Jianping Wang^*, Feifei Feng, Zhicai Guo, Hengmin Lv, Juanjuan Wang

Show more
The Department of Basic Course Education, Ji’an College, Ji’an, China.
Show less

Abstract: In this paper, the basic reproduction number is calculated for Huanglongbing (HLB) model with impulses which is a vector-borne epidemic model with impulses. For controlling HLB, farmers’ experience is replanting of healthy plants and removing infected plants. To reflect the real world, we construct an impulsive control model which considers replanting of healthy plants and removing infected plants at one fixed time. By analyzing the model, we conclude that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number Ro < 1, and we prove that the HLB is permanence if the basic reproduction number Ro > 1.

Keywords: Huanglongbing, Impulsive Control, Basic Reproduction Number, Disease-Free Periodic Solution, Permanence

Cite this paper: Wang, J. , Feng, F. , Guo, Z. , Lv, H. and Wang, J. (2019) Threshold Dynamics of a Vector-Borne Epidemic Model for Huanglongbing with Impulsive Control. Applied Mathematics, 10, 196-211. doi: 10.4236/am.2019.104015.

References

[1]   Bové, J.M. (2006) Huanglongbing: A Destructive, Newly Emerging, Century-Old Disease of Citrus. Journal of Plant Pathology, 88, 7-37.

[2]   Vilamiu, R.G.A., Ternes, S., et al. (2012) A Model for Huanglongbing Spread between Citrus Plants Including Delay Times and Human Intervention.
https://doi.org/10.1063/1.4756657

[3]   Gao, S.J., Luo, L., Yan, S.X., et al. (2018) Dynamical Behavior of a Novel Impulsive Switching Model for HLB with Seasonal Fluctuations. Complexity, 2018, Article ID: 2953623.
https://doi.org/10.1155/2018/2953623

[4]   Luo, L., Gao, S.J., et al. (2017) Transmission Dynamics of a Huanglongbing Model with Cross Protection. Advances in Difference Equations, 2017, 355.
https://doi.org/10.1186/s13662-017-1392-y

[5]   Van der Plank, J.E. (1963) Plant Diseases: Epidemics and Control. Academic Press, New York.
https://doi.org/10.1097/00010694-196410000-00018

[6]   Kranz, J. (1974) The Role and Scope of Mathematical Analysis and Modeling in Epidemiology. In: Kranz, J., Ed., Epidemics of Plant Diseases; Mathematical Analysis and Modeling, Springer, Berlin, Heidelberg, New York, 7-54.
https://doi.org/10.1007/978-3-642-96220-2_2

[7]   Chiyaka, C., Singer, B.H., Halbert, S.E., Morris, J.G. and van Bruggen, A.H.C. (2012) Modeling Huanglongbing Transmission within a Citrus Tree. Proceedings of the National Academy of Sciences of the United States of America, 109, 12213-12218.
https://doi.org/10.1073/pnas.1208326109

[8]   Wang, J.P., Gao, S.J., et al. (2014) Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments. Abstract and Applied Analysis, 2014, Article ID: 841367.
https://doi.org/10.1155/2014/841367

[9]   Fishman, S. and Marcus, R. (1984) A Model for Spread of Plant Disease with Periodic Removals. Journal of Mathematical Biology, 21, 149-158.
https://doi.org/10.1007/BF00277667

[10]   Jeger, M.J., Holt, J., van den Bosch, F. and Madden, L.V. (2004) Epidemiology of Insect-Transmitted Plant Viruses: Modelling Disease Dynamics and Control Interventions. Physiological Entomology, 29, 291-304.
https://doi.org/10.1111/j.0307-6962.2004.00394.x

[11]   Zhang, X.S. and Holt, J. (2004) Mathematical Models of Cross Protection in the Epidemiology of Plant-Virus Diseases. Phytopathology, 91, 924-934.
https://doi.org/10.1094/PHYTO.2001.91.10.924

[12]   Tang, S.Y., Xiao, Y.N. and Cheke, R.A. (2010) Dynamical Analysis of Plant Disease Models with Cultural Control Strategies and Economic Thresholds. Mathematics and Computers in Simulation, 80, 894-921.
https://doi.org/10.1016/j.matcom.2009.10.004

[13]   Ayres, A.J., Massari, C.A., Lopes, S.A., et al. (2005) Proceedings of 2nd International Citrus Canker and Huanglongbing Research Workshop, Florida Citrus Mutual, Orlando, H-4.

[14]   Gao, S.J., Tu, Y.B. and Wang, J.L. (2018) Basic Reproductive Number for a General Hybrid Epidemic Model. Advances in Difference Equations, 2018, 310.
https://doi.org/10.1186/s13662-018-1707-7

[15]   Yang, Y.P. and Xiao, Y.N. (2012) Threshold Dynamics for Compartmental Epidemic Models with Impulses. Nonlinear Analysis: Real World Applications, 13, 224-234.
https://doi.org/10.1016/j.nonrwa.2011.07.028

[16]   Wang, W. and Zhao, X.-Q. (2008) Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments. Journal of Dynamics and Differential Equations, 20, 699-717.
https://doi.org/10.1007/s10884-008-9111-8

[17]   Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S. (1989) Theory of Impulsive Differential Equations. World Scientific, Singapore.
https://doi.org/10.1142/0906

[18]   Zhao, X.-Q. (2003) Dynamical System in Population Biology. Springer-Verlag, New York.