# Analysis of a High-Dimensional Mathematical Model for Plant Virus Transmission with Continuous and Impulsive Roguing Control

Authors: Guangming Qiu; Sanyi Tang; Mengqi He
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Roguing and replanting are the most common strategies to control plant diseases and pests. How to build the mathematical models of plant virus transmission and consider the impact of roguing and replanting strategies on plant virus eradication is of great practical significance. In the present paper, we propose the mathematical models for plant virus transmission with continuous and impulsive roguing control. For the model with continuous control strategies, the threshold values for the existences and stabilities of multiple equilibria have been given, and the effect of roguing strategies on the threshold values is also addressed. Furthermore, the model with impulsive roguing control tactics is proposed, and the existence and stability of the plant-only and disease-free periodic solutions of the model are investigated by calculating several threshold values. Moreover, when selecting the design control strategy to minimize the threshold, we systematically analyze the existence of the optimal times of roguing infected plants within a replanting cycle, which is of great significance to the design and optimization of the prevention and control strategy of plant virus transmission. Finally, numerical investigations are given to reveal the main conclusions, and the biological implications of the main results are briefly discussed in the last section.


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## Body

## 1. Introduction

Plant diseases are caused by plant viruses transmitted by insect vectors [1–5], which can cause serious losses in economy and increased poverty [6]. Moreover, vector activity and behavior, especially in relation to virus transmission, are key to determining severity of plant disease transmission [7–13]. Therefore, the development of corresponding prevention and control strategies of plant infectious diseases plays a very important role in protecting plants and eradicating vectors. Note that current control tactics of plant disease mainly include gene resistance, vector insect, and culture control. Gene resistance control cannot be widely used due to its high cost, and vector control through insecticide spraying has a small risk of failure due to selection for more damaging virus strains [14–16]. However, the cultural control including roguing or removal of diseased plants has been practised and could be more effective [17–20]. Therefore, a successful plant disease control program depends on a plant production system which is closely aligned with the goals of pest management [21–23].How to evaluate the effectiveness and timeliness of these prevention and control measures needs the development of corresponding mathematical models. A large number of studies have proved that mathematical modeling approaches can help in the prediction of disease propagation in the agricultural realm [8, 13, 22, 24–27]. More recently, several mathematical models for the spread of plant diseases in populations have been analyzed mathematically and applied to control plant diseases. For example, a deterministic model with bilinear incidence was formulated to address the impact of cross protection on the spread and control of Huanglongbing (HLB) disease [22], and the main results reveal that cross protection and removing infected trees play an important role in controlling the spread of HLD. In literature [24], the authors developed an individual-based model (IBM) of vector-borne plant pathogen spread and compared the simulation results with a classical continuous model. In [25], the authors proposed a model with time delay to study the effect of both incubation delay and latent time on the dynamics of plant disease. A more realistic model has been developed in [26] which takes within-plant cell processes, vector population dynamics, behavior, and broader ecological interactions into account. In [27], considering awareness as a controlling measure, the effect of time delay and awareness campaign on the dynamics and control of the mosaic disease has been investigated.Recently, impulsive differential equations have been employed as the mathematical models for many real problems including pest control [16, 28–30], infectious diseases control, and drug administration [31]. Moreover, various plant disease models with impulsive control have been developed and studied [17–20, 32–34]. In [17], the authors proposed a mathematical model for plant population which aims to eradicate infected plants or maintain the number of infected plants below the economic threshold. In [18, 19], the plant disease models with impulsive cultural strategies were developed and analyzed, and the main results reveal that the plant disease could be eradicated or the number of infected plants could be maintained below the economic threshold by choosing proper control frequency and intensity. We note that those modelling methods have been developed and extended in various references [20, 32–34].However, most of the above works only consider (a) the transmission of diseases among a single plant population, without considering the effect of the vector population on the disease transmission; (b) the continuous human intervention strategies, without considering the instantaneity of intervention strategies; and (c) the impact of pulse control on infected plants, without considering the inevitable impact of control measures on healthy plants. Considering the above shortcomings, this paper aims to develop mathematical models, fully integrate the above three factors, develop corresponding theoretical analysis and numerical techniques, and systematically and deeply analyze their impact on plant disease control. In particular, various threshold values which guarantee the existence and stability of plant-only and disease-free periodic solutions have been obtained, and the existence of the optimal times of roguing infected plants within a replanting cycle has been also investigated in more detail.


## 2. The Model

### 2.1. System Description

In this section, based on the model proposed in [21], we develop a vector-borne plant disease model with cultural control. To do this, we denote the total plant population at time t as Nt=Xt+Yt and vector population as Mt=Ut+Vt. The population flow among those compartments is shown in the schematic diagram (i.e., Figure 1), which leads to the following model:(1)dXtdt=r−k1XtVt−gXt,dYtdt=k1XtVt−gYt−αYt,dUtdt=bUt+VT1−Ut+VtmXt+Yt−k2UtYt−cUt,dVtdt=k2UtYt−cVt,where Xt, Yt, Ut, and Vt represent the number of susceptible plants, infectious plants, susceptible vectors, and infectious vectors, respectively. The notations and definitions of all parameters are shown in Table 1.Figure 1
The transfer diagram for model (1).Table 1
Definitions of all parameters and their baseline values.
ParameterInterpretationStandard valueRangeSourcerReplanting rate0.005day−10–0.025[21]αPlant roguing rate0.003day−10–0.033[21]gPlant loss/harvesting rate0.003day−10.002–0.004[21]mMaximum vector abundance500plant−10–2500[21]bMaximum vector birth rate0.2day−10.1–0.3[21]cVector mortality0.12day−10.06–0.18[21]k1Infection rate0.008vector−1day−10.002–0.032[21]k2Acquisition rate0.008plant−1day−10.002–0.032[21]


### 2.2. Basic Properties

In this section, we show the boundedness of system (1), and it follows from system (1) that we have(2)dNtdt=r−gXt−α+gt≤r−gNt,which indicates(3)Nt≤N0exp−gt+rg1−exp−gt⟶rg,fort⟶∞.Similarly, we have(4)dMtdt=bMt1−MtmNt−cMt≤b−cMt−bgM2tmr,and consequently we get(5)Mt≤M0expb−ct1+bgM0expb−ct−1/mrb−c⟶mrb−cbg,fort⟶∞.Thus, the total plant populationNt and vector population Mt are uniformly bounded. Furthermore, the region(6)Ω=Xt,Yt,Ut,Vt∈R+4:Xt+Yt≤rg;Ut+Vt≤mrb−cbg,is positively invariant, which indicates that we can focus on the dynamics of system (1) on the set Ω in the following.


## 2.1. System Description

In this section, based on the model proposed in [21], we develop a vector-borne plant disease model with cultural control. To do this, we denote the total plant population at time t as Nt=Xt+Yt and vector population as Mt=Ut+Vt. The population flow among those compartments is shown in the schematic diagram (i.e., Figure 1), which leads to the following model:(1)dXtdt=r−k1XtVt−gXt,dYtdt=k1XtVt−gYt−αYt,dUtdt=bUt+VT1−Ut+VtmXt+Yt−k2UtYt−cUt,dVtdt=k2UtYt−cVt,where Xt, Yt, Ut, and Vt represent the number of susceptible plants, infectious plants, susceptible vectors, and infectious vectors, respectively. The notations and definitions of all parameters are shown in Table 1.Figure 1
The transfer diagram for model (1).Table 1
Definitions of all parameters and their baseline values.
ParameterInterpretationStandard valueRangeSourcerReplanting rate0.005day−10–0.025[21]αPlant roguing rate0.003day−10–0.033[21]gPlant loss/harvesting rate0.003day−10.002–0.004[21]mMaximum vector abundance500plant−10–2500[21]bMaximum vector birth rate0.2day−10.1–0.3[21]cVector mortality0.12day−10.06–0.18[21]k1Infection rate0.008vector−1day−10.002–0.032[21]k2Acquisition rate0.008plant−1day−10.002–0.032[21]


## 2.2. Basic Properties

In this section, we show the boundedness of system (1), and it follows from system (1) that we have(2)dNtdt=r−gXt−α+gt≤r−gNt,which indicates(3)Nt≤N0exp−gt+rg1−exp−gt⟶rg,fort⟶∞.Similarly, we have(4)dMtdt=bMt1−MtmNt−cMt≤b−cMt−bgM2tmr,and consequently we get(5)Mt≤M0expb−ct1+bgM0expb−ct−1/mrb−c⟶mrb−cbg,fort⟶∞.Thus, the total plant populationNt and vector population Mt are uniformly bounded. Furthermore, the region(6)Ω=Xt,Yt,Ut,Vt∈R+4:Xt+Yt≤rg;Ut+Vt≤mrb−cbg,is positively invariant, which indicates that we can focus on the dynamics of system (1) on the set Ω in the following.


## 3. Dynamics of the Model with Continuous Control

### 3.1. Boundary Equilibria and Their Stabilities

System (1) has plant-only equilibrium E01=X0,0,0,0=r/g,0,0,0 and disease-free equilibrium E02=X0,0,U0,0=r/g,0,mrb−c/bg,0 with b>c. By employing the next generation matrix method [35], we can obtain the basic reproduction number of model (1) as follows:(7)R0=R01R02=k1k2mr2b−cbcg2α+g,where R01=k1r/cg,R02=k2mrb−c/bgα+g. R0 represents the number of secondary cases that one infected individual will cause through the duration of the infective period, R01 is the expected number of plants that one vector infects throughout its infectious life time, and R02 is the expected number of vectors that one plant infects throughout its infectious life time.Theorem 1.
The plant-only equilibriumE01 is globally asymptotically stable if b≤c and R01≤1.Proof.
We introduce the following Lyapunov function:(8)V1t=X−X0−X0lnXX0+Y+M+V,and further(9)V˙1t=1−X0XXr1X−1X0−k1V+k1XV−α+gY+b−cM−bM2mX+Y+k2UY−cV=r2−XX0−X0X+k1X0−cV+k2U−α+gY+b−cM−bM2mX+Y≤r2−X0X−XX0+cR012−1V+k2M−α+gY+b−cM−bM2mX+Y≤r2−X0X−XX0+cR012−1V+α+gk2mrb−cα+gbg−1Y+b−cM−bM2mX+Y.
The arithmetical mean is great than or equal to the geometrical mean, and the function2−X0/X−X/X0 is nonpositive for all X>0, and it is equal to zero if and only if X=X0. If b≤c and R01≤1, then(10)r2−X0X−X0X+cR012−1V+α+gk2mrb−cα+gbg−1Y+b−cM−bM2mX+Y≤0.
Thus, we haveV˙1≤0. Furthermore, V˙1=0 only if Y=0, U=0, and V=0, which indicates that the maximum invariant set in X0,0,0,0:V˙=0 is the singleton E01. By LaSalle’s Invariance Principle [36], we conclude that E01 is globally asymptotically stable in Ω.Next, we show that the disease-free equilibrium is globally stable provided thatR0<1. First, the Jacobian matrix at the equilibrium E02 is given by(11)JE02=−g00−k1rg0−α+g0k1rgmb−c2bmb−c2b−k2mrb−cbg−b−c2c−b0k2mrb−cbg0−c.Stability of equilibriumE02 is determined by the eigenvalues of the Jacobian JE02 evaluated at that equilibrium. Characteristic equation at disease-free equilibrium E02 becomes(12)λ+b−cλ+gλ2+α+g+cλ+cα+g1−R02=0.It is easy to see that two roots areλ1=−b−c<0 and λ2=−g<0, and according to Routh–Hurwitz condition, λ2+α+g+cλ+cα+g1−R02=0 has roots with negative real parts when R0<1. Therefore, E02 is stable if R0<1. Furthermore, we have the following result for the global stability of E02.Theorem 2.
For system (1), if R0≤1, then the disease-free equilibrium E02 is globally asymptotically stable.Proof.
Define a Lyapunov function:(13)V2t=k2mrcgb−c2k1rY2+bgg+α2V2+k2mrb−cYV,and then the derivative of V2t with respect to system (1) is given by(14)V˙2t=k2mrcgb−ck1rYk1XV−g+αY+bgg+αVk2UY−cV+bgg+αk1XV−g+αY+bgg+αYk2UY−cV=k2mcgb−cX+k2bgg+αU−k2mrb−cg+α−k2mrb−ccYV+k22mrb−cU−k2mrcgb−cg+αk1rY2+k1k2mrb−cX−cbgg+α≤k2mrcgb−cg+αk1rR02−1Y2+bcgg+αR02−1V2.
Therefore, we haveV˙2t≤0 if R0≤1, and V˙2t=0 if R0=1 or Y=0,V=0. Substituting Y=0,V=0 into the first and third equations of (1), one has X˙t=r−gXt and U˙t=bUt1−Ut/mXt−cUt, which implies Xt⟶r/g and Ut⟶mrb−c/bg as t⟶+∞. Again, it follows from LaSalle’s Invariance Principle [36] that E02 is globally asymptotically stable in Ω.


### 3.2. The Existence of Endemic Equilibrium and Uniform Persistence

In order to determine the existence of the endemic equilibrium, we have to look for the solution of the algebraic system of equations obtained by equating the right sides of system (1) to zero.(15)r−k1X∗V∗−gX∗=0,k1X∗V∗−gY∗−αY∗=0,bU∗+V∗1−U∗+V∗mX∗+Y∗−k2U∗Y∗−cU∗=0,k2U∗Y∗−cV∗=0.From the first and second equations of (15), we have(16)X∗=rk1V∗+g,Y∗=k1rV∗α+gk1V∗+g,U∗=mb−crα+g+k1rV∗bα+gk1V∗+g−V∗.Substituting the above into the fourth equation of (15), we have(17)AV∗2+BV∗+C=0,where(18)A=−k12bα+gk2r+cα+g<0,B=k12k2mr2b−c−k1k2brgα+g−2k1bcgα+g2,C=bcg2α+g2R02−1.Equation (17) may have two roots if Δ>0, denoted by(19)V1∗=−B+Δ2A,V2∗=−B−Δ2A,with Δ=B2−4AC, and we have the following main results.Theorem 3.
For system (1), if R0>1, then there exists a unique endemic equilibrium E∗=X∗,Y∗,U∗,V∗; if R0≤1, then there is no endemic equilibrium.Next, we will employ the persistent theory developed in literature [37] to show the uniform persistence of system (1). To do this, we let E be a closed positively invariant subset of Ω, on which a continuous flow ℱ is defined. We denote the restriction ℱ to ∂E by ∂ℱ and note that ∂E is in general not positively invariant. Let N be the maximal invariant set of ∂ℱ on ∂E. Suppose that N is a closed invariant set and there exists a cover Nαα∈A of N, where A is a nonempty index set. Nα⊂∂E, N⊂∪α∈ANα, and Nαα∈A are pairwise disjoint closed invariant sets. Furthermore, we propose the following hypotheses and lemmas. H1 All Nα are isolated invariant sets of the flow ℱ. H2Nαα∈A is acyclic; that is, any finite subset of Nαα∈A does not form a cycle. H3 Any compact subset of ∂E contains, at most, finitely many sets of Nαα∈A [37].Lemma 1 (see [37]).
LetE be a closed positively invariant subset of Ω on which a continuous flow ℱ is defined. Suppose there is a constant ε>0 such that ℱ is point dissipative on S∂E,ε∩E0 and the assumptions H1−H3 hold. Then, the flow ℱ is uniformly persistent, if and only if W+Nα∩S∂E,α∩E0=ϕ for any α∈A, where W+Nα=y∈Ω,ωy⊂Nα, and E0 is interior of set E.By this lemma, we can show the uniform persistence of disease whenR0>1.Theorem 4.
In system (1), assume that R0>1 and the disease is initially present; then, the disease is uniformly persistent; i.e., there is a constant ρ>0 such that liminft⟶+∞Yt≥ρ, liminft⟶+∞Vt≥ρ.Proof.
We setE=X,Y,U,V∈R+4|0≤X+Y≤r/g,0≤U+V≤mrb−c/bg, ∂E=X,Y,U,V∈E|Y=0,V=0, and we will show that the conditions of Lemma 1 are satisfied. Clearly, Nα=E0=r/g,0,mrb−c/bg,0 is isolated. Hence, the covering is simply N=E0, which is acyclic. Thus, the conditions H1−H3 hold. We can also obtain that ℱ is point dissipative by (6). Now, we show that W+E0∩E0=ϕ. Suppose this is not true; then, there exists a solution Xt,Yt,Ut,Vt∈E0 such that limt⟶+∞Xt=r/g, limt⟶+∞Yt=0, limt⟶+∞Ut=mrb−c/bg, limt⟶+∞Vt=0. For any sufficiently small constant ε>0, there exists a positive constant T=Tε such that Xt>r/g−ε,Yt<ε,Ut>mrb−c/bg−ε,Vt<ε for all t≥T. Since R0=k1k2mr2b−c/bcg2α+g>1, for sufficiently small ε, we have(20)R0′=k1k2cα+gmrb−cbg−εrg−ε>1.
Note that(21)dYtdt≥k1rg−εVt−g+αYt,dVtdt≥k2mrb−cbg−εYt−cVt,t≥T.
Therefore, ifYt,Vt⟶0, as t⟶∞, then by a standard comparison argument and the nonnegativity, the solution nYt,nVt of(22)dnYtdt=k1rg−εnVt−α+gnYt,dnVtdt=k2mrb−cbg−εnYt−cnVt,t≥T,with initial data nYT=YT,nVT=VT, converges to 0,0 as well. Thus, limWnt=0, where Wnt>0 is defined by(23)Wnt=C1nYt+C2nVt,where C1=k2,C2=α+g/mrb−c/bg−ε.
The derivative ofWnt is given by(24)dWntdt=cα+gmrb−c/bg−εR0′2−1nVt≥0.
Therefore,Wnt goes to either infinity or some positive number as t⟶∞, which is a contradiction of limt⟶+∞Wnt=0. Thus, we have W+E0∩E0=ϕ. Then, we obtain limt⟶+∞Vt≥ρ1 for some constant ρ1>0. By the fourth equation of (1), we have ρ2=k1r/g−ερ1/α+g such that liminft⟶+∞Yt≥ρ2. Denote ρ=minρ1,ρ2; then, liminft⟶+∞Yt≥ρ and liminft⟶+∞x2t≥ρ. The proof of Theorem 4 is completed.


## 3.1. Boundary Equilibria and Their Stabilities

System (1) has plant-only equilibrium E01=X0,0,0,0=r/g,0,0,0 and disease-free equilibrium E02=X0,0,U0,0=r/g,0,mrb−c/bg,0 with b>c. By employing the next generation matrix method [35], we can obtain the basic reproduction number of model (1) as follows:(7)R0=R01R02=k1k2mr2b−cbcg2α+g,where R01=k1r/cg,R02=k2mrb−c/bgα+g. R0 represents the number of secondary cases that one infected individual will cause through the duration of the infective period, R01 is the expected number of plants that one vector infects throughout its infectious life time, and R02 is the expected number of vectors that one plant infects throughout its infectious life time.Theorem 1.
The plant-only equilibriumE01 is globally asymptotically stable if b≤c and R01≤1.Proof.
We introduce the following Lyapunov function:(8)V1t=X−X0−X0lnXX0+Y+M+V,and further(9)V˙1t=1−X0XXr1X−1X0−k1V+k1XV−α+gY+b−cM−bM2mX+Y+k2UY−cV=r2−XX0−X0X+k1X0−cV+k2U−α+gY+b−cM−bM2mX+Y≤r2−X0X−XX0+cR012−1V+k2M−α+gY+b−cM−bM2mX+Y≤r2−X0X−XX0+cR012−1V+α+gk2mrb−cα+gbg−1Y+b−cM−bM2mX+Y.
The arithmetical mean is great than or equal to the geometrical mean, and the function2−X0/X−X/X0 is nonpositive for all X>0, and it is equal to zero if and only if X=X0. If b≤c and R01≤1, then(10)r2−X0X−X0X+cR012−1V+α+gk2mrb−cα+gbg−1Y+b−cM−bM2mX+Y≤0.
Thus, we haveV˙1≤0. Furthermore, V˙1=0 only if Y=0, U=0, and V=0, which indicates that the maximum invariant set in X0,0,0,0:V˙=0 is the singleton E01. By LaSalle’s Invariance Principle [36], we conclude that E01 is globally asymptotically stable in Ω.Next, we show that the disease-free equilibrium is globally stable provided thatR0<1. First, the Jacobian matrix at the equilibrium E02 is given by(11)JE02=−g00−k1rg0−α+g0k1rgmb−c2bmb−c2b−k2mrb−cbg−b−c2c−b0k2mrb−cbg0−c.Stability of equilibriumE02 is determined by the eigenvalues of the Jacobian JE02 evaluated at that equilibrium. Characteristic equation at disease-free equilibrium E02 becomes(12)λ+b−cλ+gλ2+α+g+cλ+cα+g1−R02=0.It is easy to see that two roots areλ1=−b−c<0 and λ2=−g<0, and according to Routh–Hurwitz condition, λ2+α+g+cλ+cα+g1−R02=0 has roots with negative real parts when R0<1. Therefore, E02 is stable if R0<1. Furthermore, we have the following result for the global stability of E02.Theorem 2.
For system (1), if R0≤1, then the disease-free equilibrium E02 is globally asymptotically stable.Proof.
Define a Lyapunov function:(13)V2t=k2mrcgb−c2k1rY2+bgg+α2V2+k2mrb−cYV,and then the derivative of V2t with respect to system (1) is given by(14)V˙2t=k2mrcgb−ck1rYk1XV−g+αY+bgg+αVk2UY−cV+bgg+αk1XV−g+αY+bgg+αYk2UY−cV=k2mcgb−cX+k2bgg+αU−k2mrb−cg+α−k2mrb−ccYV+k22mrb−cU−k2mrcgb−cg+αk1rY2+k1k2mrb−cX−cbgg+α≤k2mrcgb−cg+αk1rR02−1Y2+bcgg+αR02−1V2.
Therefore, we haveV˙2t≤0 if R0≤1, and V˙2t=0 if R0=1 or Y=0,V=0. Substituting Y=0,V=0 into the first and third equations of (1), one has X˙t=r−gXt and U˙t=bUt1−Ut/mXt−cUt, which implies Xt⟶r/g and Ut⟶mrb−c/bg as t⟶+∞. Again, it follows from LaSalle’s Invariance Principle [36] that E02 is globally asymptotically stable in Ω.


## 3.2. The Existence of Endemic Equilibrium and Uniform Persistence

In order to determine the existence of the endemic equilibrium, we have to look for the solution of the algebraic system of equations obtained by equating the right sides of system (1) to zero.(15)r−k1X∗V∗−gX∗=0,k1X∗V∗−gY∗−αY∗=0,bU∗+V∗1−U∗+V∗mX∗+Y∗−k2U∗Y∗−cU∗=0,k2U∗Y∗−cV∗=0.From the first and second equations of (15), we have(16)X∗=rk1V∗+g,Y∗=k1rV∗α+gk1V∗+g,U∗=mb−crα+g+k1rV∗bα+gk1V∗+g−V∗.Substituting the above into the fourth equation of (15), we have(17)AV∗2+BV∗+C=0,where(18)A=−k12bα+gk2r+cα+g<0,B=k12k2mr2b−c−k1k2brgα+g−2k1bcgα+g2,C=bcg2α+g2R02−1.Equation (17) may have two roots if Δ>0, denoted by(19)V1∗=−B+Δ2A,V2∗=−B−Δ2A,with Δ=B2−4AC, and we have the following main results.Theorem 3.
For system (1), if R0>1, then there exists a unique endemic equilibrium E∗=X∗,Y∗,U∗,V∗; if R0≤1, then there is no endemic equilibrium.Next, we will employ the persistent theory developed in literature [37] to show the uniform persistence of system (1). To do this, we let E be a closed positively invariant subset of Ω, on which a continuous flow ℱ is defined. We denote the restriction ℱ to ∂E by ∂ℱ and note that ∂E is in general not positively invariant. Let N be the maximal invariant set of ∂ℱ on ∂E. Suppose that N is a closed invariant set and there exists a cover Nαα∈A of N, where A is a nonempty index set. Nα⊂∂E, N⊂∪α∈ANα, and Nαα∈A are pairwise disjoint closed invariant sets. Furthermore, we propose the following hypotheses and lemmas. H1 All Nα are isolated invariant sets of the flow ℱ. H2Nαα∈A is acyclic; that is, any finite subset of Nαα∈A does not form a cycle. H3 Any compact subset of ∂E contains, at most, finitely many sets of Nαα∈A [37].Lemma 1 (see [37]).
LetE be a closed positively invariant subset of Ω on which a continuous flow ℱ is defined. Suppose there is a constant ε>0 such that ℱ is point dissipative on S∂E,ε∩E0 and the assumptions H1−H3 hold. Then, the flow ℱ is uniformly persistent, if and only if W+Nα∩S∂E,α∩E0=ϕ for any α∈A, where W+Nα=y∈Ω,ωy⊂Nα, and E0 is interior of set E.By this lemma, we can show the uniform persistence of disease whenR0>1.Theorem 4.
In system (1), assume that R0>1 and the disease is initially present; then, the disease is uniformly persistent; i.e., there is a constant ρ>0 such that liminft⟶+∞Yt≥ρ, liminft⟶+∞Vt≥ρ.Proof.
We setE=X,Y,U,V∈R+4|0≤X+Y≤r/g,0≤U+V≤mrb−c/bg, ∂E=X,Y,U,V∈E|Y=0,V=0, and we will show that the conditions of Lemma 1 are satisfied. Clearly, Nα=E0=r/g,0,mrb−c/bg,0 is isolated. Hence, the covering is simply N=E0, which is acyclic. Thus, the conditions H1−H3 hold. We can also obtain that ℱ is point dissipative by (6). Now, we show that W+E0∩E0=ϕ. Suppose this is not true; then, there exists a solution Xt,Yt,Ut,Vt∈E0 such that limt⟶+∞Xt=r/g, limt⟶+∞Yt=0, limt⟶+∞Ut=mrb−c/bg, limt⟶+∞Vt=0. For any sufficiently small constant ε>0, there exists a positive constant T=Tε such that Xt>r/g−ε,Yt<ε,Ut>mrb−c/bg−ε,Vt<ε for all t≥T. Since R0=k1k2mr2b−c/bcg2α+g>1, for sufficiently small ε, we have(20)R0′=k1k2cα+gmrb−cbg−εrg−ε>1.
Note that(21)dYtdt≥k1rg−εVt−g+αYt,dVtdt≥k2mrb−cbg−εYt−cVt,t≥T.
Therefore, ifYt,Vt⟶0, as t⟶∞, then by a standard comparison argument and the nonnegativity, the solution nYt,nVt of(22)dnYtdt=k1rg−εnVt−α+gnYt,dnVtdt=k2mrb−cbg−εnYt−cnVt,t≥T,with initial data nYT=YT,nVT=VT, converges to 0,0 as well. Thus, limWnt=0, where Wnt>0 is defined by(23)Wnt=C1nYt+C2nVt,where C1=k2,C2=α+g/mrb−c/bg−ε.
The derivative ofWnt is given by(24)dWntdt=cα+gmrb−c/bg−εR0′2−1nVt≥0.
Therefore,Wnt goes to either infinity or some positive number as t⟶∞, which is a contradiction of limt⟶+∞Wnt=0. Thus, we have W+E0∩E0=ϕ. Then, we obtain limt⟶+∞Vt≥ρ1 for some constant ρ1>0. By the fourth equation of (1), we have ρ2=k1r/g−ερ1/α+g such that liminft⟶+∞Yt≥ρ2. Denote ρ=minρ1,ρ2; then, liminft⟶+∞Yt≥ρ and liminft⟶+∞x2t≥ρ. The proof of Theorem 4 is completed.


## 4. Dynamics of the Model with Pulse Perturbation

In this section, we extend model (1) by replacing the continuous removing of infected plants with a periodic pulse roguing control strategy, which is more realistic. Hence, we consider the following differential equation with pulse roguing strategy at fixed moments:(25)dXtdt=−k1XtVt−gXt,dUtdt=bUt+Vt1−Ut+VtmXt+Yt−k2UtYt−cUt,dYtdt=k1XtVt−gYt,dVtdt=k2UtYt−cVt,t≠nT,XnT+=XnT+r,UnT+=1−pUnT,YnT+=1−αYnT,VnT+=1−pVnT,t=nT,where T is a fixed positive constant and denotes the period of the impulsive effect, r is the constant replanting parameter, and n∈N which denotes the positive integer set. The parameter α denotes the proportion of the infected plants which is rouged at each pulse perturbation, and p represents vectors removal rate due to the infected plants being rouged; it is likely that some of the uninfected vectors and infected vectors will be removed incidentally.


### 4.1. The Existence and Stability of the Disease-Free Periodic Solution

IfYt=0,Vt=0, then model (25) can be reduced to the following subsystem:(26)dXtdt=−gXt,dUtdt=bUt1−UtmXt−cUt,t≠nT,XnT+=XnT+r,UnT+=1−pUnT,t=nT.From the first equation of (26), we can solve it in any impulsive interval nT,n+1T and have(27)Xt=XnT+exp−gt−nT,t∈nT,n+1T.DenoteXn=XnT+; then,(28)Xn+1=Xnexp−gT+r.There exists a steady stateX∗=r/1−exp−gT, which indicates that the system(29)dXtdt=−gXt,t≠nT,XnT+=XnT+r,t=nT,X0+=X0has a positive periodic solution X∗t=X∗exp−gt−nT, which is globally asymptotically stable.Lemma 2.
System (29) has a positive periodic solution X∗t; for any solution of (29), we have Xt⟶X∗t as t⟶∞, where X∗t=rexp−gt−nT/1−exp−gT,t∈nT,n+1T.Proof.
LetXt=XnT+exp−gt−nT for all t∈nT,n+1T be any solution of (29); then,(30)XnT+=X0+exp−ngT+r1−exp−ngT1−exp−gT,which indicates that Xt=X0+exp−ngT+r1−exp−ngT/1−exp−gTexp−gt−nT⟶rexp−gt−nT/1−exp−gT as n⟶∞ and t∈nT,n+1T.Substituting the above periodic solution into the second equation of (26), one yields(31)Ut=mg+b−cX∗UnT+expb−ct−nTmg+b−cX∗+bUnT+expg+b−ct−nT−1,t∈nT,n+1T.DenoteUn=UnT+; then,(32)Un+1=1−pmg+b−cexpb−cTX∗Unmg+b−cX∗+bUnexpg+b−cT−1.DenoteR1=1−pexpb−cT; then, we conclude that (32) always has a zero equilibrium provided that R1<1 or g+b−c=0. When R1>1 and g+b−c≠0, (32) has a stable positive equilibrium U∗=mrg+b−c1−pexpb−cT−1/bexpg+b−cT−11−exp−gT. Consequently, the system(33)dUtdt=bUt1−UtmX∗t−cUt,t≠nT,UnT+=1−pUnT,t=nT,U0+=U0has a positive periodic solution(34)U∗t=mg+b−cX∗U∗expb−ct−nTmg+b−cX∗+bU∗expg+b−ct−nT−1,t∈nT,n+1T,which is globally asymptotically stable. Furthermore, we have the following main result.Lemma 3.
IfR1>1 and g+b−c≠0, then system (33) has a positive periodic solution U∗t, and for any solution of (33) we have Ut⟶U∗t as t⟶∞.Proof.
DenoteU¯t=1/Ut and U¯t=1/U∗t; then,(35)U¯t=U¯nT+exp−b−ct−nT+b1−exp−gTexpg+b−ct−nT−1exp−b−ct−nTmrg+b−c,U¯∗t=U¯∗exp−b−ct−nT+b1−exp−gTexpg+b−ct−nT−1exp−b−ct−nTmrg+b−c.
Thus, we have(36)U¯t−U¯∗t=U¯nT+−U¯∗exp−b−ct−nT≤U¯nT+−U¯∗.
Note that ifR1>1, then we have U¯nT+−U¯∗=UnT+−U∗/UnT+U∗⟶0 as t⟶∞, and consequently the result of Lemma 2 follows.From Lemmas1 and 2, we obtain the following results.Lemma 4.
System (26) has boundary periodic solution X∗t,0, which is globally stable if R1≤1, and has a positive periodic solution X∗t,U∗t, which is globally stable if R1>1.Therefore, system (25) has a plant-only periodic solution X∗t,0,0,0 and disease-free periodic solution X∗t,U∗t,0,0, and their stabilities have been addressed in the following.Theorem 5.
The plant-only periodic solutionX∗t,0,0,0 of model (25) is globally asymptotically stable in the first quadrant if R1=1−pexpb−cT<1.Proof.
Firstly, we prove the local stability of the solutionX∗t,0,0,0 of model (25). Denote Xt=xt+X∗t,Yt=yt,Ut=ut,Vt=vt; the corresponding linear system of (25) at X∗t,0,0,0 reads as(37)dxtdt=−k1X∗tvt−gxt,dutdt=b−cut,dytdt=k1X∗tvt−gyt,dvtdt=−cvt,t≠nT,xnT+=xnT+r,unT+=1−punT,ynT+=1−αynT,vnT+=1−pvnT,t=nT.Thus, the fundamental solution matrix Φt of (37) satisfies(38)dΦtdt=−g00−k1X∗t0b−c0000−gk1X∗t000−cΦt,and Φ0=I4×4 is the identity matrix, which results in(39)Φt=e−gT00Δ140eb−cT0000e−gTΔ34000e−cT.
There is no need to give the exact forms ofΔ14 and Δ34 as they are not required in the analysis that follows. If all eigenvalues of(40)M=100001−p00001−α00001−pΦT,are less than one, then the plant-only periodic solution X∗t,0,0,0 is locally stable. It is easy to see that(41)λ1=exp−gT<1,λ2=1−pexpb−cT,λ3=1−pexp−gT<1,λ4=1−pexp−cT<1,and then λ2<1 if and only if R1<1. According to Floquet’s theory of impulsive differential equations, the plant-only periodic solution X∗t,0,0,0 is locally stable.
In the following, we prove the global attractivity of the periodic solutionX∗t,0,0,0. It follows from (25) that(42)dNtdt=−gNt,t≠nT,NnT+≤NnT+r,t=nT,N0+=N0,(43)dMtdt=bMt1−MtmNt−cMt,t≠nT,MnT+=1−pMnT,t=nT,M0+=M0.
From (42), we get(44)NnT≤N0exp−ngT+r1−exp−ngT1−exp−gT⟶r1−exp−gT,forn⟶∞.
Thus,Nt is uniformly bounded, and for ε1>0 small enough, there exists a t1>0 such that Xt,Yt≤L1, where L1=r/1−exp−gT+ε1. From (43), we get(45)Mn+1T≤1−pexpb−cTMnT1+b1−pexpb−cT−1MnT/mb−cL1+ε1≤1−pexpb−cTMnT≤1−pexpb−cTnM0⟶0,forn⟶∞.
Thus, ifR1<1, then Mt is uniformly bounded, and for ε2>0 small enough, there exists a t2>0 such that Ut,Vt≤ε2. Let ε2⟶0; then, Ut⟶0, Vt⟶0 as t⟶∞. From (25), dXt/dt≤−gT, XnT+=XnT+r. Consider the following impulsive differential equation:(46)dZ1tdt=−gZ1t,t≠nT,Z1nT+=Z1nT+r,t=nT,Z10+=X0+.
It follows from Lemma1 and the comparison theorem on impulsive differential equations [38] that we have Xt≤Z1t and Z1t⟶X∗t as t⟶∞. Hence, for ε3>0 small enough and all large t, we have(47)Xt≤Z1t<X∗t+ε3.
For simplification, we may assume that (47) holds for all t≥0. From (25), we get(48)dYtdt≤k1X∗t+ε3ε2−gYt,t≠nT,YnT+=1−αYnT,t=nT,Y0+=Y0.
Again, from the comparison theorem on impulsive differential equation, we get(49)Yn+1T≤YnT+exp−gT+exp−gT∫nTn+1Tk1X∗t+ε3ε2expgTdt=YnT+exp−gT+k1ε2∫nTn+1Trexp−gT1−exp−gT+ε3dt=YnT+exp−gT+ε3+rexp−gTk1ε2T1−exp−gT=1−αYnTexp−gT+ε3+rexp−gTk1ε2T1−exp−gTYnT≤1−αnexp−ngT+ε3+rexp−gTk1ε2T1−exp−gT11−1−αexp−gT.
Letε2⟶0; then, YnT⟶0 as n⟶∞. Therefore, Yt⟶0 as t⟶∞.
Next, we showXt⟶X∗t as t⟶∞. It follows from Vt<ε2 that for t>t2 we have(50)dXtdt≥−k1Xtε2−gXt;then, we have(51)dZ2tdt=−k1ε2+gZ2t,t≠nT,Z2nT+=Z2nT+r,t=nT,Z20+=X0+.Note that the above system has a globally stable periodic solution Z2∗t=rexp−k1ε2+gt−nT/1−exp−k1ε2+gT with Z2∗nT+=r/1−exp−k1ε2+gT. From the comparison theorem, we get Z2t≤Xt≤Z1t,Z2t⟶Z2∗t and Z1t⟶X∗t as t⟶∞. Therefore, there exists a t3 such that t3>t2, and for t>t3 we have(52)Z2∗t−ε3<Xt<X∗t+ε3,with ε3>0 small enough.
Letε2⟶0, so that Z2t⟶X∗t. Then, (52) becomes(53)X∗t−ε3<Xt<X∗t+ε3.Hence, Xt⟶X∗t as t⟶∞. This completes the proof.Next, to investigate the stability results of disease-free periodic solutionX∗t,U∗t,0,0 of model (25), we first calculate the basic reproduction number for the impulsive system (25) by using the next infection operator for the piecewise continuous periodic system proposed in [39, 40]. Denote Xt=xt+X∗t,Yt=yt,Ut=ut+U∗t,Vt=vt. The corresponding linear system (25) at X∗t,U∗t,0,0 reads as(54)W′t=GtWt,t≠nT,WnT+=HWnT,t=nT.In this system, we have Wt=xt,ut,yt,vtT,(55)Gt=G1tG2t0Ft−Vrt,H=H100H2,with(56)Ft=0k1X∗tk2U∗t0,Vrt=g00c,0=0000,(57)G1t=−g0bU∗2tmX∗2tb−c−2bU∗tmX∗t,G2t=0−k1X∗tbU∗tmX∗2t−k2U∗tb−2bU∗tmX∗t(58)H1=1001−p,H2=1−α001−p.LetΦGt=Φij,1≤i,j≤2, be the fundamental solution matrix of (54). Consequently, we have ΦG′t=GtΦGt with appropriate initial value ΦG0=I4. Further computation implies that(59)ΦGt=expG1tΦ12t0ΦFt−Vrtt;then, we obtain(60)HΦGT=H1expG1TH1Φ12T0H2ΦFt−VrtT.By [39, 40], the basic reproduction number for system (54) is given as follows:(61)R2=ρH2ΦFt−VrtT.Then, based on Floquet theory, we have the following conclusion.Theorem 6.
The disease-free periodic solutionX∗t,U∗t,0,0 of model (25) is globally asymptotically stable in the first quadrant if R1=1−p1expb−cT>1 and R2<1.Proof.
To prove the global stability ofX∗t,U∗t,0,0, we only need to show the global attractivity. Suppose R1>1; then, from the boundedness of (25), there exists a t1>0 such that Xt,Yt≤L1 and Ut,Vt≤L2 with t≥t1 for every solution Xt,Yt,Ut,Vt of (4.1), where(62)L1=r1−exp−gT+ε1,L2=b1−exp−b−cTmL1b−c1−exp−b−cT+ε.
It follows from (25) that(63)dXtdt≤−gXt,t≠nT,XnT+=XnT+r,t=nT,X0+=X0.
Consider the following system:(64)dZ3tdt=−gZ3t,t≠nT,Z3nT+=Z3nT+r,t=nT,Z30+=X0+.
From Lemma1 and the comparison theorem on impulsive differential equations [38], we have Xt≤Z3t and Z3t⟶X∗t as t⟶∞. Hence, for ε3>0 small enough and all large t, we have(65)Xt≤Z3t<X∗t+ε3.
From (25), we get(66)dYtdt=k1XtVt−gYt≐f1Yt,Vt,t≠nT,dVtdt=k2UtYt−cVt≐f2Yt,Vt,t≠nT,YnT+=1−αYnT,t=nT,VnT+=1−pVnT,t=nT,Y0+=Y0,V0+=V0.
Consider the following system:(67)dQ1tdt=k1X∗t+ε3Q2t−gQ1t≐g1Q1t,Q2t,t≠nT,dQ2tdt=k2M∗tQ1t−cQ2t≐g2Q1t,Q2t,t≠nT,Q1nT+=1−αQ1nT,t=nT,Q1nT+=1−pQ2nT,t=nT,Q10+=Y0,Q20+=V0,which is equivalent to(68)dQtdt=Ft−VrtQt+k1ε3IQt,t≠nT,QnT+=H2QnT,t=nT,where Qt=Q1t,Q2tT and I is identity matrix. By [39, 40], there exists a positive T-periodic function νt, such that Qt=expμ1tνt is a solution of system (68), where μ1=1/TlnρH2ΦF−Vr+k1ε3IT. When R2<1, we have ρH2ΦF−VrT<1. Due to the continuity of ρH2ΦF−Vr+k1ε3IT, we can choose sufficient small ε3 such that ρH2ΦF−Vr+k1ε3IT<1. Hence, we obtain μ1<0, and Qt⟶0 as t⟶∞. For any nonnegative initial value Q1t2,Q2Vt2T of system (68), there exists an η large enough such that(69)Q1t2,Q2t2T≤ην10,ν20T.
Since∂g1/∂Q2t≥0, ∂g2/∂Q1t≥0, and fi≤gi,i=1,2, system (68) is a quasimonotone increasing system [41]; by the comparison theorem [38], we obtain(70)Yt,Vt≤ηexpμ1t−t2ν1t−t2,ν2t−t2T,t≥t2.
Hence, we haveYt⟶0,Vt⟶0 as t⟶∞.
Next, we shall prove that iflimt⟶∞Ytt=0 and limt⟶∞Vt=0, then Xt⟶X∗t and Ut⟶U∗t as t⟶∞. First, we show Ut⟶U∗t as t⟶∞, since Mt=Ut+Vt⟶M∗t, when limt⟶∞Vt=0, Ut⟶M∗t=U∗t. It follows from limt⟶∞Vt=0 that, for ε3>0 small enough, there exists a t3 such that t3≥t2 and 0<Vt<ε3 for t≥t3. Thus, for t≥t3, we have(71)−k1ε3−gXt≤dXtdt≤−gXt,from which the following system is obtained:(72)dZ4tdt=−k1ε3−gZ4t,t≠nT,Z4nT+=Z4nT+r,t=nT,Z40+=X0+.
It follows from Lemma1 that system (72) has a globally stable positive periodic solution Z4∗t for t∈nT,n+1T; here, Z4∗t=rexp−k1ε3−gt−nT/1−expk1ε3−gT with Z4∗nT+=r/1−expk1ε3−gT. By the comparison theorem, we obtain Z4t≤Xt≤Z3t, Z4t⟶Z4∗t, and Z3t⟶X∗t as t⟶∞. Therefore, there is a t4 for ε4>0 small enough such that t4≥t3; for t≥t4, we have(73)Z4∗t−ε4≤Xt≤X∗t+ε4.
Letε3⟶0 in (71), so that Z2t⟶X∗t. Then,(74)X∗t−ε4≤Xt≤X∗t+ε4.
Hence,Xt⟶X∗t as t⟶∞. We have proved the global attractivity of the disease-free periodic solution X∗t,U∗t,0,0 of model (25). This completes the proof.


### 4.2. Permanence

The persistence of the system indicates that plant (susceptible and infected) and vectors (susceptible and infected) can coexist. If our goal is to eliminate infected plants and vectors, then the persistence suggests that control strategies fail to achieve it. Meanwhile, the permanent condition obtained from analyzing the system can provide scientific support for us to identify the key factors that result in failure and the effectiveness of control strategies, and then guide us to establish a good control program.Theorem 7.
System (25) is uniformly persistent if R2>1.Proof.
It follows fromR2>1 that ρH2ΦF−VrT>1, which indicates that there exists ε¯>0 such that(75)ρH2ΦF−Vr−ε¯M1T>1,where(76)M1=0k1k20.
LetXt,Ut,Yt,Vt be any solution of (25). First, we claim that we can find a positive constant η2 such that(77)limt⟶∞supYt≥η2,limt⟶∞supVt≥η2.
Otherwise, there existst1 such that Yt<η2 and Vt<η2 for all t≥t1. Based on system (25), we have(78)dXtdt=−k1VtXt−gXt≐f3Xt,Ut,t≠nT,dUtdt=bUt1−Ut+η2mXt−k2η2Ut−cUt≐f4Xt,Ut,t≠nT,XnT+=XnT+r,t=nT,UnT+=1−pUnT,t=nT,and we take the auxiliary system(79)dX˜tdt=−k1η2X˜t−gX˜t≐g3X˜,U˜t,t≠nT,dU˜tdt=bU˜t1−U˜t+ηmX˜t−k2ηU˜t−cU˜t≐g4X˜t,U˜t,t≠nT,X˜nT+=X˜nT+r,t=nT,U˜nT+=1−pU˜nT,t=nT.Since ∂f3/∂Ut=0, ∂f4/∂Xt≥0, and fi≥gi,i=3,4, system (78) is a quasimonotone increasing system [41]. By the comparison theorem [38], we have Xt≥X˜t,Ut≥U˜t and X˜t⟶X∗t,U˜t⟶U∗t as t⟶∞ and η2⟶0. Therefore, for the above ε¯>0, there exist t2≥t1 and η˜>0; for η2<η˜, we have(80)Xt≥X˜t≥X∗t−ε¯Ut≥U˜t≥U∗t−ε¯.
Again, from system (25), we have(81)dYtdt≥k1X∗t−ε¯Vt−gYt,t≠nT,dVtdt≥k2U∗t−ε¯Yt−cVt,t≠nT,YnT+=1−αYnT,t=nT,VnT+=1−pVnT,t=nT.
Further, we consider an auxiliary system(82)dY˜tdt=k1X∗tV˜t−gY˜t−ε¯V˜t,t≠nT,dV˜tdt=k2U∗tY˜t−cV˜tε¯Y˜t,t≠nT,Y˜nT+=1−αY˜nT,t=nT,V˜nT+=1−pV˜nT,t=nT,which can be rewritten as(83)dY˜tdtdV˜tdt=Ft−Vrt−ε¯M1Y˜tV˜t,t≠nT,Y˜nT+V˜nT+=H2Y˜nTV˜nT,t=nT,where Ft, Vrt, and H2 can be seen in (56) and (58). It follows from the results shown in [39, 40] that there exists a positive T-periodic function Wt=W1t,W2t such that ν2t=expμ2tWt is a solution of (83), where μ2=1/TlnρH2ΦF−Vr−ε¯M1T and ν2t=Y˜t,V˜t. From (75), μ2>0. Choose t3≥t2 and η2>0 such that Wt2≥η2ν20; we obtain(84)Wt≥η2expμ2t−t2ν2t−t2,t≥t3.
By the comparison theorem [38], we obtain that, for all t≥t3,(85)Yt≥Y˜t≥η2expμ2t−t2ν2t−t2,Vt≥V˜t≥η2expμ2t−t2ν2t−t2.
As a result, we obtainWt⟶∞ as t⟶∞. Consequently, Y˜t,V˜t⟶+∞ as t⟶∞, which conflicts with Y˜t<η2 and V˜t<η2. Thus,(86)limt⟶∞supYt≥η2,limt⟶∞supVt≥η2.Furthermore, we have the following two possibilities.(i)
Yt≥η2 and Vt≥η2 for all large t.(ii)
Yt and Vt oscillations about η2 for all large t.
If case (i) holds, then the result is obtained. Next, we consider case (ii). Becauselimt⟶∞supYt≥η2 and limt⟶∞supVt≥η2, we can choose a t1∈n1T,n1+1T such that Yt1≥η2 and Vt1≥η2. The above discussion implies that there exists t2∈n2T,n2+1T, such that Yt2≥η2 and Vt2≥η2, where n2−n1≥0 is finite. Then, we will consider the solution of system (25) in the interval t1,t2:(87)dYtdt=k1XtVt−gYt≥−gYt,which results in the following relations:(88)Yt≥Yt1exp−gt−t1≥η2exp−gt2−t1≥η2exp−gn2−n1T.
Similarly, it follows from(89)dVtdt=k2UtYt−cVt≥−cVtthat we have(90)Vt≥Vt1exp−ct−t1≥η2exp−ct2−t1≥η2exp−cn2−n1T.
Letσ1=minη2exp−gn2−n1T,η2exp−cn2−n1T; then, σ1>0 cannot be infinitely small and n2−n1≥0 is finite. We get Yt≥σ1>0 and Vt≥σ1>0.
Fort>t2, we take the same steps and get another positive σ2. As a consequence of this, we obtain the sequence σi, where σi=minη2exp−gni+1−niT,η2exp−cn2−n1T, for i=1,2,…, is noninfinitesimal when ni+1−ni≥0 is finite. In this case, the solution of system (25) satisfies Yt≥σi>0 and Vt≥σi>0 for all ti,ti+1, where ti∈niT,ni+1T. Let σ=minσi,i=1,2,…,; we obtain Yt≥σ>0 and Vt≥σ>0. This completes the proof.


## 4.1. The Existence and Stability of the Disease-Free Periodic Solution

IfYt=0,Vt=0, then model (25) can be reduced to the following subsystem:(26)dXtdt=−gXt,dUtdt=bUt1−UtmXt−cUt,t≠nT,XnT+=XnT+r,UnT+=1−pUnT,t=nT.From the first equation of (26), we can solve it in any impulsive interval nT,n+1T and have(27)Xt=XnT+exp−gt−nT,t∈nT,n+1T.DenoteXn=XnT+; then,(28)Xn+1=Xnexp−gT+r.There exists a steady stateX∗=r/1−exp−gT, which indicates that the system(29)dXtdt=−gXt,t≠nT,XnT+=XnT+r,t=nT,X0+=X0has a positive periodic solution X∗t=X∗exp−gt−nT, which is globally asymptotically stable.Lemma 2.
System (29) has a positive periodic solution X∗t; for any solution of (29), we have Xt⟶X∗t as t⟶∞, where X∗t=rexp−gt−nT/1−exp−gT,t∈nT,n+1T.Proof.
LetXt=XnT+exp−gt−nT for all t∈nT,n+1T be any solution of (29); then,(30)XnT+=X0+exp−ngT+r1−exp−ngT1−exp−gT,which indicates that Xt=X0+exp−ngT+r1−exp−ngT/1−exp−gTexp−gt−nT⟶rexp−gt−nT/1−exp−gT as n⟶∞ and t∈nT,n+1T.Substituting the above periodic solution into the second equation of (26), one yields(31)Ut=mg+b−cX∗UnT+expb−ct−nTmg+b−cX∗+bUnT+expg+b−ct−nT−1,t∈nT,n+1T.DenoteUn=UnT+; then,(32)Un+1=1−pmg+b−cexpb−cTX∗Unmg+b−cX∗+bUnexpg+b−cT−1.DenoteR1=1−pexpb−cT; then, we conclude that (32) always has a zero equilibrium provided that R1<1 or g+b−c=0. When R1>1 and g+b−c≠0, (32) has a stable positive equilibrium U∗=mrg+b−c1−pexpb−cT−1/bexpg+b−cT−11−exp−gT. Consequently, the system(33)dUtdt=bUt1−UtmX∗t−cUt,t≠nT,UnT+=1−pUnT,t=nT,U0+=U0has a positive periodic solution(34)U∗t=mg+b−cX∗U∗expb−ct−nTmg+b−cX∗+bU∗expg+b−ct−nT−1,t∈nT,n+1T,which is globally asymptotically stable. Furthermore, we have the following main result.Lemma 3.
IfR1>1 and g+b−c≠0, then system (33) has a positive periodic solution U∗t, and for any solution of (33) we have Ut⟶U∗t as t⟶∞.Proof.
DenoteU¯t=1/Ut and U¯t=1/U∗t; then,(35)U¯t=U¯nT+exp−b−ct−nT+b1−exp−gTexpg+b−ct−nT−1exp−b−ct−nTmrg+b−c,U¯∗t=U¯∗exp−b−ct−nT+b1−exp−gTexpg+b−ct−nT−1exp−b−ct−nTmrg+b−c.
Thus, we have(36)U¯t−U¯∗t=U¯nT+−U¯∗exp−b−ct−nT≤U¯nT+−U¯∗.
Note that ifR1>1, then we have U¯nT+−U¯∗=UnT+−U∗/UnT+U∗⟶0 as t⟶∞, and consequently the result of Lemma 2 follows.From Lemmas1 and 2, we obtain the following results.Lemma 4.
System (26) has boundary periodic solution X∗t,0, which is globally stable if R1≤1, and has a positive periodic solution X∗t,U∗t, which is globally stable if R1>1.Therefore, system (25) has a plant-only periodic solution X∗t,0,0,0 and disease-free periodic solution X∗t,U∗t,0,0, and their stabilities have been addressed in the following.Theorem 5.
The plant-only periodic solutionX∗t,0,0,0 of model (25) is globally asymptotically stable in the first quadrant if R1=1−pexpb−cT<1.Proof.
Firstly, we prove the local stability of the solutionX∗t,0,0,0 of model (25). Denote Xt=xt+X∗t,Yt=yt,Ut=ut,Vt=vt; the corresponding linear system of (25) at X∗t,0,0,0 reads as(37)dxtdt=−k1X∗tvt−gxt,dutdt=b−cut,dytdt=k1X∗tvt−gyt,dvtdt=−cvt,t≠nT,xnT+=xnT+r,unT+=1−punT,ynT+=1−αynT,vnT+=1−pvnT,t=nT.Thus, the fundamental solution matrix Φt of (37) satisfies(38)dΦtdt=−g00−k1X∗t0b−c0000−gk1X∗t000−cΦt,and Φ0=I4×4 is the identity matrix, which results in(39)Φt=e−gT00Δ140eb−cT0000e−gTΔ34000e−cT.
There is no need to give the exact forms ofΔ14 and Δ34 as they are not required in the analysis that follows. If all eigenvalues of(40)M=100001−p00001−α00001−pΦT,are less than one, then the plant-only periodic solution X∗t,0,0,0 is locally stable. It is easy to see that(41)λ1=exp−gT<1,λ2=1−pexpb−cT,λ3=1−pexp−gT<1,λ4=1−pexp−cT<1,and then λ2<1 if and only if R1<1. According to Floquet’s theory of impulsive differential equations, the plant-only periodic solution X∗t,0,0,0 is locally stable.
In the following, we prove the global attractivity of the periodic solutionX∗t,0,0,0. It follows from (25) that(42)dNtdt=−gNt,t≠nT,NnT+≤NnT+r,t=nT,N0+=N0,(43)dMtdt=bMt1−MtmNt−cMt,t≠nT,MnT+=1−pMnT,t=nT,M0+=M0.
From (42), we get(44)NnT≤N0exp−ngT+r1−exp−ngT1−exp−gT⟶r1−exp−gT,forn⟶∞.
Thus,Nt is uniformly bounded, and for ε1>0 small enough, there exists a t1>0 such that Xt,Yt≤L1, where L1=r/1−exp−gT+ε1. From (43), we get(45)Mn+1T≤1−pexpb−cTMnT1+b1−pexpb−cT−1MnT/mb−cL1+ε1≤1−pexpb−cTMnT≤1−pexpb−cTnM0⟶0,forn⟶∞.
Thus, ifR1<1, then Mt is uniformly bounded, and for ε2>0 small enough, there exists a t2>0 such that Ut,Vt≤ε2. Let ε2⟶0; then, Ut⟶0, Vt⟶0 as t⟶∞. From (25), dXt/dt≤−gT, XnT+=XnT+r. Consider the following impulsive differential equation:(46)dZ1tdt=−gZ1t,t≠nT,Z1nT+=Z1nT+r,t=nT,Z10+=X0+.
It follows from Lemma1 and the comparison theorem on impulsive differential equations [38] that we have Xt≤Z1t and Z1t⟶X∗t as t⟶∞. Hence, for ε3>0 small enough and all large t, we have(47)Xt≤Z1t<X∗t+ε3.
For simplification, we may assume that (47) holds for all t≥0. From (25), we get(48)dYtdt≤k1X∗t+ε3ε2−gYt,t≠nT,YnT+=1−αYnT,t=nT,Y0+=Y0.
Again, from the comparison theorem on impulsive differential equation, we get(49)Yn+1T≤YnT+exp−gT+exp−gT∫nTn+1Tk1X∗t+ε3ε2expgTdt=YnT+exp−gT+k1ε2∫nTn+1Trexp−gT1−exp−gT+ε3dt=YnT+exp−gT+ε3+rexp−gTk1ε2T1−exp−gT=1−αYnTexp−gT+ε3+rexp−gTk1ε2T1−exp−gTYnT≤1−αnexp−ngT+ε3+rexp−gTk1ε2T1−exp−gT11−1−αexp−gT.
Letε2⟶0; then, YnT⟶0 as n⟶∞. Therefore, Yt⟶0 as t⟶∞.
Next, we showXt⟶X∗t as t⟶∞. It follows from Vt<ε2 that for t>t2 we have(50)dXtdt≥−k1Xtε2−gXt;then, we have(51)dZ2tdt=−k1ε2+gZ2t,t≠nT,Z2nT+=Z2nT+r,t=nT,Z20+=X0+.Note that the above system has a globally stable periodic solution Z2∗t=rexp−k1ε2+gt−nT/1−exp−k1ε2+gT with Z2∗nT+=r/1−exp−k1ε2+gT. From the comparison theorem, we get Z2t≤Xt≤Z1t,Z2t⟶Z2∗t and Z1t⟶X∗t as t⟶∞. Therefore, there exists a t3 such that t3>t2, and for t>t3 we have(52)Z2∗t−ε3<Xt<X∗t+ε3,with ε3>0 small enough.
Letε2⟶0, so that Z2t⟶X∗t. Then, (52) becomes(53)X∗t−ε3<Xt<X∗t+ε3.Hence, Xt⟶X∗t as t⟶∞. This completes the proof.Next, to investigate the stability results of disease-free periodic solutionX∗t,U∗t,0,0 of model (25), we first calculate the basic reproduction number for the impulsive system (25) by using the next infection operator for the piecewise continuous periodic system proposed in [39, 40]. Denote Xt=xt+X∗t,Yt=yt,Ut=ut+U∗t,Vt=vt. The corresponding linear system (25) at X∗t,U∗t,0,0 reads as(54)W′t=GtWt,t≠nT,WnT+=HWnT,t=nT.In this system, we have Wt=xt,ut,yt,vtT,(55)Gt=G1tG2t0Ft−Vrt,H=H100H2,with(56)Ft=0k1X∗tk2U∗t0,Vrt=g00c,0=0000,(57)G1t=−g0bU∗2tmX∗2tb−c−2bU∗tmX∗t,G2t=0−k1X∗tbU∗tmX∗2t−k2U∗tb−2bU∗tmX∗t(58)H1=1001−p,H2=1−α001−p.LetΦGt=Φij,1≤i,j≤2, be the fundamental solution matrix of (54). Consequently, we have ΦG′t=GtΦGt with appropriate initial value ΦG0=I4. Further computation implies that(59)ΦGt=expG1tΦ12t0ΦFt−Vrtt;then, we obtain(60)HΦGT=H1expG1TH1Φ12T0H2ΦFt−VrtT.By [39, 40], the basic reproduction number for system (54) is given as follows:(61)R2=ρH2ΦFt−VrtT.Then, based on Floquet theory, we have the following conclusion.Theorem 6.
The disease-free periodic solutionX∗t,U∗t,0,0 of model (25) is globally asymptotically stable in the first quadrant if R1=1−p1expb−cT>1 and R2<1.Proof.
To prove the global stability ofX∗t,U∗t,0,0, we only need to show the global attractivity. Suppose R1>1; then, from the boundedness of (25), there exists a t1>0 such that Xt,Yt≤L1 and Ut,Vt≤L2 with t≥t1 for every solution Xt,Yt,Ut,Vt of (4.1), where(62)L1=r1−exp−gT+ε1,L2=b1−exp−b−cTmL1b−c1−exp−b−cT+ε.
It follows from (25) that(63)dXtdt≤−gXt,t≠nT,XnT+=XnT+r,t=nT,X0+=X0.
Consider the following system:(64)dZ3tdt=−gZ3t,t≠nT,Z3nT+=Z3nT+r,t=nT,Z30+=X0+.
From Lemma1 and the comparison theorem on impulsive differential equations [38], we have Xt≤Z3t and Z3t⟶X∗t as t⟶∞. Hence, for ε3>0 small enough and all large t, we have(65)Xt≤Z3t<X∗t+ε3.
From (25), we get(66)dYtdt=k1XtVt−gYt≐f1Yt,Vt,t≠nT,dVtdt=k2UtYt−cVt≐f2Yt,Vt,t≠nT,YnT+=1−αYnT,t=nT,VnT+=1−pVnT,t=nT,Y0+=Y0,V0+=V0.
Consider the following system:(67)dQ1tdt=k1X∗t+ε3Q2t−gQ1t≐g1Q1t,Q2t,t≠nT,dQ2tdt=k2M∗tQ1t−cQ2t≐g2Q1t,Q2t,t≠nT,Q1nT+=1−αQ1nT,t=nT,Q1nT+=1−pQ2nT,t=nT,Q10+=Y0,Q20+=V0,which is equivalent to(68)dQtdt=Ft−VrtQt+k1ε3IQt,t≠nT,QnT+=H2QnT,t=nT,where Qt=Q1t,Q2tT and I is identity matrix. By [39, 40], there exists a positive T-periodic function νt, such that Qt=expμ1tνt is a solution of system (68), where μ1=1/TlnρH2ΦF−Vr+k1ε3IT. When R2<1, we have ρH2ΦF−VrT<1. Due to the continuity of ρH2ΦF−Vr+k1ε3IT, we can choose sufficient small ε3 such that ρH2ΦF−Vr+k1ε3IT<1. Hence, we obtain μ1<0, and Qt⟶0 as t⟶∞. For any nonnegative initial value Q1t2,Q2Vt2T of system (68), there exists an η large enough such that(69)Q1t2,Q2t2T≤ην10,ν20T.
Since∂g1/∂Q2t≥0, ∂g2/∂Q1t≥0, and fi≤gi,i=1,2, system (68) is a quasimonotone increasing system [41]; by the comparison theorem [38], we obtain(70)Yt,Vt≤ηexpμ1t−t2ν1t−t2,ν2t−t2T,t≥t2.
Hence, we haveYt⟶0,Vt⟶0 as t⟶∞.
Next, we shall prove that iflimt⟶∞Ytt=0 and limt⟶∞Vt=0, then Xt⟶X∗t and Ut⟶U∗t as t⟶∞. First, we show Ut⟶U∗t as t⟶∞, since Mt=Ut+Vt⟶M∗t, when limt⟶∞Vt=0, Ut⟶M∗t=U∗t. It follows from limt⟶∞Vt=0 that, for ε3>0 small enough, there exists a t3 such that t3≥t2 and 0<Vt<ε3 for t≥t3. Thus, for t≥t3, we have(71)−k1ε3−gXt≤dXtdt≤−gXt,from which the following system is obtained:(72)dZ4tdt=−k1ε3−gZ4t,t≠nT,Z4nT+=Z4nT+r,t=nT,Z40+=X0+.
It follows from Lemma1 that system (72) has a globally stable positive periodic solution Z4∗t for t∈nT,n+1T; here, Z4∗t=rexp−k1ε3−gt−nT/1−expk1ε3−gT with Z4∗nT+=r/1−expk1ε3−gT. By the comparison theorem, we obtain Z4t≤Xt≤Z3t, Z4t⟶Z4∗t, and Z3t⟶X∗t as t⟶∞. Therefore, there is a t4 for ε4>0 small enough such that t4≥t3; for t≥t4, we have(73)Z4∗t−ε4≤Xt≤X∗t+ε4.
Letε3⟶0 in (71), so that Z2t⟶X∗t. Then,(74)X∗t−ε4≤Xt≤X∗t+ε4.
Hence,Xt⟶X∗t as t⟶∞. We have proved the global attractivity of the disease-free periodic solution X∗t,U∗t,0,0 of model (25). This completes the proof.


## 4.2. Permanence

The persistence of the system indicates that plant (susceptible and infected) and vectors (susceptible and infected) can coexist. If our goal is to eliminate infected plants and vectors, then the persistence suggests that control strategies fail to achieve it. Meanwhile, the permanent condition obtained from analyzing the system can provide scientific support for us to identify the key factors that result in failure and the effectiveness of control strategies, and then guide us to establish a good control program.Theorem 7.
System (25) is uniformly persistent if R2>1.Proof.
It follows fromR2>1 that ρH2ΦF−VrT>1, which indicates that there exists ε¯>0 such that(75)ρH2ΦF−Vr−ε¯M1T>1,where(76)M1=0k1k20.
LetXt,Ut,Yt,Vt be any solution of (25). First, we claim that we can find a positive constant η2 such that(77)limt⟶∞supYt≥η2,limt⟶∞supVt≥η2.
Otherwise, there existst1 such that Yt<η2 and Vt<η2 for all t≥t1. Based on system (25), we have(78)dXtdt=−k1VtXt−gXt≐f3Xt,Ut,t≠nT,dUtdt=bUt1−Ut+η2mXt−k2η2Ut−cUt≐f4Xt,Ut,t≠nT,XnT+=XnT+r,t=nT,UnT+=1−pUnT,t=nT,and we take the auxiliary system(79)dX˜tdt=−k1η2X˜t−gX˜t≐g3X˜,U˜t,t≠nT,dU˜tdt=bU˜t1−U˜t+ηmX˜t−k2ηU˜t−cU˜t≐g4X˜t,U˜t,t≠nT,X˜nT+=X˜nT+r,t=nT,U˜nT+=1−pU˜nT,t=nT.Since ∂f3/∂Ut=0, ∂f4/∂Xt≥0, and fi≥gi,i=3,4, system (78) is a quasimonotone increasing system [41]. By the comparison theorem [38], we have Xt≥X˜t,Ut≥U˜t and X˜t⟶X∗t,U˜t⟶U∗t as t⟶∞ and η2⟶0. Therefore, for the above ε¯>0, there exist t2≥t1 and η˜>0; for η2<η˜, we have(80)Xt≥X˜t≥X∗t−ε¯Ut≥U˜t≥U∗t−ε¯.
Again, from system (25), we have(81)dYtdt≥k1X∗t−ε¯Vt−gYt,t≠nT,dVtdt≥k2U∗t−ε¯Yt−cVt,t≠nT,YnT+=1−αYnT,t=nT,VnT+=1−pVnT,t=nT.
Further, we consider an auxiliary system(82)dY˜tdt=k1X∗tV˜t−gY˜t−ε¯V˜t,t≠nT,dV˜tdt=k2U∗tY˜t−cV˜tε¯Y˜t,t≠nT,Y˜nT+=1−αY˜nT,t=nT,V˜nT+=1−pV˜nT,t=nT,which can be rewritten as(83)dY˜tdtdV˜tdt=Ft−Vrt−ε¯M1Y˜tV˜t,t≠nT,Y˜nT+V˜nT+=H2Y˜nTV˜nT,t=nT,where Ft, Vrt, and H2 can be seen in (56) and (58). It follows from the results shown in [39, 40] that there exists a positive T-periodic function Wt=W1t,W2t such that ν2t=expμ2tWt is a solution of (83), where μ2=1/TlnρH2ΦF−Vr−ε¯M1T and ν2t=Y˜t,V˜t. From (75), μ2>0. Choose t3≥t2 and η2>0 such that Wt2≥η2ν20; we obtain(84)Wt≥η2expμ2t−t2ν2t−t2,t≥t3.
By the comparison theorem [38], we obtain that, for all t≥t3,(85)Yt≥Y˜t≥η2expμ2t−t2ν2t−t2,Vt≥V˜t≥η2expμ2t−t2ν2t−t2.
As a result, we obtainWt⟶∞ as t⟶∞. Consequently, Y˜t,V˜t⟶+∞ as t⟶∞, which conflicts with Y˜t<η2 and V˜t<η2. Thus,(86)limt⟶∞supYt≥η2,limt⟶∞supVt≥η2.Furthermore, we have the following two possibilities.(i)
Yt≥η2 and Vt≥η2 for all large t.(ii)
Yt and Vt oscillations about η2 for all large t.
If case (i) holds, then the result is obtained. Next, we consider case (ii). Becauselimt⟶∞supYt≥η2 and limt⟶∞supVt≥η2, we can choose a t1∈n1T,n1+1T such that Yt1≥η2 and Vt1≥η2. The above discussion implies that there exists t2∈n2T,n2+1T, such that Yt2≥η2 and Vt2≥η2, where n2−n1≥0 is finite. Then, we will consider the solution of system (25) in the interval t1,t2:(87)dYtdt=k1XtVt−gYt≥−gYt,which results in the following relations:(88)Yt≥Yt1exp−gt−t1≥η2exp−gt2−t1≥η2exp−gn2−n1T.
Similarly, it follows from(89)dVtdt=k2UtYt−cVt≥−cVtthat we have(90)Vt≥Vt1exp−ct−t1≥η2exp−ct2−t1≥η2exp−cn2−n1T.
Letσ1=minη2exp−gn2−n1T,η2exp−cn2−n1T; then, σ1>0 cannot be infinitely small and n2−n1≥0 is finite. We get Yt≥σ1>0 and Vt≥σ1>0.
Fort>t2, we take the same steps and get another positive σ2. As a consequence of this, we obtain the sequence σi, where σi=minη2exp−gni+1−niT,η2exp−cn2−n1T, for i=1,2,…, is noninfinitesimal when ni+1−ni≥0 is finite. In this case, the solution of system (25) satisfies Yt≥σi>0 and Vt≥σi>0 for all ti,ti+1, where ti∈niT,ni+1T. Let σ=minσi,i=1,2,…,; we obtain Yt≥σ>0 and Vt≥σ>0. This completes the proof.


## 5. Optimum Times of Roguing Control

Note that, in reality, replanting the susceptible and removing the infected plants are usually not at the same time, and the infected plants could be removed several times within each replanting period. The key question is whether there exist optimal times of removing the infected plants within each replanting period such that the threshold value reaches its minimum. To address this question, we extend model (25) by taking the multiple pulse controls as follows:(91)dXtdt=−k1XtVt−gXt,dUtdt=bUt+Vt1−Ut+VtmXt+Yt−k2UtYt−cUt,dYtdt=k1XtVt−gYt,dVtdt=k2UtYt−cVt,,t≠τn,tni,Xt+=Xt+r,Ut+=Ut,Yt+=Yt,Vt+=Vt,,t=τn,Xt+=Xt,Ut+=1−pUt,Yt+=1−αYt,Vt+=1−pVt,,t=tni,where τnn=0,1,2,…, and tnii=1,2,…,k are impulsive point series at which the replanting and removing actions have been taken, respectively. Further, we assume τ=τn+1−τn for all n=0,1,2,…,, where τ is the period of replanting. For this case, model (91) is called a period system with period τ if there exists a positive integer k such that tn+1k−tnk=τ, where tni∈τn,τn+1 for all i=1,2,…,k. This implies that, in each period τ, the infected plants have been removed k times. Denote Δi=tni+1−tni,i=0,1,2,…,k, where Δ0=τn,Δk+1=τn+1−tnk. Firstly, the basic properties of subsystem(92)dXtdt=−gXt,UtdtbUt1−UtmXt−cUt,,t≠τn,tni,Xt+=Xt+r,Ut+=Ut,,t=τn,Xt+=Xt,Ut+=1−pUt,,t=tniplay a key role in analyzing the disease control.In any given time intervalτn,τn+1, we integrate the first equation of system (92) and have Xt=Xτn+exp−gt−τn. At time τn+1, one replanting strategy is applied, which results in(93)Xτn+1+=Xτn+exp−gτ+r.DenoteXτn=Xτn+; then, we have the following difference equation: Xτn+1=Xτnexp−gτ+r, which has a unique steady state X¯∗=r/1−exp−gτ. Therefore, from Lemma 1, the subsystem(94)dXtdt=−gXt,t≠τn,Xt+=Xt+r,t=τn,X0+=X0has a positive periodic solution X¯∗t=X¯∗exp−gt−τn which is globally asymptotically stable.Further, substituting the above solution into the second equation of (92) and integrating the second equation of system (92) from τn to tn1 yields(95)Ut=Uτn+expb−ct−τn1+bUτn+/mg+b−cX¯∗expg+b−ct−τn−1,t∈τn,tn1.At timetn1, one removing strategy is applied, which results in(96)Utn1+=1−pUτn+expb−cΔ11+bUτn+/mg+b−cX¯∗expg+b−cΔ1−1.Again, integrating the second equation of system (92) from tn1 to tn2 yields(97)Ut=Utn1+expb−ct−tn11+bUtn1+/mg+b−cX¯∗expg+b−ct−tn1−1,t∈tn1,tn2.Further, at timetn2, one removing tactics is applied and(98)Utn2+=1−p2Uτn+expb−cΔ1+Δ21+bUτn+/mg+b−cX¯∗expg+b−cΔ1−1+1−pexpb−cΔ1expg+b−cΔ2−1.By induction, we can see that(99)Ut=1−pkUτn+exp∑i=1kΔiexpb−ct−tnk1+bUτn+/mg+b−cX¯∗∑i=1k1−pi−1expb−c∑j=1i−1Δjexpg+b−cΔi−1+1−pkexpg+b−ct−tnk−1,for all t∈tnk,τn+1. At time τn+1, we only need to replant healthy plants once without removing infected plants.(100)Uτn+1+=1−pkUτn+expb−cτ1+bUτn+/mg+b−cX¯∗∑i=1k+11−pi−1expb−c∑j=1i−1Δjexpg+b−cΔi−1.DenoteUτn=Uτn+; then, we have the following difference equation:(101)Uτn+1=1−pkUτnexpb−cτ1+bUτn/mg+b−cX¯∗∑i=1k+11−pi−1expb−c∑j=1i−1Δjexpg+b−cΔi−1.DefineR3=1−pkexpb−cτ, and we have the following: if R3≤1, then the above difference equation always has a zeros equilibrium; if R3>1, then it has a stable positive equilibrium:(102)U¯∗=1−pkexpb−cτ−1b/mg+b−cX¯∗∑i=1k+11−pi−1expb−c∑j=1i−1Δjexpg+b−cΔi−1.Thus, the system(103)dUtdt=bUt1−UtmX∗t−cUt,t≠τn,tni,Ut+=1−pUt,t=tni,U0+=U0has a positive periodic solution(104)U¯∗t=U¯∗expb−ct−τn1+bU¯∗/mg+b−cX¯∗expg+b−ct−τn−1,t∈τn,tn1,1−pU¯∗expb−cΔ1expb−ct−tn11+bU¯∗/mg+b−cX¯∗expg+b−cΔ1−1+1−pexpb−cΔ1expg+b−ct−tn1−1,t∈tn1,tn2,⋮1−pkU¯∗expb−c∑i=1kΔiexpb−ct−tnk1+bU¯∗/mg+b−cX¯∗∑i=1k1−pi−1expb−c∑j=1i−1Δjexpg+b−cΔi−1+1−pkexpb−c∑i=1kΔiexpg+b−ct−tnk−1,t∈tnk,τn+1,which indicates that we obtain the complete expression of periodic solution for system (92) over the time interval τn<t≤τn+1.Lemma 5.
System (91) has two nonnegative periodic solutions X¯∗t,0 and X¯∗t,U¯∗t. Moreover, if R3≤1, then for any solution Xt,Ut of (91), we have Xt⟶X¯∗t,Ut⟶0 as t⟶∞; if R3>1, then we have Xt⟶X¯∗t,Ut⟶U¯∗t as t⟶∞.Therefore, system (91) has a plant-only periodic solution X¯∗t,0,0,0 and disease-free positive periodic solution X¯∗t,U¯∗t,0,0. Due to the complexity of the model, it is possible but very complex to obtain the threshold conditions for the global stability of the above two periodic solutions. Therefore, we will use the numerical analysis method to discuss the influence of the control strategy on the stability of the above two boundary periodic solutions in the following.


## 6. Numerical Investigations

Due to the high dimension of the system, it is challenging to study the global dynamics of the system after integrating the pulse effect. Therefore, in this section, we employ the parameters given in Table1 as the baseline parameters for more systematic numerical analysis. This can not only help to further determine the existing main conclusions, but also help to find new dynamic behaviors and reveal the key factors of plant disease control.According to the main results related to model (1), if b≤c and R01≤1, then the plant-only equilibrium E01 is globally asymptotically stable; if b>c and R0≤1, then the disease-free equilibrium E02 is globally asymptotically stable; and if R0>1, then the disease is uniformly persistent. To confirm those results, we show the stability of the plant-only equilibrium of system (1) for α=0,b=0.1,c=0.18 in Figure 2(a), and stability of the disease-free equilibrium of system (1) for α=0.02,b=0.2,c=0.12,R0=0.7906<1 in Figure 2(b). If we choose α=0,b=0.2,c=0.12 with R0=1.9365>1, then the plant disease is persistent, as shown in Figure 2(c). The effects of roguing rate α on the basic reproduction number R0 are shown in Figure 3, from which we can see that R0 is a strictly monotonic decreasing function of α. This reveals that increasing the roguing rate can effectively reduce the threshold R0, and then the disease could be successfully controlled.Figure 2
Stabilities of the equilibriaE0ii=1,2 and E∗ for model (1). The set of parameter values is as follows: r=0.004,k1=0.003,g=0.004,m=500,k2=0.003. (a) α=0,b=0.1,c=0.18; (b) α=0.02,b=0.2,c=0.12,R0=0.7906<1; (c) α=0,b=0.2,c=0.12,R0=1.9365>1.
(a)(b)(c)Figure 3
The effects of parameterα on the R0.In Figure4(a), we show the stability of the plant-only periodic solution of model (25) with parameter set α=0.95,T=15,p=0.75, and R1<1; in Figure 4(b), we show the stability of the disease-free periodic solution of model (25) with parameter set α=0.95,T=30,p=0.85,R1<1, and R2<1. If we fix the roguing rate α=0.8, T=30, and p=0.6, then R1>1 and R2>1, and then the numerical simulations reveal that the disease is persistent (Figure 4(c)). Figure 5 shows how different period T and vectors removal rate p affect the threshold value R1. However, due to the complexity of model (25), we cannot get the analytical expression of the threshold R2; thus, we use the methods proposed in [39, 40] to calculate the threshold value R2. We fix the parameter set T=30, p=0.85, and r=0.4, and let the roguing rate α vary. It follows from Figure 6(a) that the threshold value R2 is a monotonically decreasing function with respect to roguing rate α. Similarly, for the parameter sets shown in Figure 6, we see that R2 is a monotonically decreasing function with respect to vectors removal rate p, as shown in Figure 6(b). On the other hand, R2 is a monotonically increasing function with respect to impulsive period T or replanting rate r, as shown in Figures 6(c) and 6(d), respectively. All those results indicate that increasing the roguing rate α and vectors removal rate p and reducing the impulsive period T and the replanting rate r can effectively reduce the threshold R2. Thus, it is conducive to the control of plant disease. Figure 7 further shows the effects of different parameter sets on the threshold condition R2.Figure 4
Existence and stability of periodic solutions for model (25). The set of parameter values is as follows: r=0.4,k1=0.003,g=0.003,b=0.2,m=500,k2=0.003,c=0.12. (a) T=15,p=0.75,α=0.9,R1=0.83<1; (b) T=30,p=0.85,α=0.95,R1=1.6535>1,R2=0.9331<1; (c) T=30,p=0.6,α=0.8,R1=4.4093>1,R2>1.
(a)(b)(c)Figure 5
The effects of the parameter sets on the threshold valuesR1.Figure 6
The effects of the parameter sets on the threshold levelR2. The set of parameter values is as follows: b=0.2,c=0.12,m=500,k1=0.003,k2=0.003,g=0.003. (a) T=30,p=0.85,r=0.4; (b) T=30,α=0.85,r=0.4; (c) p=0.82,α=0.82,r=0.4; (d) T=30,α=0.9,p=0.87.
(a)(b)(c)(d)Figure 7
The effects of the parameter sets on the threshold levelR2. The set of parameter values is b=0.2,c=0.12,m=500,k1=0.003,k2=0.003,g=0.003. (a) r=0.4,α=0.9; (b) r=0.4,p=0.8; (c) p=0.8,α=0.9; (d) r=0.3,T=22.
(a)(b)(c)(d)According to the theoretical analyses for model (91), we obtain two threshold values R3 and Rτk, and if R3≤1, then the plant-only periodic solution X¯∗t,0,0,0 is globally asymptotically stable. Indeed, in Figure 8(a), we capture the stability of the plant-only periodic solution X¯∗t,0,0,0. In Figure 9, we fix the replanting period τ=60 and let vectors removal rate p=0.95,0.9,0.75,0.65 vary. The simulation results indicate that the times of roguing control k is inversely proportional to vectors removal rate p. In order to maintain the threshold value R3 less than one, we find that from Figure 9 the times of roguing control k should be increased significantly as vectors removal rate p is decreasing. Figure 10 further shows the effects of different parameter sets on the threshold condition R3. Note that the value of R3 appears to be quite sensitive to small changes in removal rate p, times of roguing control k, and replanting rate r.Figure 8
The existence and stability of the periodic solutionX¯∗t,0,0,0 of model (90). The set of parameter values is as follows: k1=0.003,g=0.003,b=0.2,m=500,k2=0.003,c=0.12. (a) τ=60,Δi=Δ=10,p=0.75,r=0.4,α=0.9,k=4,R3≤1; (b) τ=240,Δi=Δ=30,p=0.75,r=0.8,α=0.95,k=3,R3>1,Rτk<1; (c) τ=240,Δi=Δ=30,p=0.75,r=0.8,α=0.95,b=0.2,k=2,R3>1Rτk>1.
(a)(b)(c)Figure 9
The effects of the times of roguing controlk and the vectors removal rate p on the threshold level R3. The set of parameter values is as follows: τ=60,b=0.2,c=0.12, and k=1,2,3,…,6. (a) p=0.95; (b) p=0.9; (c) p=0.75; (d) p=0.65.
(a)(b)(c)(d)Figure 10
The effects of the times of roguing controlk and parameter sets on the threshold level R3. The set of parameter values is as follows: b=0.2,c=0.12, and k=1,2,3,…,6. (a) τ=30; (b) p=0.9.
(a)(b)In order to discuss the influence of the control strategies on the stability of the disease-free positive periodic solutionX¯∗t,U¯∗t,0,0, similar to the calculation of R2, we can obtain a threshold Rτk=ρH2kΦF¯t−Vrtτ; here, Vrt and H2 can be seen in (56) and (58).(105)F¯t=0k1X¯∗tk2U¯∗t0.Due to the complexity of model (90) and the inability to calculate the exact expression of Rτk, we let the times of roguing control k vary and fix the other parameters. The numerical results show that if k=3 and R3>1, then the threshold value Rτk<1, which indicates that the disease-free periodic solution is globally asymptotically stable (Figure 8(b)). If k=1,2 and R3>1, then the threshold value Rτk>1, which indicates that the endemic periodic solution is persistent (Figure 8(c)).Therefore, the optimal control of plant diseases is in fact to choose suitable times of roguing controlk such that Rτk<1 and Rτk−1>1. In Figure 11, we fix the replanting period τ and the removal period Δi=Δ and let roguing rate α and the vectors removal rate p vary. The simulation results indicate that the vectors removal rate can affect the times of controls when the infected plants are removed. This shows that the vectors removal is important when removing infected plants. Figure 12 shows the effects of different parameter sets on the threshold condition Rτk. All simulation results shown in Figure 12 indicate that Rτk appears to be quite sensitive to small changes in roguing rate α and vectors removal rate p, replanting rate r, and removal period Δ.Figure 11
The effects of the times of roguing controlk and the vectors removal rate p on the threshold level Rτk. The set of parameter values is as follows: b=0.2,c=0.12,m=500,k1=0.003,k2=0.003,g=0.003,r=0.8,τ=240,Δi=Δ=30, and k=1,2,3,…,8. (a) α=0.99,p=0.99; (b) α=0.95,p=0.7; (c) α=0.95,p=0.5; (d) α=0.95,p=0.4.
(a)(b)(c)(d)Figure 12
The effects of the times of roguing controlk and parameter sets on the threshold level Rτk. The set of parameter values is as follows: b=0.2,c=0.12,k1=0.003,k2=0.003,g=0.003,τ=240, and k=1,2,3,…,6. (a) α=0.95,Δi=Δ=30,r=0.8; (b) p=0.9,Δi=Δ=30,r=0.8; (c) α=0.87,p=0.87,Δi=Δ=30; (d) α=0.9,p=0.9,r=0.8,Δi=Δ.
(a)(b)(c)(d)


## 7. Discussion

Based on the characteristics of plant diseases transmitted by insect vectors [7, 8], we proposed the mathematical models with continuous control strategy and pulse control strategy with single-pulse sequence of multipulse sequences, developed the corresponding threshold theoretical analysis and numerical techniques to address the threshold dynamics of the model, and obtained the corresponding threshold conditions [10].The main results for the model with continuous removing of infected plants reveal that we can either control the death rate of vector population (i.e.,b<c here) such that the plant-only equilibrium is globally asymptotically stable, or control the plant roguing rate such that the basic reproductive number R0 is less than one, and consequently the plant disease-free equilibrium is globally asymptotically stable. All those indicate that the infected plants and infected vectors could be completely eradicated if we implement the removal control strategies properly.According to the fact that infected plants are removed and some vectors are removed at the same time, we develop a pulse control model with fixed moments, which allows us to choose the period of impulsive control strategies such that the plant-only periodic solution and plant disease-free periodic solution are globally asymptotically stable. Based on the above purpose, we employ the relevant theory of impulsive differential equations to address the threshold dynamics of the proposed model, and the threshold values which guarantee the global stability of plant-only and plant disease-free periodic solutions have been given. The threshold conditions show that the infected plants and infected vectors can be completely eradicated if we implement the cultural control strategies relatively frequently.When considering the strategy of removing infected plants multiple times in a replanting cycle, a more realistic model has been proposed, which allows us to analyze and design the optimal control strategy such that the threshold value reaches its minimum. The optimal times of roguing infected plants within a replanting cycle can be numerically calculated, which can help us to design and optimize the prevention and control strategy of plant virus transmission. Finally, various numerical investigations have been given to confirm the main conclusions and reveal the optimal control strategies, and the biological implications of the main results have been briefly discussed in this section.As we all know, insect vectors of plant diseases live directly on the corresponding infected plants. Therefore, when the corresponding infected plants or even healthy plants are removed, a large number of insect vectors will inevitably be removed. The above factors are undoubtedly beneficial to the prevention and control of plant virus disease transmission, but how to formulate and describe the above factors, develop corresponding mathematical models, and carry out systematic analysis is challenging, which is undoubtedly one of our important work in a later stage.



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