Numerical Methods for Addressing Fractional Order Trypanosomiasis Through the Generalized Adams-Bashforth-Moulton Approach
Abstract
Trypanosomiasis remains a critical vector-borne disease burden, especially in sub-Saharan African communities where tsetse fly populations thrive and healthcare infrastructure remains limited. Conventional integer-order mathematical frameworks frequently inadequately represent the hereditary characteristics and intricate transmission patterns inherent in vector-borne disease systems. This research presents a fractional-order mathematical framework for examining the epidemiological characteristics of trypanosomiasis transmission, with particular focus on how treatment interventions at different disease stages affect overall transmission dynamics. The primary goal is to explore how variations in treatment efficacy rates and vector-human contact patterns influence disease persistence and spread within affected populations. The framework employs fractional derivatives to more accurately capture the non-Markovian properties of infection progression and immune response mechanisms. The computational results further reveal that strategically implemented treatment protocols can dramatically reduce infection prevalence, potentially driving the system toward disease elimination scenarios. The innovation of this work centers on applying fractional calculus to trypanosomiasis transmission modeling, an approach relatively unexplored in this epidemiological context, while simultaneously incorporating multi-stage treatment interventions and natural immunity waning processes. The model demonstrates superior precision in representing temporal disease progression compared to classical integer-order approaches. This investigation underscores the utility of fractional-order modeling in vector-borne disease research and emphasizes the critical importance of strengthening treatment capacity and vector control measures to effectively manage and eliminate trypanosomiasis transmission in endemic regions.
Downloads
References
Abah E., Bolaji B., Atokolo W., Amos J., Acheneje G.O., Omede B.I, Amos J.,Omeje D. (2024), Fractional mathematical model for the Transmission Dynamics and control of Diphtheria ,International Journal of mathematical Analysis and Modelling,Vol.7,ISSN:2682-5694.
Ahmed, E. & El-Sayed, A.M.A. (2007), "Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models," Journal of Mathematical Analysis and Applications, 325(1), pp. 542-553. DOI: 10.1016/j.jmaa.2006.01.087
Aksoy, S., Gibson, W.C. & Lehane, M.J. (2003), "Interactions between tsetse and trypanosomes with implications for the control of trypanosomiasis," Advances in Parasitology, 53, pp. 1-83. DOI: 10.1016/S0065-308X(03)53002-0
Amos J., Omale D., Atokolo W., Abah E., Omede B.I., Acheneje G.O., Bolaji B. (2024), Fractional mathematical model for the Transmission Dynamics and control of Hepatitis C,FUDMA Journal of Sciences,Vol.8,No.5,pp.451-463, DOI: https://doi.org/10.33003/fjs-2024-0805-2883.
Anderson, R.M. & May, R.M. (1991), "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford.
Arafa, A.A.M., Rida, S.Z. & Khalil, M. (2012), "Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection," Nonlinear Biomedical Physics, 6(1), pp. 1. DOI: 10.1186/1753-4631-6-1
Atangana, A. & Araz, S.İ. (2020), "Fractional stochastic modelling illustration with modified Chua attractor," European Physical Journal Plus, 135(11), pp. 863. DOI: 10.1140/epjp/s13360-020-00857-w
Atangana, A. & Baleanu, D. (2016), "New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model," Thermal Science, 20(2), pp. 763-769. DOI: 10.2298/TSCI160111018A
Atokolo W a, RemigiusAja .O. ,Omale .D., Ahman .Q. O.,Acheneje G. O., Amos.J, Fractional mathematical model for the transmission dynamics and control of Lassa fever Journal of journal homepage: www.elsevier. 2773-1863/© 2024com/locate/fraopehttps:// doi.org/10.1016/j.fraope.2024.100110.
Baleanu, D., Diethelm, K., Scalas, E. & Trujillo, J.J. (2012), "Fractional Calculus: Models and Numerical Methods," World Scientific, Singapore. DOI: 10.1142/8180
Baleanu, D., Jajarmi, A., Mohammadi, H. & Rezapour, S. (2020), "A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative," Chaos, Solitons & Fractals, 134, pp. 109705. DOI: 10.1016/j.chaos.2020.109705
Blum, J., Schmid, C. & Burri, C. (2006), "Clinical aspects of 2541 patients with second stage human African trypanosomiasis," Acta Tropica, 97(1), pp. 55-64. DOI: 10.1016/j.actatropica.2005.08.001
Büscher, P., Cecchi, G., Jamonneau, V. & Priotto, G. (2017), "Human African trypanosomiasis," Lancet, 390(10110), pp. 2397-2409. DOI: 10.1016/S0140-6736(17)31510-6
Büscher, P., Cecchi, G., Jamonneau, V. & Priotto, G. (2017), "Human African trypanosomiasis," Lancet, 390(10110), pp. 2397-2409. DOI: 10.1016/S0140-6736(17)31510-6
Caputo, M. & Fabrizio, M. (2015), "A new definition of fractional derivative without singular kernel," Progress in Fractional Differentiation and Applications, 1(2), pp. 73-85.
Checchi, F., Funk, S., Chandramohan, D., Haydon, D.T. & Chappuis, F. (2018), "Updated estimate of the duration of the meningo-encephalitic stage in gambiense human African trypanosomiasis," BMC Research Notes, 11(1), pp. 702. DOI: 10.1186/s13104-018-3818-7
Daftardar-Gejji, V. & Jafari, H. (2006), "An iterative method for solving nonlinear functional equations," Journal of Mathematical Analysis and Applications, 316(2), pp. 753-763. DOI: 10.1016/j.jmaa.2005.05.009
Diethelm, K. & Ford, N.J. (2004), "Multi-order fractional differential equations and their numerical solution," Applied Mathematics and Computation, 154(3), pp. 621-640. DOI: 10.1016/S0096-3003(03)00739-2
Diethelm, K. (2010), "The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type," Springer-Verlag, Berlin. DOI: 10.1007/978-3-642-14574-2
Diethelm, K., Ford, N.J. & Freed, A.D. (2002), "A predictor-corrector approach for the numerical solution of fractional differential equations," Nonlinear Dynamics, 29(1), pp. 3-22. DOI: 10.1023/A:1016592219341
El-Sayed, A.M.A., El-Mesiry, A.E.M. & El-Saka, H.A.A. (2007), "On the fractional-order logistic equation," Applied Mathematics Letters, 20(7), pp. 817-823. DOI: 10.1016/j.aml.2006.08.013
Franco, J.R., Cecchi, G., Priotto, G., Paone, M., Diarra, A., Grout, L., Mattioli, R.C. & Argaw, D. (2018), "Monitoring the elimination of human African trypanosomiasis: Update to 2016," PLoS Neglected Tropical Diseases, 12(12), pp. e0006890. DOI: 10.1371/journal.pntd.0006890
Franco, J.R., Simarro, P.P., Diarra, A. & Jannin, J.G. (2014), "Epidemiology of human African trypanosomiasis," Clinical Epidemiology, 6, pp. 257-275. DOI: 10.2147/CLEP.S39728
Funk, S., Nishiura, H., Heesterbeek, H., Edmunds, W.J. & Checchi, F. (2013), "Identifying transmission cycles at the human-animal interface: The role of animal reservoirs in maintaining gambiense human African trypanosomiasis," PLoS Computational Biology, 9(1), pp. e1002855. DOI: 10.1371/journal.pcbi.1002855
Garrappa, R. (2018), "Numerical solution of fractional differential equations: A survey and a software tutorial," Mathematics, 6(2), pp. 16. DOI: 10.3390/math6020016
Gervas, E.H., Opoku, A.A., Azangue, N. & Luboobi, L.S. (2018), "Mathematical Modelling of Human African Trypanosomiasis Using Control Measures," Computational and Mathematical Methods in Medicine, 2018, pp. 5293568. DOI: 10.1155/2018/5293568
Hargrove, J.W. (2004), "Tsetse population dynamics," in The Trypanosomiases, I. Maudlin, P.H. Holmes, and M.A. Miles, Eds. Wallingford: CABI Publishing, pp. 113-137. DOI: 10.1079/9780851994758.0113
Hargrove, J.W., Ouifki, R., Kajunguri, D., Vale, G.A. & Torr, S.J. (2012), "Modeling the control of trypanosomiasis using trypanocides or insecticide-treated livestock," PLoS Neglected Tropical Diseases, 6(5), pp. e1615. DOI: 10.1371/journal.pntd.0001615
Hashim, I., Abdulaziz, O. & Momani, S. (2009), "Homotopy analysis method for fractional IVPs," Communications in Nonlinear Science and Numerical Simulation, 14(3), pp. 674-684. DOI: 10.1016/j.cnsns.2007.09.014
Heesterbeek, H., Anderson, R.M., Andreasen, V., Bansal, S., De Angelis, D., Dye, C., Eames, K.T., Edmunds, W.J., Frost, S.D., Funk, S. & Hollingsworth, T.D. (2015), "Modeling infectious disease dynamics in the complex landscape of global health," Science, 347(6227), pp. aaa4339. DOI: 10.1126/science.aaa4339
Jalija, E., Amos, J., Atokolo, W., Omale, D., Abah, E., Alih, U., & Bolaji, B. (2025).Numerical investigations on Dengue fever model through singular and non-singular fractional operators. International Journal of Mathematical Analysis and Modelling.
Jalija, E., Amos, J., Atokolo, W., Omale, D., Abah, E., Alih, U., & Bolaji, B. (2025).Numerical Solution of fractional order Typhoid Fever and HIV/AIDS co-infection Model Via TheGeneralized Fractional Adams-Bashforth-Moulton Approach. GPH-International Journal of Mathematics, 8(5), 01-31. https://doi.org/10.5281/zenodo.15623363.
Jamonneau, V., Ilboudo, H., Kaboré, J., Kaba, D., Koffi, M., Solano, P., Garcia, A., Courtin, D., Laveissière, C., Lingue, K., Buscher, P. & Bucheton, B. (2012), "Untreated human infections by Trypanosoma brucei gambiense are not 100% fatal," PLoS Neglected Tropical Diseases, 6(6), pp. e1691. DOI: 10.1371/journal.pntd.0001691
Kennedy, P.G.E. (2013), "Clinical features, diagnosis, and treatment of human African trypanosomiasis (sleeping sickness)," Lancet Neurology, 12(2), pp. 186-194. DOI: 10.1016/S1474-4422(12)70296-X
Kennedy, P.G.E. (2019), "Update on human African trypanosomiasis (sleeping sickness)," Journal of Neurology, 266(9), pp. 2334-2337. DOI: 10.1007/s00415-019-09425-7
Khan, M.A., Ullah, S. & Okosun, K.O. (2020), "A fractional order pine wilt disease model with Caputo-Fabrizio derivative," Advances in Difference Equations, 2020(1), pp. 410. DOI: 10.1186/s13662-020-02868-1
Kilbas, A.A., Srivastava, H.M. & Trujillo, J.J. (2006), "Theory and Applications of Fractional Differential Equations," Elsevier, Amsterdam.
Kumar, S., Kumar, A. & Odibat, Z.M. (2017), "A nonlinear fractional model to describe the population dynamics of two interacting species," Mathematical Methods in the Applied Sciences, 40(11), pp. 4134-4148. DOI: 10.1002/mma.4293
Li, Y., Chen, Y. & Podlubny, I. (2010), "Mittag-Leffler stability of fractional order nonlinear dynamic systems," Automatica, 45(8), pp. 1965-1969. DOI: 10.1016/j.automatica.2009.04.003
Liana, M., Wainaina, M., Mayondi, V.G. & Mwangi, J. (2020), "African Trypanosomiasis Dynamics: Modelling the Effects of Treatment, Education, and Vector Trapping," International Journal of Mathematics and Mathematical Sciences, 2020, pp. 3690472. DOI: 10.1155/2020/3690472
Lord, J.S., Hargrove, J.W., Torr, S.J. & Vale, G.A. (2018), "Climate change and African trypanosomiasis vector populations in Zimbabwe's Zambezi Valley: A mathematical modelling study," PLoS Medicine, 15(10), pp. e1002675. DOI: 10.1371/journal.pmed.1002675
Malvy, D. & Chappuis, F. (2011), "Sleeping sickness," Clinical Microbiology and Infection, 17(7), pp. 986-995. DOI: 10.1111/j.1469-0691.2011.03536.x
Mandal, M., Jana, S., Nandi, S.K., Khatua, A., Adak, S. & Kar, T.K. (2021), "A model based study on the dynamics of COVID-19: Prediction and control," Chaos, Solitons & Fractals, 136, pp. 109889. DOI: 10.1016/j.chaos.2020.109889
Momani, S. & Odibat, Z. (2007), "Homotopy perturbation method for nonlinear partial differential equations of fractional order," Physics Letters A, 365(5-6), pp. 345-350. DOI: 10.1016/j.physleta.2007.01.046
Odibat, Z.M. & Shawagfeh, N.T. (2007), "Generalized Taylor's formula," Applied Mathematics and Computation, 186(1), pp. 286-293. DOI: 10.1016/j.amc.2006.07.102
Odiit, M., Bessell, P.R., Fèvre, E.M., Robinson, T., Kinoti, J., Coleman, P.G., Welburn, S.C., McDermott, J. & Woolhouse, M.E.J. (2006), "Using remote sensing and geographic information systems to identify villages at high risk for rhodesiense sleeping sickness in Uganda," Transactions of the Royal Society of Tropical Medicine and Hygiene, 100(4), pp. 354-362. DOI: 10.1016/j.trstmh.2005.04.022
Omede.B. I, Israel. M.,Mustapha .M. K. , Amos J. ,Atokolo .W. , and Oguntolu .F. A. (2024) Approximate solution to the fractional soil transmitted Helminth infection model using Laplace Adomian Decomposition Method.Journal of mathematics. (2024) Int. J. Mathematics. 07(04), 16-40.
Onoja.T.U., Amos J., Atokolo. W., Abah .E. , Omale .D., Acheneje .G. O. & Bolaji B. (2025) Numerical Solution of Fractional order COVID-19 Model Via the Generalized Fractional Adams-Bashforth-Moulton Approach. International Journal of Mathematical Analysis and Modelling.
Pandey, A., Atkins, K.E., Bucheton, B., Camara, M., Aksoy, S., Galvani, A.P. & Funk, S. (2015), "Evaluating long-term effectiveness of sleeping sickness control measures in Guinea," Parasites & Vectors, 8(1), pp. 550. DOI: 10.1186/s13071-015-1121-x
Philip J., Omale D., Atokolo W., Amos J., Acheneje G.O., Bolaji B. (2024), Fractional mathematical model for the Transmission Dynamics and control of HIV/AIDs,FUDMA Journal of Sciences,Vol.8,No.6,pp.451-463, DOI: https://doi.org/10.33003/fjs-2024-0805-2883.
Pinto, C.M.A. & Machado, J.A.T. (2013), "Fractional model for malaria transmission under control strategies," Computers & Mathematics with Applications, 66(5), pp. 908-916. DOI: 10.1016/j.camwa.2012.11.017
Podlubny, I. (1999), "Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications," Academic Press, San Diego.
Priotto, G., Kasparian, S., Mutombo, W., Ngouama, D., Ghorashian, S., Arnold, U., Ghabri, S. & Baudin, E. (2009), "Nifurtimox-eflornithine combination therapy for second-stage African Trypanosoma brucei gambiense trypanosomiasis: A multicentre, randomised, phase III, non-inferiority trial," Lancet, 374(9683), pp. 56-64. DOI: 10.1016/S0140-6736(09)61117-X
Rock, K.S., Stone, C.M., Hastings, I.M., Keeling, M.J., Torr, S.J. & Chitnis, N. (2015), "Mathematical models of human African trypanosomiasis epidemiology," Advances in Parasitology, 87, pp. 53-133. DOI: 10.1016/bs.apar.2014.12.003
Rock, K.S., Torr, S.J., Lumbala, C. & Keeling, M.J. (2017), "Predicting the impact of intervention strategies for sleeping sickness in two high-endemicity health zones of the Democratic Republic of Congo," PLoS Neglected Tropical Diseases, 11(1), pp. e0005162. DOI: 10.1371/journal.pntd.0005162
Simarro, P.P., Cecchi, G., Franco, J.R., Paone, M., Diarra, A., Ruiz-Postigo, J.A., Fèvre, E.M., Mattioli, R.C. & Jannin, J.G. (2012), "Estimating and mapping the population at risk of sleeping sickness," PLoS Neglected Tropical Diseases, 6(10), pp. e1859. DOI: 10.1371/journal.pntd.0001859
Simarro, P.P., Franco, J., Diarra, A., Postigo, J.A.R. & Jannin, J. (2012), "Update on field use of the available drugs for the chemotherapy of human African trypanosomiasis," Parasitology, 139(7), pp. 842-846. DOI: 10.1017/S0031182012000169
Steverding, D. (2008), "The history of African trypanosomiasis," Parasites & Vectors, 1(1), pp. 3. DOI: 10.1186/1756-3305-1-3
Sun, H., Zhang, Y., Baleanu, D., Chen, W. & Chen, Y. (2018), "A new collection of real world applications of fractional calculus in science and engineering," Communications in Nonlinear Science and Numerical Simulation, 64, pp. 213-231. DOI: 10.1016/j.cnsns.2018.04.019
Sweilam, N.H., Khader, M.M. & Al-Bar, R.F. (2007), "Numerical studies for a multi-order fractional differential equation," Physics Letters A, 371(1-2), pp. 26-33. DOI: 10.1016/j.physleta.2007.06.016
World Bank (2023), "Life expectancy at birth, total (years) - Sub-Saharan Africa," World Bank Open Data. Available: https://data.worldbank.org/indicator/SP.DYN.LE00.IN
World Health Organization (2020), "Ending the neglect to attain the Sustainable Development Goals: A road map for neglected tropical diseases 2021–2030," World Health Organization, Geneva.
World Health Organization (2023), "Human African trypanosomiasis (sleeping sickness): Epidemiological update," Weekly Epidemiological Record, 98(10), pp. 95-105.
Author(s) and co-author(s) jointly and severally represent and warrant that the Article is original with the author(s) and does not infringe any copyright or violate any other right of any third parties, and that the Article has not been published elsewhere. Author(s) agree to the terms that the GPH Journal will have the full right to remove the published article on any misconduct found in the published article.
