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

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.


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][10][11][12][13][14] and references therein).In [8], Barbour formulated a mathematical model of schistosomiasis as follows: where S h (I h ) is the susceptible (infected) host population, S v (I v ) is the susceptible (infected) vector population, a and b are infection rates, μ 1 (μ 2 ) 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 S h , infected and asymptomatic but not yet infectious E h , infectious and asymptomatic I 1 , and infectious and symptomatic I 2 , and the psyllids vector into two compartments: susceptible class S v and infected class I v .Let N h t and N v t be the total numbers of citrus trees and psyllids, respectively, at time t in a grove.That is, N h t = S h t + E h t + I 1 t + I 2 t and N v t = I v t + S v t .Assume that removed trees are immediately replaced by susceptible trees, keeping the grove size constant [19].Thus, N h t is constant and denotes N h .Inspired by the idea of Barbour's model (1), considering HLB transmission between citrus trees and psyllids, we establish the following HLB model.
where α and θ are the conversion rates, μ 1 (μ 2 ) 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: 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 σ t : where m is the number of the subsystems and σ t is a piecewise continuous switching rule such that σ t = i k ∈ P for all t ∈ t k−1 , t k .The switching times t k satisfy t k+1 > t k > 0 and lim k→∞ t k = ∞.Define the set of all switching rules by I .Motivated by above fact, we yield the switching HLB model with impulsive control: where p σ (0 ≤ p σ ≤ 1) are the killing rates of psyllids by insecticide spraying at time t k (k = 1, 2, … ).The initial conditions for system (4) satisfy 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 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 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.

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 Consider a linear impulsive switching differential system: Particularly, if σ ∈ I p , system (5) can be rewritten as follows: where A k+m t = A k t , P k+m = P k , and t k − t k−1 = ω k with ω k+m = ω k , and then ω = ∑ m k=1 ω k is the period of switch σ.Let Ψ A k t, s t ≥ s be the evolution operator of the linear ω-periodic system then there exists a positive ω-periodic vector function v t such that exp ηt v t 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 r Φ A k P k ω < 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).

R 0 for General Impulsive Periodic System with Switching
Parameters.Consider a general impulsive switching system with periodic environment: 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.
3 Complexity where Following [26], we split the compartments by two types with the first q compartments x 1 , x 2 , … , x q the infected individuals and x q+1 , x q+2 , … , x n the uninfected individuals.And denote We can rewrite system (9) as: where F k x are the newly infected rates, V k+ x are the input rates of individuals by other means, and V k− x are the rates of transfer of individuals out of compartments; then, V k x = V k− x − V k+ x represent the set transfer rates out of compartments.Thus, f k x = F k x − V k x .We assume that system (11) satisfies and g k+m = g k , 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) If x i ≥ 0, then the function F k i x , V k+ i x , and V k− i x are nonnegative and continuous on ℝ n + and continuously differential with respect to The pulse on the infected compartments must be uncoupled with the uninfected compartments; that is, h k x t k is essentially h k X t k , and h k 0 = 0.
is the fundamental solution matrix of the following system: where From (H2)-(H4), the derivatives of F k x * t and V k x * t can be parted as follows: where Furthermore, it follows from (H5) that h k are the functions of X t k .So the derivatives of ψ k x * t k can be separated as follows: where P k ∈ ℝ q×q and Γ k ∈ ℝ n−q ×q defined by In addition, from Assumption (H7) and Lemma 2, we can see that the trivial solution of the following linear switching system with impulses Complexity 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 where Y t, 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, where ϕ s is a ω-periodic function from ℝ to ℝ q + and denotes the initial distribution of infections individuals, and Now, we define the basic reproductive number R 0 for system (11) as follows: In order to calculate the implicit expression R 0 by numerical simulation, we consider the auxiliary ω-periodic switching system with impulses:
Lemma 3. Assuming that (H1)-(H7) hold, then the following statements are valid: has a positive solution λ 0 , then λ 0 is an eigenvalue of L, and so R 0 > 0.
By applying Lemma 3, one knows that R 0 for impulsive periodic switching system (11) is the solution of algebraic equation r Φ Lemma 4. Assuming that (H1)-(H7) hold, then the following statements are valid for system (11): It follows from Lemma 4 that the disease-free periodic solution x * t of system ( 11) is asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

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 > t 0 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 x * t = 0, 0, 0, 0, S * h t , S * v t , where S * h t and S * v t 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 M k , Q k , F k , V k , and P k of system (4), which are represented as the following form: By calculating, we get where There is no need to calculate the exact forms of * , as they are not required in the analysis that follows.Obviously, r Φ M k Q k ω < 1 and r Φ −V k P k ω < 1.Thus, Assumptions (H6) and (H7) hold.
Theorem 1.If R 0 < 1, then the disease-free periodic solution x * t of system (4) is globally asymptotically stable, whereas it is unstable if R 0 > 1.
Proof 1. From Lemma 4, one has that the unique disease-free periodic solution x * t is unstable if R 0 > 1, and x * t is locally stable if R 0 < 1.Therefore, one only needs to show the global attractivity of x * t for R 0 < 1.

From Lemma 4, we get r Φ
From system (4), we have that By comparison theorem in impulsive differential equations, for the abovementioned ε 1 , we have that there exists a T 1 > 0 such that According to system (4) and inequality (30), we can get that for t > T 1 , Consider the following comparison system: where J t = E h t , I 1 t , I 2 t , I v t T .In view of Lemma 1, there exists a positive ω-periodic vector function υ 1 t such that J t = υ 1 t exp ςt is a solution of system (32), where (28) that lim t→∞ E h t = 0, lim t→∞ I 1 t = 0, lim t→∞ I 2 t = 0, and lim t→∞ I v t = 0.By the comparison theorem in impulsive differential equations, we have lim t→∞ E h t = 0, lim t→∞ I 1 t = 0, lim t→∞ I 2 t = 0, and lim t→∞ I v t = 0.By the theory of asymptotic semiflows, we can get Hence, the disease-free periodic solution x * t is globally asymptotically stable.Theorem 2. If R 0 > 1, then the disease is uniformly persistent for system (4); that is, there is a positive constant ϵ > 0, such that lim inf t→∞ E h t > ϵ, lim inf t→∞ I 1 t > ϵ, lim inf t→∞ I 2 t > ϵ, and lim inf t→∞ I v t > ϵ.
6 Complexity ≥ 0 , and ∂K 0 = K \ K 0 .Let u t, t 0 , x 0 be the unique solution of system (4) with the initial value Define Poincaré map P K → K associated with system (4) as follows: Set One claims that If (37) does not hold, then there exists a point Next, for four initial values E 0 h , I 0 1 , I 0 2 , and I 0 v , 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 E 0 h = 0, I 0 1 > 0, I 0 2 > 0, and I 0 v > 0. It is obvious that S h t > 0 and I v t > 0 for any t ≥ t 0 .Then, from the first equation of system (4), one gets 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 E 0 h = I 0 1 = 0, I 0 2 > 0, and I 0 v > 0. It is obvious that S h t > 0 and I v t > 0 for any t ≥ t 0 .Using the same method as aforementioned, one can prove E h , I 1 , I 2 , I v , S h , S v ∉ ∂K 0 for 0 < t − t 0 ≪ 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.Set E 0 h = I 0 2 = I 0 v = 0 and I 0 1 > 0. It is obvious that S v t > 0 and I 1 t > 0 for any t ≥ t 0 .Then, from the fourth equation of system (4), one gets This is a contradiction.Similarly, one can prove the other cases.Thus, In the following, one proceeds by contradiction to prove that there exists ξ > 0 such that

39
where P 0 = 0, 0, 0, 0, S * h t 0 , S * v t 0 .By Lemma 4, one has r where If (39) does not hold, then for any ξ > 0, 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 ξ such that For any t ≥ t 0 , there exists an integer l ∈ ℤ + such that t = lω + t, where t ∈ t 0 , t 0 + ω .Then one has Therefore, one has From system (4) and inequality (45), one gets 7 Complexity Consider the comparison system for system (46): By Lemma 1, one knows that there exists a positive ω-periodic vector function υ 2 t such that Z t = υ 2 t exp ζt is a solution of system (47), where ζ = ln r Φ F k −V k −M ε 2 k P k ω .From (40), one can get that Z t → ∞ as t → ∞, and E h t → ∞, I 1 t → ∞, I 2 t → ∞, and I v t → ∞ as t → ∞.By the comparison theorem in impulsive differential equations, one has E h t → ∞, I 1 t → ∞, I 2 t → ∞, and I v t → ∞ 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 K 0 , ∂K 0 .
Obviously, the Poincaré map P has a global attractor P 0 .P 0 is an isolated invariant set in K and W s P 0 ∩ K 0 = ∅ and it is acyclic in M ∂ .Every solution in M ∂ converges to P .According to Zhao [30], one derives that P is uniformly persistent with respect to K 0 , ∂K 0 .This implies that the solution of system ( 4) is uniformly persistent with respect to K 0 , ∂K 0 .This completes the proof.

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.

Complexity
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 γ σ and insecticide spraying rates p σ 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 R 0 ).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 R 0 ).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, R 0 = 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.

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 R 0 with 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       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.
, and a k+m = a k , b k+m = b k , c k+m = c k , m k+m = m k , and p k+m = p k .Define I p as the set of periodic switching rule.

Figure 2 :
Figure 2: Time series plots of the total of infected trees and infected psyllids.The disease dies out with R 0 = 0 903702 < 1.

Figure 3 :
Figure 3: Time series plots of the total of infected trees and infected psyllids.The disease is persistent with R 0 = 1 801682 > 1.

Figure 4 :
Figure 4: Time series plot of the total of infected trees for the baseline scenario and Strategy I.

Figure 5 :
Figure 5: Time series plot of the total infected trees for the baseline scenario and Strategy II.

Figure 6 :
Figure 6: Time series plot of the total infected trees for the baseline scenario and without control strategies.