Asymptotic stability analysis of a fractional epidemic model for Ebola virus disease in Caputo sense

Authors

  • Samson Olaniyi
    Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria
    https://orcid.org/0000-0001-6492-1405
  • Furaha M. Chuma
    Department of Physics, Mathematics and Informatics, Dar es Salaam University College of Education, Tanzania
  • Sulaimon F. Abimbade
    Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria
    https://orcid.org/0000-0002-1992-9560

Keywords:

Mathematical model, Fractional order, Caputo derivative operator, Ebola virus disease, Lyapunov stability

Abstract

In this work, a non-integer-order epidemic system modelling the nonlinear dynamics of Ebola virus disease is formulated in the sense of Caputo-ractional derivative. The existence and uniqueness of solution of the model is established. More importantly, the theoretical analysis carried out is aimed at establishing the local and global asymptotic stability properties of the disease-free steady state of the epidemic model using the fractional Routh-Hurwitz criterion and Lyapunov functional technique, respectively. It is proved that the steady state is locally and globally asymptotically stable at the value of the key epidemiological threshold quantity lower than unity. The result is numerically validated for different values of fractional order to show the asymptotic behavior of the disease dynamics. This result is significant for fighting and preventing Ebola epidemic in the population, since the Caputo derivative operator allows for effective description of the disease dynamics with memory, where the future evolution of the disease is governed by its prior history.

Dimensions

[1] World Health Organization (WHO), “Fact sheets on Ebola virus disease”. [Online]. https://www.who.int/news-room/fact-sheets/detail/ebola-virus-disease.

[2] Centres for Diseases Control and Prevention (CDC), “Ebola virus disease”. [Online]. http://www.cdc.gov/vhf/ebola/about.html.

[3] T. Berge, J. M. S. Lubuma, G. M. Moremedi, N. Morris & R. Kondera-Shava, “A simple mathematical model for ebola in Africa”, J. Biol. Dyn. 11 (2016) 42. https://doi:10.1080/17513758.2016.1229817.

[4] I. Takaidza, O. D. Makinde & K. O. Okosun, “Computational modelling and optimal control of Ebola virus disease with non-linear incidence rate”, J. Phys. Conf. Ser. 818 (2017) 012003. https://doi.org/10.1088/1742-6596/818/1/012003.

[5] E. Bonyah, K. Badu & S. K. Asiedu-Addo, “Optimal control application to an Ebola model”, Asian Pac. J. Trop. Biomed. 6 (2016) 283. https://doi.org/10.1016/j.apjtb.2016.01.012.

[6] S. Jiang, K. Wang, C. Li, G. Hong, X. Zhang, M. Shan, H. Li & J. Wang, “Mathematical models for devising the optimal Ebola virus disease eradication”, J. Transl. Med. 15 (2017) 124. https://doi.org/10.1186/s12967-017-1224-6.

[7] A. Rachah & D. F. M. Torres, “Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa”, Discrete Dyn. Nat. Soc. 2015 (2015) 842792. https://doi.org/10.1155/2015/842792.

[8] E. Okyere, J. D. Ankamah, A. K. Hunkpe & D. Mensah, “Deterministic epidemic models for ebola infection with time-dependent controls”, Discrete Dyn. Nat. Soc. 2020 (2020) 2823816. https://doi.org/10.1155/2020/2823816.

[9] M. Rafiq, W. Ahmad, M. Abbas & D. Baleanu, “A reliable and competitive mathematical analysis of Ebola epidemic model”, Adv. Differ. Equ. 2020 (2020) 540. https://doi.org/10.1186/s13662-020-02994-2.

[10] A. Mhlanga, “Dynamical analysis and control strategies in modelling Ebola virus disease”, Adv. Differ. Equ. 2019 (2019) 458. https://doi.org/10.1186/s13662-019-2392-x.

[11] R. Verma, S. P. Tiwari & R. K. Upadhyay, “Transmission dynamics of epidemic spread and outbreak of Ebola in West Africa: fuzzy modeling and simulation”, J. Appl. Math. Comput. 60 (2019) 637. https://doi.org/10.1007/s12190-018-01231-0.

[12] T. T. Yusuf, A Abidemi, A. S. Afolabi & E. J. Dansu, “Optimal control of the coronavirus pandemic with impacts of implemented control measures”, Nigerian Society of Physical Sciences 4 (2022) 88. https:doi.org/10.4648/jnsps.2022.414.

[13] B. Z. Aga, H. G. Tasisa, T. D. Keno, A. G. Geleta, D. W. Dinsa & A. R. Geletu, “Corruption dynamics: a mathematical model analysis”, Frontiers in Applied Mathematics and Statistics 10 (2024) 1323479. https://doi.org/10.3389/fams.2024.1323479.

[14] J. K. K. Asamoah, E. Okyere, E. Yankson, A. A. Opoku, A. Adom-Konadu, E. Acheampong & Y. D. Arthur, “Non-fractional and fractional mathematical analysis and simulations for Q fever”, Chaos Solitons Fractals 156 (2021) 111821. https://doi.org/10.1016/j.chaos.2021.111821.

[15] Z. Odibat & D. Baleanu, “Nonlinear dynamics and chaos in fractional differential equations with a new generalized Caputo fractional derivative”, Chinese J. Phys. 77 (2022) 1003. https://doi.org/10.1016/j.cjph.2021.08.018.

[16] A. Atangana, “Mathematical model of survival of fractional calculus, critics and their impact: how singular is our word?”, Adv. Diff. Equ. 2021 (2021) 403. https://doi.org/10.1186/s13662-021-03494-7.

[17] I. Owusu-Mensah, L. Akinyemi, B. Oduro & O. S. Iyiola, “A fractional order approach to modeling and simulations of the novel COVID-19”, Adv. Diff. Equ. 2020 (2020) 683. https://doi.org/10.1186/s13662-020-03141-7.

[18] R. B. Ogunrinde, U. K. Nwajeri, S. E. Fadugba, R. R. Ogunrinde & K. I. Oshinubi, “Dynamic model of COVID-19 and citizens reaction using fractional derivative”, Alex. Eng. J. 60 (2021) 2001. https://doi.org/10.1016/j.aej.2020.09.016.

[19] S. Olaniyi, T. O. Alade, F. M. Chuma, A. W. Ogunsola, O. R. Aderele & S. F. Abimbade, “A fractional-order nonlinear model for a within-host chikungunya virus dynamics with adaptive immunity using Caputo derivative operator”, Healthcare Analytics 4 (2023) 100205. https://doi.org/10.1016/j.health.2023.

100205.

[20] S. Adak, S. Barman, S. Jana, S. Majee & T. K. Kar, “Modelling and analysis of fractional-order epidemic model incorporating genetic algorithm-based optimization”, Journal of Applied Mathematics and Computing (2024). https://doi.org/10.1007/s12190-024-02224-y.

[21] M. Vellappandi, P. Kumar & V. Govindaraj. “Role of fractional derivatives in the mathematical mmodelling of the transmission of Chlamydia in the United States from 1989 to 2019”, Nonlinear Dynamics 111 (2023) 4915. https://doi.org/10.1007/s11071-022-08073-3.

[22] K. S. Nisar, M. Farman, M. Abdel-Aty & J. Cao, “A review on epidemic models in sight of fractional calculus”, Alexandria Engineering Journal 75 (2023) 81. https://doi.org/10.1016j.aej.2023.05.071.

[23] A. Raza M. Farman A. Akgül & M. S. Iqbal, “Ahmad A. Simulation and numerical solution of fractional order Ebola virus model with novel technique”, AIMS Bioeng. 7 (2020) 194. https://doi.org/10.3934/bioeng.2020017.

[24] H. M. Srivastava & K. M. Saad, “Numerical simulation of the fractal-fractional Ebola virus”, Fractal Fract. 4 (2020) 49. https://doi.org/10.3390/fractalfract4040049.

[25] M. Farman A. Akgül, T. Abdeljawad P. A. Naik N. Bukhari & A. Ahmad “Modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel”, Alex. Eng. J. 61 (2022) 2062. https://doi.org/10.1016/j.aej.2021.07.040.

[26] W. Pan, T. Li & S. Ali, “A fractional order epidemic model for the simulation of outbreaks of Ebola”, Adv. Diff. Equ. 2021 (2021) 1. https://doi.org/10.1186/s13662-021-03272-5.

[27] S. Momani, R. P. Chauhan, S. Kumar & S. Hadid, “A theoretical study on fractional hemorrhagic fever model”, Fractals 30 (2022) 2240032. https://doi.org/10.1142/S0218348X22400321.

[28] M. A. Almuqrin, P. Goswami, S. Sharma, I. Khan, R. S. Dubey & A. Khan, “Fractional model of Ebola virus in population of bats in frame of Atangana-Baleanu fractional derivative”, Results Phys. 26 (2021) 104295. https://doi.org/10.1016/j.rinp.2021.104295.

[29] S. C. Mpeshe, “Fractional-order derivative model of Rift Valley fever in urban peridomestic cycle”, Discrete Dyn. Nat. Soc. 2021 (2021) 2941961. https://doi.org/10.1155/2021/2941961.

[30] S. F. Abimbade, F. M. Chuma, S. O. Sangoniyi, R. S. Lebelo, K. O. Okosun & S. Olaniyi, “Global dynamics of a social hierarchy-stratified malaria model: insight from fractional calculus”, Math. 12 (2024) 1593. https://doi.org/10.3390/math12101593.

[31] H. R. Pandey, G. R. Phaijoo & D. B. Gurung, “Dengue dynamics in Nepal: A Caputo fractional model with optimal control strategies”, Heliyon 10 (2024) e33822. https://doi.org/10.1016/j.heliyon.2024.e33822.

[32] A. Omame & F. D. Zaman, “Analytic solution of a fractional order mathematical model for tumour with polyclonality and cell mutation,” Partial Differential Equations in Applied Mathematics 8 (2023) 100545. https://doi.org/10.1016/j.padiff.2023.100545.

[33] K. Diethelm, R. Garrappa, A. Giusti & M. Stynes, “Why fractional derivatives with nonsingular kernels should not be used”, Fractional Calculus and Applied Analysis 23 (2020) 610. https://doi.org/10.1515/fac-2020-0032.

[34] A. Boukhouima, K. Hattaf & N. Yousfi, “Dynamics of a fractional order HIV infection model with specific functional response and cure rate”, Int. J. Diff. Equation 2017 (2017) 8372140. https://doi.org/10.1155/2017/8372140.

[35] S. Olaniyi, S. F. Abimbade, F. M. Chuma, O. A. Adepoju & O. D. Falowo, “A fractional-order tuberculosis model with efficient and cost-effective optimal control interventions”, Decision Analytic Journal 8 (2023) 100324. https://doi.org/10.1016/j.dajour.2023.100324.

[36] A. V. Adeyoyin, Mathematical modeling of Ebola transmission: an optimal control approach, MSc Dissertation Submitted to African Institute for Mathematical Sciences (AIMS), Tanzania (2016).

[37] S. Olaniyi, O. S. Obabiyi, K. O. Okosun, A. T. Oladipo & S. O. Adewale, “Mathematical modelling and optimal cost-effective control of COVID-19 transmission dynamics”, Eur. Phys. J. Plus. 135 (2020) 938. https://doi.org/10.1140/epjp/s13360-020-00954-z.

[38] P. van den Driessche & J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Math. Biosci. 180 (2002) 29. https://doi.org/10.1016/S00225-5564(02)00108-6.

[39] E. Ahmed, A. M. A. El-Sayed & H. A. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems”, Phys. Lett. A 358 (2006) 1. https://doi.org/10.1016/j.phsleta.2006.04.087.

[40] G. T. Tilahun, W. A. Woldegerima & N. Mohammed, “A fractional order model for the transmission dynamics of hepatitis B virus with two-age structure in the presence of vaccination”, Arab J. Basic Appl. Sci. 28 (2021) 87. https://doi.org/10.1080/25765299.2021.1896423.

[41] A. Boukhouima, K. Hattaf, E. Lotfi, M. Mahrouf, D. F. M. Torres & N. Yousfi, “Lyapunov functions for fractional-order systems in biology: methods and applications”, Chaos Soltions Fractals 140 (2020) 110224. https://doi.org/10.1016/j.chaos.2020.110224.

[42] O. S. Obabiyi & S. Olaniyi, “Global stability analysis of malaria transmission dynamics with vigilant compartment”, Electron. J. Differential Equations 2019 (2019) 1. https://www.researchgate.net/publication/330601137_Global_stability_analysis_of_malaria_transmission_dynamics_with_vigilant_compartment.

[43] S. F. Abimbade, S. Olaniyi, O. A. Ajala & M. O. Ibrahim, “Optimal control analysis of a tuberculosis model with exogenous re-infection and incomplete treatment”, Optim Control Appl Meth. 41 (2020) 2349. https://doi.org/10.1002/oca.2658.

[44] S. F. Abimbade, S. Olaniyi & O. A. Ajala, “Recurrent malaria dynamics: insight from mathematical modelling”, Eur. Phys. J. Plus 137 (2022) 292. https://doi.org/10.1140/epjp/s13360-022-02510-3.

[45] A. Abidemi, K. M. Owolabi & E. Pindza, “Modelling the transmission dynamics of Lassa fever with nonlinear incidence rate and vertical transmission”, Phys. A: Stat. Mech. Appl. 597 (2022) 1272259. https://doi.org/10.1016/j.physa.2022.127259.

[46] J. P. LaSalle, “The stability of dynamical systems”, SIAM, Philadelphia, PA, 1976. https://epubs.siam.org/doi/book/10.1137/1.9781611970432.

[47] Z. Q. Xia, S. F. Wang, S. L. Li, L. Y. Huang, W. Y. Zhang, G. Q. Sun, Z. T. Gai & Z. Jin, “Modeling the transmission dynamics of Ebola virus disease in Liberia”, Sci. Rep. 5 (2015) 13857. https://doi.org/10.1038/srep13857.

[48] Z. M. Odibat & S. Momami, “An algorithm for the numerical solution of differential equations of fractional order”, J. Appl. Math. & Informatics 26 (2008) 15. https://www.researchgate.net/publication/2264218_An_Algorithm_For_The_Numerical_Solution_Of_Differential_Equations_Of_Fractional_Order.

Published

2025-02-01

How to Cite

Asymptotic stability analysis of a fractional epidemic model for Ebola virus disease in Caputo sense. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2304. https://doi.org/10.46481/jnsps.2025.2304

Issue

Volume 7, Issue 1, February 2025

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2024 Samson Olaniyi, Fuhara M. Chuma, Sulaimon F. Abimbade

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Asymptotic stability analysis of a fractional epidemic model for Ebola virus disease in Caputo sense. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2304. https://doi.org/10.46481/jnsps.2025.2304

Similar Articles

281-290 of 291

You may also start an advanced similarity search for this article.