Semi-Analytical Solution of Fractional –Order Mathematical Model of COVID-19 Via Atangana Baleanu- Caputor Derivative
Keywords:
COVID-19, Atangana–Baleanu–Caputo derivative, Basic reproduction number, Numerical simulation, Mathematical modelling
Abstract
COVID-19, caused by the novel coronavirus SARS-CoV-2, has had a profound global impact due to its high transmission rate and evolving clinical profile. Understanding its spread and identifying effective intervention strategies remain critical for public health planning. In this study, we develop a deterministic compartmental model to explore the transmission dynamics of COVID-19, incorporating a fractional-order derivative to account for memory effects inherent in disease progression. The human population is divided into six epidemiological compartments: susceptible, exposed, infected, under treatment, deceased, and recovered individuals. To more accurately reflect the temporal and cumulative effects of infection, we applied the Atangana–Baleanu–Caputo (ABC) fractional-order derivative, which improves the model's capacity to capture long-term dependencies often overlooked in classical models. A semi-analytical solution is derived using this approach. Stability analysis reveals that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than one, and unstable when othrwise. Through numerical simulations, it showed that reducing the contact rate and enhancing treatment interventions significantly lower infection prevalence and increase recovery rate. The findings highlight the effectiveness of timely treatment and behavioral control measures in curbing COVID-19 transmission. We recommend the continued enforcement of public health strategies such as reducing human-to-human contact, improving treatment accessibility, and increasing vaccine coverage. The proposed fractional-order model provides a more realistic framework for studying infectious diseases with memory-driven dynamics and can be adapted for future epidemic preparedness.
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References
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35. Agbata, B. C., Dervishi, R., Asante Mensa, F., Kwabi, P. A., Odeh, J. O., Amoah Mensah, J., Meseda, P. K., & Obeng Deng Denteh, W. (2025). Mathematical modelling of measles disease with double dose vaccination. Journal of Basic and Applied Science Research, 3(3), 199–214. https://dx.doi.org/10.4314/jobasr.v3i3.22
2. Smith, J. R., Nguyen, H. T., Lee, S. Y., & Williams, D. (2025). Interim estimates of COVID-19 vaccine effectiveness during 2024-2025. MMWR Morbidity and Mortality Weekly Report, 74, 123–128. https://doi.org/10.15585/mmwr.mm7406a1
3. Kumar, S., Verma, A., Singh, P., & Sharma, R. (2025). COVID-19 recovery trends in Madhya Pradesh, India. International Journal of Epidemiology, 54(2), 200–208. https://doi.org/10.1093/ije/dyad015
4. Chen, L., Johnson, T. M., Garcia, R., & Thompson, M. (2025). CDC vaccine guidelines update and expert opinions in Connecticut. Vaccine, 43(5), 899–904. https://doi.org/10.1016/j.vaccine.2024.12.021
5. Jackson, T. L., Nguyen, P. T., Brown, K. A., & Miller, D. (2025). COVID-19 booster shot uptake in Australia: A public health perspective. Australian and New Zealand Journal of Public Health, 49(3), 247–254. https://doi.org/10.1111/1753-6405.13459
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8. Garcia-Beltran, W. F., Lam, E. C., St. Denis, K., Nitido, A. D., Garcia, Z. H., Hauser, B. M., Feldman, J., Pavlovic, M. N., Gregory, D. J., Poznansky, M. C., Sigal, A., Schmidt, A. G., Iafrate, A. J., Naranbhai, V., & Balazs, A. B. (2024). Effectiveness of updated COVID-19 vaccines against XBB.1.5 subvariant. Nature Medicine, 30(4), 725–733. https://doi.org/10.1038/s41591-024-01517-6
9. Baker, M. G., Fidler, D. P., & Wilson, N. (2025). Who should receive COVID-19 boosters? An evolving consensus. Health Affairs, 44(3), 399–406. https://doi.org/10.1377/hlthaff.2024.0185
10. Rogers, J. P., Watson, C. J., Badenoch, J., Cross, B., Butler, M. S., Song, J., Packer, R. J., Lewis, G., D’Mello, S., Evans, A., Pritchard, M., Hughes, C., & David, S. (2025). Organ failure linked to long COVID: Pathophysiology and clinical implications. The New England Journal of Medicine, 392(10), 987–997. https://doi.org/10.1056/NEJMoa2511323
11. Prather, K. A., Marr, L. C., Schooley, R. T., McDiarmid, M. A., Wilson, M. E., & Milton, D. K. (2023). Alternative transmission routes of COVID-19: A narrative review. Journal of Infection, 87(6), 712–723. https://doi.org/10.1016/j.jinf.2023.04.012
12. Centers for Disease Control and Prevention (CDC). (2024). Use of additional COVID-19 vaccine doses in adults aged 65 and older. MMWR Morbidity and Mortality Weekly Report, 73(49), 1118–1123. https://doi.org/10.15585/mmwr.mm7349a2
13. Harvey, W. T., Carabelli, A. M., Jackson, B., Gupta, R. K., Thomson, E. C., Harrison, E. M., Ludden, C., Reeve, R., Rambaut, A., Peacock, S. J., & Robertson, D. L. (2023). SARS-CoV-2 variants, spike mutations and immune escape. Nature Reviews Microbiology, 21, 387–401. https://doi.org/10.1038/s41579-023-00878-2
14. Goldman, E. (2023). Infectious life of COVID-19 on surfaces and fabrics: A review. Journal of Hospital Infection, 125, 10–18. https://doi.org/10.1016/j.jhin.2023.02.004
15. Liang, J., Xu, Y., Chen, C., & Liu, H. (2024). What you need to know about the 2024-2025 COVID-19 vaccines. Vaccine, 42(32), 4355–4362. https://doi.org/10.1016/j.vaccine.2024.05.004
16. Peterson, J. M., Nguyen, P., Sullivan, B., & Williams, R. (2025). Effectiveness of a single mRNA vaccine dose in previously infected individuals. Communications Medicine, 5, 151. https://doi.org/10.1038/s43856-025-00151-x
17. Nalbandian, A., Sehgal, K., Gupta, A., Madhavan, M. V., McGroder, C., Stevens, J. S., Cook, J. R., Nordvig, A. S., Shalev, D., Sehrawat, T. S., Ahluwalia, N., Bikdeli, B., Dietz, D., Der-Nigoghossian, C., Liyanage-Don, N., Rosner, G. F., Bernstein, E. J., Mohan, S., Beckley, A. A., & Wan, E. Y. (2023). Post-acute COVID-19 syndrome (Long COVID): A review of clinical characteristics and pathogenesis. Nature Reviews Disease Primers, 9(1), 41. https://doi.org/10.1038/s41572-023-00387-4
18. Corti, D., Purcell, L. A., Snell, G., & Veesler, D. (2025). The future of vaccines: Needle-free delivery systems. Nature Reviews Immunology, 25, 211–223. https://doi.org/10.1038/s41577-025-00647-9
19. Andrews, N., Stowe, J., Kirsebom, F., Toffa, S., Rickeard, T., Gallagher, E., Gower, C., Kall, M., Groves, N., O’Connell, A.-M., Simons, D., Blomquist, P. B., Zaidi, A., Nash, S., Indalao, I., Lopez-Bernal, J., Dunachie, S., Byers, C., Myers, R., & Ramsay, M. (2024). COVID-19 vaccine effectiveness against JN.1 subvariant. The Lancet Infectious Diseases, 24(3), 307–314. https://doi.org/10.1016/S1473-3099(24)00045-7
20. Murthy, S., Farooqui, M., Lee, K., & Gupta, R. (2024). Effectiveness of 2023–2024 COVID-19 vaccine formula in working-aged adults. Cleveland Clinic Journal of Medicine, 91(7), 394–402. https://doi.org/10.3949/ccjm.91a.23032
21. Anderson, R. M., & May, R. M. (2023). Early transmission dynamics and modeling of COVID-19. Preventing Chronic Disease, 20, 230089. https://doi.org/10.5888/pcd20.230089
22. Agbata, B. C., Omale, D., Ojih, P. B., & Omatola, I. U. (2019). Mathematical analysis of chickenpox transmission dynamics with control measures. Continental Journal of Applied Sciences, 14(2), 6–23. https://www.researchgate.net/publication/344234584
23. Agbata, B. C., Shior, M. M., Obeng-Denteh, W., Omotehinwa, T. O., Paul, R. V., Kwabi, P. A., & Asante-Mensa, F. (2023). A mathematical model of COVID-19 transmission dynamics with effects of awareness and vaccination program. Journal of Ghana Science Association, 21(2), 59–61. https://www.researchgate.net/publication/379485392_
24. Atangana, A., &Baleanu, D. (2016). New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model. Thermal Science, 20(3), 763-769. https://doi.org/10.2298/TSCI160111018A
25. Atangana, A., &Goufo, E. F. D. (2023). Network-based fractional-order epidemic models with impulsive control strategies. Chaos, Solitons& Fractals, 172, 113888.https://doi.org/10.1016/j.chaos.2023.113888
26. Mahmood, H., Khan, N., & Ahmad, S. (2023). Modeling tuberculosis dynamics using Caputo fractional derivatives and the homotopy perturbation method.Bulletin of the National Research Centre, 47(1), 58.https://doi.org/10.1186/s42269-023-00872-1
27. Acheneje, G. O., Omale, D., Agbata, B. C., Atokolo, W., Shior, M. M., & Bolawarinwa, B. (2024). Approximate solution of the fractional order mathematical model on the transmission dynamics of the co-infection of COVID-19 and Monkeypox using the Laplace-Adomian decomposition method. International Journal of Mathematics and Statistics Studies, 12(3), 17–51. https://eajournals.org/ijmss/vol12-issue-3-
28. Agbata, B. C., Agbebaku, D. F., Odo, C. E., Ojih, J. T., Shior, M. M., & Ezugorie, I. G. (2024). A mathematical model for the transmission dynamics of COVID-19 in Nigeria and its post-effects. International Journal of Mathematical Analysis and Modelling, 7(2), 523–547. https://tnsmb.org/journal/index.php/ijmam/article/view/191
29. Agbata, B. C., Obeng-Denteh, W., Amoah-Mensah, J., Kwabi, P. A., Shior, M. M., Asante-Mensa, F., & Abraham, S. (2024). Numerical solution of fractional order model of measles disease with double dose vaccination. Dutse Journal of Pure and Applied Sciences (DUJOPAS), 10(3b), 202–217. https://www.ajol.info/index.php/dujopas/article/view/281624
30. Abdoon, M. M., & Alzahrani, E. O. (2024). A comparative study of fractional derivatives in modeling influenza transmission. Applied Mathematics and Computation, 449, 128001. https://doi.org/10.1016/j.amc.2023.128001
31. Deressa, T. T., & Duressa, T. F. (2021). Fractional-order SEAIR epidemic model with optimal control using the Atangana–Baleanu–Caputo derivative. AIMS Mathematics, 6(12), 13267–13290. https://doi.org/10.3934/math.2021719
32. Farman, M., Khan, M. A., Shah, K., & Atangana, A. (2022). Fractal-fractional analysis of a Zika virus model via ABC Caputo derivative with real data simulation. Alexandria Engineering Journal, 61(11), 10973–10989. https://doi.org/10.1016/j.aej.2022.02.022
33. Abioye, A. I., Peter, O. J., Ogunseye, H. A., Oguntolu, F. A., Ayoola, T. A., & Oladapo, A. O. (2023). A fractional-order mathematical model for malaria and COVID-19 co-infection dynamics using the Atangana–Baleanu–Caputo derivative. Results in Applied Mathematics, 18, 100343. https://doi.org/10.1016/j.rinam.2023.100343
34. Here’s the complete APA-style reference for the specified article, including author list and link to its ResearchGate page:
35. Agbata, B. C., Dervishi, R., Asante Mensa, F., Kwabi, P. A., Odeh, J. O., Amoah Mensah, J., Meseda, P. K., & Obeng Deng Denteh, W. (2025). Mathematical modelling of measles disease with double dose vaccination. Journal of Basic and Applied Science Research, 3(3), 199–214. https://dx.doi.org/10.4314/jobasr.v3i3.22
Published
2025-06-10
How to Cite
Ezugorie, I. G., Ezugorie, M. O., & Agbata, B. C. (2025). Semi-Analytical Solution of Fractional –Order Mathematical Model of COVID-19 Via Atangana Baleanu- Caputor Derivative. GPH-International Journal of Mathematics, 8(4), 01-20. https://doi.org/10.5281/zenodo.15630427
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Copyright (c) 2025 I. G. Ezugorie, M. O. Ezugorie, B. C. Agbata
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