Numerical Solution of fractional order Typhoid Fever and HIV/AIDS co-infection Model Via The Generalized Fractional Adams-Bashforth-Moulton Approach
Keywords:
HIV/AIDs, Fractional, Adam-Bashforth-Moulton, Transmission
Abstract
We investigate the epidemiology of typhoid fever and HIV/AIDs co-infection using a fractional-order model, to understand how treatment affects the transmission of these diseases. In this study we establish situations where solutions exist and are unique and it looks at the stability of the endemic equilibrium by using the Lyapunov function. Applying the fractional Adams–Bashforth–Moulton method, numerically simulations show how the disease is controlled and spread by the chosen parameters. Numerical simulations, suggest that when people have more contacts and treatment is less effective, there is a higher chance of typhoid fever and HIV/AIDs co-infection. Optimizing how treatments are given can greatly limit the spread of the infection in the human population.
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References
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[22] Das, R., Patel, S., & Kumar, A. (2024), "Mathematical modeling of hepatitis C and COVID-19 coinfection in low- and middle-income countries: challenges and opportunities," BMC Public Health, 24(1), pp. 587.
[23] Ali.Z., Zada.A.,Shah. K., Existence and stability analysis of three-point boundary value problem, Int. J. Appl. Comput. Math.3 (2017) 651–664, http://dx.doi. org/10.1007/s40819-017-0375-8.
[24] Milici C., G. Draganescu, J.T. Machado, Introduction to Fractional Differential Equations, Springer, 2018.
[25] Bonyah. E., Zarin, R. Fatmawati, Mathematical modeling of Cancer and Hepatitis co-dynamics with non-local and nonsingular kernal, 2020, 2052–2541.https://doi.org/10.28919/ cmbn/5029.
[26] Baskonus. H.M., Bulut H., (2015) On the numerical solutions of some fractional ordinary differential equations by fractional Adams Bashforth-Moulton Method, Open Math. 13 1.
[27] Zhang.R.F.,Li. M.-C.,Gan. J.Y., Li.Q., Lan.Z.-Z., (2022). Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method, Chaos Solitons Fractals 154 (C). Results in Physics, vol. 37, article 105498.
[28] Atokolo W a, RemigiusAja .O. , Omale .D., Paul .R. V. ,Amos . J.,Ocha S. O., (2023) Mathematical modeling of the spread of vector borne diseases with influence of vertical transmission and preventive strategies FUDMA Journal of sciences: Vol. 7 No. 6, December (Special Issue), pp 75 -91 DOI: https://doi.org/10.33003/fjs-2023-0706-2174
[2] Shaikh .A.S, Nisar .K.S (2019) Transmission dynamics of fractional order typhoid fever model using Caputo-Fabrizio operator. Chaos Solitons Fractals 128:355–365.
[3] Ashcroft .M.T (1964) Basic science review: immunization against typhoid and paratyphoid fevers. Clin Pediatr 3(7):385–393.
[4] Fraser A, Goldberg E, Acosta CJ, Paul M, Leibovici L (2007) Vaccines for preventing typhoid fever. Cochrane Database Syst Rev 3.
[5] I. Chang.M.H. Hepatitis virus infection. Semen Fetal Neonatal Med, 12(2007), pp.160-167.
[6] Thornley.S., Bullen.C., Roberts.M., Hepatitis B in a high prevalence new zea land population a mathematical model applied to infection control policy J Theor Biol.254 (2008),pp.599-603.
[7] Liu .S.Q. , Wang .S.K. , Wang .L. Global dynamics of delay epidemic models with nonlinear incidence rate and relapse. Nonl Anal RWA,12(2011), pp.119-127.
[8] Liu .W.M. , Hethcote .H.W. , Levin .S.A. Dynamical behavior of epidemiological mod- els with nonlinear incidence rates. J Math Bio,25 (1987), pp.359-580.
9] Ren.J., Yang.X., Zhu.Q., Yang. L.Z., Zhang.C.; A novel computer virus model and its dynamics. Nonl Anal RWA,13(2012), pp.376-384.
[10] Boukanjime.B., Fatini .M.E. ; A stochastic hepatitis B epidemic model driven by Lvy noise. 447. Phys A. 521 (2019), pp.796-806.
[11] Odiba, P. O., G. O. Acheneje, and B. Bolaji. 2024. “A Compartmental Deterministic Epidemiological Model with Non-Linear Differential Equations for Analysing the Co-Infection Dynamics Between COVID-19, HIV, and Monkeypox Diseases.” Healthcare Analytics 5: 100311. https://doi.org/10.1016/j.health.2024.100311.
[12] Atokolo, W., Aja, R. O., Aniaku, S. E., Onah, I. S., &Mbah, G. C. (2022).Approximate solution of the fractional order sterile insect technology model via the Laplace– Adomian Decomposition Method for the spread of Zika virus disease.International Journal of Mathematics and Mathematical Sciences, 2022(1), 2297630.
[13] Atokolo W a, RemigiusAja .O. ,Omale .D., Ahman .Q. O.,Acheneje G. O., Amos . J. (2024) 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.
[14] Yunus. A.O, M.O. Olayiwola, M.A. Omolaye, A.O. Oladapo, (2023) A fractional order model of lassa fever disease using the Laplace-Adomian decomposition method, Health Care Anal. 3 100167, www.elsevier.com/locate/health.Health care Analytics.
[15] 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.
[16] 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.
[17] James P., 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.
[18] 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.
[19] Ahmed I., . Goufo E. F. D,Yusuf A., Kumam .P., Chaipanya P., and Nonlaopon K. ( 2021), “An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC fractional operator,” Alexandria Engineering Journal, vol. 60, no. 3, pp. 2979–2995.
[20] Smith, J., Johnson, A.B., & Lee, C. (2023), "Modeling the coinfection dynamics of hepatitis C and COVID-19: A systematic review," Journal of Epidemiology and Infection, 151(7), pp. 1350–1365.
[21] Ullah. A.Z. T. Abdeljawad, Z. Hammouch, K. Shah, A hybrid method for solving fuzzy Volterra integral equations of separable type kernels, J. King Saud Univ. - Sci. 33 (2020) http://dx.doi.org/10.1016/j.jksus.2020.101246.
[22] Das, R., Patel, S., & Kumar, A. (2024), "Mathematical modeling of hepatitis C and COVID-19 coinfection in low- and middle-income countries: challenges and opportunities," BMC Public Health, 24(1), pp. 587.
[23] Ali.Z., Zada.A.,Shah. K., Existence and stability analysis of three-point boundary value problem, Int. J. Appl. Comput. Math.3 (2017) 651–664, http://dx.doi. org/10.1007/s40819-017-0375-8.
[24] Milici C., G. Draganescu, J.T. Machado, Introduction to Fractional Differential Equations, Springer, 2018.
[25] Bonyah. E., Zarin, R. Fatmawati, Mathematical modeling of Cancer and Hepatitis co-dynamics with non-local and nonsingular kernal, 2020, 2052–2541.https://doi.org/10.28919/ cmbn/5029.
[26] Baskonus. H.M., Bulut H., (2015) On the numerical solutions of some fractional ordinary differential equations by fractional Adams Bashforth-Moulton Method, Open Math. 13 1.
[27] Zhang.R.F.,Li. M.-C.,Gan. J.Y., Li.Q., Lan.Z.-Z., (2022). Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method, Chaos Solitons Fractals 154 (C). Results in Physics, vol. 37, article 105498.
[28] Atokolo W a, RemigiusAja .O. , Omale .D., Paul .R. V. ,Amos . J.,Ocha S. O., (2023) Mathematical modeling of the spread of vector borne diseases with influence of vertical transmission and preventive strategies FUDMA Journal of sciences: Vol. 7 No. 6, December (Special Issue), pp 75 -91 DOI: https://doi.org/10.33003/fjs-2023-0706-2174
Published
2025-06-09
How to Cite
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 The Generalized Fractional Adams-Bashforth-Moulton Approach. GPH-International Journal of Mathematics, 8(5), 01-31. https://doi.org/10.5281/zenodo.15623363
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Copyright (c) 2025 Enejoh Jalija, Jeremiah Amos, William Atokolo, David Omale, Emmanuel Abah, Ugbede Alih, Bolarinwa Bolaji
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