Abstract
Rice yellow mottle virus causes the most important rice disease in Africa. It was first identified in 1966 in Kenya, and later it has been detected in most African countries where rice is grown. Different approaches have been carried out so far to explore the dynamics and to curtail this viral disease. Mathematical models provide a useful tool in this regard and can be used to set the appropriate controlling strategies. In this paper, we formulate a new deterministic mathematical model for the dynamics of vector-borne rice yellow mottle infection. We include control measures, namely: roguing, treatment and the use of insecticide in the model. The dynamical behaviors of such controls in the reduction of the widespread of yellow mottle virus disease in a rice field were also studied. The basic properties of the model were stated and proved accordingly. We derived the equilibrium points of the model and computed the reproduction number. The stability of disease-free equilibrium point using the Routh–Hurwitz criterion and Castillo-Chavez function was carried out separately which were found to be stable. We also determined the optimal control via Pontryagin maximum principle. For numerical simulations, we solve the optimality system using a forward-backward sweep strategy implemented in MATLAB and the results show that the use of a combination of roguing on the symptomatic infected plants and the use of insecticides on susceptible and infected grasshoppers, together with treatment of symptomatic infected plants is the most excellent technique for controlling the spread of the disease.
Similar content being viewed by others
References
Ali FH (2001) Survey of rice yellow mottle virus disease in two cropping systems in Tanzania. In: Workshop on rice research in Eastern and Southern Africa Region, Kilimanjaro Agricultural Training Institute
Allarangaye MD, Traore O, Traore EVS, Millogo RJ, Guinko S, Konate G (2007) Host range of rice yellow mottle virus in Sudano-Sahelian savannahs. Pak J Biol Sci 10(9):1414–1421
Anggriani N, Yusuf M, Supriatna AK (2017) The effect of insecticide on the vector of rice tungro disease: insight from a mathematical model. Int Inf Inst Inf 20(9A):6197–206
Anggriani N, Arumi D, Hertini E, Istifadah N, Supriatna AK (2018) Dynamical analysis of plant disease model with roguing, replanting and preventive treatment. In: 4th Icriems proceedings,Yogyakarta state university: Faculty of mathematics and Natutal science
Banwo OO, Alegbejo MD, Abo MD (2004) Rice yellow mottle virus genus sobemovirus: a continental problem in Africa. Plant Protect Sci 40(1):26–36
Blas N, David G (2017) Dynamical roguing model for controlling the spread of tungro virus via Nephotettix Virescens in a rice field. J Phys Conf Ser 983:012018
Buonomo B, Lacitignola D (2010) Analysis of a tuberculosis model with a case study in Uganda. J Biol Dyn 4(6):571–593
Castillo-Chavez C, Feng Z, Huang W (2002) On the computation of RO and its role in global stability. Math Appr Emerg Reemerg Infect Dis Introd 1:125–229
Castillo-Chavez C, Song B (2004) Dynamical models of tuberculosis and their applications. Math Biosci Eng 1(2):361–404. https://doi.org/10.3934/mbe.2004.1.361
Castillo-Chavez C, Feng Z, Huang W (2002) On the computation of and its role on global stability. Mathematical approaches for emerging and reemerging infectious disease: an introduction, IMA 125 New York: Springer- Verlag
Contreras-Medina LM, Torres-Pacheco I, Guevara-Gonza’lez RG, Romero-Troncoso RJ, Terol-Villalobos IR, Osornio-Rios RA (2009) Mathematical modeling tendencies in plant pathology. Afr J Biotech 8(25):7399–7408
Derrick NK, Grossman SL (1976) Differential equation with applications. Addison Wesley Publications Company Inc, Philippines
Guckenheimer J, Holmes P (1984) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. J Appl Mech 51(4):947
Lakshmikantham V, Leela S, Martynyuk AA (1989) Stability Analysis of Non-linear Systems. New York and Basel, New York
Luo Y, Gao S, Xie D, Dai Y (2015) A discrete plant disease model with roguing and replanting. Adv Differ Eq: Springer Open J 2015:12. https://doi.org/10.1186/s13662-014-03322-3
Madden LV, Jeger MJ, Vanden Bosch F (2000) A theoretical assessment of the effects of vector- virus transmission mechanism on plant virus disease epidemics. Phytopathology 90(6):576–942
Marsh TL, Huffaker RG, Long GE (2000) Optimal control of vector virus plant interactions: the case study of patato leafroll virus necrosis. Am Agric Econ Assoc 83(2000):556–569
Michael M, Libin M, Weimin H (2012) Convergence of the forward-backward sweep method in optimal control. Comput Optim Appl 53(1):207–226
Murwayi ALM, Onyango T, Owour B (2017) Mathematical model of plant disease dispersion model that incorporates wind strength and insect vector at equilibrium. British J Math Comput Sci 22(5):1–7
Muthuri GG, Malonza DM (2018) Mathematical modeling of TB-HIV C0 Infection, Case Study of Tigania West Sub Country, Kenya. J Adv Math Comput Sci 2227(5):1–18
Nakasuji F, Miyai S, Kawamoto H, Kiritani K (1985) Mathematical epidemiology of rice dwarf virus transmission by green rice leafhoppers: a differential equation model. J Appl Ecol 22:839–847
Okosun KO, Ouifki R, Marcus N (2011) Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems 106(2–3):136–45
Neofytou G, Kyrychko YB, Blyuss KB (2016) Mathematical model of plant virus interactions mediated by RNA interference. J Theor Biol 403(2016):129–142
Pontryaggin LS, Boltryanskii VG, Gamkrelidze RV, Mishchenco EF (1986) Mathematical theory of optimal process. Gordon and Breach Science, New York
Savary S, Willocquet L, Teng PS (1997) Modelling shealth blight epidemics on rice tillers. Agric Syst 55(3):359–384
Suzanne ML, John TW (2007) Optimal control applied to biological model. CRC Press, UK
Vander Plank JE (1963) Pant diseases: epidemics and control. Academic Press, New York
Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibra for compartmental models of disease transmission. Math Biosci 180(1–2):29–48
Zagloul SR, Khalil M, Albadrey HH, Gadellah S (2016) Mathematical model of vectorborne plant disease with memory on the host and the vector. Progress Fract Differ Appl 2(4):277–285
Zhang T, Meng X, Song Y, Li Z (2012) Dynamical analysis of delayed plant disease models with continuous or impulsive cultural control startegies. Hndawi Publish Corp 2012:8–25. https://doi.org/10.1155/2012/428453
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
None.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Momoh, A.A., Déthié, D., Isah, N.S. et al. Mathematical modeling and optimal control of a vector-borne rice yellow mottle virus disease. Int. J. Dynam. Control 12, 600–618 (2024). https://doi.org/10.1007/s40435-023-01188-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-023-01188-4