This paper proposes a vector-borne plant disease model with discontinuous treatment strategies. Constructing Lyapunov function and applying non-smooth theory to analyze discontinuous differential equations, the basic reproductive number R 0 is proved, which determines whether the plant disease will be extinct or not. If R 0 < 1 , the existence and global stability of disease-free equilibrium is established; If R 0 > 1 , there exists a unique endemic equilibrium which is globally stable. The numerical simulations are provided to verify our theoretical results, which indicate that after infective individuals reach some level, strengthening treatment measures is proved to be beneficial in controlling disease transmission.
The plants play an important role in our lives, as most of our daily food, clothing and building materials come from plants. With the change of environment, there are outbreaks of plant diseases, which seriously affect the health of plants and people’s life, such as huanglongbing [
The prevention and control of plant infectious diseases is of vital importance in agricultural production [
Treatment plays a very important role to control the spread of diseases. In recent years, many researchers [
In [
The paper is organized as follows. In the next section, we will construct the model and introduce the rational assumptions for model. In Section 3, positivity of the solution for the model will be clearly discussed. We obtain the existence of possible equilibria, the basic reproductive number, and the stability of equilibria in Section 4. In Section 5 and Section 6, we summarize our main results and main results are numerically simulated.
To construct the model, the following assumptions are being made by Shi et al. in [
(A1) The total of the insect vector population is divided into X and Y, which denotes the densities of the susceptible vector and infective vector at time t, respectively. The total of the plant host population is divided into S, I, and R, which represents the numbers of the susceptible, infective, and recovered host plant population at time t, respectively. At the same time, we assume that the number of plants in one area is fixed. The total number of plants K = S + I + R is a positive constant. In fact, when a plant has died, it would be replaced by a new plant to keep the total number of plants. Further, we assume that those new plants are susceptible, i.e., we chose the birth rate of susceptible plant host as f ( S , I ) = μ K + d I .
(A2) The susceptible plants can be infected not only by the infected insect vectors but also by the infected plants.
(A3) A susceptible vector can be infected only by an infected plant host, and after it is infected, it will hold the virus for the rest of its life. Further, there is no vertical infection being considered.
(A4) The replenishment rate of insect vectors is a positive constant, and all of the new born vectors are susceptible.
According to the principle of the compartmental model, consider the following model with discontinuous treatment:
{ S ˙ = f ( S , I ) − μ S − ( β P Y + β s I ) S , I ˙ = ( β P Y + β s I ) S − ( d + μ + γ ) I − h ( I ) , R ˙ = γ I + h ( I ) − μ R , X ˙ = Λ − β 1 I X − m X , Y ˙ = β 1 I X − m Y . (2.1)
Here the dimensionless variables and parameters (with parameter values) are given in
The function h ( I ) = φ ( I ) I represents the treatment rate. φ ( I ) satisfies the following assumptions. Obviously, the treatment rate should be nondecreasing as the number of infectious individuals is increasing. The following assumption will be needed throughout the paper.
(H1) φ : [ 0, ∞ ) → [ 0, ∞ ) is nondecreasing and has at most a finite number of jump discontinuities in every compact interval. No loss of generality, we always assume that φ is continuous at I = 0 , otherwise we define φ ( 0 ) to be φ ( 0 + ) . Here φ ( 0 + ) denotes the right limit of φ ( I ) as I → 0 + .
By adding the fourth and fifth equations of system (2.1), we get
N ˙ = Λ − m N (2.2)
where N = X + Y . From Equation (2.2), we easily get N → Λ m as t → ∞ .
Note that S + I + R = K . Since the variable R and X does not appear in the first two equations of model (2.1), meanwhile, let’s substitute X for ( Λ m − Y ) in
the fifth equation. We only need to study the first two equations and the fifth
Parameter | Description | Default value |
---|---|---|
S | number of the susceptible plant hosts | - |
I | number of the infected plant hosts | - |
R | number of the recovered plant hosts | - |
K | sum of the total plant hosts | 50 - 1000 |
X | density of the susceptible insect vectors | .. |
Y | density of the infected insect vectors | - |
N | sum of the total insect vectors density | 50 - 100 |
β1 | infection ratio between infected hosts and susceptible vectors | 0.01 - 0.02 |
βP | biting rate of an infected vector on the susceptible host plants | 0.01 - 0.02 |
βS | infection incidence between infected and susceptible hosts | 0.01 - 0.02 |
γ | the conversion rate of infected hosts to recovered hosts | 0 - 0.4 |
μ | natural death rate of plant hosts | 0 - 0.1 |
Λ | birth or immigration of insect vectors | 5 |
m | natural death rate of insect vectors | 0 - 0.5 |
d | disease-induced mortality of infected hosts | 0.1 |
equation of model (2.1), thereby lowering the order of the system to be studied, i.e.
{ S ˙ = μ ( K − S ) − ( β P Y + β s I ) S + d I , I ˙ = ( β P Y + β s I ) S − ω I − h ( I ) , Y ˙ = Λ β 1 I m − ( β 1 I + m ) Y . (2.3)
where ω = d + μ + γ . Obviously,
Ω = { ( S , I , Y ) ∈ R + 3 : 0 ≤ S + I ≤ K ,0 ≤ Y ≤ Λ m } (2.4)
is the positively invariant set for system (2.3).
According to the definition of solutions for differential equations with discontinuous right-hand sides in [
( S ( 0 ) , I ( 0 ) , Y ( 0 ) ) = ( S 0 , I 0 , Y 0 ) , S 0 , I 0 , Y 0 ≥ 0 (2.5)
of model (2.3) on [ 0 , T ) , 0 < T ≤ ∞ , if it is absolutely continuous on any compact subinterval of [ 0, T ) , and almost everywhere on [ 0, T ) (abbreviated to a.e. on [ 0, T ) ) satisfies the following differential inclusion:
{ S ˙ = μ ( K − S ) − ( β P Y + β s I ) S + d I , I ˙ ∈ ( β P Y + β s I ) S − ω I − c o ¯ [ h ( I ) ] , Y ˙ = Λ β 1 I m − ( β 1 I + m ) Y . (2.6)
where c o ¯ [ h ( I ) ] = [ h ( I − 0 ) , h ( I + 0 ) ] . Here, h ( I − 0 ) and h ( I + 0 ) denote the left limit and the right limit of the function h ( I ) at I, respectively.
From (H1), it is clear that the set map
( S , I , Y ) ↦ ( μ ( K − S ) − ( β P Y + β s I ) S + d I , ( β P Y + β s I ) S − ω I − c o ¯ [ h ( I ) ] , Λ β 1 I m − ( β 1 I + m ) Y ) (2.7)
is an upper semi-continuous set-valued map with non-empty compact convex values. By the measurable selection theorem [
{ S ˙ = μ ( K − S ) − ( β P Y + β s I ) S + d I , I ˙ ∈ ( β P Y + β s I ) S − ω I − m ( t ) , a . e . on [ 0 , T ) . Y ˙ = Λ β 1 I m − ( β 1 I + m ) Y . (2.8)
In this section, we will prove the positive of the solution to the initial condition of the model (2.3) with positive initial value. First, we will prove the following theorem.
Theorem 3.1. Suppose that assumption (H1) holds and let ( S ( t ) , I ( t ) , Y ( t ) ) be the solution with initial condition (2.5) of model (2.3) on [ 0, T ) . Then ( S ( t ) , I ( t ) , Y ( t ) ) is nonnegative on [ 0, T ) .
Proof: By the definition of a solution of (2.3) in the sense of Filippov, ( S ( t ) , I ( t ) , Y ( t ) ) must be a solution to differential inclusion (2.6). From the first equation of (2.6), we have
[ S 0 + ∫ 0 t ( μ k + d I ( u ) ) exp ( ∫ 0 u ( μ + β P Y ( ρ ) + β s I ( ρ ) ) d ρ ) d u ] ⋅ exp ( − ( ∫ 0 t ( μ + β P Y ( ρ ) + β s I ( ρ ) ) d ρ ) ) > 0 (3.1)
for all t ∈ ( 0, T ) .
Based on the previous hypothesis of (H1), we have c o ¯ [ h ( 0 ) ] = 0 and h ( I ) is continuous at I = 0 . Combining the continuity of φ at I = 0 , it may be concluded that there exists a positive constant δ such that φ ( I ) is continuous as | I | < δ . On this account, when | I | < δ the differential inclusion (2.6) becomes the following system of differential equations:
{ I ˙ = ( β P Y + β s I ) S − ( ω + φ ( I ) ) I , Y ˙ = Λ β 1 I m − ( β 1 I + m ) Y . (3.2)
We divide this into four cases to discuss the positivity of the solutions for (2.6).
1) I 0 = Y 0 = 0 .
From (3.2), we see that I ( t ) = Y ( t ) = 0 for all t ∈ [ 0, T ) .
2) I 0 > 0 , Y 0 = 0 .
By the continuity of I ( t ) at t = 0 and d Y d t | t = 0 = Λ β 1 I 0 m > 0 , we conclude I ( t ) > 0 and Y ( t ) > 0 for all t ∈ ( 0, T ) . If it is not true, then we can set
t 1 = inf { t : I ( t ) = 0 or Y ( t ) = 0 } ∈ ( 0 , T ) . (3.3)
If I ( t 1 ) = 0 , then from d I d t ≥ − ( ω + φ ( I ) ) I for 0 ≤ t ≤ t 1 , we have I ( t 1 ) ≥ I 0 exp ( − ( ω + φ ( I ) ) t 1 ) > 0 . This is a contradiction.
If I ( t 1 ) = 0 , then there is a θ such that t 1 − θ > 0 and 0 < I ( t ) < δ on [ t − θ , t 1 ) . Therefore, the second equation of (3.2) implies
d Y d t ≥ − ( β 1 I + m ) Y (3.4)
We have
Y ( t 1 ) ≥ Y ( t 1 − θ ) exp ( − ∫ t 1 − θ t 1 ( β 1 I ( ξ ) + m ) d ξ ) > 0 (3.5)
This is also a contradiction. Hence, I ( t ) and Y ( t ) are positive for all t ∈ ( 0, T ) . The same conclusion can be reached for the following two cases.
3) I 0 = 0 , Y 0 > 0 .
4) I 0 > 0 , Y 0 > 0 . This completes the proof.
In this section, we will discuss the existence of equilibria of system (2.3). First, we prove the existence of endemic equilibrium.
Let ( S ( t ) , I ( t ) , Y ( t ) ) = ( S * , I * , Y * ) is a constant solution of (2.3), where ( S * , I * , Y * ) satisfies the following system:
{ 0 = μ ( K − S * ) − ( β P Y * + β s I * ) S * + d I * , 0 ∈ ( β P Y * + β s I * ) S * − ω I * − c o ¯ [ h ( I * ) ] , 0 = β 1 I * Λ m − ( β 1 I * + m ) Y * . (4.1)
Since h ( 0 ) = 0 , there always exists a disease-free equilibrium P 0 of the model (2.3), where P 0 = ( K , 0 , 0 ) . Next, we consider that the existence of an endemic equilibrium of the model (2.3).
It follows from the first and third equations of (4.1), we conclude that
S * = d I * + μ K μ + β P Y * + β s I * , Y * = Λ β 1 I * m ( β 1 I * + m ) . (4.2)
Substituting (4.2) into the second inclusion of (4.1), we have the follows
A 1 I * 2 + B 1 I * + C 1 A 2 I * 2 + B 2 I * + C 2 − ω ∈ c o ¯ [ φ ( I ) ] = [ φ ( I * − 0 ) , φ ( I * + 0 ) ] (4.3)
where
A 1 = m d β 1 β s
B 1 = d Λ β p β 1 + m 2 d β s + μ m K β 1 β s ,
C 1 = μ K ( Λ β p β 1 + m 2 β s ) ,
A 2 = m β 1 β s ,
B 2 = μ m β 1 + Λ β 1 β p + m 2 β s ,
C 2 = μ m 2 .
Denote
g ( I * ) = A 1 I * 2 + B 1 I * + C 1 A 2 I * 2 + B 2 I * + C 2 − ω (4.4)
and let
R 0 = K ( Λ β p β 1 + β s m 2 ) m 2 ( ω + φ ( 0 ) ) (4.5)
We next claim that R 0 is the basic reproductive number for the model (2.3) which will determine the existence of an endemic equilibrium.
Theorem 4.1. Suppose that assumption (H1) holds. If R 0 ≤ 1 , then there only exists a disease-free equilibrium P 0 ( K ,0,0 ) . If R 0 > 1 , then there exists a unique positive endemic equilibrium P * ( S * , E * , I * ) except P 0 .
Proof: By R 0 ≤ 1 , we get g ( 0 ) ≤ φ ( 0 ) . Since g ( I ) is nonincreasing on I and φ ( I ) is nondecreasing on I. For this reason, the inclusion (4.3) is only valid at I = 0 . Hence, the model (2.3) has a unique disease-free equilibrium as long as R 0 ≤ 1 .
From (4.4), we have the following
A I 2 + B I + C = 0 (4.6)
where
A = m β 1 β s ( m d − ω ) < 0 ,
B = Λ d β 1 β p + m 2 d β s + μ m K β 1 β s − ω m μ β 1 − Λ ω β 1 β p − ω m 2 β s ,
C = μ ω m 2 ( R 0 − 1 ) + μ m 2 φ ( 0 ) .
If R 0 ≥ 1 , then C > 0 , and the Equation (4.6) has a unique positive root I, where
I = − B + Δ 1 2 2 A , Δ = B 2 − 4 A C . (4.7)
If R 0 > 1 , then
is bounded and non-empty. We can write
It follows easily that
We claim
From (H1), there exists a
This contradicts the definition of
which satisfy
and
Subtracting (4.13) from (4.12) gives
which implies
This is a contradiction. Hence,
Next, we prove the global stability of the disease-free equilibrium and the endemic equilibrium. We do this in several steps. We first investigate the local properties of the equilibria of system (2.3).
Theorem 4.2. Assume (H1) holds. The disease-free equilibrium
Proof: We analyze the stability of the disease-free equilibrium by investigating the eigenvalues of the Jacobian matrix of model (2.3) at
Thus, the characteristic equation at the disease-free equilibrium
It is easy to see that one of the roots with respect to
From (4.18) and Routh-Hurwitz criteria [
We have shown that there exists a positive endemic equilibrium if and only if
Theorem 4.3. Suppose that assumption (H1) holds. If
Proof: The Jacobian matrix of (2.3) at the endemic equilibrium
Replacing
The characteristic equation of
where
Since
Then
Hence, all of the Routh-Hurwitz criteria are satisfied. Thus it follows that the endemic equilibrium
Next, we will prove global stability of the disease-free equilibrium and endemic equilibrium of (2.3). We need to use the LaSalle-type invariance principle for the differential inclusion (Theorem 3 in [
Let
Set
and
For any
Hence,
When
It shows that
Furthermore, when
When
Hence, the largest weakly invariant subset of
When
From the first equation of (4.23) and x = 0, it may be concluded that
Next, we demonstrate the global stability of the endemic equilibrium
Theorem 4.4. Suppose that assumption (H1) holds. If
Proof: Let
Write
and
For any
Hence
The monotonicity of
If
Therefore,
To make our analysis more intuitive, some numerical simulations of solutions of the model (2.6) is provided which to illustrate the influence of insect vector and discontinuous treatment on the spread of plant disease. We apparent a treatment function satisfying (H1) as follows:
where
To better illustrate the effects of non-continuous healing on the spread of plant disease, the following parameters are derived from [
If we fixed all parameter values as follows:
As for the plant infectious disease model, our main object is to investigate the effect of the insect vector and discontinuous treatment function on the dynamics of spreading the plant disease. We calculated the basic reproduction number
In this paper, we studied the existence, local stability and global stability of the disease-free equilibrium and endemic equilibrium of the system (2.3) in detail. By building a suitable Lyapunov function, and the Jacobian matrix method,
employing Routh-Hurwitz criteria and LaSalle-type invariance principle, the main results as shown in Theorems 4.2, 4.3 and 4.4 have been derived. Our main results indicate that if
The research have been supported by The Natural Science Foundation of China (11561004), the Science and Technology research project of Jiangxi Provincial Education Department (171373, 171374, GJJ170815), The bidding project of Gannan Normal University (16zb02).
Lv, H.M., Fei, L.Z., Yuan, Z. and Zhang, F.M. (2018) Global Dynamic Analysis of a Vector-Borne Plant Disease Model with Discontinuous Treatment. Applied Mathematics, 9, 496-511. https://doi.org/10.4236/am.2018.95036