doi: 10.3934/dcdsb.2021302
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Global dynamics of a Huanglongbing model with a periodic latent period

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

*Corresponding author: Xiao Yu

Received  July 2021 Revised  September 2021 Early access December 2021

Fund Project: The third author is supported in part by the National Natural Science Foundation of China; and Guangdong Basic and Applied Basic Research Foundation Province (12001205, 12026602 and 2019A1515110179)

Huanglongbing (HLB) is a disease of citrus that caused by phloem-restricted bacteria of the Candidatus Liberibacter group. In this paper, we present a HLB transmission model to investigate the effects of temperature-dependent latent periods and seasonality on the spread of HLB. We first establish disease free dynamics in terms of a threshold value $ R^p_0 $, and then introduce the basic reproduction number $ \mathcal{R}_0 $ and show the threshold dynamics of HLB with respect to $ R^p $ and $ \mathcal{R}_0 $. Numerical simulations are further provided to illustrate our analytic results.

Citation: Yan Hong, Xiuxiang Liu, Xiao Yu. Global dynamics of a Huanglongbing model with a periodic latent period. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021302
References:
[1]

J. M. Bové, Huanglongbing: Adestructive, newly-emerging, century-old disease of citrus, J. Plant Pathol., 88 (2006), 7-37. 

[2]

L. CaiX. LiB. Fang and S. Ruan, Global properties of vector-host disease models with time delays, J. Math. Biol., 74 (2017), 1397-1423.  doi: 10.1007/s00285-016-1047-8.

[3]

C. ChiyakaB. H. SingerS. E. HalbertJ. G. Morris and A. H. C. van Bruggen, Modeling huanglongbing transmission within a citrus tree, PNAS, 109 (2012), 12213-12218.  doi: 10.1073/pnas.1208326109.

[4]

K.-R. Chung and R. H. Brlansky, Citrus diseases exotic to Florida: Huanglongbing (citrus greening), Institute of Food and Agricultural Sciences, 7 (2005), 210.  doi: 10.32473/edis-pp133-2005.

[5]

H. I. Freedman and J. Wu, Periodic solutions of single-spaces models with periodic delay, SIAM J. Math. Anal., 23 (1992), 689-701.  doi: 10.1137/0523035.

[6]

S. GaoD. YuX. Meng and F. Zhang, Global dynamics of a stage-structured Huanglongbing model with time delay, Chaos Solitons Fractals, 117 (2018), 60-67.  doi: 10.1016/j.chaos.2018.10.008.

[7]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[8]

D. G. Hall and M. G. Hentz, Seasonal flight activity by the Asian citrus psyllid in east central Florida, Entomologia et Applicata., 139 (2011), 75-85.  doi: 10.1111/j.1570-7458.2011.01108.x.

[9]

K. JacobsenJ. Stupiansky and S. S. Pilyugin, Mathematical modelling of citrus groves infected by huanglongbing, Math. Bioscien. Engine., 10 (2013), 705-728.  doi: 10.3934/mbe.2013.10.705.

[10]

J. A. LeeS. E. HalbertW. O. DawsonC. J. RobertsonJ. E. Keesling and B. H. Singer, Asymptomatic spread of huanglongbing and implications for disease control, PNAS, 112 (2015), 7605-7610.  doi: 10.1073/pnas.1508253112.

[11]

F. LiJ. Liu and X.-Q. Zhao, A West Nile virus model with vertical transmission and periodic time delays, J. Nonlinear Sci., 30 (2020), 449-486.  doi: 10.1007/s00332-019-09579-8.

[12]

X. LiangL. Zhang and X.-Q. Zhao, The principal eigenvalue for periodic nonlocal dispersal systems with time delay, J. Dynam. Diff. Eqns., 266 (2019), 2100-2124.  doi: 10.1016/j.jde.2018.08.022.

[13]

Y. Liu and J. H. Tsal, Effects of temperature on biology and life table parameters of the Asian citrus psyllid, Diaphorina citri Kuwayama (Homoptera: Psyllidae), Ann. appl. Biol., 137 (2000), 201-206.  doi: 10.1111/j.1744-7348.2000.tb00060.x.

[14]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573-603.  doi: 10.1007/s00332-016-9344-3.

[15]

S. A. LopesF. LuizE. C. MartinsC. G. FassiniM. C. SousaJ. C. Barbosa and A. Beattie, Candidatus liberibacter asiaticus' titers in citrus and acquisition rates by diaphorina citri are decreased by higher temperature, Plant Disease, 97 (2013), 1563-1570.  doi: 10.1094/PDIS-11-12-1031-RE.

[16]

Q. LuB. Yan and D. Zhao, Temporal and spatial variation characteristics of climate in Jiangxi Province from 1961 to 2016, Research of soil and water conservation, 26 (2019), 166-173. 

[17]

L. LuoS. GaoY. Ge and Y. Luo, Transmission dynamics of a Huanglongbing model with cross protection, Adv. Difference Equ., 355 (2017), 1-21.  doi: 10.1186/s13662-017-1392-y.

[18]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[19]

S. MunirY. LiP. HeP. HeP. HeW. CuiY. WuX. Li and Y. He, Seasonal variation and detection frequency of Candidatus Liberibacter asiaticus in Binchuan, Yunnan province China, Physiological and Molecular Plant Pathology, 106 (2019), 137-144.  doi: 10.1016/j.pmpp.2019.01.004.

[20]

T. Nakatat, Temperature-dependent development of the citrus psyllid, Diaphorina citri (Homoptera: Psylloidea), and the predicted limit of its spread based on overwintering in the nymphal stage in temperate regions of Japan, Appl. Entomol. Zool., 41 (2006), 383-387. 

[21]

R. Omori and B. Adams, Disrupting seasonality to control disease outbreaks: The case of koi herpes virus, J. Theoret. Biol., 271 (2011), 159-165.  doi: 10.1016/j.jtbi.2010.12.004.

[22]

M. ParryG. J. GibsonS. ParnellT. R. GottwaldM. S. IreyT. C. Gast and C. A. Gilligan, Bayesian inference for an emerging arboreal epidemic in the presence of control, PNAS, 111 (2014), 6258-6262.  doi: 10.1073/pnas.1310997111.

[23]

M. QasimY. LinC. K. DashB. S. BamisileK. RavindranS. U. IslamH. AliF. Wang and L. Wang, Temperature-dependent development of Asian citrus psyllid on various hosts, and mortality by two strains of Isaria, Microbial Pathogenesis, 119 (2018), 109-118.  doi: 10.1016/j.micpath.2018.04.019.

[24]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[25]

R. A. TaylorE. A. MordecaiC. A. GilliganJ. R. Rohr and L. R. Johnson, Mathematical models are a powerful method to understand and control the spread of Huanglongbing, PeerJ, 4 (2016), 2315-2319.  doi: 10.7717/peerj.2642.

[26]

R. G. d'A. VilamiuS. TernesG. A. Braga and F. F. Laranjeira, A model for Huanglongbing spread between citrus plants including delay times and human intervention, AIP Conf. Proc., 1479 (2012), 2315-2319.  doi: 10.1063/1.4756657.

[27]

F. WangR. Wu and X.-Q. Zhao, A West Nile virus transmission model with periodic incubation periods, SIAM J. Appl. Dyn. Syst., 18 (2019), 1498-1535.  doi: 10.1137/18M1236162.

[28]

X. Wang and X.-Q. Zhao, A malaria transmission model with temperature-dependent incubation period, Bull. Math. Biol., 79 (2017), 1155-1182.  doi: 10.1007/s11538-017-0276-3.

[29]

J. WangS. GaoY. Luo and D. Xie, Threshold dynamics of a huanglongbing model with logistic growth in periodic environments, Abstr. Appl. Anal., 2014 (2014), 1-10.  doi: 10.1155/2014/841367.

[30]

X. WuF. M. G. MagpantayJ. Wu and X. Zou, Stage-structured population systems with temporally periodic delay, Math. Methods Appl. Sci., 38 (2015), 3464-3481.  doi: 10.1002/mma.3424.

[31]

D. Xu and X.-Q. Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl., 311 (2005), 417-438.  doi: 10.1016/j.jmaa.2005.02.062.

[32]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

[33]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Diff. Eqns., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

show all references

References:
[1]

J. M. Bové, Huanglongbing: Adestructive, newly-emerging, century-old disease of citrus, J. Plant Pathol., 88 (2006), 7-37. 

[2]

L. CaiX. LiB. Fang and S. Ruan, Global properties of vector-host disease models with time delays, J. Math. Biol., 74 (2017), 1397-1423.  doi: 10.1007/s00285-016-1047-8.

[3]

C. ChiyakaB. H. SingerS. E. HalbertJ. G. Morris and A. H. C. van Bruggen, Modeling huanglongbing transmission within a citrus tree, PNAS, 109 (2012), 12213-12218.  doi: 10.1073/pnas.1208326109.

[4]

K.-R. Chung and R. H. Brlansky, Citrus diseases exotic to Florida: Huanglongbing (citrus greening), Institute of Food and Agricultural Sciences, 7 (2005), 210.  doi: 10.32473/edis-pp133-2005.

[5]

H. I. Freedman and J. Wu, Periodic solutions of single-spaces models with periodic delay, SIAM J. Math. Anal., 23 (1992), 689-701.  doi: 10.1137/0523035.

[6]

S. GaoD. YuX. Meng and F. Zhang, Global dynamics of a stage-structured Huanglongbing model with time delay, Chaos Solitons Fractals, 117 (2018), 60-67.  doi: 10.1016/j.chaos.2018.10.008.

[7]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[8]

D. G. Hall and M. G. Hentz, Seasonal flight activity by the Asian citrus psyllid in east central Florida, Entomologia et Applicata., 139 (2011), 75-85.  doi: 10.1111/j.1570-7458.2011.01108.x.

[9]

K. JacobsenJ. Stupiansky and S. S. Pilyugin, Mathematical modelling of citrus groves infected by huanglongbing, Math. Bioscien. Engine., 10 (2013), 705-728.  doi: 10.3934/mbe.2013.10.705.

[10]

J. A. LeeS. E. HalbertW. O. DawsonC. J. RobertsonJ. E. Keesling and B. H. Singer, Asymptomatic spread of huanglongbing and implications for disease control, PNAS, 112 (2015), 7605-7610.  doi: 10.1073/pnas.1508253112.

[11]

F. LiJ. Liu and X.-Q. Zhao, A West Nile virus model with vertical transmission and periodic time delays, J. Nonlinear Sci., 30 (2020), 449-486.  doi: 10.1007/s00332-019-09579-8.

[12]

X. LiangL. Zhang and X.-Q. Zhao, The principal eigenvalue for periodic nonlocal dispersal systems with time delay, J. Dynam. Diff. Eqns., 266 (2019), 2100-2124.  doi: 10.1016/j.jde.2018.08.022.

[13]

Y. Liu and J. H. Tsal, Effects of temperature on biology and life table parameters of the Asian citrus psyllid, Diaphorina citri Kuwayama (Homoptera: Psyllidae), Ann. appl. Biol., 137 (2000), 201-206.  doi: 10.1111/j.1744-7348.2000.tb00060.x.

[14]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573-603.  doi: 10.1007/s00332-016-9344-3.

[15]

S. A. LopesF. LuizE. C. MartinsC. G. FassiniM. C. SousaJ. C. Barbosa and A. Beattie, Candidatus liberibacter asiaticus' titers in citrus and acquisition rates by diaphorina citri are decreased by higher temperature, Plant Disease, 97 (2013), 1563-1570.  doi: 10.1094/PDIS-11-12-1031-RE.

[16]

Q. LuB. Yan and D. Zhao, Temporal and spatial variation characteristics of climate in Jiangxi Province from 1961 to 2016, Research of soil and water conservation, 26 (2019), 166-173. 

[17]

L. LuoS. GaoY. Ge and Y. Luo, Transmission dynamics of a Huanglongbing model with cross protection, Adv. Difference Equ., 355 (2017), 1-21.  doi: 10.1186/s13662-017-1392-y.

[18]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[19]

S. MunirY. LiP. HeP. HeP. HeW. CuiY. WuX. Li and Y. He, Seasonal variation and detection frequency of Candidatus Liberibacter asiaticus in Binchuan, Yunnan province China, Physiological and Molecular Plant Pathology, 106 (2019), 137-144.  doi: 10.1016/j.pmpp.2019.01.004.

[20]

T. Nakatat, Temperature-dependent development of the citrus psyllid, Diaphorina citri (Homoptera: Psylloidea), and the predicted limit of its spread based on overwintering in the nymphal stage in temperate regions of Japan, Appl. Entomol. Zool., 41 (2006), 383-387. 

[21]

R. Omori and B. Adams, Disrupting seasonality to control disease outbreaks: The case of koi herpes virus, J. Theoret. Biol., 271 (2011), 159-165.  doi: 10.1016/j.jtbi.2010.12.004.

[22]

M. ParryG. J. GibsonS. ParnellT. R. GottwaldM. S. IreyT. C. Gast and C. A. Gilligan, Bayesian inference for an emerging arboreal epidemic in the presence of control, PNAS, 111 (2014), 6258-6262.  doi: 10.1073/pnas.1310997111.

[23]

M. QasimY. LinC. K. DashB. S. BamisileK. RavindranS. U. IslamH. AliF. Wang and L. Wang, Temperature-dependent development of Asian citrus psyllid on various hosts, and mortality by two strains of Isaria, Microbial Pathogenesis, 119 (2018), 109-118.  doi: 10.1016/j.micpath.2018.04.019.

[24]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[25]

R. A. TaylorE. A. MordecaiC. A. GilliganJ. R. Rohr and L. R. Johnson, Mathematical models are a powerful method to understand and control the spread of Huanglongbing, PeerJ, 4 (2016), 2315-2319.  doi: 10.7717/peerj.2642.

[26]

R. G. d'A. VilamiuS. TernesG. A. Braga and F. F. Laranjeira, A model for Huanglongbing spread between citrus plants including delay times and human intervention, AIP Conf. Proc., 1479 (2012), 2315-2319.  doi: 10.1063/1.4756657.

[27]

F. WangR. Wu and X.-Q. Zhao, A West Nile virus transmission model with periodic incubation periods, SIAM J. Appl. Dyn. Syst., 18 (2019), 1498-1535.  doi: 10.1137/18M1236162.

[28]

X. Wang and X.-Q. Zhao, A malaria transmission model with temperature-dependent incubation period, Bull. Math. Biol., 79 (2017), 1155-1182.  doi: 10.1007/s11538-017-0276-3.

[29]

J. WangS. GaoY. Luo and D. Xie, Threshold dynamics of a huanglongbing model with logistic growth in periodic environments, Abstr. Appl. Anal., 2014 (2014), 1-10.  doi: 10.1155/2014/841367.

[30]

X. WuF. M. G. MagpantayJ. Wu and X. Zou, Stage-structured population systems with temporally periodic delay, Math. Methods Appl. Sci., 38 (2015), 3464-3481.  doi: 10.1002/mma.3424.

[31]

D. Xu and X.-Q. Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl., 311 (2005), 417-438.  doi: 10.1016/j.jmaa.2005.02.062.

[32]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

[33]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Diff. Eqns., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

Figure 1.  Tendency of citrus trees when $ \mathcal{R}_0<1 $ and $ \mathcal{R}_0>1 $
Figure 2.  Behaviors of psyllid when $ \mathcal{R}_0<1 $ and $ \mathcal{R}_0>1 $
Figure 3.  infectious psyllid under different $ p $ and $ q $
Figure 4.  $ \mathcal{R}_0 $ vs r
Table 1.  Biological interpretations for parameters in system (1)
Parameter Description
$ r $ Rate of replanting citrus tree
$ K $ Maximum citrus tree population size
$ \beta_1(t) $ Infection rate of susceptible trees
$ \mu_1 $ Natural death rate of citrus tree population
$ \delta_1 $ Death rate of infected trees
$ \delta_2 $ Rate of removal of infected trees
$ \alpha(t) $ Intrinsic growth rate of psyllid
$ m $ Maximum abundance of psyllid per tree
$ d_v(t) $ Natural death rate of psyllid population
$ \beta_2(t) $ Infection rate of susceptible psyllid
$ \theta(t) $ Killing rate of psyllid with spraying insecticide
$ \tau $ Incubation period in trees
$ \tau_v(t) $ Extrinsic incubation period(EIP) of psyllid
Parameter Description
$ r $ Rate of replanting citrus tree
$ K $ Maximum citrus tree population size
$ \beta_1(t) $ Infection rate of susceptible trees
$ \mu_1 $ Natural death rate of citrus tree population
$ \delta_1 $ Death rate of infected trees
$ \delta_2 $ Rate of removal of infected trees
$ \alpha(t) $ Intrinsic growth rate of psyllid
$ m $ Maximum abundance of psyllid per tree
$ d_v(t) $ Natural death rate of psyllid population
$ \beta_2(t) $ Infection rate of susceptible psyllid
$ \theta(t) $ Killing rate of psyllid with spraying insecticide
$ \tau $ Incubation period in trees
$ \tau_v(t) $ Extrinsic incubation period(EIP) of psyllid
Table 2.  Parameter values in simulation
Parameter Value Unit Reference
$ r $ 0.05 month$ ^{-1} $ [17]
$ K $ 2000 - [17]
$ \beta_1(t) $ to be estimated month$ ^{-1} $ see text
$ \mu_1 $ 0.00333 month$ ^{-1} $ [29]
$ \delta_1 $ 0.0015 month$ ^{-1} $ [25]
$ \delta_2 $ 0.02 month$ ^{-1} $ [25]
$ \alpha(t) $ $ 18.120952 + 14.45466475\cos(2\pi t/12) $ month$ ^{-1} $ [29]
$ m $ $ 1\times 10^{6} $ - [29]
$ d_v(t) $ to be estimated month$ ^{-1} $ see text
$ \beta_2(t) $ to be estimated month$ ^{-1} $ see text
$ \theta(t) $ to be estimated month$ ^{-1} $ see text
$ \tau $ 6 month [25]
$ \tau_v(t) $ assumed month see text
Parameter Value Unit Reference
$ r $ 0.05 month$ ^{-1} $ [17]
$ K $ 2000 - [17]
$ \beta_1(t) $ to be estimated month$ ^{-1} $ see text
$ \mu_1 $ 0.00333 month$ ^{-1} $ [29]
$ \delta_1 $ 0.0015 month$ ^{-1} $ [25]
$ \delta_2 $ 0.02 month$ ^{-1} $ [25]
$ \alpha(t) $ $ 18.120952 + 14.45466475\cos(2\pi t/12) $ month$ ^{-1} $ [29]
$ m $ $ 1\times 10^{6} $ - [29]
$ d_v(t) $ to be estimated month$ ^{-1} $ see text
$ \beta_2(t) $ to be estimated month$ ^{-1} $ see text
$ \theta(t) $ to be estimated month$ ^{-1} $ see text
$ \tau $ 6 month [25]
$ \tau_v(t) $ assumed month see text
Table 3.  Monthly mean temperature in the South of Jiangxi, 1961-2016($ ^\circ C $)
Month Jan Feb Mar Apr May June
temperature 5.7 7.5 11.6 17.3 22.4 25.0
Month Jul Aug Sep Oct Nov Dec
temperature 28.3 27.6 24.5 19.2 13.2 7.4
Month Jan Feb Mar Apr May June
temperature 5.7 7.5 11.6 17.3 22.4 25.0
Month Jul Aug Sep Oct Nov Dec
temperature 28.3 27.6 24.5 19.2 13.2 7.4
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