Transmission dynamics and optimal control of a Huanglongbing model with time delay

Zhenzhen Liao; Shujing Gao; Shuixian Yan; Genjiao Zhou; Zhenzhen Liao; Shujing Gao; Shuixian Yan; Genjiao Zhou

doi:10.3934/mbe.2021209

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 4162-4192. doi: 10.3934/mbe.2021209

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Research article Special Issues

Transmission dynamics and optimal control of a Huanglongbing model with time delay

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

Academic Editor: Masaki Ogura

Received: 24 March 2021 Accepted: 26 April 2021 Published: 12 May 2021

In this paper, a mathematical model has been formulated for the transmission dynamics of citrus Huanglongbing considering latent period as the time delay factor. Existence of the equilibria and their stability have been studied on the basis of basic reproduction number in two cases $ \tau = 0 $ and $ \tau > 0 $. The results show that stability changes occur through Hopf bifurcation in the delayed system. Optimal control theory is then applied to investigate the optimal strategy for curtailing the spread of the disease using three time-dependent control variables determined from sensitivity analysis. By using Pontryagin's Maximum Principle, we obtain the optimal integrated strategy and prove the uniqueness of optimal control solution. Analytical and numerical findings suggest that it is feasible to implement control techniques while minimizing the cost of implementation of optimal control strategies.
- basic reproduction number,
- stability,
- Hopf bifurcation,
- sensitivity analysis,
- uniqueness
Citation: Zhenzhen Liao, Shujing Gao, Shuixian Yan, Genjiao Zhou. Transmission dynamics and optimal control of a Huanglongbing model with time delay[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 4162-4192. doi: 10.3934/mbe.2021209

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Abstract

In this paper, a mathematical model has been formulated for the transmission dynamics of citrus Huanglongbing considering latent period as the time delay factor. Existence of the equilibria and their stability have been studied on the basis of basic reproduction number in two cases $ \tau = 0 $ and $ \tau > 0 $. The results show that stability changes occur through Hopf bifurcation in the delayed system. Optimal control theory is then applied to investigate the optimal strategy for curtailing the spread of the disease using three time-dependent control variables determined from sensitivity analysis. By using Pontryagin's Maximum Principle, we obtain the optimal integrated strategy and prove the uniqueness of optimal control solution. Analytical and numerical findings suggest that it is feasible to implement control techniques while minimizing the cost of implementation of optimal control strategies.

References

[1]	J. M. Bov$\acute{e}$, Huanglongbing: a destructive, newly-emerging, century-old disease of citrus, J. Plant. Pathol., 88 (2006), 7-37.
[2]	S. Gao, D. Yu, X. Meng, F. Zhang, Global dynamics of a stage-structured Huanglongbing model with time delay, Chaos. Soliton. Fract., 117 (2018), 60-67. doi: 10.1016/j.chaos.2018.10.008
[3]	K. Jacobsen, J. Stupiansky, S. S. Pilyugin, Mathematical modeling of citrus groves infected by Huanglongbing, Math. Biosci. Eng., 10 (2013), 705-728. doi: 10.3934/mbe.2013.10.705
[4]	C. Zhou, The status of citrus Huanglongbing in China, Trop. Plant. Pathol., 45 (2020), 279-284. doi: 10.1007/s40858-020-00363-8
[5]	J. M. Bov$\acute{e}$, M. E. Rogers, Huanglongbing-control workshop: summary, Acta. Hort., 1065 (2015), 55-61.
[6]	G. Fan, B. Liu, R. Wu, T. Li, Z. Cai, C. Ke, Thirty years of research on citrus Huanglongbing in China, Chin. Agri. Sci. Bull., 24 (2009), 183-190. (in Chinese)
[7]	T. R. Gottwald, Current epidemiological understanding of citrus Huanglongbing, Annu. Rev. Phytopathol., 48 (2010), 119-139. doi: 10.1146/annurev-phyto-073009-114418
[8]	S. E. Halbert, 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. doi: 10.1653/0015-4040(2004)087[0330:ACPSPA]2.0.CO;2
[9]	C. Chiyaka, B. H. Singer, S. E. Halbert, J. G. Morris, A. H. C. van Bruggen, Modeling huanglongbing transmission within a citrus tree, P. Natl. Acad. Sci. USA., 109 (2012), 12213-12218. doi: 10.1073/pnas.1208326109
[10]	F. AI Basir, S. Ray, Role of media coverage and delay in controlling infectious diseases: A mathematical model, Appl. Math. Comput., 337 (2018), 372-385.
[11]	M. Jackson, B. M. Chen-Charpentier, Modeling plant virus propagation with delays, J. Comput. Appl. Math., 309 (2017), 611-621. doi: 10.1016/j.cam.2016.06.009
[12]	F. AI Basir, Y. Takeuchi, S. Ray, Dynamics of a delayed plant disease model with Beddington-DeAngelis disease transmission, Math. Biosci. Eng., 18 (2020), 583-599.
[13]	S. Ray, F. AI Basir, Impact of incubation delay in plant-vector interaction, Math. Comput. in Simulat., 170 (2020), 16-31. doi: 10.1016/j.matcom.2019.09.001
[14]	G. Fan, J. Liu, P. V. D. Driessche, J. Wu, H. Zhu, The impact of maturation delay of mosquitoes on the transmission of West Nile virus, Math. Biosci., 228 (2010), 119-126. doi: 10.1016/j.mbs.2010.08.010
[15]	R. G. D. Vilamiu, S. Ternes, G. A. Braga, F. F. Laranjeira, A model for Huanglongbing spread between citrus plants including delay times and human intervention, Numer. Anal. Appl. Math., 1479 (2012), 2315-2319.
[16]	F. B. Agusto, M. A. Khan, Optimal control strategies for dengue transmission in pakistan, Math. Biosci., 305 (2018), 102-121. doi: 10.1016/j.mbs.2018.09.007
[17]	M. Rafikov, L. Bevilacqua, A. P. P. Wyse, Optimal control strategy of malaria vector using genetically modified mosquitoes, J. Theor. Biol., 258 (2009), 418-425. doi: 10.1016/j.jtbi.2008.08.006
[18]	K. S. Lee, A. A. Lashari, Stability analysis and optimal control of pine wilt disease with horizontal transmission in vector population, Appl. Math. Comput., 226 (2014), 793-804.
[19]	Y. Tu, S. Gao, Y. Liu, D. Chen, Y. Xu, Transmission dynamics and optimal control of stage-structured HLB model, Math. Biosci. Eng., 16 (2019), 5180-5205. doi: 10.3934/mbe.2019259
[20]	F. Zhang, Z. Qiu, B. Zhong, T. Feng, A. Huang, Modeling citrus Huanglongbing transmission within an orchard and its optimal control, Math. Biosci. Eng., 17 (2020), 2048-2069. doi: 10.3934/mbe.2020109
[21]	G. Zaman, Y. H. Kang, I. H. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, J. Biosyst., 93 (2008), 240-249. doi: 10.1016/j.biosystems.2008.05.004
[22]	Y. Pei, M. Chen, X. Liang, Z. Xia, Y. Lv, C. Li, Optimal control problem in an epidemic disease SIS model with stages and delays, Int. J. Biomath., 9 (2016), 131-152.
[23]	Z. Xu, X. Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete. Cont. Dyn-B., 17 (2012), 2615-2634.
[24]	J. Li, Z. Ma, F. Zhang, Stability analysis for an epidemic model with stage structure, J. Appl. Math. Comput., 9 (2008), 1672-1679.
[25]	J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, PA, 1976.
[26]	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. doi: 10.2307/1403510
[27]	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. doi: 10.1016/j.jtbi.2008.04.011
[28]	M. D. Mckay, R. J. Beckman, W. J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 42 (2000), 55-61. doi: 10.1080/00401706.2000.10485979
[29]	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. doi: 10.1093/oxfordjournals.aje.a009076
[30]	N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, B. Math. Biol., 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0
[31]	L. Luo, S. Gao, Y. Ge, Y. Luo, Transmission dynamics of a Huanglongbing model with cross protection, Adv. Differ. Equ-NY., 2017 (2017), 1-21. doi: 10.1186/s13662-016-1057-2
[32]	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.
[33]	X. Deng, Formming process and basis and technological points of the theory emphasis on control citrus psylla for integrated control Huanglongbing, Chin. Agri. Sci. Bull., 25 (2009), 358-363. (in Chinese)
[34]	J. Hale, Theory of Functional Sifferential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1977.
[35]	R. Gamkrelidze, L. S. Pontryagin, V. G. Boltjanskij, The Mathematical Theory of Optimal Processes, Macmillan Company, 1964.
[36]	W. H. Fleming, R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York/Berlin, 1975.
[37]	C. T. H. Baker, C. A. H. Paul, D. R. Will, Issues in the numerical solution of evolutionary delay differential equations, Adv. Comput. Math., 3 (1995), 171-196. doi: 10.1007/BF03028370
[38]	S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC, Boca Raton, FL, 2007.
[39]	A. Naeem, M. B. S. Afzal, S. Freed, F. Hafeez, S. M. Zaka, Q. Ali, et al., First report of thiamethoxam resistance selection, cross resistance to various insecticides and realized heritability in Asian citrus psyllid Diaphorina citri from Pakistan, Crop Prot., 121 (2019), 11-17. doi: 10.1016/j.cropro.2019.03.004
[40]	F. Tian, X. Mo, S. A. Rizvi, C. Li, X. Zeng, Detection and biochemical characterization of insecticide resistance in field populations of Asian citrus psyllid in Guangdong of China, Sci. Rep., 8 (2018), 12587. doi: 10.1038/s41598-018-30674-5
[41]	E. Wang, D. Li, Study on monitoring and control technology of citrus Huanglongbing in citrus orchards, Chin. Agri. Sci. Bull., 28 (2012), 278-282. (in Chinese)

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