Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Global dynamical analysis of plant-disease models with nonlinear impulsive cultural control strategy

1 School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, P.R. China
2 Department of Mathematics and Faculty of Medicine, The University of Ottawa, 150 Louis-Pasteur Pvt, Ottawa, ON, K1N 6N5, Canada

Special Issues: Non-smooth biological dynamical systems and applications

To eradicate plant diseases and maintain the number of infected plants below an economic threshold, two impulsive plant-disease models with periodic and state-dependent nonlinear cultural control are established. We focus on saturated nonlinear roguing (identifying and removing infected plants), with three situations for healthy plants: constant replanting, proportional replanting and proportional incidental removal. The global dynamics of the model with periodic impulsive effects are investigated. We establish conditions for the existence and stability of the disease-free periodic solution, the existence of a positive periodic solution and permanence. Latin Hypercube Sampling is used to perform a sensitivity analysis on the threshold of disease extinction to determine the significance of each parameter. Disease extinction will follow from increasing the harvesting rate, increasing the intervention period or decreasing the replanting number. The global behavior of the model with an economic threshold is established, including the existence and global stability of periodic solutions. The results imply that the control methods have an important effect on disease development, and the density-dependent parameter can decelerate extinction and accelerate the growth of healthy plants. These findings suggest that we can successfully eradicate the disease or maintain the infections below a certain level under suitable control measures. The analytic methods developed here provide a general framework for exploring plant-disease models with nonlinear impulsive control strategies.
  Figure/Table
  Supplementary
  Article Metrics

References

1. R. W. Gibson, J. P. Legg and G. W. Otim-Nape, Unusually severe symptoms are a characteristic of the current epidemic of mosaic virus disease of cassava in Uganda, Ann. Appl. Biol., 128 (1996), 479–490.

2. T. Iljon, J. Stirling and R. J. Smith?, A mathematical model describing an outbreak of Fire Blight, in Understanding the dynamics of emerging and re-emerging infectious diseases using mathematical models (eds. S. Mushayabasa and C.P. Bhunu), Transworld Research network, (2012), 91–104.

3. R. A. C. Jones, Using epidemiological information to develop effective integrated virus disease management strategies, Virus Res., 100 (2004), 5–30.

4. C. A. Gilligan, Sustainable agriculture and plant diseases: an epidemiological perspective, Phil. Trans. R. Soc. B, 363 (2008), 741–759.

5. L. M. C. Medina, I. T. Pacheco, R. G. G. Gonzalez, et al., Mathematical modeling tendencies in plant pathology, Afr. J. Biotechnol., 8 (2009), 7399–7408.

6. R. A. C. Jones, Determining threshold levels for seed-borne virus infection in seed stocks, Virus Res., 71 (2000), 171–183.

7. T. T. Zhao and S. Y. Tang, Plant disease control with Economic Threshold, J. Bioma., 24 (2009), 385–396.

8. F. Van den Bosch and A. M. Roos, The dynamics of infectious diseases in orchards with roguing and replanting as control strategy, J. Math. Biol., 35 (1996), 129–157.

9. M. S. Chan and M. J. Jeger, An analytical model of plant virus disease dynamics with roguing, J. Appl. Ecol., 31 (1994), 413–427.

10. H. R. Thieme and J. A. P. Heesterbeek, How to estimate the efficacy of periodic control of an infectious plant disease, Math. Biosci. 93 (1989), 15–29.

11. R. W. Gibson and V. Aritua, The perspective of sweet potato chlorotic stunt virus in sweet potato production in Africa, a review, Afr. Crop Sci. J., 10 (2002), 281–310.

12. J. M. Thresh and R. J. Cooter, Strategies for controlling cassava mosaic disease in Africa, Plant Pathol., 54 (2005), 587–614.

13. J. M. Thresh, The origins and epidemiology of some important plant virus diseases, Appl. Biol., 5 (1980), 1–65.

14. S. Fishman, R. Marcus, H. Talpaz, et al., Epidemiological and economic models for the spread and control of citrus tristeza virus disease, Phytoparasitica, 11 (1983), 39–49.

15. J. M. Thresh and G. K. Owusu, The control of cocoa swollen shoot disease in Ghana: an evaluation of eradication procedures, Crop Prot., 5 (1986) 41–52.

16. A. N. Adams, The incidence of plume pox virus in England and its control in orchards, in Plant Disease Epidemiology (ed. P.R. Scott and A. Bainbridge), Blackwell Scientific Publications, (1978), 213–219.

17. G. Hughes, N. McRoberts, L. V. Madden, et al., Validating mathematical models of plant-disease progress in space and time, Math. Med. Biol., 14 (1997), 85–112.

18. J. Holt and T. C. B. Chancellor, A model of plant virus disease epidemics in asynchronously-planted cropping systems, Plant Pathol., 46 (1997), 490–501.

19. F. van den Bosch, N. McRoberts, F. van den Berg, et al., The basic reproduction number of plant pathogens: matrix approaches to complex dynamics, Phytopathology, 98 (2008), 239–249.

20. C. Chen, Y. Kang and R. J. Smith?, Sliding motion and global dynamics of a Filippov fire-blight model with economic thresholds, Nonlinear Anal.-Real, 39 (2018), 492–519.

21. A. Wang, Y. Xiao and R. J. Smith?, Using non-smooth models to determine thresholds for microbial pest management, J. Math. Biol., 78 (2019), 1389–1424.

22. L. V. Madden, G. Hughes and F. van den Bosch, The study of plant disease epidemics, The American Phytopathological Society, St. Paul, 2007.

23. F. Van den Bosch, M. J. Jeger and C. A. Gilligan, Disease control and its selection for damaging plant virus strains in vegetatively propagated staple food crops; a theoretical assessment, Proc. R. Soc. B, 274 (2007), 11–18.

24. S. Fishman and R. Marcus, A model for spread of plant disease with periodic removals, J. Math. Biol., 21 (1984), 149–158.

25. Y. N. Xiao, D. Z. Cheng and H. S. Qin, Optimal impulsive control in periodic ecosystem, Syst. Contr. Lett., 55 (2006), 558–565.

26. X. Y. Zhang, Z. S. Shuai and K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Anal.-Real, 4 (2003), 639–651.

27. 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.

28. M. C. Smith, J. Holt, L. Kenyon, et al., Quantitative epidemiology of banana bunchy top virus disease and its control, Plant Pathol., 47 (1998), 177–187.

29. S. Y. Tang, Y. N. Xiao and R. A. Cheke, Multiple attractors of host-parasitoid models with integrated pest management strategies: eradication, persistence and outbreak, Theor. Popul. Biol., 73 (2008), 181–197.

30. T. T. Zhao, Y. N. Xiao and R. J. Smith?, Non-smooth plant disease models with economic thresholds, Math. Biosci., 241 (2013), 34–48.

31. S. Y. Tang, R. A. Cheke and Y. N. Xiao, Optimal impulsive harvesting on non-autonomous Beverton–Holt difference equations, Nonlinear Anal.-Theor., 65 (2006), 2311–2341.

32. D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, New York, 1993.

33. J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2 (2005), 281–293.

34. J. Li, D. Blakeley and R. J. Smith?, The Failure of R0, Comp. Math. Meth. Med., 2011 (2011), Article ID 527610.

35. A. Lakmech and O. Arino, Bifurcation of nontrival periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn. Cont. Discr. Impul. Syst., 7 (2000), 265–287.

36. M. D. McKay, W. J. Conover and R. J. Beckman, A comparison of three models for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21 (1979), 239–245 .

37. R. M. Corless, G. H. Gonnet, D. E. G. Hare, et al., On the Lambert W Function, Adv. Comput. Math., 5 (1996), 329–359.

38. S. Y. Tang and Y. N. Xiao, Dynamics System of Single Species, Science Press, Beijing, 2008.

39. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved