Transmission dynamics of a Huanglongbing model with cross protection
Luo et al. Advances in Difference Equations
Transmission dynamics of a Huanglongbing model with cross protection
Lei Luo
Shujing Gao
Yangqiu Ge
Youquan Luo
Huanglongbing (HLB) is one of the most common widespread vector-borne 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 R0 is established to measure whether or not the disease is uniformly persistent. The existence of a backward bifurcation presents a further sub-threshold condition below R0 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.
Huanglongbing model; cross protection; backward bifurcation; sub-threshold 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” []. 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
[]. The infected trees are usually destroyed or become unproductive in to years [].
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) [].
In order to control HLB effectively, most of growers usually take the following measures:
pesticides, tree removal, antibiotics [], changes to tree spacing, natural enemies of psyllid.
A few new intervention strategies are explored, including heat treatment [], new
tolerant or resistant tree stocks [], nutrient additions [], cross protection, intercropping [].
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 [] 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 []. In [], 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
M CTV strains (protecting) in citrus aurantif olia, Cui et al. [] proved the obvious
effect of cross protection on Bendizao mandarin. Van Vuuren et al. [] studied the effect
of cross protection on HLB of Africa by vaccinating multiple citrus recession viral strains.
Hartung [], who improved T 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 [–]). Meng and Li [] 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 [], 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. [] proposed
and compared two different control strategies in the model. In [], 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 [, , ]). Gao et al. [] 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. [] proposed a model to study cross protection
between the viruses in . Zhang and Holt [] improved the model in [], 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. Recen (...truncated)