Research Article

Year 2024,
Volume: 4 Issue: 3, 296 - 334, 30.09.2024
### Abstract

### References

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- [4] Savini, H., Janvier, F., Karkowski, L., Billhot, M., Aletti, M., Bordes, J. et al. Occupational exposures to Ebola virus in Ebola treatment center, Conakry, Guinea. Emerging Infectious Diseases, 23(8), 1380-1383, (2017).
- [5] Madelain, V., Nguyen, T.H.T., Olivo, A., De Lamballerie, X., Guedj, J., Taburet, A. and Mentré, F. Ebola virus infection: review of the pharmacokinetic and pharmacodynamic properties of drugs considered for testing in human efficacy trials. Clinical Pharmacokinetics, 55, 907-923, (2016).
- [6] Billioux, B.J., Smith, B. and Nath, A. Neurological complications of Ebola virus infection. Neurotherapeutics, 13(3), 461-470, (2016).
- [7] Adu, I.K., Wireko, F.A., Sebil, C. and Asamoah, J.K.K. A fractal-fractional model of Ebola with reinfection. Results in Physics, 52, 106893, (2023).
- [8] Thom, R., Tipton, T., Strecker, T., Hall, Y., Bore, J.A., Maes, P. et al. Longitudinal antibody and T cell responses in Ebola virus disease survivors and contacts: an observational cohort study. The Lancet Infectious Diseases, 21(4), 507-516, (2021).
- [9] Rugarabamu, S., Mboera, L., Rweyemamu, M., Mwanyika, G., Lutwama, J., Paweska, J. and Misinzo, G. Forty-two years of responding to Ebola virus outbreaks in Sub-Saharan Africa: a review. BMJ Global Health, 5(3), e001955, (2020).
- [10] Karaagac, B., Owolabi, K.M. and Pindza, E. A computational technique for the Caputo fractal fractional diabetes mellitus model without genetic factors. International Journal of Dynamics and Control, 11, 2161-2178, (2023).
- [11] Wireko, F.A., Adu, I.K., Gyamfi, K.A. and Asamoah, J.K.K. Modelling the transmission behavior of Measles disease considering contaminated environment through a fractal-fractional Mittag-Leffler kernel. Physica Scripta,, 99, 075025, (2024).
- [12] Wireko, F.A., Adu, I.K., Sebil, C. and Asamoah, J.K.K. A fractal-fractional order model for exploring the dynamics of Monkeypox disease. Decision Analytics Journal, 8, 100300, (2023).
- [13] Nana-Kyere, S., Boateng, F.A., Jonathan, P., Donkor, A., Hoggar, G.K., Titus, B.D. et al. Global analysis and optimal control model of COVID-19. Computational and Mathematical Methods in Medicine, 2022, 9491847, (2022).
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- [15] Asamoah, J.K.K., Okyere, E., Yankson, E., Opoku, A.A., Adom-Konadu, A., Acheampong, E. and Arthur, Y.D. Non-fractional and fractional mathematical analysis and simulations for Q fever. Chaos, Solitons & Fractals, 156, 111821, (2022).
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- [26] Singh, H. Analysis for fractional dynamics of Ebola virus model. Chaos, Solitons & Fractals, 138, 109992, (2020).
- [27] Farman, M., Akgül, A., Abdeljawad, T., Naik, P.A., Bukhari, N. and Ahmad, A. Modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel. Alexandria Engineering Journal,, 61(3), 2062-2073, (2022).
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- [35] Samet, B., Vetro, C. and Vetro, P. Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), 2154-2165, (2012).
- [36] Jiang, S., Zhang, J., Zhang, Q. and Zhang, Z. Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Communications in Computational Physics, 21(3), 650-678, (2017).
- [37] Padder, A., Almutairi, L., Qureshi, S., Soomro, A., Afroz, A., Hincal, E. and Tassaddiq, A. Dynamical analysis of generalized tumor model with Caputo fractional-order derivative. Fractal and Fractional, 7(3), 258, (2023).
- [38] Sikora, B. Remarks on the Caputo fractional derivative. Minut, 5, 76-84, (2023).
- [39] Baba, I.A., Ahmed, I., Al-Mdallal, Q. M., Jarad, F. and Yunusa, S. Numerical and theoretical analysis of an awareness COVID-19 epidemic model via generalized Atangana-Baleanu fractional derivative. Journal of Applied Mathematics and Computational Mechanics, 21(1), 7-18, (2022).
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- [42] Ahmed, I., Yusuf, A., Tariboon, J., Muhammad, M., Jarad, F. and Mikailu, B.B. A Dynamical and sensitivity analysis of the Caputo fractional-order Ebola virus model: implications for control measures. Science & Technology Asia, 28(4), 26-37, (2023).
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- [47] Asamoah, J.K.K., Owusu, M.A., Jin, Z., Oduro, F.T., Abidemi, A. and Gyasi, E.O. Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: using data from Ghana. Chaos, Solitons & Fractals, 140, 110103, (2020).
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In this article, we seek to formulate a robust mathematical model to study the Ebola disease through fractal-fractional operators. The study thus incorporates the transmission rate in the treatment centers and the relapse rate, since the Ebola virus persists or mostly hides in the immunologically protected sites of survivors. The Ebola virus disease (EVD) is one of the infectious diseases that has recorded a high death rate in countries where it is endemic, and Uganda is not an exception. The world at large has suffered from this deadly disease since 1976 when it was declared epidemic by the World Health Organization. The study employed fractal-fractional operators to identify the epidemiological patterns of EVD, especially in treatment centers and relapse. Memory loss and relapse are mostly observed in EVD survivors and this justifies the use of fractional operators to capture the true dynamics of the disease. Through dynamical analysis, the model is proven to be positive and bounded in the region. The model is further explicitly shown to have a solution that is unique and stable. The reproduction number was duly computed by using the next-generation matrix approach. By taking EVD epidemic cases in Uganda, the study fitted all parameters to real data. It has been shown through sensitivity index analysis that the transmission rate outside treatment centers and relapse have a significant effect on the endemic state of the disease, as they lead to an increase in the basic reproduction ratio.

- [1] Adu, I.K., Wireko, F.A., Nana-Kyere, S., Appiagyei, E., Osman, M.A.L. and Asamoah, J.K.K. Modelling the dynamics of Ebola disease transmission with optimal control analysis. Modeling Earth Systems and Environment, 10, 4731-4757, (2024).
- [2] Adu, I.K., Wireko, F.A., Osman, M.A.L. and Asamoah, J.K.K. A fractional order Ebola transmission model for dogs and humans. Scientific African, 24, e02230, (2024).
- [3] Gonzalez, A., Nikparvar, B., Matson, M.J., Seifert, S.N., Ross, H.D., Munster, V. and Bharti, N. Human movement and transmission dynamics early in Ebola outbreaks. medRxiv, (2023).
- [4] Savini, H., Janvier, F., Karkowski, L., Billhot, M., Aletti, M., Bordes, J. et al. Occupational exposures to Ebola virus in Ebola treatment center, Conakry, Guinea. Emerging Infectious Diseases, 23(8), 1380-1383, (2017).
- [5] Madelain, V., Nguyen, T.H.T., Olivo, A., De Lamballerie, X., Guedj, J., Taburet, A. and Mentré, F. Ebola virus infection: review of the pharmacokinetic and pharmacodynamic properties of drugs considered for testing in human efficacy trials. Clinical Pharmacokinetics, 55, 907-923, (2016).
- [6] Billioux, B.J., Smith, B. and Nath, A. Neurological complications of Ebola virus infection. Neurotherapeutics, 13(3), 461-470, (2016).
- [7] Adu, I.K., Wireko, F.A., Sebil, C. and Asamoah, J.K.K. A fractal-fractional model of Ebola with reinfection. Results in Physics, 52, 106893, (2023).
- [8] Thom, R., Tipton, T., Strecker, T., Hall, Y., Bore, J.A., Maes, P. et al. Longitudinal antibody and T cell responses in Ebola virus disease survivors and contacts: an observational cohort study. The Lancet Infectious Diseases, 21(4), 507-516, (2021).
- [9] Rugarabamu, S., Mboera, L., Rweyemamu, M., Mwanyika, G., Lutwama, J., Paweska, J. and Misinzo, G. Forty-two years of responding to Ebola virus outbreaks in Sub-Saharan Africa: a review. BMJ Global Health, 5(3), e001955, (2020).
- [10] Karaagac, B., Owolabi, K.M. and Pindza, E. A computational technique for the Caputo fractal fractional diabetes mellitus model without genetic factors. International Journal of Dynamics and Control, 11, 2161-2178, (2023).
- [11] Wireko, F.A., Adu, I.K., Gyamfi, K.A. and Asamoah, J.K.K. Modelling the transmission behavior of Measles disease considering contaminated environment through a fractal-fractional Mittag-Leffler kernel. Physica Scripta,, 99, 075025, (2024).
- [12] Wireko, F.A., Adu, I.K., Sebil, C. and Asamoah, J.K.K. A fractal-fractional order model for exploring the dynamics of Monkeypox disease. Decision Analytics Journal, 8, 100300, (2023).
- [13] Nana-Kyere, S., Boateng, F.A., Jonathan, P., Donkor, A., Hoggar, G.K., Titus, B.D. et al. Global analysis and optimal control model of COVID-19. Computational and Mathematical Methods in Medicine, 2022, 9491847, (2022).
- [14] Qureshi, S. and Atangana, A. Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data. Chaos, Solitons & Fractals, 136, 109812, (2020).
- [15] Asamoah, J.K.K., Okyere, E., Yankson, E., Opoku, A.A., Adom-Konadu, A., Acheampong, E. and Arthur, Y.D. Non-fractional and fractional mathematical analysis and simulations for Q fever. Chaos, Solitons & Fractals, 156, 111821, (2022).
- [16] Alzahrani, E.O. and Khan, M.A. Modeling the dynamics of Hepatitis E with optimal control. Chaos, Solitons & Fractals, 116, 287-301, (2018).
- [17] Liana, Y.A. and Chuma, F.M. Mathematical modeling of giardiasis transmission dynamics with control strategies in the presence of carriers. Journal of Applied Mathematics, 2023, 1562207, (2023).
- [18] Eikenberry, S.E. and Gumel, A.B. Mathematical modeling of climate change and malaria transmission dynamics: a historical review. Journal of Mathematical Biology, 77, 857-933, (2018).
- [19] Byamukama, M., Kajunguri, D. and Karuhanga, M. Optimal control analysis of pneumonia and HIV/AIDS co-infection model. Mathematics Open, 3, 2450006, (2024).
- [20] Cetin, M.A. and Araz, S.I. Prediction of COVID-19 spread with models in different patterns: A case study of Russia. Open Physics, 22(1), 20240009, (2024).
- [21] Arik, I.A., Sari, H.K. and Araz, S.˙I. Numerical simulation of Covid-19 model with integer and non-integer order: The effect of environment and social distancing. Results in Physics, 51, 106725, (2023).
- [22] Djiomba Njankou, S.D. and Nyabadza, F. Modelling the role of human behaviour in Ebola virus disease (EVD) transmission dynamics. Computational and Mathematical Methods in Medicine, 2022, 150043, (2022).
- [23] Rafiq, M., Ahmad, W., Abbas, M. and Baleanu, D. A reliable and competitive mathematical analysis of Ebola epidemic model. Advances in Difference Equations, 2020, 540, (2020).
- [24] Nazir, A., Ahmed, N., Khan, U., Mohyud-Din, S.T., Nisar, K.S. and Khan, I. An advanced version of a conformable mathematical model of Ebola virus disease in Africa. Alexandria Engineering Journal, 59(5), 3261-3268, (2020).
- [25] Rachah, A. and Torres, D.F.M. Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa. Discrete Dynamics in Nature and Society, 2015, 842792, (2015).
- [26] Singh, H. Analysis for fractional dynamics of Ebola virus model. Chaos, Solitons & Fractals, 138, 109992, (2020).
- [27] Farman, M., Akgül, A., Abdeljawad, T., Naik, P.A., Bukhari, N. and Ahmad, A. Modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel. Alexandria Engineering Journal,, 61(3), 2062-2073, (2022).
- [28] Adu, I.K. and Wireko, F.A. On SITR theoretical model of Ebola virus propagation with relapse and reinfection. International Journal of Innovation and Development, 1(3), (2023).
- [29] Addai, E., Zhang, L., Preko, A.K. and Asamoah, J.K.K. Fractional order epidemiological model of SARS-CoV-2 dynamism involving Alzheimer’s disease. Healthcare Analytics, 2, 100114, (2022).
- [30] Qureshi, A.I., Chughtai, M., Loua, T.O., Pe Kolie, J., Camara, H.F.S., Ishfaq, M.F. et al. Study of Ebola virus disease survivors in Guinea. Clinical Infectious Diseases, 61(7), 1035-1042, (2015).
- [31] Kengne, J.N. and Tadmon, C. Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment. Infectious Disease Modelling, 9(3), 775-804, (2024).
- [32] MacIntyre, C.R. and Chughtai, A.A. Recurrence and reinfection a new paradigm for the management of Ebola virus disease. International Journal of Infectious Diseases, 43, 58-61, (2016).
- [33] Rezapour, S., Asamoah, J.K.K., Hussain, A., Ahmad, H., Banerjee, R., Etemad, S. and Botmart, T. A theoretical and numerical analysis of a fractal-fractional two-strain model of meningitis. Results in Physics, 39, 105775, (2022).
- [34] Atangana, A. Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons & Fractals, 102, 396-406, (2017).
- [35] Samet, B., Vetro, C. and Vetro, P. Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), 2154-2165, (2012).
- [36] Jiang, S., Zhang, J., Zhang, Q. and Zhang, Z. Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Communications in Computational Physics, 21(3), 650-678, (2017).
- [37] Padder, A., Almutairi, L., Qureshi, S., Soomro, A., Afroz, A., Hincal, E. and Tassaddiq, A. Dynamical analysis of generalized tumor model with Caputo fractional-order derivative. Fractal and Fractional, 7(3), 258, (2023).
- [38] Sikora, B. Remarks on the Caputo fractional derivative. Minut, 5, 76-84, (2023).
- [39] Baba, I.A., Ahmed, I., Al-Mdallal, Q. M., Jarad, F. and Yunusa, S. Numerical and theoretical analysis of an awareness COVID-19 epidemic model via generalized Atangana-Baleanu fractional derivative. Journal of Applied Mathematics and Computational Mechanics, 21(1), 7-18, (2022).
- [40] Ahmed, I., Yusuf, A., Ibrahim, A., Kumam, P. and Ibrahim, M.J. A mathematical model of the ongoing coronavirus disease (COVID-19) pandemic: a case study in Turkey. Science & Technology Asia, 27(4), 248-258, (2022).
- [41] Hussain, A., Ahmed, I., Yusuf, A. and Ibrahim, M.J. Existence and stability analysis of a fractional-order COVID-19 model. Bangmod International Journal of Mathematical and Computational Science, 7, 102-125, (2021).
- [42] Ahmed, I., Yusuf, A., Tariboon, J., Muhammad, M., Jarad, F. and Mikailu, B.B. A Dynamical and sensitivity analysis of the Caputo fractional-order Ebola virus model: implications for control measures. Science & Technology Asia, 28(4), 26-37, (2023).
- [43] Granas, A. and Dugundji, J. Fixed Point Theory. Springer: New York, (2003).
- [44] Hyers, D.H. On the stability of the linear functional equation. Proceedings of the National Academy of Sciences, 27(4), 222-224, (1941).
- [45] Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society, 72, 297-300, (1978).
- [46] Gopal, K., Lee, L.S. and Seow, H.V. Parameter estimation of compartmental epidemiological model using harmony search algorithm and its variants. Applied Sciences, 11(3), 1138, (2021).
- [47] Asamoah, J.K.K., Owusu, M.A., Jin, Z., Oduro, F.T., Abidemi, A. and Gyasi, E.O. Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: using data from Ghana. Chaos, Solitons & Fractals, 140, 110103, (2020).
- [48] Asamoah, J.K.K., Jin, Z., Sun, G.Q., Seidu, B., Yankson, E., Abidemi, A. et al. Sensitivity assessment and optimal economic evaluation of a new COVID-19 compartmental epidemic model with control interventions. Chaos, Solitons & Fractals, 146, 110885, (2021).
- [49] Allahamou, A., Azroul, E., Hammouch, Z. and Alaoui, A.L. Modeling and numerical investigation of a conformable co-infection model for describing Hantavirus of the European moles. Mathematical Methods in the Applied Sciences, 45(5), 2736-2759, (2022).
- [50] Hamou, A.A., Rasul, R.R.Q., Hammouch, Z. and Özdemir, N. Analysis and dynamics of a mathematical model to predict unreported cases of COVID-19 epidemic in Morocco. Computational and Applied Mathematics, 41, 289, (2022).
- [51] Alla Hamou, A., Azroul, E. and Lamrani Alaoui, A. Fractional model and numerical algorithms for predicting COVID-19 with isolation and quarantine strategies. International Journal of Applied and Computational Mathematics, Springer, 7, 142, (2021).
- [52] Hamou, A.A., Azroul, E., Hammouch, Z. and Alaoui, A.L. A fractional multi-order model to predict the COVID-19 outbreak in Morocco. Applied and Computational Mathematics, 20(1), 177-203, (2020).
- [53] Martcheva, M. An Introduction to Mathematical Epidemiology (Vol. 61). Springer: New York, (2015).
- [54] Asamoah, J.K.K. A fractional mathematical model of heartwater transmission dynamics considering nymph and adult amblyomma ticks. Chaos, Solitons & Fractals, 174, 113905, (2023).
- [55] Branda, F. and Maruotti, A. 2022 Uganda Ebola outbreak: Early descriptions and open data. Journal of Medical Virology, 95, e28344, (2023).
- [56] Branda, F., Mahal, A., Maruotti, A., Pierini, M. and Mazzoli, S. The challenges of open data for future epidemic preparedness: The experience of the 2022 Ebolavirus outbreak in Uganda. Frontiers in Pharmacology, 14, 1101894, (2023).
- [57] Worldometer. Population of Uganda, (2022). World Population Prospects: The 2022 Revision, Frontiers Media SA 1, (2022). https://www.worldometers.info/world-population/uganda-population.
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There are 63 citations in total.

Primary Language | English |
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Subjects | Biological Mathematics, Complex Systems in Mathematics, Dynamical Systems in Applications |

Journal Section | Research Articles |

Authors | |

Publication Date | September 30, 2024 |

Submission Date | July 11, 2024 |

Acceptance Date | September 14, 2024 |

Published in Issue | Year 2024 Volume: 4 Issue: 3 |

The published articles in MMNSA are licensed under a Creative Commons Attribution 4.0 International License