Artificial neural networks for stability analysis and simulation of delayed rabies spread models

Ramsha Shafqat; Ateq Alsaadi; Ramsha Shafqat; Ateq Alsaadi

doi:10.3934/math.20241599

AIMS Mathematics

2024, Volume 9, Issue 12: 33495-33531. doi: 10.3934/math.20241599

Previous Article Next Article

Research article Special Issues

Artificial neural networks for stability analysis and simulation of delayed rabies spread models

Ramsha Shafqat ^{1
,
,},
Ateq Alsaadi 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: 26 September 2024 Revised: 11 November 2024 Accepted: 19 November 2024 Published: 26 November 2024
MSC : 34D20, 34K20, 34K60, 92C60, 92D45

Rabies remains a significant public health challenge, particularly in areas with substantial dog populations, necessitating a deeper understanding of its transmission dynamics for effective control strategies. This study addressed the complexity of rabies spread by integrating two critical delay effects—vaccination efficacy and incubation duration—into a delay differential equations model, capturing more realistic infection patterns between dogs and humans. To explore the multifaceted drivers of transmission, we applied a novel framework using piecewise derivatives that incorporated singular and non-singular kernels, allowing for nuanced insights into crossover dynamics. The existence and uniqueness of solutions was demonstrated using fixed-point theory within the context of piecewise derivatives and integrals. We employed a piecewise numerical scheme grounded in Newton interpolation polynomials to approximate solutions tailored to handle singular and non-singular kernels. Additionally, we leveraged artificial neural networks to split the dataset into training, testing, and validation sets, conducting an in-depth analysis across these subsets. This approach aimed to expand our understanding of rabies transmission, illustrating the potential of advanced mathematical tools and machine learning in epidemiological modeling.
- rabies spread model,
- piecewise derivative,
- Caputo derivative,
- Atangana-Baleanu-Caputo derivative,
- newton polynomials numerical method,
- artificial neural network
Citation: Ramsha Shafqat, Ateq Alsaadi. Artificial neural networks for stability analysis and simulation of delayed rabies spread models[J]. AIMS Mathematics, 2024, 9(12): 33495-33531. doi: 10.3934/math.20241599

Related Papers:

Abstract

Rabies remains a significant public health challenge, particularly in areas with substantial dog populations, necessitating a deeper understanding of its transmission dynamics for effective control strategies. This study addressed the complexity of rabies spread by integrating two critical delay effects—vaccination efficacy and incubation duration—into a delay differential equations model, capturing more realistic infection patterns between dogs and humans. To explore the multifaceted drivers of transmission, we applied a novel framework using piecewise derivatives that incorporated singular and non-singular kernels, allowing for nuanced insights into crossover dynamics. The existence and uniqueness of solutions was demonstrated using fixed-point theory within the context of piecewise derivatives and integrals. We employed a piecewise numerical scheme grounded in Newton interpolation polynomials to approximate solutions tailored to handle singular and non-singular kernels. Additionally, we leveraged artificial neural networks to split the dataset into training, testing, and validation sets, conducting an in-depth analysis across these subsets. This approach aimed to expand our understanding of rabies transmission, illustrating the potential of advanced mathematical tools and machine learning in epidemiological modeling.

References

[1]	M. S. Shenoy, A. Santra, A. K. Giri, Rabies elimination policy guidelines: Where do we stand, Indian J. Community He., 35 (2023), 258–263. http://doi.org/10.47203/IJCH.2023.v35i03.002 doi: 10.47203/IJCH.2023.v35i03.002
[2]	S. Abdulmajid, A. S. Hassan, Analysis of time delayed Rabies model in human and dog populations with controls, Afr. Mat., 32 (2021), 1067–1085. http://doi.org/10.1007/s13370-021-00882-w doi: 10.1007/s13370-021-00882-w
[3]	J. Chen, L. Zou, Z. Jin, S. G. Ruan, Modeling the geographic spread of rabies in China, PLoS Negl. Trop. Dis., 9 (2015), e0003772. https://doi.org/10.1371/journal.pntd.0003772 doi: 10.1371/journal.pntd.0003772
[4]	M. R. A. Nurdiansyah, Kasbawati, S. Toaha, Stability analysis and numerical simulation of rabies spread model with delay effects, AIMS Mathematics, 9 (2024), 3399–3425. http://doi.org/10.3934/math.2024167 doi: 10.3934/math.2024167
[5]	V. Bay, M. R. Shirzadi, M. J. Sirizi, I. M. Asl, Animal bites management in Northern Iran: Challenges and solutions, Heliyon, 9 (2023), e18637. http://doi.org/10.1016/j.heliyon.2023.e18637 doi: 10.1016/j.heliyon.2023.e18637
[6]	J. Zhang, Z. Jin, G. Q. Sun, T. Zhou, S. G. Ruan, Analysis of rabies in China: Transmission dynamics and control, PLoS One, 6 (2011), e20891. https://doi.org/10.1371/journal.pone.0020891 doi: 10.1371/journal.pone.0020891
[7]	K. Tohma, M. Saito, C. S. Demetria, D. L. Manalo, B. P. Quiambao, T. Kamigaki, et al., Molecular and mathematical modeling analyses of inter-island transmission of rabies into a previously rabies-free island in the Philippines, Infect. Genet. Evol., 38 (2016), 22–28. http://doi.org/10.1016/j.meegid.2015.12.001 doi: 10.1016/j.meegid.2015.12.001
[8]	Y. H. Huang, M. T. Li, Application of a mathematical model in determining the spread of the rabies virus: simulation study, JMIR Med. Inform., 8 (2020), e18627. http://doi.org/10.2196/18627 doi: 10.2196/18627
[9]	B. Pantha, S. Giri, H. R. Joshi, N. K. Vaidya, Modeling transmission dynamics of rabies in Nepal, Infectious Disease Modelling, 6 (2021), 284–301. http://doi.org/10.1016/j.idm.2020.12.009 doi: 10.1016/j.idm.2020.12.009
[10]	J. K. K. Asamoah, F. T. Oduro, E. Bonyah, B. Seidu, Modelling of rabies transmission dynamics using optimal control analysis, J. Appl. Math., 2017 (2017), 2451237. https://doi.org/10.1155/2017/2451237 doi: 10.1155/2017/2451237
[11]	M. J. Carroll, A. Singer, G. C. Smith, D. P. Cowan, G. Massei, The use of immunocontraception to improve rabies eradication in urban dog populations, Wildlife Res., 37 (2010), 676–687. http://doi.org/10.1071/WR10027 doi: 10.1071/WR10027
[12]	C. S. Bornaa, B. Seidu, M. I. Daabo, Mathematical analysis of rabies infection, J. Appl. Math., 2020 (2020), 1804270. https://doi.org/10.1155/2020/1804270 doi: 10.1155/2020/1804270
[13]	E. K. Renalda, D. Kuznetsov, K. Kreppel, Desirable Dog-Rabies control methods in an urban setting in Africa–A mathematical model, International Journal of Mathematical Sciences and Computing(IJMSC), 6 (2020), 49–67. http://doi.org/10.5815/ijmsc.2020.01.05 doi: 10.5815/ijmsc.2020.01.05
[14]	R. Haberman, Mathematical models: mechanical vibrations, population dynamics, and traffic flow, New Jersey: Society for Industrial and Applied Mathematics, 1998.
[15]	A. Columbu, R. D. Fuentes, S. Frassu, Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction–repulsion chemotaxis models with logistics, Nonlinear Anal.-Real, 79 (2024), 104135. https://doi.org/10.1016/j.nonrwa.2024.104135 doi: 10.1016/j.nonrwa.2024.104135
[16]	Z. Jiao, I. Jadlovská, T. X. Li, Global existence in a fully parabolic attraction-repulsion chemotaxis system with singular sensitivities and proliferation, J. Differ. Equations, 411 (2024), 227–267. https://doi.org/10.1016/j.jde.2024.07.005 doi: 10.1016/j.jde.2024.07.005
[17]	T. X. Li, S. Frassu, G. Viglialoro, Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption, Z. Angew. Math. Phys., 74 (2023), 109. https://doi.org/10.1007/s00033-023-01976-0 doi: 10.1007/s00033-023-01976-0
[18]	C. X. Huang, B. W. Liu, H. D. Yang, J. D. Cao, Positive almost periodicity on SICNNs incorporating mixed delays and D operator, Nonlinear Anal.-Model., 27 (2022), 1–21. https://doi.org/10.15388/namc.2022.27.27417 doi: 10.15388/namc.2022.27.27417
[19]	B. W. Liu, Finite-time stability of CNNs with neutral proportional delays and time-varying leakage delays, Math. Method. Appl. Sci., 40 (2017), 167–174. https://doi.org/10.1002/mma.3976 doi: 10.1002/mma.3976
[20]	X. Long, S. H. Gong, New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays, Appl. Math. Lett., 100 (2020), 106027. https://doi.org/10.1016/j.aml.2019.106027 doi: 10.1016/j.aml.2019.106027
[21]	Z. Sabir, S. B. Said, Q. Al-Mdallal, An artificial neural network approach for the language learning model, Sci. Rep., 13 (2023), 22693. https://doi.org/10.1038/s41598-023-50219-9 doi: 10.1038/s41598-023-50219-9
[22]	Z. Sabir, S. B. Said, Q. Al-Mdallal, M. R. Ali, A neuro swarm procedure to solve the novel second order perturbed delay Lane-Emden model arising in astrophysics, Sci. Rep., 12 (2022), 22607. https://doi.org/10.1038/s41598-022-26566-4 doi: 10.1038/s41598-022-26566-4
[23]	P. Kumar, A. Felicita, B. Nagaraja, A. R. Ajaykumar, Q. Al-Mdallal, Neural network model using Levenberg Marquardt backpropagation algorithm for the prandtl fluid flow over stratified curved sheet, IEEE Access, 12 (2024), 102242–102260. https://doi.org/10.1109/ACCESS.2024.3422099 doi: 10.1109/ACCESS.2024.3422099
[24]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, 2016, arXiv: 1602.03408.
[25]	I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386.
[26]	L. Zhang, M. Ur Rahman, H. D. Qu, M. Arfan, Adnan, Fractal-fractional Anthroponotic Cutaneous Leishmania model study in sense of Caputo derivative, Alex. Eng. J., 61 (2022), 4423–4433. https://doi.org/10.1016/j.aej.2021.10.001 doi: 10.1016/j.aej.2021.10.001
[27]	C. J. Xu, M. X. Liao, P. L. Li, L. Y. Yao, Q. W. Qin, Y. L. Shang, Chaos control for a fractional-order jerk system via time delay feedback controller and mixed controller, Fractal Fract., 5 (2021), 257. https://doi.org/10.3390/fractalfract5040257 doi: 10.3390/fractalfract5040257
[28]	B. W. Zhou, X. B. Shu, F. Xu, F. Y. Yang, Y. Wang, Exponential synchronization of dynamical complex networks via random impulsive scheme, Nonlinear Anal.-Model., 29 (2024), 816–832. http://doi.org/10.1090/S0894-0347-1992-1124979-1 doi: 10.1090/S0894-0347-1992-1124979-1
[29]	S. Li, L. X. Shu, X. B. Shu, F. Xu, Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics, 91 (2019), 857–872. http://doi.org/10.1080/17442508.2018.1551400 doi: 10.1080/17442508.2018.1551400
[30]	X. Zhu, P. Xia, Q. He, Z. Ni, L. Ni, Ensemble classifier design based on perturbation binary salp swarm algorithm for classification, Comput. Model. Eng. Sci,, 135 (2023), 653-671. https://doi.org/10.32604/cmes.2022.022985
[31]	B. Li, Z. Eskandari, Dynamical analysis of a discrete-time SIR epidemic model, J. Franklin I., 360 (2023), 7989-8007. https://doi.org/10.1016/j.jfranklin.2023.06.006 doi: 10.1016/j.jfranklin.2023.06.006
[32]	B. Li, T. X. Zhang, C. Zhang, Investigation of financial bubble mathematical model under fractal-fractional Caputo derivative, Fractals, 31 (2023), 2350050. https://doi.org/10.1142/S0218348X23500500 doi: 10.1142/S0218348X23500500
[33]	X. H. Zhu, P. F. Xia, Q. Z. He, Z. W. Ni, L. P. Ni, Coke price prediction approach based on dense GRU and opposition-based learning salp swarm algorithm, Int. J. Bio-Inspir. Com., 21 (2023), 106–121. http://doi.org/10.1504/IJBIC.2022.10047653 doi: 10.1504/IJBIC.2022.10047653
[34]	B. Li, Z. Eskandari, Z. Avazzadeh, Dynamical behaviors of an SIR epidemic model with discrete time, Fractal Fract., 6 (2022), 659. https://doi.org/10.3390/fractalfract6110659 doi: 10.3390/fractalfract6110659
[35]	K. Abuasbeh, R. Shafqat, A. Alsinai, M. Awadalla, Analysis of controllability of fractional functional random integroevolution equations with delay, Symmetry, 15 (2023), 290. http://doi.org/10.3390/sym15020290 doi: 10.3390/sym15020290
[36]	K. Abuasbeh, R. Shafqat, Fractional Brownian motion for a system of fuzzy fractional stochastic differential equation, J. Math., 2022 (2022), 3559035. https://doi.org/10.1155/2022/3559035 doi: 10.1155/2022/3559035
[37]	A. Atangana, S. İ. Araz, New concept in calculus: Piecewise differential and integral operators, Chaos. Soliton. Fract., 145 (2021), 110638. https://doi.org/10.1016/j.chaos.2020.110638 doi: 10.1016/j.chaos.2020.110638
[38]	Y. N. Anjam, R. Shafqat, I. E. Sarris, M. Ur Rahman, S. Touseef, M. Arshad, A fractional order investigation of smoking model using Caputo-Fabrizio differential operator, Fractal Fract., 6 (2022), 623. https://doi.org/10.3390/fractalfract6110623 doi: 10.3390/fractalfract6110623
[39]	A. Sami, A. Ali, R. Shafqat, N. Pakkaranang, M. Ur Rahmamn, Analysis of food chain mathematical model under fractal fractional Caputo derivative, Math. Biosci. Eng., 20 (2023), 2094–2109. http://doi.org/10.3934/mbe.2023097 doi: 10.3934/mbe.2023097
[40]	K. Abuasbeh, R. Shafqat, A. Alsinai, M. Awadalla, Analysis of the mathematical modelling of COVID-19 by using mild solution with delay Caputo operator, Symmetry, 15 (2023), 286. https://doi.org/10.3390/sym15020286 doi: 10.3390/sym15020286
[41]	A. Turab, R. Shafqat, S. Muhammad, M. Shuaib, M. F. Khan, M. Kamal, Predictive modeling of hepatitis B viral dynamics: A caputo derivative-based approach using artificial neural networks, Sci. Rep., 42 (2024), 21853. http://doi.org/10.1038/s41598-024-70788-7 doi: 10.1038/s41598-024-70788-7
[42]	H. D. Qu, S. Saifullah, J. Khan, A. Khan, M. Ur Rahman, G. Z. Zheng, Dynamics of leptospirosis disease in context of piecewise classical-global and classical-fractional operators, Fractals, 30 (2022), 2240216. https://doi.org/10.1142/S0218348X22402162 doi: 10.1142/S0218348X22402162
[43]	C. J. Xu, Z. X. Liu, P. L. Li, J. L. Yan, L. Y. Yao, Bifurcation mechanism for fractional-order three-triangle multi-delayed neural networks, Neural Process. Lett., 55 (2023), 6125–6151. http://doi.org/10.1007/s11063-022-11130-y doi: 10.1007/s11063-022-11130-y
[44]	Q. Z. He, P. F. Xia, C. Hu, B. Li, Public information, actual intervention and inflation expectations, Transform. Bus. Econ., 21 (2022), 644–666.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)