Threshold Dynamics of a Vector-Borne Epidemic Model for Huanglongbing with Impulsive Control

In this paper, the basic reproduction number is calculated for Huanglongbing (HLB) model with impulses which is a vector-borne epidemic model with impulses. For controlling HLB, farmers’ experience is replanting of healthy plants and removing infected plants. To reflect the real world, we construct an impulsive control model which considers replanting of healthy plants and removing infected plants at one fixed time. By analyzing the model, we conclude that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number 0 1 R < , and we prove that the HLB is permanence if the basic reproduction number 0 1 R > .


Introduction
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 [1].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 [2].
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 [1].As 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 [8] [9] [10] [11] [12].
Periodic replanting of healthy plants or removing (rouging) infected plants in plant-virus disease epidemics is widely used to minimize losses and maximize returns [12].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 [13].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. [8].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.
The model is satisfied with the following assumptions.

( ) ( )
> are the nature death rate and disease induced death rate of citrus, respectively.
> are the infected rate and nature death rate of psyllid, respec- tively.
• 0, 0 1 δ φ ≥ < < are the recruitment rate of citrus and removing rate of in- fected citrus by impulses, respectively.• 0 T > is the interpulse time, i.e., the time between two consecutive pulse replanting and removing.
The following lemma is obvious.
( ) 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).

N S I N S I = +
= + .By system (2.1), we have By the first and third equations of (2.3), we get 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.

The Basic Reproduction Number of (2.1)
Let ( ) R R + be the standard ordered n-dimensional Euclidean space with a norm  .For ,

( )
A t be cooperative, irreducible and periodic n n × matrix function with period ω (>0), P be a n n × constant matrix, T be a pulse period satisfying , T q q N ω = ∈ .Then ( ) is the fundamental solution matrix of the linear differential equation is the spectral radius of ( ) ) in the sense that it is simple and admits an eigenvector * 0 v  .
Firstly, we introduce some lemmas which will be useful for our further arguments.( ) , .

y t A t y t kT k N t y t Py t t kT k N
In what follows, we give the basic reproduction number 0 R for system (2.1).
Similar to Yang and Xiao [15].An impulsive periodic differential mathematical model in which impulses occur at fixed times may be described as follows: x t f t x t kT t x t x t I t t kT where :  ( ) be the input rate of newly infected individuals in the i-th compartment, and x kt x kt I kt x kt , , , , where T A denotes the transpose of A, and h g h g , , , .
2) can be written as . , Define s X to be the set of all disease-free states: , , f t x t , x and ψ , respectively.
We make the following assumptions, which are the same biological meanings as those by Wang and Zhao [16] and Yang and Xiao [15].
(H5) The pulses on the infected compartments must be uncoupled with the uninfected compartments; that is, ( ) ( ) (H6) It holds that ( ) 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 [16].

Let ( )
, , Y t s t s ≥ be the evolution operator of the linear impulsive periodic system

y t V t y t nT t y t Py t t nT
where the explicit expression of ( ) , Y t s can be found in [17], 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 C ω is defined as the ordered Banach space of all ω-periodic functions from R to m R , 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 The proof in detail is similar to periodic systems in [15].Lemma 3.4.If 0 1 R < the disease-free periodic solution ( ) x t  is asymptotically stable, and unstable if 0 1 R > .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 ( ) ( ) . By assumption (H7), that is, ( ) ( ) < , it then follows that the disease-free periodic solution is asymptomatically stable if Following, we demonstrate the existence of the disease-free periodic solution.Set ( ) ( ) for all 0 t > .Under this condition, we have the follow- ing system: From the first and third equations of system (3.6), we have ,  d , .
h h h h

S t d S t t kT t S t S t t kT
Then, over the k-th impulsive interval, ( ) Accordingly, the impulsive periodic solution of the system (3.7) is ( )

vI t d I t t x t t x t S t I t d S t S t I t d S t
λ β . By [15], suppose that ( ) x t 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 , where ( ) , f t x t , x and ψ , respectively.

Global Stability of the Disease-Free Equilibrium
In this section, we prove that the disease-free periodic solution is globally asymptotically stable, if 0 1 R < and hence, the disease extinct.
Firstly, we need to prove the following lemma.

= − =
In similar method, we can prove Hence, the proof is completed.Theorem 4.1.For any solution of system (2.1), if 0 1 R < , then the dis- ease-free periodic solution By the second, fourth, sixth and eighth equations of system (2.1), we have

I t S t I t d v I t t t kT I t S t I t d I t t I t I t t kT I t I t
Let us consider the following system

I t S t I t d v I t t t kT I t S t I t d I t t I t I t t kT I t I t
Moreover, we obtain that Hence, the disease-free periodic solution is globally attractive.This completes the proof.

Permanence
In this section, we show that if 0 1 R > , then the disease persists.Let X  be a matrix space, : f X X →   be a continuous map, and 0 are disjoint, compact, and invariant subsets of 0 X ∂ , and each of them is isolated in We present persistence theory [18] as follows: Lemma 5.1.Assume that 1) ( ) empty, has an acyclic covering M  and where Then, f is uniformly persistent with respect to ( ) . Now, we denote Theorem 5.1.Suppose that 0 1 R > , then system (2.1) exists a positive con- stant 0 >  such that for all ( ) ( ) ( ) ( ) ( ) ( ) Proof: Firstly, we prove that P  is uniformly persistent with respect to ( ) 0 0 , X X ∂ .From Theorem 2.1, it is obvious that X  and 0 X are positively invariant.We also know that P  is point dissipative on 4 R + from Lemma 4.1.

Denote
Next, we need to show that  t > .This implies that ( ) , , , In the following, we need to prove We write ( ) x S I S I X = ∈ .By the continuity of the solutions with respect to the initial conditions, 0 ∀ >  , there exist 0 0 δ > , such that for all Without loss of the generality, we can assume that ( ) ( ) Thus, we obtain that the greatest integer less than or equal to 1 t ω .So, we have that Then, by the first, third, fifth and seventh equations of system (2.1), we have Using the same method as aforementioned, we have that (5.3) admits a positive periodic solution there exists a small enough ε such that ( ) ( ) As before, we have that Therefore, the Lemma 5.1 is satisfied for system (2.1).Furthermore, we obtain that the disease is permanence, when 0 1 R > .

Conclusion
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 0 1 R < , the disease-free equilibrium is global stability, and Huanglongbing is uniformly persistent if 0 1 R > .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.
By the impulsive condition, we have 1 and the standard comparison principle, there exists a positive T-periodic function

(
On the other hand, the standard comparison theorem implies that there exist  , which implies each orbit in M ∂ converges to 0 E , and hence 0 E is acyclic in M ∂ .