An optimal control strategy and Grünwald-Letnikov finite-difference numerical scheme for the fractional-order COVID-19 model




Caputo fractional derivative, optimal control strategy, Grünwald-Letnikov numerical method, stability analysis


In this article, a mathematical model of the COVID-19 pandemic with control parameters is introduced. The main objective of this study is to determine the most effective model for predicting the transmission dynamic of COVID-19 using a deterministic model with control variables. For this purpose, we introduce three control variables to reduce the number of infected and asymptomatic or undiagnosed populations in the considered model. Existence and necessary optimal conditions are also established. The Grünwald-Letnikov non-standard weighted average finite difference method (GL-NWAFDM) is developed for solving the proposed optimal control system. Further, we prove the stability of the considered numerical method. Graphical representations and analysis are presented to verify the theoretical results.


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Bashier, H., Khader, Y., Al-Souri, R., & Abu-Khader, I. A Novel Coronavirus Outbreak: A Teaching Case-Study. The Pan African Medical Journal, 36(11), (2020).

Ndaïrou, F., Area, I., Nieto, J.J., & Torres, D.F. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos, Solitons & Fractals, 135, 109846, (2020).

Khan, M.A., & Atangana, A. Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alexandria Engineering Journal, 59(4), 2379-2389, (2020).

Machado, J.A., & Lopes, A.M. Rare and extreme events: the case of COVID-19 pandemic. Nonlinear dynamics, 100(3), 2953-2972, (2020).

Chen, T.M., Rui, J., Wang, Q.P., Zhao, Z.Y., Cui, J.A., & Yin, L. A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infectious diseases of poverty, 9(1), 1-8, (2020).

Ivorra, B., Ferrández, M.R., Vela-Pérez, M., & Ramos, A.M. Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China. Communications in nonlinear science and numerical simulation, 88, 105303, (2020).

Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).

Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A., & Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a Sars-Cov-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 56-66, (2021).

Özköse, F., Yavuz, M., Şenel, M.T., & Habbireeh, R. Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom. Chaos, Solitons & Fractals, 157, 111954, (2022).

Dietz, K., & Heesterbeek, J.A.P. Bernoulli was ahead of modern epidemiology. Nature, 408(6812), 513-514, (2000).

Khan, H., Gómez-Aguilar, J.F., Alkhazzan, A., & Khan, A. A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law. Mathematical Methods in the Applied Sciences, 43(6), 3786-3806, (2020).

Sweilam, N.H., Al-Mekhlafi, S.M., & Hassan, A.N. Numerical treatment for solving the fractional two-group influenza model. Progr Fract Differ Appl, 4, 1-15, (2018).

Kumar, S., Ghosh, S., Lotayif, M. S., & Samet, B. A model for describing the velocity of a particle in Brownian motion by Robotnov function based fractional operator. Alexandria Engineering Journal, 59(3), 1435-1449, (2020).

Rihan, F.A., Baleanu, D., Lakshmanan, S., & Rakkiyappan, R. On fractional SIRC model with salmonella bacterial infection. Abstract and Applied Analysis (2014).

Hammouch, Z., Yavuz, M., & Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).

Özköse, F., Şenel, M.T., & Habbireeh, R. Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83, (2021).

Kostylenko, O., Rodrigues, H.S., & Torres, D.F. The risk of contagion spreading and its optimal control in the economy. arXiv preprint arXiv:1812.06975, (2018).

Lemos-Paião, A.P., Silva, C.J., Torres, D.F., & Venturino, E. Optimal control of aquatic diseases: A case study of Yemen’s cholera outbreak. Journal of Optimization Theory and Applications, 185(3), 1008-1030, (2020).

Ammi, M.R.S., & Torres, D.F. Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives. Computers & Mathematics with Applications, 78(5), 1507-1516, (2019).

Brandeau, M.L., Zaric, G.S., & Richter, A. Resource allocation for control of infectious diseases in multiple independent populations: beyond cost-effectiveness analysis. Journal of health economics, 22(4), 575-598, (2003).

Ball, F., & Becker, N.G. Control of transmission with two types of infection. Mathematical biosciences, 200(2), 170-187, (2006).

Castilho, C. Optimal control of an epidemic through educational campaigns. Electronic Journal of Differential Equations (EJDE), 2006.

Rihan, F.A., Lakshmanan, S., & Maurer, H. Optimal control of tumour-immune model with time-delay and immuno-chemotherapy. Applied Mathematics and Computation, 353, 147-165, (2019).

Sweilam, N., Rihan, F., & Seham, A.M. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete & Continuous Dynamical Systems-S, 13(9), 2403, (2020).

Zaky, M.A., & Machado, J.T. On the formulation and numerical simulation of distributed-order fractional optimal control problems. Communications in Nonlinear Science and Numerical Simulation, 52, 177-189, (2017).

Agrawal, O.P. A formulation and a numerical scheme for fractional optimal control problems. IFAC Proceedings Volumes, 39(11), 68-72, (2006).

Agrawal, O.P., Defterli, O., & Baleanu, D. Fractional optimal control problems with several state and control variables. Journal of Vibration and Control, 16(13), 1967-1976, (2010).

Sweilam, N.H., Al-Mekhlafi, S.M., & Baleanu, D. Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains. Journal of advanced research, 17, 125-137, (2019).

Sweilam, N.H., Al-Mekhlafi, S.M., Alshomrani, A.S., & Baleanu, D. Comparative study for optimal control nonlinear variable-order fractional tumor model. Chaos, Solitons & Fractals, 136, 109810, (2020).

Sweilam, N.H., & AL–Mekhlafi, S.M. Optimal control for a time delay multi-strain tuberculosis fractional model: a numerical approach. IMA Journal of Mathematical Control and Information, 36(1), 317-340, (2019).

Liu, Z., Magal, P., Seydi, O., & Webb, G. Predicting the cumulative number of cases for the COVID-19 epidemic in China from earlydata. arXiv preprint arXiv:2002.12298, (2020).

Lakshmikantham, V., & Vatsala, A.S. Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications, 69(8), 2677-2682, (2008).

Arenas, A.J., Gonzalez-Parra, G., & Chen-Charpentier, B.M. Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order. Mathematics and Computers in Simulation, 121, 48-63, (2016).

Heimann, B., Fleming, W.H., Rishel, R.W., Deterministic and Stochastic Optimal Control. New York-Heidelberg-Berlin. SpringerVerlag. 1975. XIII, 222 S, DM 60, 60. Zeitschrift Angewandte Mathematik und Mechanik, 59(9), 494-494, (1979).

Kouidere, A., Youssoufi, L.E., Ferjouchia, H., Balatif, O., & Rachik, M. Optimal control of mathematical modeling of the spread of the COVID-19 pandemic with highlighting the negative impact of quarantine on diabetics people with cost-effectiveness. Chaos, Solitons & Fractals, 145, 110777, (2021).

Sweilam, N.H., Al-Ajami, T.M., & Hoppe, R.H. Numerical solution of some types of fractional optimal control problems. The Scientific World Journal, 2013, (2013).

Scherer, R., Kalla, S.L., Tang, Y., & Huang, J. The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications, 62(3), 902-917, (2011).

Li, L., & Wang, D. Numerical stability of Grünwald–Letnikov method for time fractional delay differential equations. BIT Numerical Mathematics, 1-33, (2021).

Chakraborty, M., Maiti, D., Konar, A., & Janarthanan, R. A study of the Grunwald-Letnikov definition for minimizing the effects of random noise on fractional order differential equations. In 2008 4th International Conference on Information and Automation for Sustainability, 449-456, IEEE, (2008).



DOI: 10.53391/mmnsa.2022.009

How to Cite

Haq, I. U., Ali, N., & Nisar, K. S. (2022). An optimal control strategy and Grünwald-Letnikov finite-difference numerical scheme for the fractional-order COVID-19 model. Mathematical Modelling and Numerical Simulation With Applications, 2(2), 108–116.



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