Huanglongbing (HLB) is one of the most serious problems of citrus worldwide which caused by the bacteria Candidatus Liberibacter spp., whose name in Chinese means “yellow dragon disease’’, was first reported from southern China in 1919 and is now known to occur in next to 40 different Asian, African, Oceanian, South and North American countries  . HLB has no cure and affects all citrus varieties, reducing the productivity of orchards because the fruits of infected plants have poor quality and, in extreme cases, infection leads to plant death  . HLB symptoms are virtually the same wherever the disease occurs. Infected trees show a blotchy mottle condition of the leaves that result in the development of yellow shoots, the early and very characteristic symptom of the disease  . As we all know, HLB can be spread efficiently by vector psyllids to all commercial cultivars of citrus   .
Mathematical models play an important role in understanding the epidemiology of vector-transmitted plant diseases. Applications of mathematical approach to plant epidemics were reviewed by Van der Plank  and Kranz  . There are many authors establish continuous mathematical models to describe the transmission of HLB. Chiyaka et al.  proposed a compartmental model of ordinary differential equations for the HLB transmission dynamics within a citrus tree considering 10 dimensions. In  , Raphael et al. constructed a 6 dimensions model of ordinary differential equations with delay time. However the dynamic behaviors of these models are studied only by using computer simulations.
But, in our real world, farmers’ experiences have led to development of integrated management concepts for virus diseases that combine available host resistance, cultural, chemical and biological control measures. A cultural control strategy including replanting, and/or removing (rouging) diseased plants is a widely accepted treatment for plant epidemics which involves periodic inspections followed by removal of the detected infected plants      . Periodic replanting of healthy plants or removing (rouging) infected plants in plant-virus disease epidemics is widely used to minimize losses and maximize returns  . There are only a few countries have been able to control Asian HLB. São Paulo State (SPS) might be one of the first to be successful. In SPS, encouraging results have been obtained in the control of HLB by tree removal and insecticide treatments against psyllids  . Monocrotophos has a short residual effect on psyllid, repeated application is often required to suppress psyllid, which can cause pesticide resistance. Pesticides pollution is also recognized as a major health hazard to human beings and beneficial insects. To deal with these questions, we propose model dealing in detail with the killing efficiency rate and decay rate of pesticides. The residual effects of pesticides (i.e. killing efficiency rate and decay rate) on the threshold conditions are also addressed.
A model for the temporal spread of an epidemic in a closed plant population with periodic removals of infected plants has been considered by Fishman et al.  . Integrated management has been found to be more effective at eliminating epidemics. In this paper, according to the above biological background, we develop a hybrid impulsive control model, in which replanting of healthy plants and removing infected plants at one fixed moment and pesticide spraying at another fixed moment are considered, to propose optimal control strategy.
The paper is organized as follows. In Section 2, we formulate the impulsive epidemic model and also simplify the original system (2.1). In Section 3, we introduce some useful lemmas and the basic reproduction number of the model. In Sections 4 and 5, we proved the global stability of the disease-free equilibrium and permanence of the model, respectively. In the finally section, a brief discussion is given.
2. Model Formulation
Let denote susceptible citrus host and infected citrus host, respectively, and represent susceptible psyllid and infected psyllid, respectively. We give the following system:
with initial condition
The model is satisfied with the following assumptions.
・ and are left continuous, that is, , , and for all .
・ is the infected rate of citrus host. are the nature death rate and disease induced death rate of citrus, respectively.
・ is constant recruitment rate of psyllid.
・ are the infected rate and nature death rate of psyllid, respectively.
・ are the recruitment rate of citrus and removing rate of infected citrus by impulses, respectively.
・ is the interpulse time, i.e., the time between two consecutive pulse replanting and removing.
The following lemma is obvious.
Lemma 2.1. If , , and , then , , and for every .
Denote , where , .
Theorem 2.1. The solutions of system (2.1) with initial condition (2.2) eventually enter into G and G is positively invariant for system (2.1).
Proof: Let . By system (2.1), we have
By the first and third equations of (2.3), we get
Thus, we have
From the second and fourth equations of (2.3), we have
Then, we have .
Then, from the above analysis, which implies that G is positively invariant.
3. The Basic Reproduction Number of (2.1)
Let be the standard ordered n-dimensional Euclidean space with a norm . For , we write if , if , if , respectively.
Set be cooperative, irreducible and periodic matrix function with period (>0), P be a constant matrix, T be a pulse period satisfying . Then is the fundamental solution matrix of the linear differential equation
and is the spectral radius of . By Perron-Frobenius theorem, is the principal eigenvalue of in the sense that it is simple and admits an eigenvector .
Firstly, we introduce some lemmas which will be useful for our further arguments.
Lemma 3.1.  Let . Then there exists a positive, ω-periodic function such that is a solution of
In what follows, we give the basic reproduction number for system (2.1). Similar to Yang and Xiao  .
An impulsive periodic differential mathematical model in which impulses occur at fixed times may be described as follows:
where is ω-periodic function and , for , and is an open set.
Let be the input rate of newly infected individuals in the i-th compartment, and where be the input rate of individuals by other means, and be the rate of transfer of individuals out of compartment i; then denotes the net transfer rate out of compartments. We suppose that immediately after pulses equals
where denotes the transpose of A, and are n homogeneous compartments in a heterogeneous population, with each being the number of individuals in each compartment. Assume that the compartments sort by two types, with the first m compartments the infected individual, and the uninfected individuals. Denote
Now, system (3.2) can be written as
Define to be the set of all disease-free states:
Furthermore, assume that
be a disease-free periodic solution over the k-th time interval with , , for all .
and , where , , , and are the i-th component of , , , x and , respectively.
We make the following assumptions, which are the same biological meanings as those by Wang and Zhao  and Yang and Xiao  .
(H1) If , then for .
(H2) If , then . In particular, if , then for .
(H3) if .
(H4) If , then and for .
(H5) The pulses on the infected compartments must be uncoupled with the uninfected compartments; that is, is essentially .
(H6) It holds that .
(H7) , where is the fundamental solution matrix of the system
In the following, we study the threshold dynamics of system (2.1) and show that its basic reproduction number can be defined as the spectral radius of the so-called next infection operator as that in impulsive and periodic environment  .
Let be the evolution operator of the linear impulsive periodic system
where the explicit expression of can be found in  , we omit it here. By assumption (H1)-(H8), we also know that the periodic solution of system (3.4) is asymptotically stable.
Now, we define the so-called next infection operator L as follows:
where is defined as the ordered Banach space of all ω-periodic functions from R to , equipped with the maximum norm , and the positive cone ; is the initial distribution of infectious individuals.
The limit as a goes to infinity does exist, and the next infection operator L is well defined, continuous, positive and compact on the domain. We now define the basic reproductive number as the spectral radius of L is
From above discussion, we have the following results.
Lemma 3.3. Assume that (H1)-(H8) hold, Then the following statements are valid:
1) if and only if .
2) if and only if .
3) if and only if .
The proof in detail is similar to periodic systems in  .
Lemma 3.4. If the disease-free periodic solution is asymptotically stable, and unstable if .
Proof: Observe that the linearized system of system (3.3) at the disease-free periodic solution is
Then the monodromy matrix of the impulsive system (3.5) equals
where represents a non-zero block matrix. Then the Floquet multipliers of system (3.3) are the eigenvalues of and . By assumption (H7), that is, , it then follows that the disease-free periodic solution is asymptomatically stable if , and unstable if . This completes the proof.
Following, we demonstrate the existence of the disease-free periodic solution. Set for all . Under this condition, we have the following system:
From the first and third equations of system (3.6), we have
Then, over the k-th impulsive interval, . By the impulsive condition, we have . The unique fixed point of this system equals .
Accordingly, the impulsive periodic solution of the system (3.7) is
Obviously, is globally asymptomatically stable.
From system (3.6), we know that is not affected by impulse, and we have . Hence, system (2.1) has a unique disease-free periodic solution .
Obviously, by Lemma 3.4, we have that of system (2.4) is asymptotically stable if , and unstable if .
We denote , then for system (2.1), we have
Furthermore, we denote . By  , suppose that immediately after pulses equals
For the system (2.1), we have
Clearly, conditions (H1)-(H6) are satisfied for system (2.1). There are only (H7) and (H8) should be verified in the following.
is the disease-free periodic solution for system (2.1). We define ,
and , where , and are the i-th component of , x and , respectively.
Then, from (3.8) and (3.9), we obtain
From (3.10), we have
and hence, . Therefore, (H7) holds.
We further denote and P are matrices defined by
, and , where and are the i-th component of and , respectively. Then from (3.8), (3.9) and (3.10), it follows that
It is easy to see that satisfied. (H8) is hold.
Thus, the Lemma 3.3 is right for system (2.1).
4. Global Stability of the Disease-Free Equilibrium
In this section, we prove that the disease-free periodic solution is globally asymptotically stable, if and hence, the disease extinct.
Firstly, we need to prove the following lemma.
Lemma 4.1. For the system (2.1), it holds that
Proof: Let , from Theorem 2.1, we have
Obviously, by (4.1), (4.2) and the comparison principle of impulsive differential equations in  , we have
In similar method, we can prove
Hence, the proof is completed.
Theorem 4.1. For any solution of system (2.1), if , then the disease-free periodic solution is globally asymptotically stable and if , then it is unstable.
Proof: By Lemma 3.3, if , then is unstable and if , then is locally stable. Hence, it is sufficient to show that the global attractivity of for .
Now, we prove the global attractivity of the disease-free solution.
From Lemma 4.1, there exist a and a positive constant such that , .
By the second, fourth, sixth and eighth equations of system (2.1), we have
Set be the matrix function such that
By Lemma 3.3, we have , we restrict , such that . Let us consider the following system
By Lemma 3.1 and the standard comparison principle, there exists a positive T-periodic function such that where and . Then, we see that and .
Moreover, we obtain that , . Hence, the disease-free periodic solution is globally attractive. This completes the proof.
In this section, we show that if , then the disease persists.
Let be a matrix space, be a continuous map, and be an open set. Define
is a maximal compact invariant set of f in . A finite sequence are disjoint, compact, and invariant subsets of , and each of them is isolated in .
We present persistence theory  as follows:
Lemma 5.1. Assume that
1) and f has a global attractor A;
2) The maximal compact invariant set of f in , possibly empty, has an acyclic covering and where with the following properties:
a) is isolated in ; b) for each .
Then, f is uniformly persistent with respect to , i.e., there is such that for any compact internally chain transitive set L with for all , .
Define Poincaré map associated with system (2.1), satisfying , where is the unique solution of system (2.1) with . Now, we denote , and .
Theorem 5.1. Suppose that , then system (2.1) exists a positive constant such that for all ,
Proof: Firstly, we prove that is uniformly persistent with respect to . From Theorem 2.1, it is obvious that and are positively invariant. We also know that is point dissipative on from Lemma 4.1.
Next, we need to show that .
Obviously, . We now need to prove that . Suppose it’s not hold. For any . For the case , it is obvious that and for all . From second and sixth equations of system (2.4), we have
then it hold that for all from Lemma 3.2, where . In the similar method, for the case , then we have and for all . This implies that for sufficiently small. It follows that . Thus, . It is clear that is a unique fixed point of in .
In the following, we need to prove .
We write . By the continuity of the solutions with respect to the initial conditions, , there exist , such that for all with , it hold that
Now, we show that
Suppose not hold, then for some . Without loss of the generality, we can assume that . Thus, we obtain that and .
For any , let , where and . is the greatest integer less than or equal to . So, we have that
It follows that
Then, by the first, third, fifth and seventh equations of system (2.1), we have
Consider an auxiliary system
Using the same method as aforementioned, we have that (5.3) admits a positive periodic solution . Since holds. Then, there exists a small enough such that , and is continuous for small , where
As before, we have that is globally asymptotically stable, and meanwhile , , thus there exist small enough and a constant , such that
On the other hand, the standard comparison theorem implies that there exist and such that
for all . Then, for all , we have
By the second, fourth, sixth and eighth equations of system (2.1), we have
Set be the matrix function such that
where is small enough.
By Lemma 3.1 and the standard comparison principle, it follows that there exists a positive T-periodic function such that is a solution of system (5.4), where
. Since , and
is continuous for small . So we can choose small enough, such that . It follows that , we can choose such that
By the comparison principle we have
for all . Then, we obtain that and , which contradicts to the boundedness of , . Thus we have proved , which implies each orbit in converges to , and hence is acyclic in .
Therefore, the Lemma 5.1 is satisfied for system (2.1). Furthermore, we obtain that the disease is permanence, when .
In this paper, a vector-borne epidemic model for Huanglongbing with impulsive control is established. Under the reasonable assumptions (H1)-(H8), one studied the threshold dynamics behavior of the model. Based on comparison theorem of impulsive differential equation and method of enlarging and reducing, we proved that if the , the disease-free equilibrium is global stability, and Huanglongbing is uniformly persistent if . We only consider replanting susceptible and rouging infective in model, spraying insecticides to kill psyllid is not. It’s a lot of room for us to improve.
The research has been supported by the Science and Technology Plan Projects of Jiangxi Provincial Education Department (GJJ151491, GJJ171373), Guidance Project of Ji’an Science and Technology Bureau, the Natural Science Foundation of Ji’an College (16JY103).
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