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 AM  Vol.9 No.5 , May 2018
Global Dynamic Analysis of a Vector-Borne Plant Disease Model with Discontinuous Treatment
Abstract:
This paper proposes a vector-borne plant disease model with discontinuous treatment strategies. Constructing Lyapunov function and applying non-smooth theory to analyze discontinuous differential equations, the basic reproductive number R0 is proved, which determines whether the plant disease will be extinct or not. If R0 < 1 , the existence and global stability of disease-free equilibrium is established; If R0 > 1 , there exists a unique endemic equilibrium which is globally stable. The numerical simulations are provided to verify our theoretical results, which indicate that after infective individuals reach some level, strengthening treatment measures is proved to be beneficial in controlling disease transmission.
Cite this paper: Lv, H. , Fei, L. , Yuan, Z. and Zhang, F. (2018) Global Dynamic Analysis of a Vector-Borne Plant Disease Model with Discontinuous Treatment. Applied Mathematics, 9, 496-511. doi: 10.4236/am.2018.95036.
References

[1]   Lee, J.A., Halbert, S.E., Dawson, W.O., et al. (2015) Asymptomatic Spread of Huanglongbing and Implications for Disease Control. Proceedings of the National Academy of Sciences, 112, 7605-7610.
https://doi.org/10.1073/pnas.1508253112

[2]   Smith, A.B., Beeck, C.P., Cowling, W.A., et al. (2013) A Bivariate Mixed Model Approach for the Analysis of Plant Survival Data. Euphytica, 190, 371-383.
https://doi.org/10.1007/s10681-012-0791-0

[3]   Xia, L., Gao, S., Zou, Q., et al. (2013) Analysis of a Nonautonomous Plant Disease Model with Latent Period. Applied Mathematics and Computation, 223, 147-159.
https://doi.org/10.1016/j.amc.2013.08.011

[4]   Wang, J., Zhang, F., Wang, L., et al. (2016) Equilibrium, Pseudoequilibrium and Sliding-Mode Heteroclinic Orbit in a Filippov-Type Plant Disease Model. Nonlinear Analysis: Real World Applications, 31, 308-324.
https://doi.org/10.1016/j.nonrwa.2016.01.017

[5]   Zhao, W., Li, J., Zhang, T., et al. (2017) Persistence and Ergodicity of Plant Disease Model with Markov Conversion and Impulsive Toxicant Input. Communications in Nonlinear Science and Numerical Simulation, 48, 70-84.
https://doi.org/10.1016/j.cnsns.2016.12.020

[6]   Jackson, M. and Chen-Charpentier, B.M. (2016) Modeling Plant Virus Propagation with Delays. Journal of Computational and Applied Mathematics, 2016, 1-14.

[7]   Xue, Y. and Wang, J. (2012) Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment. Abstract and Applied Analysis, 2012, Article ID: 647853.
https://doi.org/10.1155/2012/647853

[8]   Hussaini, N. and Winter, M. (2010) Travelling Waves for an Epidemic Model with Non-Smooth Treatment Rates. Journal of Statistical Mechanics, 11, Article ID: 11019.
https://doi.org/10.1088/1742-5468/2010/11/P11019

[9]   Brauer, F. (2008) Epidemic Models with Heterogeneous Mixing and Treatment. Bulletin of Mathematical Biology, 70, 1869-1885.
https://doi.org/10.1007/s11538-008-9326-1

[10]   Hu, Z., Liu, S. and Wang, H. (2008) Backward Bifurcation of an Epidemic Model with Standard Incidence Rate and Treatment Rate. Nonlinear Analysis: Real World Applications, 9, 2302-2312.
https://doi.org/10.1016/j.nonrwa.2007.08.009

[11]   Wang, W. and Ruan, S. (2004) Bifurcations in an Epidemic Model with Constant Removal Rate of the Infectives. Journal of Mathematical Analysis and Applications, 291, 775-793.
https://doi.org/10.1016/j.jmaa.2003.11.043

[12]   Wang, W. (2006) Backward Bifurcation of an Epidemic Model with Treatment. Mathematical Biosciences, 201, 58-71.
https://doi.org/10.1016/j.mbs.2005.12.022

[13]   Guo, Z. Huang, L. and Zou, X. (2012) Impact of Discontinuous Treatments on Disease Dynamics in an SIR Epidemic Model. Mathematical Biosciences and Engineering, 9, 97-110.
https://doi.org/10.3934/mbe.2012.9.97

[14]   Zhang, T., Kang, R., Wang, K. and Liu, J. (2015) Global Dynamics of an SEIR Epidemic Model with Discontinuous Treatment. Advances in Difference Equations, 361, 1-16.

[15]   Huang, L., Guo, Z. and Wang, J. (2011) Theory and Applications of Differential Equations with Discontinuous Righthand Sides. Science Press, Beijing. (In Chinese)

[16]   Shi, R., Zhao, H. and Tang, S. (2014) Global Dynamic Analysis of a Vector-Borne Plant Disease Model. Advances in Difference Equations, 59, 1-16.
https://doi.org/10.1186/1687-1847-2014-59

[17]   Filippov, A.F. (1988) Differential Equations with Discontinuous Righthand Sides. Mathematics and Its Applications (Soviet Series). Kluwer Academic, Boston.
https://doi.org/10.1007/978-94-015-7793-9

[18]   Ma, Z. and Zhou, C. (2016) Methods of Qualitative and Stability of Ordinary Differential Equations. Science Press, Beijing. (In Chinese)

[19]   Baciotti, A. and Ceragioli, F. (1999) Stability and Stabilization of Discontinuous Systems and Non-Smooth Lyapunov Function. ESAIM: Control, Optimisation and Calculus of Variations, 4, 361-376.
https://doi.org/10.1051/cocv:1999113

 
 
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