Transmission dynamics and optimal control of stage-structured HLB model

Yunbo Tu; Shujing Gao; Yujiang Liu; Di Chen; Yan Xu; Yunbo Tu; Shujing Gao; Yujiang Liu; Di Chen; Yan Xu

doi:10.3934/mbe.2019259

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5180-5205. doi: 10.3934/mbe.2019259

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Research article Special Issues

Transmission dynamics and optimal control of stage-structured HLB model

1.
Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou 341000, China
2.
National Research Center of Navel Orange Engineering and Technology, Gannan Normal University, Ganzhou 341000, China

Received: 25 March 2019 Accepted: 30 May 2019 Published: 06 June 2019

Citrus Huanglongbing (HLB) is one of severe quarantine diseases affecting citrus production both in abroad and domestic. Based on the mechanism and characteristics of citrus HLB transmission, we establish a vector-borne model with stage structure and integrated strategy and investigate the effect of the strategy in controlling the spread of HLB. By calculating, we obtain the basic reproductive number

$R_0$ , and prove that the disease can be eradicated if

$R_0 < 1$ , whereas the disease will persist if

$R_0>1$ . Meanwhile, we apply the optimal control theory to obtain an optimal integrated strategy. Finally, we use our model to simulate the data of the numbers of inspected and infected citrus trees in "Yuan Orchard", located in Ganzhou City, Jiangxi Province in the southeast of P.R China. We also give some numerical simulations for our theoretical findings.

Keywords:

Citation: Yunbo Tu, Shujing Gao, Yujiang Liu, Di Chen, Yan Xu. Transmission dynamics and optimal control of stage-structured HLB model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5180-5205. doi: 10.3934/mbe.2019259

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Abstract

$R_0$ , and prove that the disease can be eradicated if

$R_0 < 1$ , whereas the disease will persist if

References

[1]	D. Farnsworth, K. A. Grogan, A. H. C. V. Bruggen, et al., The potential economic cost and response to greening in Florida citrus, Agric. Appl. Econ. Assoc., 29 (2014): 1–6.
[2]	J. M. Bove, Huanglongbing: A destructive, newly-emerging, century-old disease of citrus, J. Plant. Pathol., 88 (2006): 7–37.
[3]	S. Alvarez, D. Solis and M. Thomas, Can Florida's citrus industry be saved while preserving the environment? An economic analysis for the bio-control of the Asian Citrus Psyllid, S. Agric. Econ. Assoc., 2015.
[4]	H. Su, Research and health management of citrus Huanglongbing in Taiwan, In: Proc Intern Research Conference on Huanglongbing, Orlando, Florida, USA, (2008): 57–92.
[5]	A. J. Ayres, J. B. Jr and J. M. Bové, The experience with Huanglongbing management in Brazil, Acta. Hortic., 1065 (2015): 55–62.
[6]	W. Li, Y. N. Yao, L. Wu, et al., Detection and seasonal variations of Huanglongbing disease in navel orange trees using direct lonization mass spectrometry, J. Agric. Food. Chem. , 67 (2019): 2265–2271.
[7]	C. Yang, H. Chen, H. Chen, et al., Antioxidant and anticancer activities of essential oil from Gannan navel orange peel, Molecules, 22 (2017): 1931–1400.
[8]	G. Rao, L. Huang, M. H. Liu, et al., Identification of Huanglongbing-infected navel oranges based on laser-induced breakdown spectroscopy combined with different Chemometric methods, Appl. Optics., 57 (2018): 8738–8742.
[9]	Y. H. Liu and J. H. Tsai, 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.
[10]	T. H. Hung, S. C. Hung, C. N. Chen, et al., Detection by PCR of Candidatus Liberibacter asiaticus,the bacterium causing citrus huanglongbing in vector psyllids: application to the study of vector pathogen relationships, Plant Pathol., 53 (2004): 96–102.
[11]	F. Wu, Y. J. Cen, X. L. Deng, et al., Movement of Diaphorina citri(Hemiptera: Liviidae) adults between huanglongbing infected and healthy citrus, Flor. Entomol., 98 (2015): 410–416.
[12]	W. Shen, S. E. Halbert, E. Dickstein, et al., Occurrence and ingrove distribution of citrus huang- longbing in north central Florida, J. Plant. Pathol., 95 (2003): 361–371.
[13]	K. L. Manjunath, S. E. Halbert, C. Ramadugu, et al., Detection of Candidatus Liberibacter asiaticus in Diaphorina citri and its importance in the management of citrus huanglongbing in Florida, Phytopathology, 98 (2008): 387–396.
[14]	E. E. G. Cardwell, L. L. Stelinski and P. A. Stansly, Biology and man agement of Asian Citrus Psyllid,vector of the Huanglongbing pathogens, Annu. Rev. Entomol., 58 (2013): 413–432.
[15]	X. Z. Meng and Z. Q. Li, The dynamics of plant disease models with continuous and impulsive cultural control strategies, J. Theor. Biol., 266 (2010): 29–40.
[16]	X. S. Zhang and J. Holt, Mathematical models of cross protection in the epidemiology of plant- virus disease, Anal. Theo. Plant. Pathol., 91 (2001): 924–934.
[17]	S. J. Gao, L. J. Xia, Y. Liu, et al., A plant virus disease model with periodic environment and pulse roguing, Stud. Appl. Math., 136 (2016): 357–381.
[18]	S. Y. Tang, Y. N. Xiao and R. A. Cheke, Dynamical analysis of plant disease models with cultural control strategies and economic thresholds, Math. Comput. Simul., 80 (2010): 894–921.
[19]	M. S. Chan and M. J. Jeger, An analytical model of plant virus disease dynamics with roguing and replanting, J. Appl. Ecol., 31 (1994): 413–427.
[20]	C. Chiyaka, B. H. Singer, S. E. Halbert, et al., Modeling huanglongbing transmission within a citrus tree, PNAS, 109 (2012): 12213–12218.
[21]	R. A. Taylor, E. A. Mordecai, C. A. Gilligan, et al., Mathematical models are a powerful method to understand and control the spread of Huanglongbing, Peer J, 4 (2016): DOI 10.7717/peerj.2642.
[22]	K. Jacobsen, J. Stupiansky and S. S. Pilyugin, Mathematical modeling of citrus groves infected by Huanglongbing, Math. Biosci. Eng, 10 (2013): 705–728.
[23]	R. G. dA. Vilamiu, S. Ternes, G. A. Braga, et al., A model for Huanglongbing spread between citrus plants including delay times and human intervention, AIP. Conf. Proc., 1479 (2012): 2315–2319.
[24]	S. J. Gao, L. Luo, S. X. Yan, et al., Dynamical behavior of a novel impulsive switching model for HLB with seasonal fluctuations, Complexity, 2018 (2018): 1–11.
[25]	J.A. Lee, S. E. Halbert, W. O. Dawson, et al., Asymptomatic spread of Huanglongbing and implications for disease control, Proc. Natl. Acad. Sci. USA., 112 (2015): 7605–7610.
[26]	M. Parry, G. J. Gibson, S. Parnell, et al., Bayesian inference for an emerging arboreal epidemic in the presence of control, Proc. Natl. Acad. Sci. USA., 111 (2014): 6258–6262.
[27]	S. E. Halbert and K. L. Manjunath, Asian citrus psyllids (Sternorrhyncha: Psyllidae) and greening disease of citrus: A literature review and assessment of risk in Florida, Flor. Entomol., 87 (2004): 330–353.
[28]	W. D. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008): 699–717.
[29]	P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002): 29–48.
[30]	H. R. Thieme, Convergence results and poincare-bendixson trichotomy for asymptotically au-tonomous differential equations, J. Math. Biol., 30 (1992): 755–763.
[31]	P. V. D. Driessche and J. Watmough, Reproductive numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002): 29–48.
[32]	D. Xu and X. Q. Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl., 311 (2005): 417–438.
[33]	X. Zhao, Dynamical Systems in Population Biology, Springer, New York, NY, USA, 2003.
[34]	T. K. Kar and A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems, 104 (2011): 127–135.
[35]	L. Y. Pang, S. G. Ruan, S. H. Liu, et al. , Transmission dynamics and optimal control of measles epidemics, Appl. Math. Comput., 256 (2015): 131–147.
[36]	M. A. Khan, R. Khan, Y. Khan, et al., A mathematical analysis of Pine Wilt disease with variable population size and optimal control strategies, Chaos, Solitons and Fract., 108 (2018): 205–217.
[37]	W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, Berlin, 1975.
[38]	D. L. Lukes, Differential equations: Classical to Controlled, Mathematics in Science and Engi-neering, Academic Press, New York, 1982.
[39]	S. Lenhart and J. T. Workman, Optimal control applied to biological models, Mathematical and Computational Biology Series, Chapman & Hall/CRC Press, London/Boca Raton, 2007.
[40]	X. Zhang, Y. Zhao and A. Neumann, Partial immunity and vaccination for influenza, J. Comput. Biol., 17 (2009): 1689–1696.
[41]	T. Gottwald, Current epidemiological understanding of citrus Huanglongbing, Annu. Rev. Phy-topathol., 48 (2010): 119–139.
[42]	K. S. P. Stelinski, R. H. Brlansky, T. A. Ebert, et al., Transmission parameters for Candidatus Liberibacter asiaticus by Asian citrus psyllid (Hemiptera: Psyllidae), J. Econ. Entomol., 103 (2010): 1531–1541.
[43]	M. E. Rogers, General pest management considerations, Citrus. Ind., 89 (2008): 12–15.

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