Exploring COVID-19 model with general fractional derivatives: novel Physics-Informed-Neural-Networks approach for dynamics and order estimation
Authors
H. Aghdaoui
- MAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, P.O. Box 509, Errachidia 52000, Morocco.
A. A. Raezah
- Department of Mathematics, Faculty of Science, King Khalid University, Abha 62529, Saudi Arabia.
M. Tilioua
- MAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, P.O. Box 509, Errachidia 52000, Morocco.
Y. Sabbar
- MAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, P.O. Box 509, Errachidia 52000, Morocco.
Abstract
In this paper, a fractional coronavirus disease model including unreported cases is suggested. The considered model includes a general fractional derivative incorporating well-known types, specifically Caputo-Fabrizio, Atangana-Baleanu and Weighted Atangana-Baleanu. Our theoretical results are two-fold. First, under suitable assumptions, the existence of a solution for the considered system is proven. Moreover, the local stability of the free-disease and endemic equilibrium points is addressed in terms of \(R_{0}\). Secondly, a particular example is considered where the fractional derivative has two varying parameters, and an approach allowing for their estimation is proposed, with the aim of providing the best approximation of the real COVID-19 dynamics. The main novelty of our proposed approach is its use of physics-informed-neural-networks (PINNs) for estimating the fractional orders. On the other hand, to validate our results, a numerical simulations are conducted to illustrate the local stability of the disease dynamics, as well as the effectiveness of our proposed method in providing the best approximation of the two fractional derivative parameters.
Share and Cite
ISRP Style
H. Aghdaoui, A. A. Raezah, M. Tilioua, Y. Sabbar, Exploring COVID-19 model with general fractional derivatives: novel Physics-Informed-Neural-Networks approach for dynamics and order estimation, Journal of Mathematics and Computer Science, 36 (2025), no. 2, 142--162
AMA Style
Aghdaoui H. , Raezah A. A. , Tilioua M., Sabbar Y., Exploring COVID-19 model with general fractional derivatives: novel Physics-Informed-Neural-Networks approach for dynamics and order estimation. J Math Comput SCI-JM. (2025); 36(2):142--162
Chicago/Turabian Style
Aghdaoui, H. , Raezah, A. A. , Tilioua, M., Sabbar, Y.. "Exploring COVID-19 model with general fractional derivatives: novel Physics-Informed-Neural-Networks approach for dynamics and order estimation." Journal of Mathematics and Computer Science, 36, no. 2 (2025): 142--162
Keywords
- Epidemic model
- incidence rate
- equilibrium points
- optimal control
- numerical analysis
MSC
References
-
[1]
Q. T. Ain, Nonlinear stochastic cholera epidemic model under the influence of noise, J. Math. Techniques Model., 1 (2024), 52–74
-
[2]
H. Aghdaoui, M. Tilioua, K. S. Nisar, I. Khan, A Fractional Epidemic Model with Mittag-Leffler Kernel for COVID-19, Math. Biol. Bioinform., 16 (2021), 39–56
-
[3]
E. A. Antonelo, E. Camponogara, L. O. Seman, J. P. Jordanou, E. R. de Souza, J. F. Hübner, Physics-informed neural nets for control of dynamical systems, Neurocomputing, 579 (2024),
-
[4]
L. Boujallal, Stability analysis of fractional order mathematical model of leukemia, Int. J. Math. Model. Comput., 11 (2021), 15–27
-
[5]
E. Dahy, A. M. Elaiw, A. A. Raezah, H. Z. Zidan, A. E. A. Abdellatif, Global Properties of Cytokine-Enhanced HIV-1 Dynamics Model with Adaptive Immunity and Distributed Delays, Computation, 11 (2023), 1–36
-
[6]
O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382
-
[7]
A. Din, Y. Li, A. Yusuf, Delayed hepatitis B epidemic model with stochastic analysis, Chaos Solitons Fractals, 146 (2021), 16 pages
-
[8]
A. Haleem, R. Vaishya, M. Javaid, I. H. Khan, Artificial Intelligence (AI) applications in orthopaedics: an innovative technology to embrace, J. Clin. Orthop. Trauma, 11 (2020), S80–S81
-
[9]
S. Han, L. Stelz, H. Stoecker, L. Wang, K. Zhou, Approaching epidemiological dynamics of COVID-19 with physicsinformed neural networks, J. Frank. Inst., 361 (2023),
-
[10]
F. Heldmann, S. Berkhahn, M. Ehrhardt, K. Klamroth, PINN training using biobjective optimization: the trade-off between data loss and residual loss, J. Comput. Phys., 488 (2023), 21 pages
-
[11]
A. Jafarian, M. Mokhtarpour, D. Baleanu, Artificial neural network approach for a class of fractional ordinary differential equation, Neural Comput. Appl., 28 (2017), 765–773
-
[12]
A. D. Jagtap, K. Kawaguchi, G. E. Karniadakis, Adaptive activation functions accelerate convergence in deep and physicsinformed neural networks, J. Comput. Phys., 404 (2020), 23 pages
-
[13]
P. Jiang, X. Fu, Y. Van Fan, J. J. Klemeš, P. Chen, S. Ma, W. Zhang, Spatial-temporal potential exposure risk analytics and urban sustainability impacts related to COVID-19 mitigation: A perspective from car mobility behaviour, J. Clean. Prod., 279 (2021), 1–15
-
[14]
H. Khalid, A New Generalized Definition of Fractional Derivative with Non-Singular Kernel, Computation, 8 (2020), 1–9
-
[15]
L. Li, L. Qin, Z. Xu, Y. Yin, X. Wang, B. Kong, J. Bai, Y. Lu, Z. Fang, Q. Song, K. Cao, D. Liu, G. Wang, Q. Xu, X. Fang, S. Zhang, J. Xia, J. Xia, Artificial intelligence distinguishes COVID-19 from community acquired pneumonia on chest CT, Radiology, (2020)
-
[16]
Z. Liu, P. Magal, O. Seydi, G. Webb, Predicting the cumulative number of cases for the COVID - 19 epidemic in China from early data, arXiv:2002.12298v1, (2020), 1–10
-
[17]
H. Luo, Q.-L. Tang, Y.-X. Shang, S.-B. Liang, M. Yang, N. Robinson, J.-P. Liu, Can Chinese medicine be used for prevention of corona virus disease 2019 (COVID-19)? A review of historical classics, research evidence and current prevention programs, Chin. J. Integr. Med., 26 (2020), 243–250
-
[18]
P. Magal, G. Webb, Predicting the number of reported and unreported cases for the COVID - 19 epidemic in South Korea, Italy, France and Germany, MedRxiv, (2020), 1–16
-
[19]
S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theoret. Biol., 254 (2008), 178–196
-
[20]
X. Ning, J. Guan, X.-A. Li, Y. Wei, F. Chen, Physics-informed neural networks integrating compartmental model for analyzing COVID-19 transmission dynamics, Viruses, 15 (2023), 1–16
-
[21]
X. Ning, L. Jia, Y. Wei, X.-A. Li, F. Chen, Epi-DNNs: Epidemiological priors informed deep neural networks for modeling COVID-19 dynamics, Comput. Biol. Med., 158 (2023), 1–9
-
[22]
M. Pakdaman, A. Ahmadian, S. Effati, S. Salahshour, D. Baleanu, Solving differential equations of fractional order using an optimization technique based on training artificial neural network, Appl. Math. Comput., 293 (2017), 81–95
-
[23]
G. Pang, L. Lu, G. E. Karniadakis, fPINNs: fractional physics-informed neural networks, SIAM J. Sci. Comput., 41 (2019), A2603–A2626
-
[24]
B. Rahnsch, L. Taghizadeh, Network-based uncertainty quantification for mathematical models in epidemiology, J. Theoret. Biol., 577 (2024), 12 pages
-
[25]
M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations, arXiv preprint arXiv:1711.10561, (2017),
-
[26]
M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707
-
[27]
F. Rostami, A. Jafarian, A new artificial neural network structure for solving high-order linear fractional differential equations, Int. J. Comput. Math., 95 (2018), 528–539
-
[28]
S. M. A. Shah, H. Tahir, A. Khan, W. A. Khan, A. Arshad, Stochastic Model on the Transmission of Worms in Wireless Sensor Network, J. Math. Techniques Model., 1 (2024), 75–88
-
[29]
S. M. Sivalingam, P. Kumar, V. Govindaraj, The hybrid average subtraction and standard deviation based optimizer, Adv. Eng. Softw., 176 (2023),
-
[30]
S. M. Sivalingam, P. Kumar, V. Govindaraj, A neural networks-based numerical method for the generalized Caputo-type fractional differential equations, Math. Comput. Simulation, 213 (2023), 302–323
-
[31]
C. Sohrabi, Z. Alsafi, N. O’Neill, M. Khan, A. Kerwan, A. Al-Jabir, C. Iosifidis, R. Agha, World Health Organization declares global emergency: A review of the 2019 novel coronavirus (COVID-19), Int. J. Surg., 76 (2020), 71–76
-
[32]
M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 1–16
-
[33]
R. Verma, S. K. Gawre, N. P. Patidar, S. Nandanwar, A state of art review on the opportunities in automatic generation control of hybrid power system, Electric Power Systems Research, 226 (2024), 1–17
-
[34]
M. Wu, J. Zhang, Z. Huang, X. Li, Y. Dong, Numerical solutions of wavelet neural networks for fractional differential equations, Math. Methods Appl. Sci., 46 (2023), 3031–3044
-
[35]
C. Yang, Y. Deng, J. Yao, Y. Tu, H. Li, L. Zhang, Fuzzing automatic differentiation in deep-learning libraries, In: 2023 IEEE/ACM 45th International Conference on Software Engineering (ICSE), IEEE, (2023), 1174–1186
-
[36]
D. Zhang, L. Lu, L. Guo, G. E. Karniadakis, Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems, J. Comput. Phys., 397 (2019), 19 pages
-
[37]
J. Zhang, Y. Zhao, Y. Tang, Adaptive loss weighting auxiliary output fPINNs for solving fractional partial integrodifferential equations, Phys. D, 460 (2024), 13 pages