Threshold Dynamics of a Vector-Borne Epidemic Model for Huanglongbing with Impulsive Control ()
1. 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 we all know, HLB can be spread efficiently by vector psyllids to all commercial cultivars of citrus [3] [4] .
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 [5] and Kranz [6] . There are many authors establish continuous mathematical models to describe the transmission of HLB. Chiyaka et al. [7] proposed a compartmental model of ordinary differential equations for the HLB transmission dynamics within a citrus tree considering 10 dimensions. In [2] , 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 [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.
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:
(2.1)
with initial condition
(2.2)
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
(2.3)
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. [14] Let
. Then there exists a positive, ω-periodic function
such that
is a solution of
(3.1)
In what follows, we give the basic reproduction number
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:
(3.2)
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
.
Denote
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
(3.3)
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
.
Let
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 [16] and Yang and Xiao [15] .
(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
(H8)
.
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
be the evolution operator of the linear impulsive periodic system
(3.4)
where the explicit expression of
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
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 [15] .
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
(3.5)
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:
(3.6)
From the first and third equations of system (3.6), we have
(3.7)
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
(3.8)
(3.9)
and
.
Furthermore, we denote
. By [15] , suppose that
immediately after pulses equals
(3.10)
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
where
.
Proof: Let
, from Theorem 2.1, we have
(4.1)
and
(4.2)
for
.
Obviously, by (4.1), (4.2) and the comparison principle of impulsive differential equations in [17] , we have
In similar method, we can prove
for
.
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
for
.
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.
5. Permanence
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
(5.1)
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 [18] 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.
Denote
.
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
(5.2)
Consider an auxiliary system
(5.3)
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
for
.
On the other hand, the standard comparison theorem implies that there exist
and
such that
for all
. Then, for all
, we have
where
.
By the second, fourth, sixth and eighth equations of system (2.1), we have
(5.4)
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
.
6. 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
, 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.
Acknowledgements
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).