Analysis and optimal control of a fractional MSD model

Document Type : Research Article

Authors

• M. Bagherpoorfard ¹
• F. Akhavan Ghassabzade ²

¹ Department of Mathematics, Fasa Branch, Islamic Azad University, Fasa, Iran.

² Department of Mathematics, Faculty of sciences, University of Gonabad, Gonabad, Iran.

10.22067/ijnao.2023.79093.1189

Abstract

In this research, we aim to analyze a mathematical model of Maize streak virus disease as a problem of fractional optimal control. For dynamical analysis, the boundedness and uniqueness of solutions have been investi-gated and proven. Also, the basic reproduction number is obtained, and local stability conditions are given for the equilibrium points of the model. Then, an optimal control strategy is proposed for the purpose of examining the best strategy to fight the maize streak disease. We solve the fractional optimal control problem by a forward-backward sweep iterative algorithm. In this algorithm, the state variable is obtained in a forward and co-state variable by a backward method where an explicit Runge-Kutta method is used to solve differential equations arising from fractional optimal control problems. Some comparative results are presented in order to verify the model and show the efficacy of the fractional optimal control treatments.

Keywords

• Fractional differential equation
• Maize streak virus
• Fractional-order optimal control
• Sweep method
• Numerical simulation

20.1001.1.24236977.2023.13.3.7.2

Main Subjects

• Optimal Control

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References

[1] Agrawal, O.P., Defterli, O. and Baleanu, D. Fractional optimal control problems with several state and control variables, J. Vib. Control 16 (2010), 1967–1976.
[2] Akhavan Ghassabzadeh, F., Tohidi, E., Singh, H. and Shateyi, S. RBF collocation approach to calculate numerically the solution of the nonlinear system of qFDEs, J. King Saud. Univ. Sci. 33(2) (2021), 101288.
[3] Alemneh, H.T., Kassa, A.S. and Godana, A.A. An optimal control model with cost effectiveness analysis of Maize streak virus disease in maize plant, Infect. Dis. Model. 6 (2020), 169–182.
[4] Ameen, I.G., Baleanu, D. and Mohamed Ali, H. Different strategies to confront maize streak disease based on fractional optimal control formu-lation, Chaos, Solitons & Fractals, 164 (2022), 112699.
[5] Baleanu, D., Akhavan Ghassabzade, F., Nieto, J.J. and Jajarmi, A. On a new and generalized fractional model for a real cholera outbreak, Alexandria Eng. J. 61(11) (2022), 9175–9186.
[6] Bozkurt, F., Yousef, A., Abdeljawad, T., Kalinli, A. and Al Mdallal, Q. A fractional-order model of COVID-19 considering the fear effect of the media and social networks on the community, Chaos Solitons Fract. 152 (2021), 111403.
[7] Choi, S.K., Kang, B. and Koo, N. Stability for Caputo fractional differ-ential systems, Abstr. Appl. Anal. (2014), Art. ID 631419, 6 pp.
[8] Collins, O.C. and Duffy, K.J. Optimal control of maize foliar diseases using the plants population dynamics, Acta Agriculturae Scandinavica, Section B– Soil and Plant Science 66(1) (2016), 20–26.

[9] Cunniffe, N.J. and Gilligan, C.A. A theoretical framework for biological control of soil-borne plant pathogens: identifying effective strategies, J. Theor. Biol. 278 (2011), 32–43.
[10] Diethelm, K., Siegmund, S. and Tuan, H.T. Asymptotic behavior of solutions of linear multi-order fractional differential systems, Fract. Calc. Appl. Anal. 20(5) (2017), 1165–1195.
[11] Evirgen, F., Uçar, S. and Özdemir, N. System analysis of HIV infection model with CD4+T under non-singular kernel derivative, Appl. Math. Nonlinear Sci. 5(1) (2020), 13–146.
[12] Fuller, C. Mealie variegation In: 1st Report of the Government Ento-mologist, Natal, 1899-1900, Pietermaritzburg, Natal, South Africa: P. Davis & Sons, Government Printers, 1901.
[13] Hristov, J. Transient heat diffusion with a non-singular fading mem-ory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Sci. 20(2) (2016), 757–762.
[14] Hugo, A., Lusekelo, E.M. and Kitengeso, R. Optimal control and cost effectiveness analysis of tomato yellow leaf curl virus disease epidemic model, Applied Mathematics, 9(3) (2019), 82–88.
[15] Ionescu, C., Lopes, A., Copot, D., Machado, J.A.T. and Bates, J.H.T. The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul. 51 (2017), 141–159.
[16] Kilbas, A.A., Srivastava, H.H. and Trujillo, J.J. Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
[17] Li, H.L., Zhang, L., Hu, C., Jiang, Y.-L. and Teng, Z. Dynamical analy-sis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput. 54 (2017), 435–449.
[18] Matignon, D. Stability results on fractional differential equations to control processing, In Proceedings of the Computational Engineering in Systems and Application Multiconference; IMACS, IEEE-SMC: Lille, France, 2 (1996), 963–968.
[19] Saeedian, M., Khalighi, M., Azimi-Tafreshi, N., Jafari, G.R. and Aus-loos, M. Memory effects on epidemic evolution: the susceptible-infected-recovered epidemic model, Phys. Rev. 95(2) (2017), 0224091–0224099.
[20] Sandhu, K.S., Singh, N. and Malhi, N.S. Some properties of corn grains and their flours I: Physicochemical, functional and chapati-making prop-erties of flours, Food Chem. 101(3) (2007), 938–946.

[21] Sene, N. Integral balance methods for Stokes’ first, equation described by the left generalized fractional derivative, Physics, 1(1) (2019), 154–166.
[22] Sene, N. Second-grade fluid model with Caputo-Liouville generalized fractional derivative , Chaos, Solit. Fractals, 133 (2020), 109631.
[23] Shatanawi, W., Abdo, M.S., Abdulwasaa, M.A., Shah, K., Panchal, S.K., Kawale, S.V. and Ghadle K.P. A fractional dynamics of tuberculo-sis (TB) model in the frame of generalized Atangana- Baleanu derivative, Res. Phys. 29 (2021), 104739.
[24] Shepherd, D.N., Martin, D.P., Van Der Walt, E., Dent, K., Varsani, A. and Rybicki, E.P. Maize streak virus: an old and complex ’emerging’ pathogen, Mol. Plant Pathol. 11(1) (2010), 1–12.
[25] Shi, R., Zhao, H. and Tang, S. Global dynamic analysis of a vector-borne plant disease model, Adv. Difference Equ. (59) (2014), 16.
[26] Traore, A. and Sene, N. Model of economic growth in the context of fractional derivative, Alex. Eng. J. 59(6) (2020), 4843–4850.