Deep learning for parameter estimation in a typhoid fever model

Ramsha Shafqat; Mohammed M. Alshamrani; Ramsha Shafqat; Mohammed M. Alshamrani

doi:10.3934/math.2026616

AIMS Mathematics

2026, Volume 11, Issue 5: 14984-15007. doi: 10.3934/math.2026616

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Deep learning for parameter estimation in a typhoid fever model

Ramsha Shafqat ^{1
,
,},
Mohammed M. Alshamrani ²

1.
Department of Mathematics and Statistics, The University of Lahore, Sargodha 40100, Pakistan
2.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 14 April 2026 Revised: 18 May 2026 Accepted: 22 May 2026 Published: 28 May 2026
MSC : 34D20, 34K20, 34K60, 92C60, 92D45

Typhoid fever remains a major public-health problem, particularly in regions with poor sanitation, unsafe water, and limited healthcare. Motivated by persistent transmission, treatment failure, and relapse, this study proposes a deterministic typhoid fever model with direct and environmental transmission, treatment, and relapse mechanisms. The model is formulated in the modified Atangana–Baleanu–Caputo (mABC) fractional framework to incorporate memory effects. Its main qualitative properties, including positivity, boundedness, existence, and uniqueness of solutions, are established. Numerically, the Laplace–Adomian decomposition method (LADM) is applied to obtain approximate solutions for different fractional orders. The results show that the fractional order strongly affects transient disease dynamics while preserving biologically meaningful behavior. A deep neural network (DNN) surrogate is also trained on the generated trajectories and assessed through standard diagnostics. In addition, parameter estimation is performed using clean and noisy synthetic data, showing good agreement between fitted and reference dynamics.
- typhoid fever,
- mABC fractional derivative,
- qualitative analysis,
- Laplace–Adomian decomposition,
- parameter estimation,
- model calibration,
- deep neural network
Citation: Ramsha Shafqat, Mohammed M. Alshamrani. Deep learning for parameter estimation in a typhoid fever model[J]. AIMS Mathematics, 2026, 11(5): 14984-15007. doi: 10.3934/math.2026616

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Abstract

Typhoid fever remains a major public-health problem, particularly in regions with poor sanitation, unsafe water, and limited healthcare. Motivated by persistent transmission, treatment failure, and relapse, this study proposes a deterministic typhoid fever model with direct and environmental transmission, treatment, and relapse mechanisms. The model is formulated in the modified Atangana–Baleanu–Caputo (mABC) fractional framework to incorporate memory effects. Its main qualitative properties, including positivity, boundedness, existence, and uniqueness of solutions, are established. Numerically, the Laplace–Adomian decomposition method (LADM) is applied to obtain approximate solutions for different fractional orders. The results show that the fractional order strongly affects transient disease dynamics while preserving biologically meaningful behavior. A deep neural network (DNN) surrogate is also trained on the generated trajectories and assessed through standard diagnostics. In addition, parameter estimation is performed using clean and noisy synthetic data, showing good agreement between fitted and reference dynamics.

References

[1]	S. Mushayabasa, C. P. Bhunu, E. T. Ngarakana-Gwasira, Mathematical analysis of a typhoid model with carriers, direct and indirect disease transmission, International J. Math. Sci. Eng. Appl., 7 (2013), 79–90.
[2]	World Health Organization, Typhoid vaccines: WHO position paper–March 2018, Weekly Epidemiological Record, 93 (2018), 153–172.
[3]	G. T. Tilahun, O. D. Makinde, D. Malonza, Modelling and optimal control of typhoid fever disease with cost‐effective strategies, Comput. Math. Meth. Med., 2017 (2017), 2324518. https://doi.org/10.1155/2017/2324518 doi: 10.1155/2017/2324518
[4]	M. K. Bhan, R. Bahl, S. Bhatnagar, Typhoid and paratyphoid fever, Lancet, 366 (2005), 749–762. https://doi.org/10.1016/S0140-6736(05)67181-4 doi: 10.1016/S0140-6736(05)67181-4
[5]	World Health Organization, Typhoid vaccines: WHO position paper, Weekly Epidemiological Record, 83 (2008), 49–59.
[6]	A. K. Schemmer, Heterogeneity of inflammation and host metabolism in a thyphoid fever model, PhD diss., Verlag nicht ermittelbar, 2012.
[7]	T. Butler, Treatment of typhoid fever in the 21st century: Promises and shortcomings, Clin. Microbiol. Infect., 17 (2011), 959–963. https://doi.org/10.1111/j.1469-0691.2011.03552.x doi: 10.1111/j.1469-0691.2011.03552.x
[8]	K. A. Tijani, C. E. Madubueze, R. I. Gweryina, Modelling typhoid fever transmission with treatment relapse response: Optimal control and cost-effectiveness analysis, Math. Models Comput. Simul., 16 (2024), 457–485. https://doi.org/10.1134/S2070048224700169 doi: 10.1134/S2070048224700169
[9]	B. A. Connor, E. Schwartz, Typhoid and paratyphoid fever in travellers, Lancet Infect. Dis., 5 (2005), 623–628. https://doi.org/10.1016/S1473-3099(05)70239-5 doi: 10.1016/S1473-3099(05)70239-5
[10]	R. B. Kiplang'at, W. Kirui, L. O. Olwamba, B. Tonui, Prey-predator model on the interaction of pathogenic bacteria and bacteriophages in the presence of medication, J. Adv. Math. Comput. Sci., 39 (2024), 71–84. https://doi.org/10.9734/jamcs/2024/v39i91928 doi: 10.9734/jamcs/2024/v39i91928
[11]	H. Abboubakar, R. Racke, Mathematical modeling, forecasting, and optimal control of typhoid fever transmission dynamics, Chaos Solitons Fract., 149 (2021), 111074. https://doi.org/10.1016/j.chaos.2021.111074 doi: 10.1016/j.chaos.2021.111074
[12]	I. Petráš, Fractional-order nonlinear systems: Modeling, analysis and simulation, Berlin: Springer Science & Business Media, 2011.
[13]	P. Okolo, O. Abu, On optimal control and cost-effectiveness analysis for typhoid fever model, Fudma J. Sci., 4 (2020), 437–445. https://doi.org/10.33003/fjs-2020-0403-258 doi: 10.33003/fjs-2020-0403-258
[14]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Amsterdam: Elsevier, 1998.
[15]	A. Zeb, G. Nazir, K. Shah, E. Alzahrani, Theoretical and semi-analytical results to a biological model under Atangana–Baleanu–Caputo fractional derivative, Adv. Diff. Equ., 2020 (2020), 654. https://doi.org/10.1186/s13662-020-03117-7 doi: 10.1186/s13662-020-03117-7
[16]	C. Milici, G. Drăgănescu, J. T. Machado, Introduction to fractional differential equations, Berlin: Springer, 2018.
[17]	M. N. Jan, T. Khan, G. Zaman, Analytical approximate solution of hepatitis B epidemic model comparison with vaccination, Punjab Uni. J. Math., 51 (2019), 53–69.
[18]	A. I. Abioye, M. O. Ibrahim, O. J. Peter, S. Amadiegwu, F. A. Oguntolu, Differential transform method for solving mathematical model of SEIR and SEI spread of malaria, Verlag nicht ermittelbar, 2018.
[19]	A. Ahmad, R. Ali, I. Ahmad, F. A. Awwad, E. A. Ismail, Global stability of fractional order HIV/AIDS epidemic model under caputo operator and its computational modeling, Fractal Fract., 7 (2023), 643. https://doi.org/10.3390/fractalfract7090643 doi: 10.3390/fractalfract7090643
[20]	A. Ahmad, R. Ali, I. Ahmad, M. Ibrahim, Fractional view analysis of the transmission dynamics of norovirus with contaminated food and water, Int. J. Biomath., 17 (2024), 2350072. https://doi.org/10.1142/S1793524523500729 doi: 10.1142/S1793524523500729
[21]	R. Ali, Z. Zhang, H. Ahmad, M. M. Alam, The analytical study of soliton dynamics in fractional coupled Higgs system using the generalized Khater method, Opt. Quant. Electron., 56 (2024), 1067. https://doi.org/10.1007/s11082-024-06924-4 doi: 10.1007/s11082-024-06924-4
[22]	R. Ali, Z. Zhang, H. Ahmad, Exploring soliton solutions in nonlinear spatiotemporal fractional quantum mechanics equations: an analytical study, Opt. Quant. Electron., 56 (2024), 838. https://doi.org/10.1007/s11082-024-06370-2 doi: 10.1007/s11082-024-06370-2
[23]	A. Al-Quran, R. Shafqat, A. Alsaadi, A. M. Djaouti, Well‐posedness and ulam–Hyers stability of a normalized Caputo–Fabrizio fractional model for ischemic heart disease progression, J. Math., 2026 (2026), 9509503. https://doi.org/10.1155/jom/9509503 doi: 10.1155/jom/9509503
[24]	R. Shafqat, K. Abuasbeh, S. Trabelsi, M. Balti, Epidemic dynamics prediction using fractional SIRD and deep learning, Sci. Rep., 16 (2025), 3043. https://doi.org/10.1038/s41598-025-34299-3 doi: 10.1038/s41598-025-34299-3
[25]	R. Shafqat, Imran, A. Al-Quran, A. M. Djaouti, Enhancing rabies epidemic modeling with neural networks and fractional calculus, Sci. Rep., 16 (2026), 10409. https://doi.org/10.1038/s41598-026-40853-4 doi: 10.1038/s41598-026-40853-4
[26]	A. Al-Quran, R. Shafqat, A. Alsaadi, A. M. Djaouti, A hybrid fractal‐fractional and machine learning framework for Zika virus spread prediction, J. Math., 2026 (2026), 9999891. https://doi.org/10.1155/jom/9999891 doi: 10.1155/jom/9999891
[27]	S. M. Alamry, R. Shafqat, A. Alsaadi, Poliomyelitis dynamics under the normalized caputo–fabrizio operator: Analysis, stability, and numerics, J. Taibah Uni. Sci., 20 (2026), 2659460. https://doi.org/10.1080/16583655.2026.2659460 doi: 10.1080/16583655.2026.2659460
[28]	T. Li, S. Frassu, G. Viglialoro, Combining effects ensuring boundedness in an attraction-repulsion chemotaxis model with production and consumption, preprint paper, 2022. https://doi.org/10.48550/arXiv.2211.03608
[29]	T. Li, D. Acosta‐Soba, A. Columbu, G. Viglialoro, Dissipative gradient nonlinearities prevent $\delta$‐formations in local and nonlocal attraction–repulsion chemotaxis models, Stud. Appl. Math., 154 (2025), e70018. https://doi.org/10.1111/sapm.70018 doi: 10.1111/sapm.70018
[30]	J. Mushanyu, F. Nyabadza, G. Muchatibaya, P. Mafuta, G. Nhawu, Assessing the potential impact of limited public health resources on the spread and control of typhoid, J. Math. Biol., 77 (2018), 647–670. https://doi.org/10.1007/s00285-018-1219-9 doi: 10.1007/s00285-018-1219-9
[31]	S. Rashid, A. A. El-Deeb, M. Inc, A. Akgül, M. Zakarya, W. Weera, Stochastic dynamical analysis of the co-infection of the fractional pneumonia and typhoid fever disease model with cost-effective techniques and crossover effects, Alex. Eng. J., 69 (2023), 35–55. https://doi.org/10.1016/j.aej.2023.01.027 doi: 10.1016/j.aej.2023.01.027
[32]	H. Abboubakar, R. K. Regonne, K. S. Nisar, Fractional dynamics of typhoid fever transmission models with mass vaccination perspectives, Fractal Fract., 5 (2021), 149. https://doi.org/10.3390/fractalfract5040149 doi: 10.3390/fractalfract5040149
[33]	M. Al-Refai, D. Baleanu, On an extension of the operator with Mittag-Leffler kernel, Fractals, 30 (2022), 2240129. https://doi.org/10.1142/S0218348X22401296 doi: 10.1142/S0218348X22401296
[34]	I. A. Adetunde, Mathematical models for the dynamics of typhoid fever in kassena-nankana district of upper east region of Ghana, J. Mod. Math. Stat., 2 (2008), 45–49.
[35]	J. M. Mutua, F. B. Wang, N. K. Vaidya, Modeling malaria and typhoid fever co-infection dynamics, Math. Biosci., 264 (2015), 128–144. https://doi.org/10.1016/j.mbs.2015.03.014 doi: 10.1016/j.mbs.2015.03.014
[36]	H. R. Amick, Typhoid and paratyphoid fever, In: CDC Yellow Book 2020: Health Information for International Travel, 177 (2019), 364.
[37]	M. Ghosh, P. Chandra, P. Sinha, J. B. Shukla, Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population, Nonlinear Anal.: Real World Appl., 7 (2006), 341–363. https://doi.org/10.1016/j.nonrwa.2005.03.005 doi: 10.1016/j.nonrwa.2005.03.005
[38]	K. A. Tijani, C. E. Madubueze, R. I. Gweryina, Typhoid fever dynamical model with cost-effective optimalcontrol, J. Niger. Soc. Phys. Sci., 5 (2023), 1579. https://doi.org/10.46481/jnsps.2023.1579 doi: 10.46481/jnsps.2023.1579

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