Modeling Citrus Huanglongbing transmission within an orchard and its optimal control

Fumin Zhang; Zhipeng Qiu; Balian Zhong; Tao Feng; Aijun Huang; Fumin Zhang; Zhipeng Qiu; Balian Zhong; Tao Feng; Aijun Huang

doi:10.3934/mbe.2020109

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2048-2069. doi: 10.3934/mbe.2020109

Previous Article Next Article

Research article

Modeling Citrus Huanglongbing transmission within an orchard and its optimal control

1.
School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
2.
Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques/National Research Center of Navel Orange Engineering and Technology, Gannan Normal University, Ganzhou 341000, China

Received: 30 October 2019 Accepted: 23 December 2019 Published: 31 December 2019

Citrus Huanglongbing (HLB) is the most devastating citrus disease worldwide. In this paper, a deterministic dynamical model is proposed to explore the transmission dynamics of HLB between citrus tree and Asian citrus psyllid (ACP). Using the theory of dynamical system, the dynamics of the model are rigorously analyzed. The results show that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number $\mathscr{R}_0 < 1$, and when $\mathscr{R}_0 > 1$ the system is uniformly persistent. Applying the global sensitivity analysis of $\mathscr{R}_0$, some parameters that have the greatest impact on HLB transmission dynamics are obtained. Furthermore, the optimal control theory is applied to the model to study the corresponding optimal control problem. Both analytical and numerical results show that: (1) the infected ACP plays a decisive role in the transmission of HLB in citrus trees, and eliminating the ACP will be helpful to curtail the spread of HLB; (2) optimal control strategy is superior to the constant control strategy in decreasing the prevalence of the diseased citrus trees, and the cost of implementing optimal control is much lower than that of the constant control strategy; and (3) spraying insecticides is more effective than other control strategies in reducing the number of ACP in the early phase of the transmission of HLB. These theoretical and numerical results may be helpful in making public policies to control HLB in orchards more effectively.
- Citrus Huanglongbing,
- global stability,
- persistence,
- sensitivity analysis,
- optimal control
Citation: Fumin Zhang, Zhipeng Qiu, Balian Zhong, Tao Feng, Aijun Huang. Modeling Citrus Huanglongbing transmission within an orchard and its optimal control[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2048-2069. doi: 10.3934/mbe.2020109

Related Papers:

Abstract

Citrus Huanglongbing (HLB) is the most devastating citrus disease worldwide. In this paper, a deterministic dynamical model is proposed to explore the transmission dynamics of HLB between citrus tree and Asian citrus psyllid (ACP). Using the theory of dynamical system, the dynamics of the model are rigorously analyzed. The results show that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number $\mathscr{R}_0 < 1$, and when $\mathscr{R}_0 > 1$ the system is uniformly persistent. Applying the global sensitivity analysis of $\mathscr{R}_0$, some parameters that have the greatest impact on HLB transmission dynamics are obtained. Furthermore, the optimal control theory is applied to the model to study the corresponding optimal control problem. Both analytical and numerical results show that: (1) the infected ACP plays a decisive role in the transmission of HLB in citrus trees, and eliminating the ACP will be helpful to curtail the spread of HLB; (2) optimal control strategy is superior to the constant control strategy in decreasing the prevalence of the diseased citrus trees, and the cost of implementing optimal control is much lower than that of the constant control strategy; and (3) spraying insecticides is more effective than other control strategies in reducing the number of ACP in the early phase of the transmission of HLB. These theoretical and numerical results may be helpful in making public policies to control HLB in orchards more effectively.

References

[1]	J. M. Bové, Huanglongbing: a destructive, newly-emerging, century-old disease of citrus, J. Plant. Pathol., 88 (2006), 7-37.
[2]	R. A. Taylor, E. A. Mordecai, C. A. Gilligan, J. R. Rohr, L. R. Johnson, Mathematical models are a powerful method to understand and control the spread of Huanglongbing, Peer J., 19 (2016), 2642.
[3]	J. M. Bové, M. E. Rogers, Huanglongbing-control workshop: summary, Acta. Hort., 1065 (2015), 869-889.
[4]	A. J. Ayres, J. B. Jr, J. M. Bové, The experience with Huanglongbing management in Brazil, Acta. Hort., 1065 (2015), 55-61.
[5]	K. Chen, C. Li, Survey of citrus Huanglongbing research, J. Zhejiang Agr. Sci., 56 (2015), 1048-1050. (in Chinese)
[6]	C. Chiyaka, B. H. Singer, S. E. Halbert, J. G. M. Jr, A. H. C. van Bruggen, Modeling Huanglongbing transmission within a citrus tree, Proc. Natl. Acad. Sci. USA, 109 (2012), 12213-12218.
[7]	M. S. Chan, M. J. Jeger, An analytical model of plant virus disease dynamics with roguing and replanting, J. Appl. Ecol., 31 (1994), 413-427.
[8]	M. J. Jeger, L. V. Madden, F. van den Bosch, The effect of transmission route on plant virus epidemic development and disease control, J. Theor. Biol., 258 (2009), 198-207.
[9]	M. S. Sisterson, Effects of insect-vector preference for healthy or infected plants on pathogen spread: Insights from a model, J. Econ. Entomol., 101 (2008), 1-8.
[10]	X. Z. Meng, Z. Q. Li, The dynamics of plant disease models with continuous and impulsive cultural control strategies, J. Theor. Biol., 266 (2010), 29-40.
[11]	X. Z. Meng, Z. T. Song, L. S. Chen, A new mathematical model for optimal control strategies of integrated pest management, J. Biol. Syst., 15, (2007), 219-234.
[12]	W. C. Zhao, J. Li, T. Q. Zhang, X. Z. Meng, T. H. Zhang, Persistence and ergodicity of plant disease model with Markov conversion and impulsive toxicant input, Commun. Nonlinear Sci., 48 (2017), 70-84.
[13]	L. J. Xia, S. J. Gao, Q. Zou, J. P. Wang, Analysis of a nonautonomous plant disease model with latent period, Appl. Math. Comput., 223 (2013), 147-159.
[14]	S. J. Gao, L. J. Xia, J. L. Wang, Z. J. Zhang, Modelling the effects of cross protection control in plant disease with seasonality, Int. J. Biomath., 10 (2017), 1-24.
[15]	S. J. Gao, D. Yu, X. Z. Meng, F. M. Zhang, Global dynamics of a stage-structured Huanglongbing model with time delay, Chaos. Soliton. Fract., 117 (2018), 60-67.
[16]	X. S. Zhang, J. Holt, J. Colvin, Mathematical models of host plant infection by helper-dependent virus complexes: why are helper viruses always avirulent, Phytopathology, 90 (2000), 85-93.
[17]	A. U. Awan, M. Ozair, Stability analysis of pine wilt disease model by periodic use of insecticides, J. Biol. Dynam., 10 (2016), 506-524.
[18]	M. A. Khan, R. Khan, Y. Khan, S. Islam, A mathematical analysis of Pine Wilt disease with variable population size and optimal control strategies, Chaos Soliton Fract., 108 (2018), 205-217.
[19]	K. Jacobsen, J. Stupiansky, S. S. Pilyugin, Mathematical modeling of citrus groves infected by Huanglongbing, Math. Biosci. Eng., 10 (2013), 705-728.
[20]	X. S. Zhang, J. Holt, J. Colvin, Synergism between plant viruses: a mathematical analysis of the epidemiological implications, Plant Pathol., 50 (2001), 732-746.
[21]	K. S. Lee, D. Kim, Global dynamics of a pine wilt disease transmission model with nonlinear incidence rates, Appl. Math. Model, 37 (2013), 4561-4569.
[22]	M. Ozair, X. Y. Shi, T. Hussain, Control measures of pine wilt disease, Comp. Appl. Math., 35 (2016), 519-531.
[23]	M. A. Khan, K. Ali, E. Bonyah, K. O. Okosun, S. Islam, A. Khan, Mathematical modeling and stability analysis of Pine Wilt Disease with optimal control, Sci. Rep., 7 (2017), 3115.
[24]	M. A. Khan, R. Khan, Y. Khan, S. Islam, A mathematical analysis of Pine Wilt disease with variable population size and optimal control strategies, Chaos Soliton Fract., 108 (2018), 205-217.
[25]	J. A. Lee, S. E. Halbert, W. O. Dawson, C. J. Robertson, J. E. Keesling, B. H. Singer, Asymptomatic spread of Huanglongbing and implications for disease control, Proc. Natl. Acad. Sci. USA, 112 (2015), 7605-7610.
[26]	R. G. dA. Vilamiu, S. Ternes, G. A. Braga, F. F. Laranjeira, A model for Huanglongbing spread between citrus plants including delay times and human intervention, AIP Conf. Proc., 1479 (2012), 2315-2319.
[27]	Wikipedia, Available from: https://en.wikipedia.org/wiki/Citrus_greening-disease.
[28]	M. X. Deng, Forming process and basis and technological points of the theory emphasis on control citrus psylla for integrated control Huanglongbing, Chin. Agric. Sci. Bull., 25 (2009), 358-363.
[29]	O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathscr{R}_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
[30]	P. Dreessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
[31]	X. Q. Zhao, Dynamical Systems in Population Biology (Second Edition), Springer-Verlag, New York, 2017.
[32]	H. L. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, 1995.
[33]	C. Castillo-Chavez, H. Thieme, Asymptotically autonomous epidemic models, Math. Popul. Dyn.: Anal. Heterog., 1 (1995), 33-50.
[34]	H. R. Thieme, Persistence under relaxed point-dissipativity with an application to an epidemic model, SIAM J. Math. Anal., 24 (1993), 407-435.
[35]	R. Gamkrelidze, L. S. Pontrjagin, V. G. Boltjanskij, The Mathematical Theory of Optimal Processes, Macmillan Company, 1964.
[36]	S. M. Blower, H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, Int. Stat. Rev., 62 (1994), 229-243.
[37]	J. C. Helton, J. D. Johnson, C. J. Sallaberry, C. B. Storlie, Survey of sampling-based methods for uncertainty and sensitivity analysis, Reliab. Eng. Syst. Saf., 91 (2006), 1175-1209.
[38]	S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196.
[39]	M. A. Sanchez, S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate: tuberculosis as an example, Am. J. Epidemiol., 145 (1997), 1127-1137.
[40]	S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC, Boca Raton, FL, 2007.
[41]	X. F. Yan, Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Comput. Model., 47 (2008), 235-245.
[42]	F. B. Agusto, M. A. Khan, Optimal control strategies for dengue transmission in pakistan, Math. Biosci., 305 (2018), 102-121.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)