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

Authors: Shujing Gao; Lei Luo; Shuixian Yan; Xinzhu Meng
Journal: Complexity
 (2018)

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## 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.


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## Body

## 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:
(1)dIhtdt=aShIv−μ1Ih,dIvtdt=bSvIh−μ2Iv,dShtdt=μ1−aShIv−μ1Sh,dSvtdt=μ2−bSvIh−μ2Sv,where Sh (Ih) is the susceptible (infected) host population, Sv (Iv) is the susceptible (infected) vector population, a and b are infection rates, μ1 (μ2) 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 Sh, infected and asymptomatic but not yet infectious Eh, infectious and asymptomatic I1, and infectious and symptomatic I2, and the psyllids vector into two compartments: susceptible class Sv and infected class Iv. Let Nht and Nvt be the total numbers of citrus trees and psyllids, respectively, at time t in a grove. That is, Nht=Sht+Eht+I1t+I2t and Nvt=Ivt+Svt. Assume that removed trees are immediately replaced by susceptible trees, keeping the grove size constant [19]. Thus, Nht is constant and denotes Nh. Inspired by the idea of Barbour’s model (1), considering HLB transmission between citrus trees and psyllids, we establish the following HLB model.
(2)dEhtdt=aShIvNh−μ1Eh−αEh−dEh,dI1tdt=αEh−μ1I1−θI1−dI1,dI2tdt=θI1−μ1I2−dI2,dIvtdt=cSvI1Nh+bSvI2Nh−μ2Iv,dShtdt=μ1Nh−μ1Sh−aShIvNh+dEh+dI1+dI2,dSvtdt=Λ−cSvI1Nh−bSvI2Nh−μ2Sv,where α and θ are the conversion rates, μ1 (μ2) is the natural mortality rate of the citrus tree (psyllids), a is the infection rate from infected psyllids to susceptible trees, b is the infection rate from infectious and symptomatic trees to psyllids, c=kb means the infection rate from infectious and asymptomatic trees to psyllids, and k (0<k≤1) is the proportional coefficient, d is the mortality rate of citrus trees due to illness, and Λ 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:
(3)dEhtdt=aShIvNh−μ1Eh−αEh−dEh,dI1tdt=αEh−μ1I1−θI1−dI1,dI2tdt=θI1−μ1I2−dI2−γI2,dIvtdt=cSvI1Nh+bSvI2Nh−μ2Iv,dShtdt=μ1Nh−μ1Sh−aShIvNh+dEh+dI1+dI2+γI2,dSvtdt=Λ−cSvI1Nh−bSvI2Nh−μ2Sv.t∈tk−1,tk,Ivt+=1−pIvt,Svt+=1−pSvt,t=tk.where γ is the removal rate of infected symptomatic trees and p 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 σt:tk−1,tk→1,2,…,m=P,k=1,2,…, where m is the number of the subsystems and σt is a piecewise continuous switching rule such that σt=ik∈P for all t∈tk−1,tk. The switching times tk satisfy tk+1>tk>0 and limk→∞tk=∞. Define the set of all switching rules by I. Motivated by above fact, we yield the switching HLB model with impulsive control:
(4)dEhtdt=aσShIvNh−μ1Eh−αEh−dEh,dI1tdt=αEh−μ1I1−θI1−dI1,dI2tdt=θI1−μ1I2−dI2−γσI2,dIvtdt=cσSvI1Nh+bσSvI2Nh−μ2Iv,dShtdt=μ1Nh−μ1Sh−aσShIvNh+dEh+dI1+dI2+γσI2,dSvtdt=Λ−cσSvI1Nh−bσSvI2Nh−μ2Sv,t∈tk−1,tk,Ivt+=1−pσIvt,Svt+=1−pσSvt,t=tk,where pσ (0≤pσ≤1) are the killing rates of psyllids by insecticide spraying at time tk (k=1,2,…). The initial conditions for system (4) satisfy Eht0+=Eht0≥0, I1t0+=I1t0≥0, I2t0+=I2t0≥0, Ivt0+=Ivt0≥0, Sht0+=Sht0>0, and Svt0+=Svt0>0. The model flow diagram is depicted in Figure 1.Figure 1
A schematic of model (4) showing transitions to different categories for trees and psyllids. Black arrows show the transitions between compartments. Orange dashed arrows show the necessary interactions between trees and psyllids to obtain transmission.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 σ satisfies tk−tk−1=ωk with ωk+m=ωk, and then ω=∑k=1mωk is the period of switch σ. Assume that aik=ak, bik=bk, cik=ck, mik=mk, and pik=pk for t∈tk−1,tk, and ak+m=ak, bk+m=bk, ck+m=ck, mk+m=mk, and pk+m=pk. Define Ip 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 (m=1), then model (4) yields to model (3). Further, if control strategies are not in use, in the case where the killing rate of psyllids (p) and the removal rate of infected symptomatic trees (γ) 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 Section2, 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.Defineℝ+=x∈ℝ∣x≥0,ℝ+n=x1,…,xn∈ℝn∣xi≥0,i=1,2,…,n. Let rB be the spectral radius of matrix B.Consider a linear impulsive switching differential system:(5)ẋt=Aσtxt,t∈tk−1,tk,xtk+=Pσxtk,t=tk,k∈ℕ,xt0+=x0,t0≥0,where x=x1,x2,…,xnT, Aσ,Pσ∈ℝn×n, σ∈I.Particularly, ifσ∈Ip, system (5) can be rewritten as follows:
(6)ẋt=Aktxt,t∈tk−1,tk,xtk+=Pkxtk,t=tk,k∈ℕ,xt0+=x0,t0≥0,where Ak+mt=Akt, Pk+m=Pk, and tk−tk−1=ωk with ωk+m=ωk, and then ω=∑k=1mωk is the period of switch σ.LetΨAkt,st≥s be the evolution operator of the linear ω-periodic system
(7)ẋt=Aktxt,x∈ℝn.Denote(8)ΦAkPkω≔∏k=1mPm−k+1ΨAktm−k+1,tm−k.Lemma 1.
Ifη=1/ωlnrΦAkPkω, then there exists a positive ω-periodic vector function vt such that expηtvt is a solution of system (6).
Since the proof is similar to that of Lemma 1 in [24], so one omits it.Lemma 2.
IfrΦAkPkω<1, 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.R0 for General Impulsive Periodic System with Switching Parameters

Consider a general impulsive switching system with periodic environment:(9)ẋt=fkx,t∈tk−1,tk,xt+=ψkxt,t=tk,k∈ℕ,xt0+=x0,t0≥0,where fk:ℝ+n→ℝn, ψk:ℝ+n→ℝ+n, fk+m=fk, and ψk+m=ψk.Following [26], we split the compartments by two types with the first q compartments x1,x2,…,xq the infected individuals and xq+1,xq+2,…,xn the uninfected individuals. And denote X=x1,x2,…,xq, Y=xq+1,xq+2,…,xn, ψk=hk,gkT, hk=ψ1k,ψ2k,…,ψqk, and gk=ψq+1k,…,ψnk. Define
(10)Xs=x∈ℝ+n∣xi=0,i=1,…,q.We can rewrite system (9) as:
(11)ẋt=Fkxt−Vkxt,t∈tk−1,tk,Xt+=hkxt,Yt+=gkxt,t=tk,k∈ℕ,xt+=x0,t0≥0,where Fkx are the newly infected rates, Vk+x are the input rates of individuals by other means, and Vk−x are the rates of transfer of individuals out of compartments; then, Vkx=Vk−x−Vk+x represent the set transfer rates out of compartments. Thus, fkx=Fkx−Vkx. We assume that system (11) satisfies Fk+m=Fk, Vk+m=Vk, hk+m=hk, and gk+m=gk, and system (11) has a disease-free periodic solution x∗t.We make the following assumptions, which share the same biological meanings as those by Wang and Zhao [27] and Yang and Xiao [28].
(H1)
Ifxi≥0, then the function Fikx, Vik+x, and Vik−x are nonnegative and continuous on ℝ+n and continuously differential with respect to x for i=1,…,n.
(H2)
Ifxi=0, then Vik−x=0. Particularly, if x∈Xs, then Vik−x=0 for i=1,…,q.
(H3)
Fikx=0 for q+1,…,n.
(H4)
Ifx∈Xs, then Fikx=Vik+x=0 for i=1,…,q.
(H5)
The pulse on the infected compartments must be uncoupled with the uninfected compartments; that is,hkxtk is essentially hkXtk, and hk0=0.
(H6)
rΦMkQkω<1, where ΦMkQkω=∏k=1mQm−k+1ΨMktm−k+1,tm−k, and ΦMkQkt is the fundamental solution matrix of the following system:
(12)Żt=MktZt,t∈tk−1,tk,Ztk+=QkZtk,t=tk,k∈ℕ,where
(13)Mkt=∂fikx∗t∂xjq+1≤i,j≤n,Qk=∂ψikx∗tk∂xjq+1≤i,j≤n.From (H2)–(H4), the derivatives ofFkx∗t and Vkx∗t can be parted as follows:
(14)DFkx∗t=Fkt000,DVkx∗t=Vkt0Jk−Mkt,where
(15)Fkt=∂Fikx∗t∂xj1≤i,j≤q,Vkt=∂Vikx∗t∂xj1≤i,j≤q.Furthermore, it follows from (H5) thathk are the functions of Xtk. So the derivatives of ψkx∗tk can be separated as follows:
(16)Dψkx∗tk=Pk0ΓkQk,where Pk∈ℝq×q and Γk∈ℝn−q×q defined by
(17)Pk=∂ψikx∗tk∂xj1≤i,j≤q,Γk=∂ψikx∗tk∂xjq+1≤i≤n,1≤j≤q.
(H7)
rΦ−VktPkω<1.In addition, from Assumption (H7) and Lemma2, we can see that the trivial solution of the following linear switching system with impulses
(18)ẏt=−Vkyt,t∈tk−1,tk,yt+=Pkyt,t=tk,k∈ℕ,is asymptotically stable. According to Remark 3.5 in Sect. III. 7 of [29], we have that there exist constants K>0 and ρ>0, such that
(19)∥Yt,s∥≤Kexp−ρt−s,∀t≥s,s∈ℝ,where Yt,s is the evolution operator of system (18).Similar to the notation and definition of [24], we define the so-called next infection operator L,
(20)Lϕt=∫−∞tYt,sFsϕsds=∫0∞Yt,t−aFt−aϕt−ada,∀t∈ℝ+,where ϕs is a ω-periodic function from ℝ to ℝ+q and denotes the initial distribution of infections individuals, and Ft=Fkt when t∈tk−1,tk.Now, we define the basic reproductive numberR0 for system (11) as follows:
(21)R0=rL.In order to calculate the implicit expressionR0 by numerical simulation, we consider the auxiliary ω-periodic switching system with impulses:
(22)U̇t=−Vkt+FktλUt,t∈tk−1,tk,Ut+=PkUt,t=tk,k∈ℕ,where λ∈0,∞. Set Ut,s,λt≥s,s∈ℝ to be the evolution operator of system (22), then Uω,0,λ=ΦFk/λ−VkPkω. Following the idea in [28], we have following results.Lemma 3.
Assuming that (H1)–(H7) hold, then the following statements are valid:(i)
IfrΦFk/λ−VkPkω=1 has a positive solution λ0, then λ0 is an eigenvalue of L, and so R0>0.
(ii)
IfR0>0, then λ=R0 is the unique solution of rΦFk/λ−VkPkω=1.
(iii)
R0=0 if and only if rΦFk/λ−VkPkω<1 for all λ>0.
By applying Lemma3, one knows that R0 for impulsive periodic switching system (11) is the solution of algebraic equation rΦFk/λ−VkPkω=1.Lemma 4.
Assuming that (H1)–(H7) hold, then the following statements are valid for system (11):
(i)
R0=1 if and only if rΦFk−VkPkω=1.
(ii)
R0>1 if and only if rΦFk−VkPkω>1.
(iii)
R0<1 if and only if rΦFk−VkPkω<1.
It follows from Lemma4 that the disease-free periodic solution x∗t of system (11) is asymptotically stable if R0<1 and unstable if R0>1.


## 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.Defineℝ+=x∈ℝ∣x≥0,ℝ+n=x1,…,xn∈ℝn∣xi≥0,i=1,2,…,n. Let rB be the spectral radius of matrix B.Consider a linear impulsive switching differential system:(5)ẋt=Aσtxt,t∈tk−1,tk,xtk+=Pσxtk,t=tk,k∈ℕ,xt0+=x0,t0≥0,where x=x1,x2,…,xnT, Aσ,Pσ∈ℝn×n, σ∈I.Particularly, ifσ∈Ip, system (5) can be rewritten as follows:
(6)ẋt=Aktxt,t∈tk−1,tk,xtk+=Pkxtk,t=tk,k∈ℕ,xt0+=x0,t0≥0,where Ak+mt=Akt, Pk+m=Pk, and tk−tk−1=ωk with ωk+m=ωk, and then ω=∑k=1mωk is the period of switch σ.LetΨAkt,st≥s be the evolution operator of the linear ω-periodic system
(7)ẋt=Aktxt,x∈ℝn.Denote(8)ΦAkPkω≔∏k=1mPm−k+1ΨAktm−k+1,tm−k.Lemma 1.
Ifη=1/ωlnrΦAkPkω, then there exists a positive ω-periodic vector function vt such that expηtvt is a solution of system (6).
Since the proof is similar to that of Lemma 1 in [24], so one omits it.Lemma 2.
IfrΦAkPkω<1, 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.R0 for General Impulsive Periodic System with Switching Parameters

Consider a general impulsive switching system with periodic environment:(9)ẋt=fkx,t∈tk−1,tk,xt+=ψkxt,t=tk,k∈ℕ,xt0+=x0,t0≥0,where fk:ℝ+n→ℝn, ψk:ℝ+n→ℝ+n, fk+m=fk, and ψk+m=ψk.Following [26], we split the compartments by two types with the first q compartments x1,x2,…,xq the infected individuals and xq+1,xq+2,…,xn the uninfected individuals. And denote X=x1,x2,…,xq, Y=xq+1,xq+2,…,xn, ψk=hk,gkT, hk=ψ1k,ψ2k,…,ψqk, and gk=ψq+1k,…,ψnk. Define
(10)Xs=x∈ℝ+n∣xi=0,i=1,…,q.We can rewrite system (9) as:
(11)ẋt=Fkxt−Vkxt,t∈tk−1,tk,Xt+=hkxt,Yt+=gkxt,t=tk,k∈ℕ,xt+=x0,t0≥0,where Fkx are the newly infected rates, Vk+x are the input rates of individuals by other means, and Vk−x are the rates of transfer of individuals out of compartments; then, Vkx=Vk−x−Vk+x represent the set transfer rates out of compartments. Thus, fkx=Fkx−Vkx. We assume that system (11) satisfies Fk+m=Fk, Vk+m=Vk, hk+m=hk, and gk+m=gk, and system (11) has a disease-free periodic solution x∗t.We make the following assumptions, which share the same biological meanings as those by Wang and Zhao [27] and Yang and Xiao [28].
(H1)
Ifxi≥0, then the function Fikx, Vik+x, and Vik−x are nonnegative and continuous on ℝ+n and continuously differential with respect to x for i=1,…,n.
(H2)
Ifxi=0, then Vik−x=0. Particularly, if x∈Xs, then Vik−x=0 for i=1,…,q.
(H3)
Fikx=0 for q+1,…,n.
(H4)
Ifx∈Xs, then Fikx=Vik+x=0 for i=1,…,q.
(H5)
The pulse on the infected compartments must be uncoupled with the uninfected compartments; that is,hkxtk is essentially hkXtk, and hk0=0.
(H6)
rΦMkQkω<1, where ΦMkQkω=∏k=1mQm−k+1ΨMktm−k+1,tm−k, and ΦMkQkt is the fundamental solution matrix of the following system:
(12)Żt=MktZt,t∈tk−1,tk,Ztk+=QkZtk,t=tk,k∈ℕ,where
(13)Mkt=∂fikx∗t∂xjq+1≤i,j≤n,Qk=∂ψikx∗tk∂xjq+1≤i,j≤n.From (H2)–(H4), the derivatives ofFkx∗t and Vkx∗t can be parted as follows:
(14)DFkx∗t=Fkt000,DVkx∗t=Vkt0Jk−Mkt,where
(15)Fkt=∂Fikx∗t∂xj1≤i,j≤q,Vkt=∂Vikx∗t∂xj1≤i,j≤q.Furthermore, it follows from (H5) thathk are the functions of Xtk. So the derivatives of ψkx∗tk can be separated as follows:
(16)Dψkx∗tk=Pk0ΓkQk,where Pk∈ℝq×q and Γk∈ℝn−q×q defined by
(17)Pk=∂ψikx∗tk∂xj1≤i,j≤q,Γk=∂ψikx∗tk∂xjq+1≤i≤n,1≤j≤q.
(H7)
rΦ−VktPkω<1.In addition, from Assumption (H7) and Lemma2, we can see that the trivial solution of the following linear switching system with impulses
(18)ẏt=−Vkyt,t∈tk−1,tk,yt+=Pkyt,t=tk,k∈ℕ,is asymptotically stable. According to Remark 3.5 in Sect. III. 7 of [29], we have that there exist constants K>0 and ρ>0, such that
(19)∥Yt,s∥≤Kexp−ρt−s,∀t≥s,s∈ℝ,where Yt,s is the evolution operator of system (18).Similar to the notation and definition of [24], we define the so-called next infection operator L,
(20)Lϕt=∫−∞tYt,sFsϕsds=∫0∞Yt,t−aFt−aϕt−ada,∀t∈ℝ+,where ϕs is a ω-periodic function from ℝ to ℝ+q and denotes the initial distribution of infections individuals, and Ft=Fkt when t∈tk−1,tk.Now, we define the basic reproductive numberR0 for system (11) as follows:
(21)R0=rL.In order to calculate the implicit expressionR0 by numerical simulation, we consider the auxiliary ω-periodic switching system with impulses:
(22)U̇t=−Vkt+FktλUt,t∈tk−1,tk,Ut+=PkUt,t=tk,k∈ℕ,where λ∈0,∞. Set Ut,s,λt≥s,s∈ℝ to be the evolution operator of system (22), then Uω,0,λ=ΦFk/λ−VkPkω. Following the idea in [28], we have following results.Lemma 3.
Assuming that (H1)–(H7) hold, then the following statements are valid:(i)
IfrΦFk/λ−VkPkω=1 has a positive solution λ0, then λ0 is an eigenvalue of L, and so R0>0.
(ii)
IfR0>0, then λ=R0 is the unique solution of rΦFk/λ−VkPkω=1.
(iii)
R0=0 if and only if rΦFk/λ−VkPkω<1 for all λ>0.
By applying Lemma3, one knows that R0 for impulsive periodic switching system (11) is the solution of algebraic equation rΦFk/λ−VkPkω=1.Lemma 4.
Assuming that (H1)–(H7) hold, then the following statements are valid for system (11):
(i)
R0=1 if and only if rΦFk−VkPkω=1.
(ii)
R0>1 if and only if rΦFk−VkPkω>1.
(iii)
R0<1 if and only if rΦFk−VkPkω<1.
It follows from Lemma4 that the disease-free periodic solution x∗t of system (11) is asymptotically stable if R0<1 and unstable if R0>1.


## 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 t>t0 and ultimately bounded.The proof of Lemma5 is simple; we omit it.Referring to [21], we can get that system (4) has a unique disease-free periodic solution x∗t=0,0,0,0,Sh∗t,Sv∗t, where Sh∗t and Sv∗t are the unique periodic solution of the following systems, respectively:
(23)dShtdt=μ1Nh−μ1Sht,and
(24)dSvtdt=Λ−μ2Svt,t≠tk,Svt+=1−pkSvt,t=tk.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 Mk, Qk, Fk, Vk, and Pk of system (4), which are represented as the following form:
(25)Mkt=−μ100−μ2,Qk=1001−pk,Fkt=000ak000000000ckSv∗tNhbkSv∗tNh0,Vkt=μ1+α+d000−αμ1+θ+d000−θμ1+d+γk0000μ2,Pk=1000010000100001−pk.By calculating, we get(26)ΦMkQkω=exp−μ1ω00∏k=1m1−pkexp−μ2ω,and
(27)Φ−VkPkω=J1000∗J200∗∗J30000J4,where J1=exp−μ1+d+αω, J2=exp−μ1+d+θω, J3=exp∑k=1m−μ1+d+γkωk, and J4=∏k=1m1−pkexp−μ2ω. There is no need to calculate the exact forms of ∗, as they are not required in the analysis that follows. Obviously, rΦMkQkω<1 and rΦ−VkPkω<1. Thus, Assumptions (H6) and (H7) hold.Theorem 1.
IfR0<1, then the disease-free periodic solution x∗t of system (4) is globally asymptotically stable, whereas it is unstable if R0>1.Proof 1.
From Lemma4, one has that the unique disease-free periodic solution x∗t is unstable if R0>1, and x∗t is locally stable if R0<1. Therefore, one only needs to show the global attractivity of x∗t for R0<1.From Lemma4, we get rΦFk−VkPkω<1 since R0<1. So we can choose a sufficiently small ε1>0 such that
(28)rΦFk−Vk+Mε1kPkω<1,whereMε1k=0000000000000ckε1Nhbkε1Nh0.From system (4), we have that
(29)dSvtdt≤Λ−μ2Svt,t≠tk,Svt+=1−pkSvt,t=tk.By comparison theorem in impulsive differential equations, for the abovementionedε1, we have that there exists a T¯1>0 such that
(30)Svt≤Sv∗t+ε1,fort>T¯1.According to system (4) and inequality (30), we can get that for t>T¯1,
(31)dEhtdt≤akIv−μ1Eh−αEh−dEh,dI1tdt=αEh−μ1I1−θI1−dI1,dI2tdt=θI1−μ1I2−dI2−γkI2,dIvtdt≤ckSv∗t+ε1I1Nh+bkSv∗t+ε1I2Nh−μ2Iv,t∈tk−1,tk,Eht+=Eht,I1t+=I1t,I2t+=I2t,Ivt+=1−pkIvt,t=tk.Consider the following comparison system:(32)dJ¯tdt=Fkt−Vkt+Mε1kJ¯t,t≠tk,J¯t+=PkJ¯t,t=tk,where J¯t=E¯ht,I¯1t,I¯2t,I¯vtT.In view of Lemma1, there exists a positive ω-periodic vector function υ1t such that J¯t=υ1texpςt is a solution of system (32), where ς=lnrΦFk−Vk+Mε1kPkω<0. So J¯t→0, as t→∞. It follows (28) that limt→∞E¯ht=0, limt→∞I¯1t=0, limt→∞I¯2t=0, and limt→∞I¯vt=0. By the comparison theorem in impulsive differential equations, we have limt→∞Eht=0, limt→∞I1t=0, limt→∞I2t=0, and limt→∞Ivt=0. By the theory of asymptotic semiflows, we can get
(33)limt→∞Sht=Sh∗t,limt→∞Svt=Sv∗t.Hence, the disease-free periodic solutionx∗t is globally asymptotically stable.Theorem 2.
IfR0>1, then the disease is uniformly persistent for system (4); that is, there is a positive constant ϵ>0, such that liminft→∞Eht>ϵ, liminft→∞I1t>ϵ, liminft→∞I2t>ϵ, and liminft→∞Ivt>ϵ.Proof 2.
DenoteK~=Eh,I1,I2,Iv,Sh,Sv∈ℝ+6, K0=Eh,I1,I2,Iv,Sh,Sv∈K~:Eh>0,I1>0,I2>0,Iv>0,Sh≥0,Sv≥0, and ∂K0=K~\K0. Let ut,t0,x0 be the unique solution of system (4) with the initial value x0=Eh0,I10,I20,Iv0,Sh0,Sv0 at time t0.
Define Poincaré mapP:K~→K~ associated with system (4) as follows:
(34)Px0=ut0+ω+,x0,∀x0∈K~,∀t0∈ℝ+.
Set(35)M∂=Eh0,I10,I20,Iv0,Sh0,Sv0∈∂K0∣PmEh0,I10,I20,Iv0,Sh0,Sv0∈∂K0,∀m∈ℤ+.
One claims that(36)M∂=0,0,0,0,Sh,Sv∣Sh≥0,Sv≥0.
Obviously,0,0,0,0,Sh,Sv∣Sh≥0,Sv≥0⊆M∂. Next, one wants to show
(37)M∂\0,0,0,0,Sh,Sv∣Sh≥0,Sv≥0=∅.
If (37) does not hold, then there exists a point Eh0,I10,I20,Iv0,Sh0,Sv0∈M∂\0,0,0,0,Sh,Sv∣Sh≥0,Sv≥0. Next, for four initial values Eh0,I10,I20, and Iv0, three cases should be discussed.
Case (i).
One initial value equals zero, and the others are larger than zero. Without loss of generality, one choosesEh0=0, I10>0, I20>0, and Iv0>0. It is obvious that Sht>0 and Ivt>0 for any t≥t0. Then, from the first equation of system (4), one gets dEht/dt∣t=t0=a1Sht0Ivt0/Nh>0. Thus, Eh,I1,I2,Iv,Sh,Sv∉∂K0 for 0<t−t0≪1. This is a contradiction. Other cases are similarly proved.
Case (ii).
Two initial values equal zero, and the others are larger than zero. One letsEh0=I10=0, I20>0, and Iv0>0. It is obvious that Sht>0 and Ivt>0 for any t≥t0. Using the same method as aforementioned, one can prove Eh,I1,I2,Iv,Sh,Sv∉∂K0 for 0<t−t0≪1. This is a contradiction. Other cases can be proved similarly.
Case (iii).
Three initial values equal zero, and the other is larger than zero. SetEh0=I20=Iv0=0 and I10>0. It is obvious that Svt>0 and I1t>0 for any t≥t0. Then, from the fourth equation of system (4), one gets dIvt/dt∣t=t0=b1Svt0I1t0/Nh>0. Thus, Eh,I1,I2,Iv,Sh,Sv∉∂K0 for 0<t−t0≪1. This is a contradiction. Similarly, one can prove the other cases.
Thus,(38)M∂=0,0,0,0,Sh,Sv∣Sh≥0,Sv≥0.
In the following, one proceeds by contradiction to prove that there existsξ^>0 such that
(39)limsupt→∞dPmEh0,I10,I20,Iv0,Sh0,Sv0,P0≥ξ^,∀x0∈K~,m∈ℤ+.where P0=0,0,0,0,Sh∗t0,Sv∗t0.
By Lemma4, one has rΦFk−VkPkω>1 if R0>1. So one can choose ε2>0 sufficiently small such that
(40)rΦFk−Vk−Mε2kPkω>1,whereMε2k=000akε2Nh000000000ckε2Nhbkε2Nh0.
If (39) does not hold, then for any ξ^>0, one obtains
(41)limsupt→∞dPmEh0,I10,I20,Iv0,Sh0,Sv0,P0<ξ^,forsomeEh0,I10,I20,Iv0,Sh0,Sv0∈K~.
Without loss of generality, one supposes that(42)dPmEh0,I10,I20,Iv0,Sh0,Sv0,P0<ξ^,∀ξ^>0,m∈ℤ+.
By the continuity of the solution with respect to initial values, one has that there exists sufficiently smallξ^ such that
(43)∥ut,PmEh0,I10,I20,Iv0,Sh0,Sv0−ut,P0∥≤ε2,∀t∈t0,t0+ω,∀m∈ℤ+.
For anyt≥t0, there exists an integer l∈ℤ+ such that t=lω+t^, where t^∈t0,t0+ω. Then one has
(44)ut,PmEh0,I10,I20,Iv0,Sh0,Sv0−ut,P0=ut^,PmEh0,I10,I20,Iv0,Sh0,Sv0−ut^,P0≤ε2.
Therefore, one has(45)Sht≥Sh∗t−ε2,Svt≥Sv∗t−ε2,forallt≥t0.
From system (4) and inequality (45), one gets
(46)dEhtdt≥akSh∗t−ε2IvNh−μ1Eh−αEh−dEh,dI1tdt=αEh−μ1I1−θI1−dI1,dI2tdt=θI1−μ1I2−dI2−γkI2,dIvtdt≥ckSv∗t−ε2I1Nh+bkSv∗t−ε2I2Nh−μ2Iv,t∈tk−1,tk,Eht+=Eht,I1t+=I1t,I2t+=I2t,Ivt+=1−pkIvt,t=tk,k∈ℕ.
Consider the comparison system for system (46):
(47)dZ˘tdt=Fkt−Vkt−Mε2kZ˘t,t≠tkZ˘t+=PkZ˘t,t=tkwhere Z˘t=E˘ht,I˘1t,I˘2t,I˘vtT.
By Lemma1, one knows that there exists a positive ω-periodic vector function υ2t such that Z˘t=υ2texpζt is a solution of system (47), where ζ=lnrΦFk−Vk−Mε2kPkω. From (40), one can get that Z˘t→∞ as t→∞, and E˘ht→∞, I˘1t→∞, I˘2t→∞, and I˘vt→∞ as t→∞. By the comparison theorem in impulsive differential equations, one has Eht→∞, I1t→∞, I2t→∞, and Ivt→∞ as t→∞. This contradicts with the boundedness of the solutions. Thus, one has proved that (39) holds and P is weakly uniformly persistent with respect to K0,∂K0.
Obviously, the Poincaré mapP has a global attractor P0. P0 is an isolated invariant set in K~ and WsP0∩K0=∅ and it is acyclic in M∂. Every solution in M∂ converges to P0. According to Zhao [30], one derives that P is uniformly persistent with respect to K0,∂K0. This implies that the solution of system (4) is uniformly persistent with respect to K0,∂K0. 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.Table 1
Parameter values for system (4).
Parameter
Value
Unit
Reference
Λ
6,028,433
year−1
[31]
Nh
2000
—
[31]
μ1
0.04
year−1
[32]
μ2
3.2
year−1
[33]
d
0.025
year−1
[21]
α
24.33
year−1
[15]
θ
1.8
year−1
[16, 17]
ω
1
year
—
k
0.6
—
EstimationIn [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
(48)σt=1,ift∈k,k+0.25,winter,2,ift∈k+0.25,k+0.5,spring,3,ift∈k+0.5,k+0.75,summer,4,ift∈k+0.75,k+1,autumn,k=0,1,2,….Consider dynamical behavior of system (4) with initial conditions Eh0=100, I10=40, I20=60, Iv0=330, Sh0=1800, and Sv0=10028. The switched parameter values are used as follows: a1=0.002, a2=0.078, a3=0.05, a4=0.078, b1=0.006, b2=0.156, b3=0.106, and b4=0.156. For the control switched parameter values, we set γ1=0.2, γ2=0.8, γ3=0.2, γ4=0.8, p1=0, p2=0.8, p3=0.8, and p4=0.8. According to Lemma 3, we can get R0=0.903702<1 by numerical calculation, which shows that the disease dies out (see Figure 2). Set γ1=0.2, γ2=0.4, γ3=0.2, γ4=0.35, p1=0, p2=0.2, p3=0.15, and p4=0.5. We get R0=1.801682>1; the disease is uniformly persistent by Theorem 2, which is showed from Figure 3.Figure 2
Time series plots of the total of infected trees and infected psyllids. The disease dies out withR0=0.903702<1.Figure 3
Time series plots of the total of infected trees and infected psyllids. The disease is persistent withR0=1.801682>1.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 ratesγσ and insecticide spraying rates pσ 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.Table 2
The effect of parameterspσ and γσ on the disease control.
p1
p2
p3
p4
γ1
γ2
γ3
γ4
R0
Baseline scenario
0.3
0.8
0.3
0.8
0.2
0.8
0.2
0.8
0.878143
Strategy I
0.3
0.8
0.3
0.8
0.8
0.2
0.8
0.2
0.868525
Strategy II
0.8
0.3
0.8
0.3
0.2
0.8
0.2
0.8
0.994301Figures4 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 R0). 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 R0). 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.Figure 4
Time series plot of the total of infected trees for the baseline scenario and Strategy I.Figure 5
Time series plot of the total infected trees for the baseline scenario and Strategy II.By calculating,R0=5.023193 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.Figure 6
Time series plot of the total infected trees for the baseline scenario and without control strategies.


## 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 valueR0 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.



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