We consider a nonlinear fractional-order Covid-19 model in a sense of the Atagana–Baleanu fractional derivative used for the analytic and computational studies. The model consists of six classes of persons, including susceptible, protected susceptible, asymptomatic infected, symptomatic infected, quarantined, and recovered individuals. The model is studied for the existence of solution with the help of a successive iterative technique with limit point as the solution of the model. The Hyers–Ulam stability is also studied. A numerical scheme is proposed and tested on the basis of the available literature. The graphical results predict the curtail of spread within the next 5000 days. Moreover, there is a gradual increase in the population of protected susceptible individuals.
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10 June 2023
A Correction to this paper has been published: https://doi.org/10.1007/s10958-023-06501-2
References
A. E. Gorbalenya, S. C. Baker, R. S. Baric, R. J. de Groot, C. Drosten, A. A. Gulyaeva, B. L. Haagmans, C. Lauber, A. M. Leontovich, A. W. Neuman, and D. Penzar, “Coronaviridae study group of the international committee on taxonomy of viruses. The species severe acute respiratory syndrome-related coronavirus: classifying 2019-nCoV and naming it SARS-CoV-2,” Nat. Microbiol., 5, No. 4, 536–544 (2020).
S. A. Morrison, J. Gregor, and S. Gregor, “Responding to a global pandemic: Republic of Slovenia on maintaining physical activity during self-isolation,” Scand. J. Med. Sci. Sports, 30, No. 8 (2020).
N. Kokudo and H. Sugiyama, “Call for international cooperation and collaboration to effectively tackle the Covid-19 pandemic,” Global Health Med., 30, 2(2), 2–60 (2020).
S. Omer and S. Ali, “Preventive measures and management of Covid-19 in pregnancy,” Drugs Therapy Perspect., 36, No. 6 (2020).
K. F. Owusu, E. F. Goufo, and S. Mugisha, “Modelling intracellular delay and therapy interruptions within Ghanaian HIV population,” Adv. Difference Equat., 1, 1–9 (2020).
M. Anderson, M. Mckee, and E. Mossialos, “Developing a sustainable exit strategy for Covid-19: health, economic and public policy implications,” J. R. Soc. Med., 113(5), 8–176 (2020).
E. D. Goufo and R. Maritz, “A note on ebolas outbreak and human migration dynamic,” J. Human Ecol., 51, No. 3, 257–263 (2015).
P. T. Djomegni, A.Tekle, and M. Y. Dawed, “Pre-exposure prophylaxis HIV/AIDS mathematical model with non classical isolation,” Jap. J. Ind. Appl. Math., 37, 781–801 (2020).
K. F. Owusu, E. F. Goufo, and S. Mugisha, “Modelling intracellular delay and therapy interruptions within Ghanaian HIV population,” Adv. Difference Equat., 1, 1–9 (2020).
Z. A. Khan, F. Jarad, A. Khan, and H. Khan, “Nonlinear discrete fractional sum inequalities related to the theory of discrete fractional calculus with applications,” Adv. Difference Equat., 1, 1–3 (2021).
A. Khan, H. M. Alshehri, T. Abdeljawad, Q. M. Al-Madlal, and H. Khan, “Stability analysis of fractional nabla difference Covid-19 model,” Results Phys., 4, 103–888 (2021).
A. Shah, R. A. Khan, A. Khan, H. Khan, and J. F. Gomez-Aguilar, “Investigation of a system of nonlinear fractional order hybrid differential equations under usual boundary conditions for existence of solution,” Math. Meth. Appl. Sci., 30, No. 2, 1628–38 (2021).
T. Abdeljawad, “A Lyapunov type inequality for fractional operators with non singular Mittag-Leffler kernel,” J. Inequal Appl., 130, (2017); https://doi.org/10.1186/s13660-017-1400-5.
T. Abdeljawad, “Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals,” Chaos, 29, No. 2, 023102 (2019).
T. Abdeljawad and D. Baleanu, “Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels,” Adv. Difference Equat., Paper No. 232 (2016).
T. Abdeljawad, Q. M. Al-Mdallal, and F. Jarad, “Fractional logistic models in the frame of fractional operators generated by conformable derivatives,” Chaos Solitons Fractals, 119, 94–101 (2019).
T. Abdeljawad, M. A. Hajji, Q. M. Al-Mdallal, and F. Jarad, “Analysis of some generalized ABC-fractional logistic models,” Alexandria Eng. J., 59, No. 4, 8–2141 (2020).
B. Acay, E. Bas, and T. Abdeljawad, “Fractional economic models based on market equilibrium in the frame of different type kernels,” Chaos Solitons Fractals, 130, 109438 (2020).
A. Atangana and S. I. Araz, “Mathematical model of Covid-19 spread in Turkey and South Africa: theory, methods and applications,” Adv. Difference Equat., Paper No. 659 (2020).
A. Atangana and S. I. Araz, “Nonlinear equations with global differential and integral operators: existence, uniqueness with application to epidemiology,” Results Phys., 20, 103593 (2021).
A. Atangana and D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model,” Thermal Sci., 20(00), (2016); https://doi.org/10.2298/TSCI160111018A.
I. V. Atamas’ and V. I. Slinko, “Stability of the fixed points of a class of quasilinear cascades in the space conv ℝn,” (Russian. English, Ukrainian Summary) Ukr. Mat. Zh., 69, No. 8, 1166–1179 (2017).
M. Arfan, K. Shah, T. Abdeljawad, N. Mlaiki, and A. Ullah, “A Caputo power law model predicting the spread of the Covid-19 outbreak in Pakistan,” Alexandria Eng. J., 1, 60(1), 56–447 (2021).
R. Begum, O. Tunç, H. Khan, H. Gulzar, and A. Khan, “A fractional order Zika virus model with Mittag-Leffler kernel,” Chaos Solitons Fractals, 146, No. 110 898 (2021).
M. Bohner, O. Tunç and C. Tunç, “Qualitative analysis of Caputo fractional integro-differential equations with constant delays,” Comput. Appl. Math., 40, No. 6, Paper No. 214 (2021).
A. I. Dvirnyi and V. I. Slyn’ko, “Stability of solutions of pseudolinear differential equations with impulse action,” Math. Notes, 93, No. 5-6, 691–703 (2013).
J. F. Gomez, L. Torres, and R. F. Escobar, “Fractional derivatives with Mittag-Leffler kernel. Trends and applications in science and engineering,” Studies in Systems, Decision and Control, 194, Springer, Cham (2019).
F. Jarad, T. Abdeljawad, and Z. Hammouch, “On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative,” Chaos Solitons Fractals, 1, No. 117, 16–20 (2018).
K. M. Owolabi and A. Atangana, “On the formulation of Adams–Bashforth scheme with Atangana–Baleanu–Caputo fractional derivative to model chaotic problems,” Chaos, 29, No 2, 023111 (2019).
K. Shah, M. A. Alqudah, F. Jarad, and T. Abdeljawad, “Semi-analytical study of pine wilt disease model with convex rate under Caputo–Febrizio fractional order derivative,” Chaos Solitons Fractals, 135, 109754 (2020).
O. Tunç, Ö. Atan, C. Tunç, and J. C. Yao, “Qualitative analyses of integro-fractional differential equations with Caputo derivatives and retardations via the Lyapunov–Razumikhin method,” Axioms, 10, No. 2 (2021); https://doi.org/10.3390/axioms10020058.
E. Tunç and O. Tunç, “On the oscillation of a class of damped fractional differential equations,” Miskolc Math. Notes, 17, No. 1, 647–656 (2016).
M. Yavuz and T. Abdeljawad, “Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel,” Adv. Difference Equat., Paper No. 367 (2020).
F. B. Yousef, A. Yousef, T. Abdeljawad, and A. Kalinli, “Mathematical modeling of breast cancer in a mixed immune-chemotherapy treatment considering the effect of ketogenic diet,” Eur. Phys. J. Plus., 135, No. 12, 1–23 (2020).
M. A. Khan, A. Atangana, and E. Alzahrani, “The dynamics of Covid-19 with quarantined and isolation,” Adv. Difference Equat., 2020, No. 1, 1–22 (2020).
P. M. Djomegni, M. D. Haggar, W. T. Adigo, “Mathematical model for Covid-19 with “protected susceptible” in the post-lockdown era,” Alexandria Eng. J., 1, 60(1), 35–527 (2021).
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Published in Neliniini Kolyvannya, Vol. 24, No. 3, pp. 378–400, July–September, 2021.
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Khan, H., Ibrahim, M., Khan, A. et al. A Fractional Order Covid-19 Epidemic Model with Mittag–Leffler Kernel. J Math Sci 272, 284–306 (2023). https://doi.org/10.1007/s10958-023-06417-x
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DOI: https://doi.org/10.1007/s10958-023-06417-x