[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Modelling and Analysis of Pathogens Impact on the Plant Disease Transmission with Optimal Control

Journal of Applied Nonlinear Dynamics 11(3) (2022) 499--521 | DOI:10.5890/JAND.2022.09.001

Abdisa Shiferaw Melese$^{1}$, Oluwole Daniel Makinde$^{2}$, Legesse Lemecha Obsu$^{1}$

$^{1}$ Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia

$^{2}$ Faculty of Military Science, Stellenbosch University, Stellenbosch, South Africa

Download Full Text PDF

 

Abstract

In this study, a nonlinear deterministic mathematical model for the impact of pathogens on plant disease transmission with optimal control is proposed. Mathematically, we analyzed the dynamics of the system properties including boundedness of the solutions, existence of pathogen free and coexistence equilibria, local and global stability of equilibrium points. Furthermore, we computed the basic reproduction number $R_{0}$ and studied its sensitivity with respect to model parameters to identify the most affecting parameters. The local stability of the equilibria is also established via the Jacobian matrix and Routh Hurwitz criteria while the global stability of the equilibria is proved by using an appropriate Lyapunov function. Using center manifold theory, we proved that the model exhibits forward bifurcation. An optimal control problem is proposed which minimizes costs incurred due to the disease and applied controls. The characterization of optimal paths is analytically derived using Pontryagin's Minimum Principle. We then numerically studied the optimal control problem. A comparative study is conducted by considering control strategies: implementation of only quarantine, chemical, execution of both quarantine and chemical. We observe that the inclusive use of quarantine and chemical controls can reduce the infected host population with minimum implementation cost.

References

  1. [1] Mikkelsen, L., Elphinstone, J., and Jensen, D.F. (2006), Literature review on detection and eradication of plant pathogens in sludge, soils and treated biowaste, Desk study on bulk density. Bruxelles: The European Commission DG RTD under the Framework, 6.
  2. [2]  Abdulkhair, W.M. and Alghuthaymi, M.A. (2016), Plant pathogens, Plant Growth, 49. DOI: 10.5772/65325.
  3. [3]  Belachew, K., Teferi, D., and Livelihood, E. (2015), Climatic variables and impact of coffee berry diseases (Colletotrichum kahawae) in Ethiopian coffee production, Journal of Biology, Agriculture and Healthcare, 5(2015), 55-64.
  4. [4]  Cunniffe, N.J. and Gilligan, C.A. (2010), Invasion, persistence and control in epidemic models for plant pathogens: the effect of host demography, Journal of the Royal Society Interface, 7(44), 439-451.
  5. [5]  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(3), 291-304.
  6. [6]  Murwayi, A.L.M., Onyango, T., and Owour, B. (2017), Mathematical analysis of plant disease dispersion model that incorporates wind strength and insect vector at equilibrium, Journal of Advances in Mathematics and Computer Science, 22(5), 1-17.
  7. [7]  Jacobsen, K., Stupiansky, J., and Pilyugin, S.S. (2013), Mathematical modeling of citrus groves infected by huanglongbing, Mathematical Biosciences $\&$ Engineering, 10(3), 705-728.
  8. [8]  Kinene, T., Luboobi, L.S., Nannyonga, B., and Mwanga, G.G. (2015), A mathematical model for the dynamics and cost effectiveness of the current controls of cassava brown streak disease in Uganda, J. Math. Comput. Sci., 5(4), 567-600.
  9. [9]  Alemneh, H.T., Makinde, O.D., and Theuri, D.M. (2019), Mathematical modelling of msv pathogen interaction with pest invasion on maize plant, Global Journal of Pure and Applied Mathematics, 15(1), 55-79.
  10. [10]  Allen, L.J., Brauer, F., Van den Driessche, P., and Wu, J. (2008), Mathematical epidemiology, 1945, Berlin: Springer.
  11. [11] Kung'aro, M., Massawe, E.S., and Makinde, O.D. (2013), Transmission dynamics of HIV/AIDS with screening and non-linear incidence, African Journal of Science and Technology, 12(2), 31-43.
  12. [12]  Gilligan, C.A. (1990), Mathematical modeling and analysis of soilborne pathogens, In Epidemics of plant diseases, 96-142. Springer, Berlin, Heidelberg.
  13. [13]  Diekmann, O., Heesterbeek, J.A.P., and Metz, J.A. (1990), On the definition and the computation of the basic reproduction ratio $ R_0$ in models for infectious diseases in heterogeneous populations, Journal of mathematical biology, 28(4), 365-382.
  14. [14]  Van den Driessche, P. and Watmough, J. (2002), Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180(1-2), 29-48.
  15. [15]  Heffernan, J.M., Smith, R.J., and Wahl, L.M. (2005), Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2(4), 281-293.
  16. [16]  Chitnis, N., Cushing, J.M., and Hyman, J.M. (2006), Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal on Applied Mathematics, 67(1), 24-45.
  17. [17] Iddi, A.J. (2011), Modelling the impact of infected immigrants on vector-borne diseases with direct transmission, (Doctoral dissertation, University of Dar es Salaam).
  18. [18] Ullah, S. and Khan, M.A. (2020), Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study, Chaos, Solitons $\&$ Fractals, 139, 110075.
  19. [19] Asamoah, J.K.K., Owusu, M.A., Jin, Z., Oduro, F.T., Abidemi, A., and Gyasi, E.O. (2020), Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: using data from Ghana, Chaos, Solitons $\&$ Fractals, 140, 110103.
  20. [20]  Britton, N. (2005), Essential mathematical biology, Springer Science \& Business Media.
  21. [21]  LaSalle, J.P. (1976), The stability of dynamical systems, regional conference series in appl, Math., SIAM, Philadelphia.
  22. [22]  Castillo-Chavez, C. and Song, B. (2004), Dynamical models of tuberculosis and their applications, Mathematical Biosciences $\&$ Engineering, 1(2), 361.
  23. [23]  Hall, R.J., Gubbins, S., and Gilligan, C.A. (2004), Invasion of drug and pesticide resistance is determined by a trade-off between treatment efficacy and relative fitness, Bulletin of Mathematical Biology, 66(4), 825-840.
  24. [24]  Cook, R.J. and Baker, K.F. (1983), The nature and practice of biological control of plant pathogens, American Phytopathological Society.
  25. [25]  Takaidza, I., Makinde, O.D., and Okosun, O.K. (2017), Computational modelling and optimal control of Ebola virus disease with non-linear incidence rate, In Journal of Physics: Conference Series, 818(1), 012003. IOP Publishing.
  26. [26]  Boltyanskiy, V.G., Gamkrelidze, R.V., Mishchenko, Y., and Pontryagin, L.S. (1962), Mathematical theory of optimal processes, John Wiley and Sons, London, UK.
  27. [27]  Pang, L., Ruan, S., Liu, S., Zhao, Z., and Zhang, X. (2015), Transmission dynamics and optimal control of measles epidemics, Applied Mathematics and Computation, 256, 131-147.
  28. [28]  Lenhart, S. and Workman, J.T. (2007), Optimal control applied to biological models, CRC press, New York.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates