown. And also Figures 5–9 represent the dynamics of T^max in terms of p1,ϑ3,ν^1,η^1,ν^2, and η^2.Figure 1
The contour plot ofTmax in terms of p1 and ϑ1.Figure 2
The contour plot ofTmax in terms of p1 and ν^1.Figure 3
The contour plot ofTmax in terms of p1 and η^1.Figure 4
The contour plot ofTmax in terms of p1 and ν^2.Figure 5
The contour plot ofT^max in terms of p1 and ϑ3.Figure 6
The contour plot ofT^max in terms of p1 and ν^1.Figure 7
The contour plot ofT^max in terms of p1 and η^1.Figure 8
The contour plot ofT^max in terms of p1 and ν^2.Figure 9
The contour plot ofT^max in terms of p1 and η^2.


## 6. Comparison with Other Models

This paper considered two models. The first model deals with two pest control techniques and the second deals with three pest control techniques, respectively. If we consider the other authors’ models, they discussed biological controls in different manners. Xiang et al. in 2009 made a study on model (1), in which natural enemy is not considered as an impulsive control. And the TLIP is given by(89)Tmax=ϑ1η^1rδ1=3.64.Song in 2010 formulated an SEI model using infected pests, whose functioning is the same as a microbial pesticide. These infected pests can also attack healthy pests. For the extinction of pests, the authors only considered the impulsive releasing of infected pests and did not consider the importance of natural enemies. Mathur et al. in 2016 formulate an eco-epidemiology mathematical model (2) and (3) which includes natural enemies also as a control input. And the TLIP is(90)Tmax=1rϑ1η^1δ1+ϑ2ν^2δ4=6.64.The impulsive releasing of both natural enemies and infected pests affects the extinction period. And in case of Xiang et al. and Song, releasing the number of infected pests only matters. In our model, we consider natural enemies has two phases of life and this gives a more accurate result and also introduces one more control input as virus particle. If there is a shortage in the availability of infected pest, we manage the pest by spraying virus particle or by releasing natural enemies in large quantity. Even then, it is not possible to optimize the pest population after a limit without natural enemies. This forces us to produce large amounts of virus particles and infected pests. That is to say, we need to introduce more biological controls to make models more practical. Also, the TLIP is more in our model and impulsive strategy with virus particle gives more fruitful results than the other models. This study can be described as in Table5.Table 5
Comparison with other models: TLIP.
ModelXiang et al. (2009)Mathur et al. (2016)(8) and (9)(10) and (11)Tmax approx. (week)3.646.649.5516.893


## 7. Conclusion

This paper deals with the dynamic behavior of two integrated pest management models with and without periodic releasing of virus particles at fixed times. This work is attempting for a comparative analysis of models which provide pest control methods in the sense of IPM. IPM constitutes an eco-friendly blend of several control methods like incorporating infected pests, virus particles, and predation through natural enemies. Microbial pesticides and virus spray have the same functions. They can affect healthy pests and weaken this pest function till death. The behavior of models is analyzed using numerical methods. The influences of natural enemy populations, infected pests, and virus particles are measured numerically. For instance, a higher rate of predation decreases the size of the population of pests. The numerical analysis shows that the virus particle is more effective and this is because of the interaction between the pest and virus particle.Further research on our models will contribute to the improvement and evaluation of pest control methods in ecosystems. How does the stochastic release of natural enemies and infected pests affect the model dynamics? Answering these questions will give good results.



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