Numerical Solution of Fractional order Malaria Model via the Generalized Fractional Adams-Bashforth-Moulton Approach
Keywords:
malaria, Fractional, Adam-Bashforth-Moulton, Transmission, Control, strategies
Abstract
A fractional-order mathematical model studies the effects of contact rate and recovery rate on Malaria transmission dynamics as this paper investigates different epidemiological characteristics of malaria infection. We put forward conditions to ensure the model solution uniqueness and performs an endemic equilibrium stability assessment through Lyapunov function application. Numerical simulations running the fractional Adams–Bashforth–Moulton method reveal how fractional-order values together with model parameters affect malaria control and dynamics. Numerical surface and contour plots reveal that Malaria prevalence rises when both contact rates and recovery rate increase but the recovery rate enhances the population's resistance against the disease spread. Decreasing contact rate in the population results in lower prevalence rates of malaria in the population.
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References
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[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] Joseph .E., Akor .L. F., Atokolo .W., Jeremiah Amos .J. (2025), Numerical Solution of Fractionalorder Chlamydia Model Via the Generalized Fractional Adams-Bashforth-Moulton Approach,International Journal of mathematics ,Vol.8 ,ISSN (01) :01-23.doi:https//doi.org/10.5281/zenodo.14716850.
[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.
[25Bonyah. 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.
[27] 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] World Health Organization, World Malaria Report, 2011, Website: http:// www.rbm.who.int, Accessed January 2013.
[3] World Health Organization, World Malaria Report, 2012, Website: http:// www.rbm.who.int, Accessed February 2013.
[4] Chiyaka .C.,Mukandavire.A.,Das.P, Global dynamics of a malaria model with partial immunity and two discrete time delays, Biomathematics 67 (1) (2006) 24.
[5] Esteva .L., Gumel .A.B.,Vargas.C., Leon. Qualitative study of transmission dynamics of drug-resistant malaria, Math. Comput. Modell. 50 (2009) 611.
[6] Fillinger .U., Ndenga .B., Githeko .A., Lindsay .S.W., Integrated malaria vector control with microbial larvicides and insecticide-treated nets in western Kenya: a controlled trial, Bull. World Health Organ. 87 (2009) 655.
[7] Rhee .M., Sissoko .M., Perry Sh.,Mcfarland.W., J. Parsonnet, O. Doumbo, Use of insecticide-treated nets (ITNs) following a malaria education intervention in Piron, Mali: a control trial with systematic allocation of households, Malaria J. 4 (35) (2005) 1.
[8] Roberts .D.R., Laughlin .L.L., Hsheih .P., DDT, global strategies, and a malaria control crisis in south America, Emerg. Infect. Dis. 3 (3) (1997) 295.
[9] Zi .S., Zhipeng .Q., Qingkai.K., Yun .Z., Assessment of vector control and pharmaceutical treatment in reducing malaria burden: a sensitivity and optimal control analysis, J. Biol. Syst. 20 (1) (2012) 67.
[10] Diethelm .K., (2022) The Frac PECE subroutine for the numerical solution of differential equations of fractional order,.DOI: https://doi.org/10.33003/fjs-2023-0706-2174.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] Joseph .E., Akor .L. F., Atokolo .W., Jeremiah Amos .J. (2025), Numerical Solution of Fractionalorder Chlamydia Model Via the Generalized Fractional Adams-Bashforth-Moulton Approach,International Journal of mathematics ,Vol.8 ,ISSN (01) :01-23.doi:https//doi.org/10.5281/zenodo.14716850.
[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.
[25Bonyah. 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.
[27] 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
2025-04-13
How to Cite
Nuhu, A., Omale, D., Atokolo, W., Amos, J., Onuche Acheneje, G., Abah, E., Abah, E., Egbemhenghe, J., & Bolaji, B. (2025). Numerical Solution of Fractional order Malaria Model via the Generalized Fractional Adams-Bashforth-Moulton Approach. GPH-International Journal of Mathematics, 8(03), 01-25. https://doi.org/10.5281/zenodo.15205492
Section
Articles
Copyright (c) 2025 Augustine Nuhu, David Omale, William Atokolo, Jeremiah Amos, Godwin Onuche Acheneje, Emmanuel Abah, Joseph Egbemhenghe, Bolarinwa Bolaji
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