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.

1. Introduction

Plant diseases are caused by plant viruses transmitted by insect vectors [15], 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 [713]. 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 [1416]. However, the cultural control including roguing or removal of diseased plants has been practised and could be more effective [1720]. Therefore, a successful plant disease control program depends on a plant production system which is closely aligned with the goals of pest management [2123].

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, 2427]. 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, 2830], infectious diseases control, and drug administration [31]. Moreover, various plant disease models with impulsive control have been developed and studied [1720, 3234]. 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, 3234].

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 as and vector population as . The population flow among those compartments is shown in the schematic diagram (i.e., Figure 1), which leads to the following model:where , , , and 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.

2.2. Basic Properties

In this section, we show the boundedness of system (1), and it follows from system (1) that we havewhich indicates

Similarly, we haveand consequently we get

Thus, the total plant population and vector population are uniformly bounded. Furthermore, the regionis positively invariant, which indicates that we can focus on the dynamics of system (1) on the set in the following.

3. Dynamics of the Model with Continuous Control

3.1. Boundary Equilibria and Their Stabilities

System (1) has plant-only equilibrium and disease-free equilibrium with . By employing the next generation matrix method [35], we can obtain the basic reproduction number of model (1) as follows:where . represents the number of secondary cases that one infected individual will cause through the duration of the infective period, is the expected number of plants that one vector infects throughout its infectious life time, and is the expected number of vectors that one plant infects throughout its infectious life time.

Theorem 1. The plant-only equilibrium is globally asymptotically stable if and .

Proof. We introduce the following Lyapunov function:and furtherThe arithmetical mean is great than or equal to the geometrical mean, and the function is nonpositive for all , and it is equal to zero if and only if . If and , thenThus, we have . Furthermore, only if , , and , which indicates that the maximum invariant set in is the singleton . By LaSalle’s Invariance Principle [36], we conclude that is globally asymptotically stable in .

Next, we show that the disease-free equilibrium is globally stable provided that . First, the Jacobian matrix at the equilibrium is given by

Stability of equilibrium is determined by the eigenvalues of the Jacobian evaluated at that equilibrium. Characteristic equation at disease-free equilibrium becomes

It is easy to see that two roots are and , and according to Routh–Hurwitz condition, has roots with negative real parts when . Therefore, is stable if . Furthermore, we have the following result for the global stability of .

Theorem 2. For system (1), if , then the disease-free equilibrium is globally asymptotically stable.

Proof. Define a Lyapunov function:and then the derivative of with respect to system (1) is given byTherefore, we have if , and if or . Substituting into the first and third equations of (1), one has and , which implies and as . Again, it follows from LaSalle’s Invariance Principle [36] that is globally asymptotically stable in .

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.

From the first and second equations of (15), we have

Substituting the above into the fourth equation of (15), we havewhere

Equation (17) may have two roots if , denoted bywith , and we have the following main results.

Theorem 3. For system (1), if , then there exists a unique endemic equilibrium ; if , 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 be a closed positively invariant subset of , on which a continuous flow is defined. We denote the restriction to by and note that is in general not positively invariant. Let be the maximal invariant set of on . Suppose that is a closed invariant set and there exists a cover of , where is a nonempty index set. , , and are pairwise disjoint closed invariant sets. Furthermore, we propose the following hypotheses and lemmas. All are isolated invariant sets of the flow . is acyclic; that is, any finite subset of does not form a cycle. Any compact subset of contains, at most, finitely many sets of [37].

Lemma 1 (see [37]). Let be a closed positively invariant subset of on which a continuous flow is defined. Suppose there is a constant such that is point dissipative on and the assumptions hold. Then, the flow is uniformly persistent, if and only if for any , where , and is interior of set .

By this lemma, we can show the uniform persistence of disease when .

Theorem 4. In system (1), assume that and the disease is initially present; then, the disease is uniformly persistent; i.e., there is a constant such that , .

Proof. We set , , and we will show that the conditions of Lemma 1 are satisfied. Clearly, is isolated. Hence, the covering is simply , which is acyclic. Thus, the conditions hold. We can also obtain that is point dissipative by (6). Now, we show that . Suppose this is not true; then, there exists a solution such that , , , . For any sufficiently small constant , there exists a positive constant such that for all . Since , for sufficiently small , we haveNote thatTherefore, if , as , then by a standard comparison argument and the nonnegativity, the solution ofwith initial data , converges to as well. Thus, , where is defined bywhere .
The derivative of is given byTherefore, goes to either infinity or some positive number as , which is a contradiction of . Thus, we have . Then, we obtain for some constant . By the fourth equation of (1), we have such that . Denote ; then, and . 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:where is a fixed positive constant and denotes the period of the impulsive effect, is the constant replanting parameter, and which denotes the positive integer set. The parameter denotes the proportion of the infected plants which is rouged at each pulse perturbation, and 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

If , then model (25) can be reduced to the following subsystem:From the first equation of (26), we can solve it in any impulsive interval and have

Denote ; then,

There exists a steady state , which indicates that the systemhas a positive periodic solution , which is globally asymptotically stable.

Lemma 2. System (29) has a positive periodic solution ; for any solution of (29), we have as , where .

Proof. Let for all be any solution of (29); then,which indicates that as and .

Substituting the above periodic solution into the second equation of (26), one yields

Denote ; then,

Denote ; then, we conclude that (32) always has a zero equilibrium provided that or . When and , (32) has a stable positive equilibrium . Consequently, the systemhas a positive periodic solutionwhich is globally asymptotically stable. Furthermore, we have the following main result.

Lemma 3. If and , then system (33) has a positive periodic solution , and for any solution of (33) we have as .

Proof. Denote and ; then,Thus, we haveNote that if , then we have as , and consequently the result of Lemma 2 follows.

From Lemmas 1 and 2, we obtain the following results.

Lemma 4. System (26) has boundary periodic solution , which is globally stable if , and has a positive periodic solution , which is globally stable if .

Therefore, system (25) has a plant-only periodic solution and disease-free periodic solution , and their stabilities have been addressed in the following.

Theorem 5. The plant-only periodic solution of model (25) is globally asymptotically stable in the first quadrant if .

Proof. Firstly, we prove the local stability of the solution of model (25). Denote ; the corresponding linear system of (25) at reads asThus, the fundamental solution matrix of (37) satisfiesand is the identity matrix, which results inThere is no need to give the exact forms of and as they are not required in the analysis that follows. If all eigenvalues ofare less than one, then the plant-only periodic solution is locally stable. It is easy to see thatand then if and only if . According to Floquet’s theory of impulsive differential equations, the plant-only periodic solution is locally stable.
In the following, we prove the global attractivity of the periodic solution . It follows from (25) thatFrom (42), we getThus, is uniformly bounded, and for small enough, there exists a such that , where . From (43), we getThus, if , then is uniformly bounded, and for small enough, there exists a such that . Let ; then, , as . From (25), , . Consider the following impulsive differential equation:It follows from Lemma 1 and the comparison theorem on impulsive differential equations [38] that we have and as . Hence, for small enough and all large , we haveFor simplification, we may assume that (47) holds for all . From (25), we getAgain, from the comparison theorem on impulsive differential equation, we getLet ; then, as . Therefore, as .
Next, we show as . It follows from that for we havethen, we haveNote that the above system has a globally stable periodic solution with . From the comparison theorem, we get and as . Therefore, there exists a such that , and for we havewith small enough.
Let , so that . Then, (52) becomesHence, as . This completes the proof.

Next, to investigate the stability results of disease-free periodic solution 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 . The corresponding linear system (25) at reads asIn this system, we have ,with

Let be the fundamental solution matrix of (54). Consequently, we have with appropriate initial value . Further computation implies thatthen, we obtain

By [39, 40], the basic reproduction number for system (54) is given as follows:

Then, based on Floquet theory, we have the following conclusion.

Theorem 6. The disease-free periodic solution of model (25) is globally asymptotically stable in the first quadrant if and .

Proof. To prove the global stability of , we only need to show the global attractivity. Suppose ; then, from the boundedness of (25), there exists a such that and with for every solution of (4.1), whereIt follows from (25) thatConsider the following system: