Stochastic analysis and ANN-based approximation of measles transmission dynamics under Lévy noise

Anwarud Din; Anwarud Din

doi:10.3934/math.2026761

AIMS Mathematics

2026, Volume 11, Issue 6: 18715-18745. doi: 10.3934/math.2026761

Previous Article Next Article

Research article

Stochastic analysis and ANN-based approximation of measles transmission dynamics under Lévy noise

Anwarud Din ^,

School of Mechanical and Electrical Engineering, Chengdu University of Technology, Chengdu 610059, China

Received: 20 April 2026 Revised: 01 June 2026 Accepted: 10 June 2026 Published: 25 June 2026
MSC : 92D30, 60H10, 60G51, 68T07, 65C30

We develop a rigorous mathematical framework for measles transmission that integrates the impact of Lévy noise to account for environmental disturbances. First, the model is formulated, and the uniqueness and existence of a positive solution are investigated. A stochastic threshold is derived to ensure sufficient conditions for the persistence and extinction of the disease. It is demonstrated that the solution curves of the system fluctuate in the neighborhood of the disease-free state of the underlying deterministic model when $\mathbb{R}_D < 1$. We establish sufficient conditions for both extinction and persistence of the disease in the proposed stochastic system. The obtained results demonstrate the ability of the model to characterize the long-term dynamics of measles transmission. Artificial neural networks (ANNs) are employed to approximate the dynamics of the system with Lévy noise. The ANN model, trained on time-series data from stochastic simulations, accurately predicts transitions among the susceptible, exposed, infected, first-dose vaccinated, second-dose vaccinated, and recovered populations. The model's ability to capture stochastic dynamics driven by Lévy noise highlights the potential of ANNs in approximating complex epidemic models. The results validate the theoretical findings and demonstrate that stochastic environmental perturbations can significantly influence the transmission of epidemic diseases. As noise levels increase, a population's ability to sustain the spread of illness is limited, underscoring the critical role of the environment in shaping epidemic dynamics.
- Lévy noise,
- artificial neural networks,
- disease transmission,
- persistence,
- noise-induced dynamics
Citation: Anwarud Din. Stochastic analysis and ANN-based approximation of measles transmission dynamics under Lévy noise[J]. AIMS Mathematics, 2026, 11(6): 18715-18745. doi: 10.3934/math.2026761

Related Papers:

Abstract

We develop a rigorous mathematical framework for measles transmission that integrates the impact of Lévy noise to account for environmental disturbances. First, the model is formulated, and the uniqueness and existence of a positive solution are investigated. A stochastic threshold is derived to ensure sufficient conditions for the persistence and extinction of the disease. It is demonstrated that the solution curves of the system fluctuate in the neighborhood of the disease-free state of the underlying deterministic model when $\mathbb{R}_D < 1$. We establish sufficient conditions for both extinction and persistence of the disease in the proposed stochastic system. The obtained results demonstrate the ability of the model to characterize the long-term dynamics of measles transmission. Artificial neural networks (ANNs) are employed to approximate the dynamics of the system with Lévy noise. The ANN model, trained on time-series data from stochastic simulations, accurately predicts transitions among the susceptible, exposed, infected, first-dose vaccinated, second-dose vaccinated, and recovered populations. The model's ability to capture stochastic dynamics driven by Lévy noise highlights the potential of ANNs in approximating complex epidemic models. The results validate the theoretical findings and demonstrate that stochastic environmental perturbations can significantly influence the transmission of epidemic diseases. As noise levels increase, a population's ability to sustain the spread of illness is limited, underscoring the critical role of the environment in shaping epidemic dynamics.

References

[1]	Measles, World Health Organization, 2018. Available from: https://www.who.int/zh/news-room/fact-sheets/detail/measles.
[2]	C. J. Fraser-Bell, Global re-emergence of measles–2019 update, Global Biosecur., 2019.
[3]	R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford: Oxford University Press, 1991.
[4]	M. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton: Princeton University Press, 2008.
[5]	B. T. Grenfell, O. N. Bjornstad, J. Kappey, Traveling waves and spatial hierarchies in measles epidemics, Nature, 414 (2002), 716–723. https://doi.org/10.1038/414716a doi: 10.1038/414716a
[6]	N. C. Grassly, C. Fraser, Seasonal infectious disease epidemiology, Proc. Biol. Soc., 273 (2006), 2541–2550. https://doi.org/10.1098/rspb.2006.3604 doi: 10.1098/rspb.2006.3604
[7]	K. Xu, B. Gao, J. Li, Y. Xiang, L. Cao, M. Zhao, Clinical features, diagnosis, and management of COVID-19 vaccine-associated Vogt-Koyanagi-Harada disease, Hum. Vacc. Immunother., 19 (2023), 2220630. https://doi.org/10.1080/21645515.2023.2220630 doi: 10.1080/21645515.2023.2220630
[8]	Y. Jia, B. Fu, L. Dong, M. Zhao, Sweet syndrome induced by SARS-CoV-2 vaccines: A systematic review of patient-report studies, Hum. Vacc. Immunother., 19 (2023), 2217076. https://doi.org/10.1080/21645515.2023.2217076 doi: 10.1080/21645515.2023.2217076
[9]	X. Li, J. Xiang, F. Wu, M. Li, A dual ranking algorithm based on the multiplex network for heterogeneous complex disease analysis, IEEE/ACM Trans. Comput. Biol. Bioinform., 19 (2022), 1993–2002. https://doi.org/10.1109/TCBB.2021.3059046 doi: 10.1109/TCBB.2021.3059046
[10]	F. Brauer, C. Castillo-Chavez, Mathematical Models in Epidemiology, Berlin: Springer, 2008.
[11]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
[12]	L. J. S. Allen, Stochastic Population and Epidemic Models, Berlin: Springer, 2017.
[13]	H. Andersson, T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Berlin: Springer, 2000.
[14]	A. Mozokhina, I. Popravka, M. Saade, V. Volpert, Epidemic model with time delays and fertility/mortality rates, AIMS Math., 10 (2025), 25849–25878. https://doi/10.3934/math.20251143 doi: 10.3934/math.20251143
[15]	S. Ouyang, G. Chen, Y. Xia, G. Shi, Y. Feng, Stochastic dynamics of influenza epidemic model with vaccination strategy, AIMS Math., 10 (2025), 20412–20442. https://doi.org/10.3934/math.2025912 doi: 10.3934/math.2025912
[16]	S. P. Rajasekar, M. Manju Priya, Dynamics on stationary distribution and probability density function of a $(2n+2)$ multi-strain stochastic epidemic model, In: Biology and Sustainable Development Goals: Applications of Mathematical Methods, Singapore: Springer Nature Singapore, 2025, 33–83.
[17]	S. P. Rajasekar, R. Ramesh, Y. Sabbar, Based on epidemiological parameter data, probe into a stochastically perturbed dominant variant of the COVID-19 pandemic model, Gene, 926 (2024), 148608. https://doi.org/10.1016/j.gene.2024.148608 doi: 10.1016/j.gene.2024.148608
[18]	A. Khan, A. Din, Stochastic analysis for measles transmission with Lévy noise: A case study, AIMS Math., 8 (2023), 18696–18716. http://doi.org/10.3934/math.2023952 doi: 10.3934/math.2023952
[19]	Weekly epidemiological record–Relevé épidémiologique hebdomadaire, World Health Organization, 15 (1940), 231.
[20]	O. J. Peter, M. M. Ojo, R. Viriyapong, F. A. Oguntolu, Mathematical model of measles transmission dynamics using real data from Nigeria, J. Differ. Equ. Appl., 28 (2022), 753–770. https://doi.org/10.1080/10236198.2022.2079411 doi: 10.1080/10236198.2022.2079411
[21]	O. J. Peter, H. S. Panigoro, M. A. Ibrahim, O. M. Otunuga, T. A. Ayoola, A. O. Oladapo, Analysis and dynamics of measles with control strategies: A mathematical modeling approach, Int. J. Dyn. Control, 11 (2023), 2538–2552. https://doi.org/10.1007/s40435-022-01105-1 doi: 10.1007/s40435-022-01105-1
[22]	M. A. Ibrahim, A. Dénes, Stability and threshold dynamics in a seasonal mathematical model for measles outbreaks with double-dose vaccination, Mathematics, 11 (2023), 1791. https://doi.org/10.3390/math11081791 doi: 10.3390/math11081791
[23]	H. T. Alemneh, A. M. Belay, Modelling, analysis, and simulation of measles disease transmission dynamics, Discrete Dyn. Nat. Soc., 2023 (2023), 9353540. https://doi.org/10.1155/2023/9353540 doi: 10.1155/2023/9353540
[24]	M. El Hajji, A. H. Albargi, A mathematical investigation of an SVEIR epidemic model for the measles transmission, Math. Biosci. Eng., 19 (2022), 2853–2875. http://doi.org/10.3934/mbe.2022131 doi: 10.3934/mbe.2022131
[25]	M. A. Kuddus, M. Mohiuddin, A. Rahman, Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination, Sci. Rep., 11 (2021), 16571. https://doi.org/10.1038/s41598-021-95913-8 doi: 10.1038/s41598-021-95913-8
[26]	Z. Sabir, M. A. Z. Raja, A. S. Alnahdi, M. B. Jeelani, M. A. Abdelkawy, Numerical investigations of the nonlinear smoke model using the Gudermannian neural networks, Math. Biosci. Eng., 19 (2022), 351–370. https://doi.org/10.3934/mbe.2022018 doi: 10.3934/mbe.2022018
[27]	Z. Sabir, M. A. Z. Raja, J. L. Guirao, T. Saeed, Meyer wavelet neural networks to solve a novel design of fractional order pantograph Lane-Emden differential model, Chaos Solitons Fract., 152 (2021), 111404. https://doi.org/10.1016/j.chaos.2021.111404 doi: 10.1016/j.chaos.2021.111404
[28]	Z. Sabir, M. A. Z. Raja, M. Umar, M. Shoaib, D. Baleanu, FMNSICS: Fractional Meyer neuro-swarm intelligent computing solver for nonlinear fractional Lane-Emden systems, Neural Comput. Appl., 34 (2022), 4193–4206. https://doi.org/10.1007/s00521-021-06452-2 doi: 10.1007/s00521-021-06452-2
[29]	J. Jin, W. Chen, A. Ouyang, F. Yu, H. Liu, A time-varying fuzzy parameter zeroing neural network for the synchronization of chaotic systems, IEEE Trans. Emerg. Topics Comput. Intell., 8 (2024), 364–376. https://doi.org/10.1109/TETCI.2023.3301793 doi: 10.1109/TETCI.2023.3301793
[30]	X. Hu, Y. Lu, G. Tian, P. Bing, B. Wang, B. He, Predicting herb-disease associations through graph convolutional network, Curr. Bioinform., 18 (2023), 610–619. https://doi.org/10.2174/1574893618666230504143647 doi: 10.2174/1574893618666230504143647
[31]	J. Xiang, J. Zhang, R. Zheng, X. Li, M. Li, NIDM: Network impulsive dynamics on multiplex biological network for disease-gene prediction, Brief. Bioinform., 22 (2021), bbab080. https://doi.org/10.1093/bib/bbab080 doi: 10.1093/bib/bbab080
[32]	K. Guedri, R. Zarin, M. Oreijah, S. K. Alharbi, H. A. E. W. Khalifa, Artificial neural network-driven modeling of Ebola transmission dynamics with delays and disability outcomes, Comput. Biol. Chem., 115 (2025), 108350. https://doi.org/10.1016/j.compbiolchem.2025.108350 doi: 10.1016/j.compbiolchem.2025.108350
[33]	M. Farhan, Z. Ling, S. Ullah, M. Alsubhi, M. Asiri, M. B. Riaz, A novel physics-informed neural network approach to assess the impact of double-dose vaccination on measles transmission, Eur. Phys. J. Plus, 139 (2024), 1059. https://doi.org/10.1140/epjp/s13360-024-05838-0 doi: 10.1140/epjp/s13360-024-05838-0
[34]	H. Tahir, W. A. Khan, M. ur Rahman, Numerical investigation of a wireless sensor network epidemic model with $\alpha$-stable noise: A case study, AIMS Math., 10 (2025), 20312–20345. https://doi.org/10.3934/math.2025908 doi: 10.3934/math.2025908
[35]	S. M. A. Shah, Y. Nie, A. Alkhazzan, C. Tunç, A stochastic hepatitis B model with media coverage and Lévy noise, J. Taibah Univ. Sci., 18 (2024), 2414523. https://doi.org/10.1080/16583655.2024.2414523 doi: 10.1080/16583655.2024.2414523
[36]	A. Din, Y. Li, Lévy noise impact on a stochastic hepatitis B epidemic model under real statistical data and its fractal-fractional Atangana-Baleanu order model, Phys. Scr., 96 (2021), 124008. https://doi.org/10.1088/1402-4896/ac1c1a doi: 10.1088/1402-4896/ac1c1a
[37]	M. El Fatini, I. Sekkak, Lévy noise impact on a stochastic delayed epidemic model with Crowly-Martin incidence and crowding effect, Phys. A, 541 (2020), 123315. https://doi.org/10.1016/j.physa.2019.123315 doi: 10.1016/j.physa.2019.123315
[38]	X. B. Zhang, X. D. Wang, H. F. Huo, Extinction and stationary distribution of a stochastic SIRS epidemic model with standard incidence rate and partial immunity, Phys. A, 531 (2019), 121548. https://doi.org/10.1016/j.physa.2019.121548 doi: 10.1016/j.physa.2019.121548
[39]	M. Zahri, Multidimensional Milstein scheme for solving a stochastic model for prebiotic evolution, J. Taibah Univ. Sci., 8 (2014), 186–198. https://doi.org/10.1016/j.jtusci.2013.12.002 doi: 10.1016/j.jtusci.2013.12.002
[40]	D. Desta, A. Abebe, J. Brannan, Dynamical systems framework for modeling COVID-19 with Lévy noise, Stoch. Dyn., 26 (2026), 2550040. https://doi.org/10.1142/S0219493725500406 doi: 10.1142/S0219493725500406
[41]	X. N. Leng, T. Feng, X. Z. Meng, Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps, J. Inequal. Appl., 2017 (2017), 138. https://doi.org/10.1186/s13660-017-1418-8 doi: 10.1186/s13660-017-1418-8
[42]	J. Danane, K. Allali, Z. Hammouch, K. S. Nisar, Mathematical analysis and simulation of a stochastic COVID-19 Lévy jump model with isolation strategy, Results Phys., 23 (2021), 103994. https://doi.org/10.1016/j.rinp.2021.103994 doi: 10.1016/j.rinp.2021.103994
[43]	X. Mao, F. Wei, T. Wiriyakraikul, Positivity preserving truncated Euler-Maruyama method for stochastic Lotka-Volterra competition model, J. Comput. Appl. Math., 394 (2021), 113566. https://doi.org/10.1016/j.cam.2021.113566 doi: 10.1016/j.cam.2021.113566

Reader Comments

Your name:*

Email:*
© 2026 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)