Submit your papersSubmit Now
For Enquiries: [email protected]
IIARD LogoIIARD

An Approximate Solution of a Fractional-Order Epidemic Model Using the Homotopy Perturbation Method

P. C. Agina, B. E. Chuka-Obidiegwu, E. I. Chukwuma, K. K. Ibeh, V. B. Otugene, G.C.E. Mbah

Abstract

Ebola virus disease is a highly infectious and life-threatening illness that poses a major public health challenge. In this study, a fractional-order Ebola virus transmission model is developed to describe the spread of the disease within a population. The model uses fractional differential equations to incorporate memory effects in the transmission dynamics. Because the resulting system is nonlinear and difficult to solve exactly, the homotopy perturbation method is applied to obtain approximate analytical solutions. Numerical simulations are presented to illustrate the behavior of the model and to examine the influence of the semi-analytical method on disease spread. The results show that the homotopy perturbation method is an effective technique for solving fractional epidemic models and provides useful insights into Ebola virus transmission dynamics.

Keywords

fractional-order modelhomotopy perturbation methodEbola virus transmissionnon- linear system.

References

[1] Breman, J.G., Heymann, D.L., Loyd, G., McCormick, J.B., Miatudila, M., Murphy, F.A., ... & Johnson, K.M. (2016). Discovery and description of Ebola Zaire virus in 1976 and relevance to the West African epidemic during 2013 – 2016. Journal of Infectious Diseases, 214, 93–101. [2] Ndanusa, A., Abdulrahman, S., & Abdulmalik, A. (2015). A mathematical model for controlling the spread of Ebola virus disease in Nigeria. International Journal of Humanities and Management Sciences, 3(3). [3] Andrawus, J., Abdulrahman, S., Singh, R.V.K., & Manga, S. S. (2022). Sensitivity analysis of a mathematical modeling of Ebola virus population dynamics in the presence of vaccine. Dutse Journal of Pure and Applied Sciences, 8. https://dx.doi.org/10.4314/dujopas.v8i2a.5 [4] Faïçal Ndaïrou, Moein Khalighi, & Leo Lahti. (2023). Ebola epidemic model with dynamic population and memory. Chaos, Solitons & Fractals, 170, 113361 [5] Ahmad, M.D., Usman, M., Khan, A., & Imran, M. (2016). Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination. Infectious Diseases of Poverty, 5(72), DOI10.1186/s40249-016-0161-6 [6] Riaz, M., Khan, Z. A., Ahmad, S., & Ateya, A. A. (2024). Fractional order dynamics in epidemic disease modeling with advanced perspectives of fractional calculus. Fractal and fractional, 8(5), 291. https://doi.org/10.3390/fractalfract8050291 [7] Ali, Z., Rabiei, F., Rashidi, M. M., & Khodadadi, T. (2022). A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions. The European Physical Journal Plus, 137(395). https://doi.org/10.1140/epjp/s13360-022-02603-z [8] Pokhrel, S., Sharma, N., & Thakuri, R. (2026). Mathematical Models and Their Applications in Understanding the Dynamics of Infectious Diseases. Journal of Applied Mathematics, 2026, 3007179, 6 pages. https://doi.org/10.1155/jama/3007179 [9] Angstmann, C.N., Henry, B.I., & McGann, A.V. (2016). A Fractional Order Recovery SIR Model from a Stochastic Process. Bulletin of Mathematical Biology, 78(3), 468 – 499. [10] Kabli, K., Moujaddid, S., Niri, K., & Tridane, A. (2018). Cooperative system analysis of the Ebola virus epidemic model. Infectious Disease Modelling, 3, 145–159. https://doi.org/10.1016/j.idm.2018.09.004 [11] Abboubakar, H., Fandio, R., Sofack, B.S., & Fouda, H.P.E. (2022). Fractional Dynamics of a Measles Epidemic Model. Axioms, 11(8), 363. https://doi.org/10.3390/axioms11080363 [12] Farman M., Akgul, A., Abdeljawad, T., Ahmad Naik, P., Bukhari, N., & Ahmad, A. (2022). Modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel. Alexandria Engineering Journal, 61(3), 2062 – 2073. [13] Alraqad, T., Almalahi, M. A., Mohammed, N., Alahmade, A., Khaled A. Aldwoah, K. A., & Saber, H. (2024). Modeling Ebola Dynamics with a Φ-Piecewise Hybrid Fractional Derivative Approach. Fractal and Fractional. 8(10), 596. https://doi.org/10.3390/fractalfract8100596 [14] Mouaouine, A., Boukhouima, A., Hattaf, K. & Yousfi, N. (2018). A fractional order SIR epidemic model with nonlinear incidence rate. Advances in Difference Equations, 2018(160). https://doi.org/10.1186/s13662-018-1613-z [15] Nandi, T. R., Saha, A. K., & Roy, S. (2024). Analysis of a fractional order epidemiological model for tuberculosis transmission with vaccination and reinfection. Scientific Reports, 14(28290). https://doi.org/10.1038/s41598-024-73392-x E- ISSN 2489-009X , [16] Sidi-Ammi, M. R., Tahiri, M., & Torres, D.F.M. (2021). Global stability of a Caputo fractional SIRS model with general incidence rate. Mathematics in Computer Science ,15(9), 91-105. https://doi.org/10.1007/s11786-020-00467-z [17] Otugene, V.B., Agina, P.C., Edogbanya, H., & Digwo, D.C. (2022). Existence and uniqueness analysis of Atangana-Baleanu fractional order network model of Noro virus. International Journal of Mathematical Analysis and Modelling, 5(2), 63–75. [18] Olaniyi, S., Chuma, F.M., & Abimbade, S.F. (2025). Asymptotic stability analysis of a fractional epidemic model for Ebola virus disease in Caputo sense. Journal of the Nigerian Society of Physical Sciences, 7. DOI:10.46481/jnsps.2025.2304 [19] Farman, M., Bin Rasheed, Q., Saleem, M.U., & Ahmad, A. (2020). Modelling and Analysis of the Fractional Order Ebola Virus Model with Caputo-Fabrizio Derivative. Punjab University Journal of Mathematics, 52(10), 25-45. [20] Agina, P.C., Chukwuma, E.I., Chuka-Obidiegwu, B.E., Otugene, V.B., Ibeh, K.K., & Mbah, G.C.E. (2026). Local and Global Stability Analysis of the Disease-Free Equilibrium in a Fractional-Order Ebola Virus Transmission Model. International Journal of Applied Science and Mathematical Theory, 12(3), 30 – 48. [21] Raza, A., Farman, M., Akgul, A., Iqbal, M.S., & Ahmad, A. (2020). Simulation and numerical solution of fractional order Ebola virus model with novel technique. AIMS Bioengineering, 7(4), 194–207. [22] Rosa, S., & Ndairou, F. (2024). Optimal Control Applied to Piecewise Fractional Ebola Model. Mathematics, 12(7). https://doi.org/10.3390/math12070985 [23] Pan, W., Li, T., & Ali, S. (2021). A fractional order epidemic model for the simulation of outbreaks of Ebola. Advances in Difference Equations, 161, 1- 21. [24] Agina, P.C., Otugene, V.B., Adedoyin, J.D., Chukwuma, E.I., Chuka-Obidiegwu, B.E., Ezeora, D.I., & Mbah, G.C.E. (2025). Derivation of the basic reproduction number using the next generation matrix for a non-linear Ebola Virus transmission model. International Journal of Mathematical Analysis and Modelling, 8(2), 555 – 573. [25] Lolika, P.O., Helikumi, M., Jomah, S.A.S., Bakhet, M.Y.A., Galla, K.C., & Kheiralla, A.H. (2024). Global Stability Analysis of a Fractional-Order Ebola Epidemic Model with Control Strategies. Journal of Advances in Mathematics and Computer Science, 39(2), 20- 51. [26] Yadav, P., Jahan, S., & Nisar, K.S. (2023). Fractional order mathematical model of Ebola virus under Atangana–Baleanu–Caputo operator. Results in Control and Optimization, 13. [27] Shekari, P., Jajarmi, A., Torkzadeh, L., & Nourim, K. (2025). Fractional-order modeling of human behavior in infections: analysis using real data from Liberia. Computer Methods in Biomechanics and Biomedical, 6, 1–15. doi: 10.1080/10255842.2024.2448559. [28] Agina, P.C. & Echezona, G.N. (2022). Comparative analysis of the Gauss-Seidel and Jacobi methods for solving linear systems. COOU Journal of Physical Sciences, 5(1), 60–79. [29] Kolawole, M.K., Popoola, A.O., Odeyemi, K.A. & Bashiru, K.A. (2023). An Approximate Solution of Fractional Order Epidemic Model of Typhoid using the Homotopy Perturbation Method. UNIOSUN Journal of Engineering and Environmental Sciences, 5(1), 98 – 106. DOI: 10.36108/ujees/3202.50.0180 [30] He, J.H. (1999). Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. http://dx.doi.org/10.1016/S0045- 7825(99)00018-3 E- ISSN 2489-009X , [31] WHO (2023). Ebola virus disease. Retrieved from https://www.who.int/news-room/fact- sheets/detail/ebola-virus-disease. [32] Otugene, V.B., Mbah, G.C.E, Owhenagbo, P., Idoko, P.I., Adedoyin, J.D., Enyejo, L.A., & Agina, P.C. (2024). Mathematical Analysis of Hepatitis B Virus Transmission Dynamics in the Absence of Therapy with Atangana-Baleanu Fractional-order SPQWXY Model. Journal of Advances in Mathematics and Computer Science, 39(11), 1–28. [33] Ghorbal, K., & Sogokan, A. (2021). Characterizing positively invariant sets: Inductive and topological methods. arXiv:2009.09797 [cs.CG]. https://doi.org/10.48550/arXiv.2009.09797 [34] Yildirim, A. (2009). Application of He’s homotopy perturbation method for solving the Cauchy reaction-diffusion problem. Computers & Mathematics with Applications, 57(4), 612 – 618. https://doi.org/10.1016/j.camwa.2008.11.003 [35] He, J.H. (2012). Homotopy Perturbation Method with an Auxiliary Term. Abstract and Applied Analysis, 857612, 7 pages. https://doi.org/10.1155/2012/857612 [36] Didigwu, N. Mbah, G. & Abonyi, M. (2021). Solution of Ebola virus disease transmission dynamics mathematical model using homotopy perturbation method. Academic Journal of Statistics and Mathematics, 7(1), 1 – 8. [37] Madubueze, C.E., Kimbir, A.R., & Aboiyar, T. (2018). Global stability of Ebola virus disease model with contact tracing and quarantine. Applications and Applied Mathematics, 13(1), 382–403. [38] Statista. (2024). Africa: mortality rate 2000–2027 / Statista. Retrieved from https://www.statista.com/statistics/1227851/crude-death-rate-in-africa/. [39] Ahman, Q.O., Omale, D., Asogwa, C.C., Nnaji, D.U., & Mbah, G.C.E. (2021). Transmission dynamics of Ebola virus disease with vaccine, condom use, quarantine, isolation and treatment drug. African Journal of Infectious Diseases, 15(1), 10–23. https: //doi.org/10.21010/ajidv15i1.2 [40] Mbah, G.C.E., Onah, I.S., Ahman, Q.O., Collins, O.C., Asogwa, C.C., & Okoye, C. (2023). Mathematical modeling approach of the study of Ebola virus disease transmission dynamics in a developing country. African Journal of Infectious Diseases, 17(1), 10–26. https://doi.org/10.21010/Ajidv17il.2 [41] Gomes, M.F.C., Piontti, A.P., Rossi, L., Chao, D., Longini, I., Hallorans, M.E., & Vespignani, A. (2014). Assessing the international spreading risk associated with the 2014 West African Ebola outbreak. PLOS Currents Outbreak, Sep 2, Edition 1. doi: 10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5. [42] Legrand, J., Grais, R.F., Boelle, P.Y., Valleron, A.J., & Flahault, A. (2007). Understanding the dynamics of Ebola epidemics. Epidemiology & Infection, 135(5), 610–621. doi:10.1017/S0950268806007217 [43] Rivers, C.M., Lofgren, E.T., Marathe, M., Eubank, S., & Lewis, B.L. (2014). Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia. PLOS Currents, 6(6). doi: 10.1371/currents.outbreaks.fd38dd85078565450b0be3fcd78f5ccf