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 Open Access
Transmission dynamics of a Huanglongbing model with cross protection
 Lei Luo^{1},
 Shujing Gao^{1, 2}Email author,
 Yangqiu Ge^{1} and
 Youquan Luo^{1}
https://doi.org/10.1186/s136620171392y
© The Author(s) 2017
 Received: 10 May 2017
 Accepted: 7 October 2017
 Published: 6 November 2017
Abstract
Huanglongbing (HLB) is one of the most common widespread vectorborne transmission diseases through psyllid, which is called a kind of cancer of plant disease. In recent years, biologists have focused on the role of cross protection strategy to control the spread of HLB. In this paper, according to transmission mechanism of HLB, a deterministic model with cross protection is formulated. A threshold value \(R_{0}\) is established to measure whether or not the disease is uniformly persistent. The existence of a backward bifurcation presents a further subthreshold condition below \(R_{0}\) for the spread of the disease. We also discuss the effects of cross protection and removing infected trees in spreading the disease. Numerical simulations suggest that cross protection is a promotion control measure, and replanting trees is bad for HLB control.
Keywords
 Huanglongbing model
 cross protection
 backward bifurcation
 subthreshold value
1 Introduction
Nowadays, Huanglongbing (HLB) is one of the most serious problems of citrus worldwide caused by the bacteria Candidatus Liberibacter spp., whose name in Chinese means “yellow dragon disease” [1]. The main symptoms on HLB include chlorosis of leaves, dieback and, in extreme cases, tree death. Additionally, infected trees develop fruit that is of poor quality and drops early, reducing yields of edible and marketable fruit from diseased trees [2]. The infected trees are usually destroyed or become unproductive in 5 to 8 years [1].
Most of the known plant viruses are transmitted by insect vectors. HLB, a destructive disease of citrus, can be transmitted by grafting from psyllid to citrus. The primary vector of the spread of the disease is the psyllid (Diaphorina Citri Kuwayama) [3].
In order to control HLB effectively, most of growers usually take the following measures: pesticides, tree removal, antibiotics [4], changes to tree spacing, natural enemies of psyllid. A few new intervention strategies are explored, including heat treatment [5], new tolerant or resistant tree stocks [6], nutrient additions [7], cross protection, intercropping [8]. Cross protection is one of biological methods. In recent years, cross protection is widely considered and applied in prevention and control of plant diseases.
Cross protection, first shown by McKinney [9] with tobacco mosaic virus (TMV), is a phenomenon whereby prior infection with one (protecting) plant virus will prevent or interfere with superinfection by another, usually related (challenging) virus [10]. In [11], the authors explored the cross protection between MAV (protecting) virus and PAV (challenging) virus in cereal, which belong to barley yellow dwarf viruses (BYDVs). By vaccinating M6 CTV strains (protecting) in citrus aurantif olia, Cui et al. [12] proved the obvious effect of cross protection on Bendizao mandarin. Van Vuuren et al. [13] studied the effect of cross protection on HLB of Africa by vaccinating multiple citrus recession viral strains. Hartung [14], who improved T36 CTV strains, described the resistance effect of cross protection on citrus HLB.
In recent years, some mathematical models on plant disease have been studied by many researchers (see [15–19]). Meng and Li [15] discussed the effect of cultural control on the healthy growth of the host plant. Local stability for the free periodic solution and persistence of the disease are key issues in the study of epidemic models. In fact, these issues are solved. In [16], Meng et al. illustrated that biological control may be a better way for pest management strategies by adopting a new mathematical model. Zhang et al. [17] proposed and compared two different control strategies in the model. In [18], Zhao et al. proposed a plant disease model with Markov conversion and impulsive toxicant input. Then thresholds of extinction and persistence in mean were obtained.
To the best of the authors’ knowledge, there has been little work on plant disease models with cross protection (see [10, 20, 21]). Gao et al. [20] took seasonality into account and put forward a nonautonomous plant disease model with cross protection. The results showed that cross protection played an important role in controlling the spread of the challenging virus in plants. Zhang et al. [21] proposed a model to study cross protection between the viruses in 1999. Zhang and Holt [10] improved the model in [21], in which cross protection can occur both naturally and through artificial intervention. Our main purpose is to investigate the transmission of HLB between citrus tree and psyllid populations with cross protection and evaluate the effect of cross protection in controlling the spread of HLB.
To achieve the above goals, we formulate a HLB model with cross protection and analyze the dynamical behavior theoretically including a backward bifurcation. Recently, there have been a number of studies on a backward bifurcation in the epidemic literature, for example, see [22–26]. Garba et al. [22] considered a dengue model with standard incidence formulation undergoing the phenomenon of backward bifurcation. Ahmed et al. [23] modeled the spread and control of dengue with limited public health resources, which exhibited the phenomenon of backward bifurcation. Li et al. [24] constructed an SIR epidemic model with nonlinear incidence and treatment. The results show that a backward bifurcation occurs if the capacity is small, and there exist bistable endemic equilibria if the capacity is low [27].
The paper is organized as follows. We formulate our HLB model with cross protection in Section 2. In Section 3, we determine the existence and stability of equilibrium point of the model. Moreover, we prove the existence of a backward bifurcation around the diseasefree equilibrium. In Section 4, we discuss the persistence of the disease. Numerical simulation and discussion are given in Section 5.
2 Model formulation
We will study the transmission dynamics of HLB disease in the rest of this paper. Before giving the main result, we present the following lemmas.
Lemma 2.1
Suppose \((X(t),Y(t),Z(t),U(t),S(t),I(t))\) is a solution of system (2.1) with initial conditions (2.2), then \((X(t),Y(t),Z(t),U(t),S(t),I(t))\geq0\) for all \(t\geq0\).
Lemma 2.2
Proof
3 Existence and stability of equilibrium points
To better organize the analysis, in the following we denote \(k_{1}=d+\alpha\), \(k_{2}=d+\alpha_{1}\), \(k_{3}=d+m\), \(k_{4}=d+m+ \alpha_{1}\), \(k_{5}=d+\alpha+\alpha_{1}\), and \(k_{6}=d+m+\mu\).
3.1 Existence of equilibrium points
In this subsection, we determine the existence of the equilibrium points of model (2.1). It is straightforward to establish that there is, for all parameter values, a diseasefree equilibrium \(E_{0}=(X_{0}, Y_{0},Z _{0},U_{0},S_{0},I_{0})=(\frac{\alpha_{1}Kd}{k_{1}k_{2}},\frac{\alpha \alpha_{1}K}{k_{1}k_{2}},0,0,\frac{\Lambda}{\mu},0)\).
 (1)
Assume \(R_{0}>1\), then \(c<0\) and thus \(\Delta>0\). It is clear that \(h(I)=0\) has a simple positive root, which we denote by \(I_{2}\). In this case system (2.1) has a unique endemic equilibrium, which we denote by \(E_{2}\).
 (2)
Suppose \(R_{0}=1\), then \(c=0\), and the equation \(h(I)=0\) has two roots, which are 0 and \(\frac{b}{a}\). Hence system (2.1) has a unique endemic equilibrium, denoted by \(E_{2}\), if and only if \(b<0\).
 (3)Suppose \(R_{0}<1\), then \(c>0\), and there are two possible subcases:
 (i)
If \(b>0\), \(h(I)=0\) does not have any positive root.
 (ii)
\(h(I)=0\) has two positive roots \(I_{1}\) and \(I_{2}\) if and only if \(\Delta>0\) and \(b<0\); further, \(h(I)=0\) has a double positive root \(I^{*}\) if and only if \(\Delta=0\) and \(b<0\).
 (i)
We can summarize the previous calculations in the following theorem.
Theorem 3.1
 (1)
if \(R_{0}>1\), there exists a unique endemic equilibrium \(E_{2}\);
 (2)
if \(R_{0}=1\), there exists a unique endemic equilibrium \(E_{2}\) if and only if \(b<0\); otherwise, there is no endemic equilibrium;
 (3)if \(R_{0}<1\), and
 (i)
if \(b>0\), there is no endemic equilibrium;
 (ii)
system (2.1) has two endemic equilibria \(E_{1}\) and \(E_{2}\) if and only if \(\Delta>0\) and \(b<0\); and these two equilibria coalesce into \(E^{*}\) if and only if \(\Delta=0\) and \(b<0\); otherwise, there is no endemic equilibrium.
 (i)
3.2 Stability of the diseasefree equilibrium point
In this subsection, we show the stability of the diseasefree equilibrium for model (2.1).
Theorem 3.2
For system (2.1), the diseasefree equilibrium \(E_{0}\) is locally asymptotically stable if \(R_{0}<1\) and unstable if \(R_{0}>1\).
Proof
Theorem 3.3
Proof
Obviously, \(R_{0}<\hat{R}\). It follows from Theorem 3.2 that the diseasefree equilibrium is locally asymptotically stable if \(\hat{R}<1\). Thus we only show that it attracts all nonnegative solutions of model (2.1).
Let \(0< Z(0)< y_{1}(0)\), \(0< U(0)< y_{2}(0)\), \(0< I(0)< y_{3}(0)\). If \((y_{1}(t),y _{2}(t),y_{3}(t))\) is any solution of system (3.9) which passes a nonnegative initial value \((y_{1}(0),y_{2}(0),y_{3}(0))\), according to the comparison principle of differential equations, we can obtain that \(Z(t)\leq y_{1}(t)\), \(U(t)\leq y_{2}(t)\), \(I(t)\leq y_{3}(t)\), for all \(t\geq0\).
3.3 Existence of a backward bifurcation
In most epidemiological models, the diseasefree equilibrium loses its local stability through a forward bifurcation at \(R_{0}=1\), at the same time a stable endemic equilibrium appears at this parameter value. This phenomenon can also be described as a supercritical transcritical bifurcation. However, under certain circumstances, such as nonlinear incidence, nonlinear recovery rate, and vectorborne transmission, a backward bifurcation may occur although \(R_{0}<1\) (see [31]). The system exhibits an endemic equilibrium along with a stable diseasefree equilibrium. There is then a subcritical transcritical bifurcation at \(R_{0}=1\).
We can use Theorem 4.1 in [32] to explore when system (2.1) undergoes either a forward or a backward bifurcation when \(R_{0}=1\). So two quantities, labeled Ã and B̃, need to be computed. In order to compute Ã and B̃, a change of coordinates involving the right and left eigenvectors of the Jacobian matrix \(J(E_{0})\) associated with the eigenvalue \(\lambda=0\) is required. We will express Ã and B̃ in terms of parameters.
Theorem 3.4
System (2.1) undergoes, at \(R_{0}=1\), a backward bifurcation if and only if \({\Delta^{\prime}>0}\), \(A_{2}<0\), and \(\alpha^{*}< \alpha<\alpha^{**}\); otherwise, system (2.1) undergoes a forward bifurcation.
Remark 3.1
Remark 3.2
From Remark 3.1, we know that the backward bifurcation gives a further subthreshold condition beyond the reproduction number for the control of HLB, i.e., \(R_{0}< R^{c}_{0}<1\). The existence of the backward bifurcation illustrates that the long term HLB activity in a citrus orchard depends on the initial population sizes of citrus trees and psyllids.
4 Permanence
In this section, we demonstrate the permanence of system (2.1). We first give some notations and a lemma.
Let K̃ be a matrix space, \(f:\tilde{K} \rightarrow\tilde{K}\) be a continuous map, and \(K_{0}\subseteq\tilde{K}\) be an open set.
Lemma 4.1
(See [33])
 (I)
\(f(K_{0})\subset K_{0}\) and f has a global attractor A.
 (II)The maximal compact invariant set \(A_{\partial}=A\bigcap M_{ \partial}\) of f in \(\partial K_{0}\), possibly empty, has an acyclic covering \(\tilde{M}=\{M_{1},\ldots,M_{k}\}\) with the following properties:
 (a)
\(M_{i}\) is isolated in K̃.
 (b)
\(W^{s}(M_{i})\bigcap K_{0}=\emptyset\) for each \(1\leq i\leq k\).
 (a)
Theorem 4.1
If \(R_{0}>1\), system (2.1) is permanent.
Proof
To apply Lemma 4.1, for system (2.1), we define \(\tilde {K}=\{(X,Y,Z,U,S,I) \in R^{6}_{+}\}\), \(K_{0}=\{(X,Y,Z,U,S,I)\in\tilde{K} : X\geq0,Y \geq0,Z>0,U>0,S\geq0,I>0\}\), and \(\partial K_{0}= \tilde{K} \backslash K_{0}\), and denote \(u(t,x^{0})\) as the unique solution of system (2.1) with the initial value \(x^{0}=(X^{0},Y^{0},Z^{0},U^{0}, S^{0},I^{0})\).
Case (i). \(Z^{0}=0\), \(U^{0}>0\), \(I^{0}>0\). It is obvious that \(X(t)>0\), \(I(t)>0\) for any \(t>0\). Then, from the third equation of system (2.1), \(\frac{dZ}{dt}_{t=0}=\beta_{1}X(0)I(0)>0\) holds. It follows that \((X,Y,Z,U,S,I)\notin\partial K_{0}\) for \(0< t\ll1\).This is a contradiction.
Similarly, we can prove the other cases: (a) \(Z^{0}>0\), \(U^{0}=0\), \(I^{0}>0\), and (b) \(Z^{0}>0\), \(U^{0}>0\), \(I^{0}=0\).
Case (ii). \(Z^{0}=U^{0}=0\), \(I^{0}>0\). It is obvious that \(X(t)>0\), \(Y(t)>0\), \(I(t)>0\) for any \(t>0\). Then, from the third, fourth equations of system (2.1), we get \(\frac{dZ}{dt}_{t=0}=\beta _{1}X(0)I(0)>0\), \(\frac{dU}{dt}_{t=0}=\beta_{2}Y(0)I(0)>0\) hold. It follows that \((X,Y,Z,U,S,I)\notin\partial K_{0}\) for \(0< t\ll1\). This is a contradiction.
Similarly, we can prove the other cases: (c) \(Z^{0}=I^{0}=0\), \(U^{0}>0\), and (d) \(U^{0}=I^{0}=0\), \(Z^{0}>0\).
We can easily obtain that P has a global attractor \(E_{0}\). It is easy to obtain that \(E_{0}\) is an isolated invariant set in K̃ and \(W^{s}(E_{0})\cap K_{0}=\emptyset\). We know that \(E_{0}\) is acyclic in \(M_{\partial}\), and every solution in \(M_{\partial}\) converges to \(E_{0}\). According to Zhao [33], we have that P is uniformly persistent with respect to \((K_{0},\partial K_{0})\). This implies that the solution of system (2.1) is uniformly persistent with respect to \((K_{0},\partial K_{0})\). This completes the proof. □
5 Numerical simulation

The average life expectancy of trees was from 20 years to 30 years [35]. We can take 25 as the current average life expectancy. Thus, the natural death rate of citrus trees \(d=\frac{1}{25}=0.04\).

The annual average temperature of Jiangxi Province is from 8.6 to 20.6 degree centigrade in 2010. So we can choose the average temperature \(T=17^{\circ}\mathrm{C}\).

It follows from the literature [36–38] that the natural mortality of psyllids is taken as the form \(\mu=\frac{1}{L}365\), in which \(L=0.14221\ast T^{2}+4.31998\ast T+31.25498\). Thus, we can get \(\mu=5.7394\).

The maximum number of trees that can be planted in the grove is 2,000, i.e., \(K=2{,}000\). According to the implementation of control measures for HLB in the South of Jiangxi, we take the replanting rate \(\alpha_{1}=0.6\).

Using the same transmission forms in [36], we get \(\beta_{1}=\frac {0.45625(d+ \alpha_{1})}{\alpha_{1}K}=0.000243333\), \(\beta_{3}=\frac{0.365(d+ \alpha_{1})}{\alpha_{1}K}=0.000194667\). Further, we take \(\beta_{2}=0.015 \beta_{1}\) and \(\beta_{4}=1.1\beta_{3}\), then \(\beta_{2}=0.00000365\) and \(\beta_{4}=0.000214133\).

From literature [36] and [37], we know the recruitment rate of psyllid \(\Lambda=\frac{3\alpha_{1} K\ast \mathit{EFD}\ast P_{ea} \ast \mathit{MDR}}{\mu(d+\alpha_{1})}\), where \(\mathit{EFD}=0.0107\ast365\ast T\ast(T13) \ast\sqrt{30.8T}\), \(P_{ea}=0.47192+0.0109\ast T\), \(\mathit{MDR}=5.286 \ast10^{5}\ast365\ast T\ast(T10.02)\ast\sqrt{34.17T}\), then we can calculate \(\Lambda=6{,}028{,}433\).
Parameter values used for numerical simulations of the HLB model
Parameters  Values  Unit  References 

d  0.04  year^{−1}  [35] 
μ  5.7394  year^{−1}  [36] 
K  2,000    [36] 
\(\alpha_{1}\)  0.6  year^{−1}  Estimation 
\(\beta_{1}\)  0.000243333  year^{−1}  [36] 
\(\beta_{2}\)  0.00000365  year^{−1}  Estimation 
\(\beta_{3}\)  0.000194667  year^{−1}  [36] 
\(\beta_{4}\)  0.000214133  year^{−1}  Estimation 
Λ  6,028,433  year^{−1}  [36] 
α  0 ∼ 1  year^{−1}   
m  0 ∼ 1  year^{−1}   
Finally, numerical simulations are carried out to illustrate the effectiveness of the obtained results. For the simulations that follow, we applied this set of parameters shown in Table 1 unless otherwise stated.
6 Conclusions
In this paper, a deterministic model with bilinear incidence is formulated to study the impact of cross protection on the spread and control of HLB. When we choose appropriate parameters, there exists a backward bifurcation. If \(R_{0}>1\), then there is a unique endemic equilibrium and the disease is uniformly persistent. If \(R_{0}<1\), there may be two endemic equilibria, and the endemic equilibrium can coexist with the diseasefree equilibrium. This illustrates that \(R_{0}<1\) cannot ensure the eradication of the disease, and decreasing \(R_{0}\) below the subthreshold \(R^{c}_{0}\) would be a propositional control strategy. If \(R^{c}_{0}< R_{0}<1\), only when the numbers of infected cases are small enough, it is a sufficient condition to eliminate HLB. Numerical examples are given to demonstrate the effectiveness of the theoretical results.
Our investigations suggest that cross protection and removing infected trees play an important role in controlling the spread of HLB. Cross protection also dramatically affects the disease transmission dynamics. Moreover, increasing the replanting rate is bad for disease control. The result strongly suggests and supports the previous observations [39, 40].
Declarations
Acknowledgements
The research has been supported by the National Natural Science Foundation of China (11561004) and the Natural Science Foundation of Jiangxi Province (20171BAB201006).
Authors’ contributions
The main idea and theoretical proof of this paper were proposed by LL and SG. Programming of numerical simulation was completed by YL and YG. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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