Modeling and Simulation of Typhoid Fever Using a Fractional-Order Approach with the Generalized Adams–Bashforth–Moulton Method
Abstract
We consider the epidemiological characteristics of Typhoid fever infection in this paper in the equation of a fractional-order mathematical model in Caputo derivative. The interventions that are employed in the model to control the disease include treatment and vaccination to investigate the impact of the controls on the dynamics of the disease. The existence and uniqueness of solutions under the frame of the fractional order and the stability of the endemic equilibrium point are defined and tested by the theory of Lyapunov functions. The model is numerically determined by using the fractional Adams-Bashforth-Moulton algorithm to point out the modification of the model parameters and the fractional orders of the model parameters into the influence of each of the above parameters on the disease progression. It has been demonstrated by the use of simulation that increased treatment and vaccination of the disease reduces the prevalence of Typhoid fever, and indicates the high degree of flexibility and realism of the fractional-order models compared to the classical integer order equations. The significance of fractional modeling in the description of the interactions between the effects of memory and nonlocal interaction between the biological systems is identified in the paper, and this improves the comprehension and management of infectious diseases. The model however presupposes that the population is homogeneous mixed and hypothetical values of the parameters therefore inhibits empirical validation. In order to render the model more predictive and applicable in practice in the development of effective control strategies on Typhoid fever, future investigations should be able to incorporate the spatial heterogeneity, stochastic effects.
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 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.
Ali.Z., Zada.A.,Shah. K., (2017) Existence and stability analysis of three-point boundary value problem, Int. J. Appl. Comput. Math.3 651–664, http://dx.doi. org/10.1007/s40819-017-0375-8.
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.
Ashcroft .M.T (1964) Basic science review: immunization against typhoid and paratyphoid fevers. Clin Pediatr 3(7):385–393.
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.
Atokolo W a, RemigiusAja .O. ,Omale .D., Ahman .Q. O.,Acheneje G. O., 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.
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.
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.
Bonyah. E., Zarin, R. Fatmawati, (2020), Mathematical modeling of Cancer and Hepatitis co-dynamics with non-local and nonsingular kernal, , 2052–2541.https://doi.org/10.28919/ cmbn/5029.
Boukanjime.B., Fatini .M.E. (2019),; A stochastic hepatitis B epidemic model driven by Lvy noise. 447. Phys A. 521 pp.796-806.
Boukanjime.B., Fatini .M.E. ; A stochastic hepatitis B epidemic model driven by Lvy noise. 447. Phys A. 521 (2019), pp.796-806.
Boukanjime.B., Fatini .M.E. ; A stochastic hepatitis B epidemic model driven by Lvy noise. 447. Phys A. 521 (2019), pp.796-806.
Chang.I .M.H. Hepatitis virus infection. Semen Fetal Neonatal Med, 12(2007), pp.160-167.
Chang.M.H. Hepatitis virus infection. Semen Fetal Neonatal Med, 12(2007), pp.160-167.
Chuanqing Xu , YuWang , Kedeng Cheng , Xin Yang , Xiaojing Wang , Songbai Guo , Maoxing Liu and Xiaoling Liu (2023) A Mathematical Model to Study the Potential Hepatitis B Virus Infections and Effects of Vaccination Strategies in China, Vaccines 11, 1530. https://doi.org/10.3390/vaccines11101530.
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.
Fraser A, Goldberg E, Acosta CJ, Paul M, Leibovici L (2007) Vaccines for preventing typhoid fever. Cochrane Database Syst Rev 3.
Ghanbari .B., Nisar.K. S., (2020), Some effective numerical techniques for chaotic systems involving fractal-fractional derivatives with different laws, Front. Phys., 8 192. https://doi.org/10.3389/fphy.2020.00192.
Granas.A., Dugundji .J., Fixed point theory, Springer: New York, 2003. https://doi.org/10.1007/978-0-387-21593-8.
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.
Khan.T., Ullah.Z., Ali.N., Zaman.G. Modeling and control of the hepatitis B virus spreading using an epidemic model. Chaos, Solitions Fractals, 124 (2019), pp.1-9.
Lavanchy.D. Hepatitis B virus epidemiology, disease burden, treatment and current and emerging prevention and control measuresJ Viral Hepat, 11 (2004), pp. 97-107.
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.
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.
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.
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.
Mann.J., Roberts.M. Modelling the epidemiology of hepatitis B in New Zealand. J Theor Biol.269 (2011), pp.266-272.
Mc Mahon.B.J. Epidemiology and natural history of hepatitis B.Semin Liver Dis, 25 (Suppl 1) (2005), pp. 3-8.h.
Milici C., G. Draganescu, J.T. Machado,( 2018) Introduction to Fractional Differential Equations, Springer,
Nyarko .C. C., Nsowa-Nuamah .N., Nicodemus .N., Nyarko .P. K., Wiah .E. N., Buabeng .A. (2021), Modelling Chlamydia trachomatis infection among Young women in Ghana: A case study at Tarkwa NsuaemMunicipalityAm. J. Appl. Math., 9 (3) pp. 75-85
Odionyenma U.B., Omame A., Ukanwoke N.O. (2022),, NometaI.Optimal control of Chlamydia model with vaccinationInt. J. Dyn. Control, 10 (1) pp. 332-348.
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.
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.
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.
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.
Samanta G.P., Sharma S., (2014),Analysis of a delayed Chlamydia epidemic model with pulse vaccination Appl. Math. Comput., 230 pp. 555-569.
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.
Sharma .S., Samanta. G.P. (2014), Analysis of a Chlamydia epidemic modelJ. Biol. Syst., 22 (04) pp. 713-744.
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.
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.
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.
Tilahun.G.T, Makinde.O.D, Malonza D (2017) Modeling and optimal control of typhoid fever disease with cost-effective strategies. In: Computational and mathematical methods in medicine, 2017.
Ullah. A.Z. T. Abdeljawad, Z. Hammouch, K. Shah, (2020) A hybrid method for solving fuzzy Volterra integral equations of separable type kernels, J. King Saud Univ. - Sci. 33 http://dx.doi.org/10.1016/j.jksus.2020.101246.
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.
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.
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.
