Dynamical Behavior of a Novel Impulsive Switching Model for HLB with Seasonal Fluctuations

Gao, Shujing; Luo, Lei; Yan, Shuixian; Meng, Xinzhu

doi:https://doi.org/10.1155/2018/2953623

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Mathematical Modeling and Dynamic Analysis of Complex Biological Systems

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Research Article | Open Access

Volume 2018 | Article ID 2953623 | https://doi.org/10.1155/2018/2953623

Dynamical Behavior of a Novel Impulsive Switching Model for HLB with Seasonal Fluctuations

Shujing Gao,^1,2Lei Luo,¹Shuixian Yan,¹and Xinzhu Meng³

Academic Editor: Alejandro Vargas

Received16 Nov 2017

Accepted31 May 2018

Published26 Jul 2018

Abstract

This paper studies a new model for Huanglongbing with seasonal fluctuations. Switching coefficients and switching control schemes are considered in this model. The main purpose of this paper is to study the effects of switching control schemes on dynamics of the model. Firstly, we theoretically investigate the basic reproductive number and its computation formulae for general impulsive switching model with periodic environment. Secondly, the basic reproductive number and global dynamics of the impulsive switching model for Huanglongbing are analyzed. Finally, numerical results indicate that spring and autumn are the optimum seasons for killing psyllids, and winter is the optimum season for removing infected trees.

1. Introduction

Citrus Huanglongbing (HLB), also known as citrus greening, is one of the most devastating diseases of citrus worldwide [1]. The Asiatic citrus psyllid and Diaphorina citri Kuwayama are the only two known vectors of the debilitating citrus HLB [2]. Nearly 50 countries are affected by this disease especially in Asian, African, and American countries, such as Brazil, USA, and China. It was estimated by the University of Florida in 2012 that, in Florida, HLB had resulted in the loss of 6611 jobs from 2006 throughout 2011, 1.3 billion in revenue to growers, and 3.63 billion in economic activity [3]. In São Paulo, 64.1% of the commercial citrus blocks and 6.9% of the citrus trees were affected by HLB in 2012 [4]. Till now, in China, the damaged area of citrus is more than 80% of the total cultivated area [5]. Unfortunately, there currently is no cure for HLB nor is there any naturally occurring citrus cultivar that is resistant to HLB.

HLB is a vector-transmitted bacterial infection through psyllids [6]. Since the pioneering work of MacDonald and Barbour on schistosomiasis [7, 8], many mathematical models have been proposed in analyzing the spread and control of vector-borne diseases, such as malaria, dengue fever, schistosomiasis, West Nile disease, HLB (see [9–14] and references therein). In [8], Barbour formulated a mathematical model of schistosomiasis as follows: where () is the susceptible (infected) host population, () is the susceptible (infected) vector population, and are infection rates, () is the natural mortality rate of the host (vector) population.

As we know, young flush (initial infection of newly developing cluster of young leaves) become infectious within 15 days after receiving an inoculum of bacteria [15], and symptoms of HLB do not appear on leaves for months to years after initial infection. The survey results from [16, 17] indicated that the incubation period from grafting to development of HLB symptoms is 3 to 12 months under greenhouse conditions. For large trees in a field situation, the incubation period may be much longer, up to more than 5 years. This means that HLB has a long incubation period during which the plant is asymptomatic but infectious [18]. Therefore, in this paper, we classify the citrus tree into four compartments: susceptible , infected and asymptomatic but not yet infectious , infectious and asymptomatic , and infectious and symptomatic , and the psyllids vector into two compartments: susceptible class and infected class . Let and be the total numbers of citrus trees and psyllids, respectively, at time in a grove. That is, and . Assume that removed trees are immediately replaced by susceptible trees, keeping the grove size constant [19]. Thus, is constant and denotes . Inspired by the idea of Barbour’s model (1), considering HLB transmission between citrus trees and psyllids, we establish the following HLB model. where and are the conversion rates, () is the natural mortality rate of the citrus tree (psyllids), is the infection rate from infected psyllids to susceptible trees, is the infection rate from infectious and symptomatic trees to psyllids, means the infection rate from infectious and asymptomatic trees to psyllids, and () is the proportional coefficient, is the mortality rate of citrus trees due to illness, and is the constant recruitment rate of psyllids.

In general, spraying insecticides over entire groves as well as eliminating infected symptomatic trees have always been implemented in controlling the spread of HLB. In Thailand, 3–6 sprays per year was required during flush periods to rehabilitate citrus production in a HLB-infected area [20]. However, the common assumption about the continuity of control activities is contradictory from the reality that the control behavior usually occurs in regular pulses [21]. Spraying insecticides is generally applied at a fixed time, and the effect of pesticide spraying depends on the time of initial spraying and frequency. By considering impulsive control strategies, system (2) can be described by impulsive differential equations as follows: where is the removal rate of infected symptomatic trees and is the killing rate of psyllids by insecticide spraying.

Furthermore, in endemic areas, removing of citrus trees is always based predominantly on the presence of visible symptoms [22]. All of the trees showing HLB symptoms should be removed 3 times in each year [20]. These imply that the infection rates and the removal rate vary with season fluctuations. Thus, it is necessary to consider that some coefficients of model (3) are time-varying and switching in time. Suppose that some parameters are modeled as switching parameters and governed by a switching rule where is the number of the subsystems and is a piecewise continuous switching rule such that for all . The switching times satisfy and . Define the set of all switching rules by . Motivated by above fact, we yield the switching HLB model with impulsive control: where () are the killing rates of psyllids by insecticide spraying at time (). The initial conditions for system (4) satisfy , , , , , and . The model flow diagram is depicted in Figure 1.

The spread of infectious diseases is influenced by many factors, such as the behavior of the human population and the environment in which it spread [23]. Consequently, it is more realistic to consider the periodic switching rule. Following the idea of [23], we assume that the switching rule satisfies with , and then is the period of switch . Assume that , , , , and for , and , , , , and . Define as the set of periodic switching rule.

Note that models (2) and (3) can be considered as special cases of model (4). If the parameters of model (4) are constant and not switching in time, that is, there is only one independent subsystem (), then model (4) yields to model (3). Further, if control strategies are not in use, in the case where the killing rate of psyllids () and the removal rate of infected symptomatic trees () are zero, then model (3) reduces to model (2). Our main purpose is to explore the effects of switching control schemes on the dynamics properties of model (4).

The rest of this paper is organized as follows: In Section 2, some basic notations and useful results are given. In Section 3, the threshold condition and global asymptotic stability of the disease-free periodic solution of system (4) are studied. Furthermore, sufficient condition for persistence of the disease is derived. Numerical simulations are given in Section 4. Brief discussion and conclusion are presented in Section 5.

2. Some Useful Results

2.1. Some Useful Results for Linear Impulsive Switching System

Before investigating system (4), we will present some notations and state some results for linear impulsive switching system with periodic environment.

Define . Let be the spectral radius of matrix .

Consider a linear impulsive switching differential system: where , , .

Particularly, if , system (5) can be rewritten as follows: where , , and with , and then is the period of switch .

Let be the evolution operator of the linear -periodic system

Denote

Lemma 1. If , then there exists a positive -periodic vector function such that is a solution of system (6).
Since the proof is similar to that of Lemma 1 in [24], so one omits it.

Lemma 2. If , then the trivial solution of system (6) is asymptotically stable.
Using the similar method in [25], this result can be easily proved (not shown in this paper).

2.2. for General Impulsive Periodic System with Switching Parameters

Consider a general impulsive switching system with periodic environment: where , , , and .

Following [26], we split the compartments by two types with the first compartments the infected individuals and the uninfected individuals. And denote , , , , and . Define

We can rewrite system (9) as: where are the newly infected rates, are the input rates of individuals by other means, and are the rates of transfer of individuals out of compartments; then, represent the set transfer rates out of compartments. Thus, . We assume that system (11) satisfies , , , and , and system (11) has a disease-free periodic solution .

We make the following assumptions, which share the same biological meanings as those by Wang and Zhao [27] and Yang and Xiao [28]. (H1)If , then the function , , and are nonnegative and continuous on and continuously differential with respect to for .(H2)If , then . Particularly, if , then for .(H3) for .(H4)If , then for .(H5)The pulse on the infected compartments must be uncoupled with the uninfected compartments; that is, is essentially , and .(H6), where , and is the fundamental solution matrix of the following system:where

From (H2)–(H4), the derivatives of and can be parted as follows: where

Furthermore, it follows from (H5) that are the functions of . So the derivatives of can be separated as follows: where and defined by (H7).

In addition, from Assumption (H7) and Lemma 2, we can see that the trivial solution of the following linear switching system with impulses is asymptotically stable. According to Remark 3.5 in Sect. III. 7 of [29], we have that there exist constants and , such that where is the evolution operator of system (18).

Similar to the notation and definition of [24], we define the so-called next infection operator , where is a -periodic function from to and denotes the initial distribution of infections individuals, and when .

Now, we define the basic reproductive number for system (11) as follows:

In order to calculate the implicit expression by numerical simulation, we consider the auxiliary -periodic switching system with impulses: where . Set to be the evolution operator of system (22), then . Following the idea in [28], we have following results.

Lemma 3. Assuming that (H1)–(H7) hold, then the following statements are valid: (i)If has a positive solution , then is an eigenvalue of , and so .(ii)If , then is the unique solution of .(iii) if and only if for all .By applying Lemma 3, one knows that for impulsive periodic switching system (11) is the solution of algebraic equation .

Lemma 4. Assuming that (H1)–(H7) hold, then the following statements are valid for system (11): (i) if and only if .(ii) if and only if .(iii) if and only if .It follows from Lemma 4 that the disease-free periodic solution of system (11) is asymptotically stable if and unstable if .

3. Main Results

In this section, we are going to explore the threshold condition which leads to the extinction and persistence of the disease for impulsive switching model (4) for HLB with seasonal fluctuations.

Lemma 5. All solutions of system (4) with nonnegative initial conditions are nonnegative for all and ultimately bounded.

The proof of Lemma 5 is simple; we omit it.

Referring to [21], we can get that system (4) has a unique disease-free periodic solution , where and are the unique periodic solution of the following systems, respectively: and

We can easily obtain that Assumptions (H1)–(H5) hold for system (4). Next, we will show that Assumptions (H6) and (H7) hold. By (13), (15), and (17), we can calculate , , , , and of system (4), which are represented as the following form:

By calculating, we get and where , , , and . There is no need to calculate the exact forms of , as they are not required in the analysis that follows. Obviously, and . Thus, Assumptions (H6) and (H7) hold.

Theorem 1. If , then the disease-free periodic solution of system (4) is globally asymptotically stable, whereas it is unstable if .

Proof 1. From Lemma 4, one has that the unique disease-free periodic solution is unstable if , and is locally stable if . Therefore, one only needs to show the global attractivity of for .

From Lemma 4, we get since . So we can choose a sufficiently small such that

From system (4), we have that

By comparison theorem in impulsive differential equations, for the abovementioned , we have that there exists a such that

According to system (4) and inequality (30), we can get that for ,

Consider the following comparison system: where .

In view of Lemma 1, there exists a positive -periodic vector function such that is a solution of system (32), where . So , as . It follows (28) that , , , and . By the comparison theorem in impulsive differential equations, we have , , , and . By the theory of asymptotic semiflows, we can get

Hence, the disease-free periodic solution is globally asymptotically stable.

Theorem 2. If , then the disease is uniformly persistent for system (4); that is, there is a positive constant , such that , , , and .

Proof 2. Denote , , and . Let be the unique solution of system (4) with the initial value at time .
Define Poincaré map associated with system (4) as follows: Set One claims that Obviously, . Next, one wants to show If (37) does not hold, then there exists a point . Next, for four initial values and , three cases should be discussed. Case (i).One initial value equals zero, and the others are larger than zero. Without loss of generality, one chooses , , , and . It is obvious that and for any . Then, from the first equation of system (4), one gets . Thus, for . This is a contradiction. Other cases are similarly proved.Case (ii).Two initial values equal zero, and the others are larger than zero. One lets , , and . It is obvious that and for any . Using the same method as aforementioned, one can prove for . This is a contradiction. Other cases can be proved similarly.Case (iii).Three initial values equal zero, and the other is larger than zero. Set and . It is obvious that and for any . Then, from the fourth equation of system (4), one gets . Thus, for . This is a contradiction. Similarly, one can prove the other cases.Thus, In the following, one proceeds by contradiction to prove that there exists such that where .
By Lemma 4, one has if . So one can choose sufficiently small such that If (39) does not hold, then for any , one obtains Without loss of generality, one supposes that By the continuity of the solution with respect to initial values, one has that there exists sufficiently small such that For any , there exists an integer such that , where . Then one has Therefore, one has From system (4) and inequality (45), one gets Consider the comparison system for system (46): where .
By Lemma 1, one knows that there exists a positive -periodic vector function such that is a solution of system (47), where . From (40), one can get that as , and , , , and as . By the comparison theorem in impulsive differential equations, one has , , , and as . This contradicts with the boundedness of the solutions. Thus, one has proved that (39) holds and is weakly uniformly persistent with respect to .
Obviously, the Poincaré map has a global attractor . is an isolated invariant set in and and it is acyclic in . Every solution in converges to . According to Zhao [30], one derives that is uniformly persistent with respect to . This implies that the solution of system (4) is uniformly persistent with respect to . This completes the proof.

4. Numerical Simulations

In this section, we first provide results from numerical simulations of model (4) that demonstrate and support our theoretical results. For these simulations, part of parameters values for model (4) are outlined in Table 1.

In [20], Zhao revealed that 1-2 sprays should be done in the period after picking and before spring sprout, in spring, summer growth, and in autumn growth. So we assume that the system is composed of four subsystems, and the switching law is periodic and satisfies

Consider dynamical behavior of system (4) with initial conditions , , , , , and . The switched parameter values are used as follows: , , , , , , , and . For the control switched parameter values, we set , , , , , , , and . According to Lemma 3, we can get by numerical calculation, which shows that the disease dies out (see Figure 2). Set , , , , , , , and . We get ; the disease is uniformly persistent by Theorem 2, which is showed from Figure 3.

Switching parameters have an effect on the peak size of infected individuals for switching epidemic models. Next, we consider the effect of varying switching removal rates and insecticide spraying rates to evaluate the effectiveness of various control measures, while holding the other switched parameters constant. In Table 2, we give two different control projects to compare with the baseline scenario, which is denoted by Strategy I and Strategy II.

Figures 4 and 5 show the numerical simulations of the baseline scenario, Strategy I, and Strategy II. If we compare the baseline scenario and Strategy I (see rows 1 and 2), the evaluation implies that the baseline scenario is worse than Strategy I (larger final and peak sizes and ). If we compare the baseline scenario and Strategy II (see rows 1 and 3), the evaluation suggests that the baseline scenario is better than Strategy II (lower final and peak sizes and ). This illustrates that Strategy I is the best control project, and the most effective control strategy is spraying in spring and autumn and removing in winter.

By calculating, in the absence of control strategies. We can observe from Figure 6 that the disease breaks out rapidly. This illustrates that removing infected trees and spraying pesticides play an important role in controlling the spread of HLB.

5. Conclusions

By introducing switching parameters into a general impulsive HLB model, a novel impulsive switching model for HLB with seasonal fluctuations has been constructed and a threshold value with switching effect has been established to measure whether the disease is uniformly persistent. The modeling and analytic methods presented in this paper improve the classical results for the systems with impulsive interventions. Numerical examples have been given to demonstrate the effectiveness of the results obtained.

Our numerical investigations demonstrate that the most effective season of spraying insecticide is in spring and autumn and the most effective season of removing infected trees is winter. The result strongly suggests and supports the previous observations [19, 34]. This can serve as an integrating measure to design an appropriate strategy to control HLB spread.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The research has been supported by the National Natural Science Foundation of China (11561004) and the Natural Science Foundation of Jiangxi Province (20171BAB201006).

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