A Radial Basis Scale Conjugate Gradient Deep Neural Network for the Monkeypox Transmission System
Abstract
:1. Introduction
- The stochastic RB-SCGDNN is presented to find the precise solutions of the nonlinear dynamics of MTS.
- The solutions of MTS are obtained through the RB-SCGDNN by taking twelve and twenty neurons in the hidden layers.
- A dataset is provided through the Adams method that is refined further by using the training, validation, and testing process with the statics of 0.15, 0.13, and 0.72.
- The exactness of RB-SCGDNN is presented by comparison of reference and achieved results, which is further validated using the negligible values of the absolute error together with different statistical measures to solve the MTS.
2. Methodology
2.1. RB-SCGDNN Procedures
2.2. Scale Conjugate Gradient (SCG)
3. Results and Discussion
4. Conclusions
- The nonlinear system of differential equations based on the MTS is successfully solved by using the stochastic approaches;
- The deep learning neural network along with the SCG and radial basis is used to present the numerical solutions of the MTS;
- Twelve and twenty neurons in the structure of hidden layers have been used in the deep learning process;
- A dataset was constructed using the Adams method, which is refined further through the process of train, validation, and test by taking 0.15, 0.13 and 0.72 values;
- 850 epochs, activation radial basis function, test performances through MSE and optimization-based SCG were used throughout the process for solving the MTS;
- The exactness of RB-SCGDNN was performed through the comparison of proposed and reference results. Moreover, negligible AE further enhanced the correctness of the scheme.
- The reliability of RB-SCGDNN procedure was verified through different statistical configuration using regression, correlation, ToS, and EHs.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Arden, M.A.; Chilcot, J. Health psychology and the coronavirus (COVID-19) global pandemic: A call for research. Br. J. Health Psychol. 2020, 25, 231–232. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Xiang, Y.; White, A. Monkeypox virus emerges from the shadow of its more infamous cousin: Family biology matters. Emerg. Microbes Infect. 2022, 11, 1768–1777. [Google Scholar] [CrossRef] [PubMed]
- Realegeno, S.; Puschnik, A.S.; Kumar, A.; Goldsmith, C.; Burgado, J.; Sambhara, S.; Olson, V.A.; Carroll, D.; Damon, I.; Hirata, T.; et al. Monkeypox Virus Host Factor Screen Using Haploid Cells Identifies Essential Role of GARP Complex in Extracellular Virus Formation. J. Virol. 2017, 91, e00011-17. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Guarner, J.; del Rio, C.; Malani, P.N. Monkeypox in 2022—What Clinicians Need to Know. JAMA 2022, 328, 139–140. [Google Scholar] [CrossRef]
- Durski, K.N.; McCollum, A.M.; Nakazawa, Y.; Petersen, B.W.; Reynolds, M.G.; Briand, S.; Djingarey, M.H.; Olson, V.; Damon, I.K.; Khalakdina, A. Emergence of monkeypox in West Africa and Central Africa, 1970–2017/Emergence de l’orthopoxvirose simienne en Afrique de l’Ouest et en Afrique centrale, 1970–2017. Wkly. Epidemiol. Rec. 2018, 93, 125–133. [Google Scholar]
- Durski, K.N.; McCollum, A.M.; Nakazawa, Y.; Petersen, B.W.; Reynolds, M.G.; Briand, S.; Djingarey, M.H.; Olson, V.; Damon, I.K.; Khalakdina, A. Emergence of monkeypox—West and central Africa, 1970–2017. Morb. Mortal. Wkly. Rep. 2018, 67, 306. [Google Scholar] [CrossRef]
- Pastula, D.M.; Tyler, K.L. An Overview of Monkeypox Virus and Its Neuroinvasive Potential. Ann. Neurol. 2022, 92, 527–531. [Google Scholar] [CrossRef]
- Paluku, K.; Jezek, Z.; Szczeniowski, M. Human monkeypox: Confusion with chickenpox. Acta Trop. 1988, 45, 297–307. [Google Scholar] [CrossRef]
- Minhaj, F.S.; Ogale, Y.P.; Whitehill, F.; Schultz, J.; Foote, M.; Davidson, W.; Hughes, C.M.; Wilkins, K.; Bachmann, L.; Chatelain, R.; et al. Monkeypox outbreak—Nine states, May 2022. Morb. Mortal. Wkly. Rep. 2022, 71, 764. [Google Scholar] [CrossRef]
- Huhn, G.D.; Bauer, A.M.; Yorita, K.; Graham, M.B.; Sejvar, J.; Likos, A.; Damon, I.K.; Reynolds, M.; Kuehnert, M.J. Clinical Characteristics of Human Monkeypox, and Risk Factors for Severe Disease. Clin. Infect. Dis. 2005, 41, 1742–1751. [Google Scholar] [CrossRef]
- Rizk, J.G.; Lippi, G.; Henry, B.M.; Forthal, D.N.; Rizk, Y. Prevention and treatment of monkeypox. Drugs 2022, 82, 957–963. [Google Scholar] [CrossRef] [PubMed]
- Karan, A.; Styczynski, A.R.; Huang, C.; Sahoo, M.K.; Srinivasan, K.; Pinsky, B.A.; Salinas, J.L. Human Monkeypox without Viral Prodrome or Sexual Exposure, California, USA, 2022. Emerg. Infect. Dis. 2022, 28, 2121–2123. [Google Scholar] [CrossRef] [PubMed]
- Bunge, E.M.; Hoet, B.; Chen, L.; Lienert, F.; Weidenthaler, H.; Baer, L.R.; Steffen, R. The changing epidemiology of human monkeypox—A potential threat? A systematic review. PLoS Neglected Trop. Dis. 2022, 16, e0010141. [Google Scholar] [CrossRef] [PubMed]
- Statistics Sweden. Design Your Questions Right: How to Develop, Test, Evaluate and Improve Questionnaires 2004. Available online: http://www.scb.se/statistik/_publikationer/OV9999_2004A01_BR_X97OP0402.pdf (accessed on 5 June 2020).
- Nguyen, P.Y.; Ajisegiri, W.S.; Costantino, V.; Chughtai, A.A.; MacIntyre, C.R. Reemergence of human monkeypox and declining population Immunity in the context of urbanization, Nigeria, 2017–2020. Emerg. Infect. Dis. 2021, 27, 1007. [Google Scholar] [CrossRef] [PubMed]
- Alakunle, E.; Moens, U.; Nchinda, G.; Okeke, M. Monkeypox Virus in Nigeria: Infection Biology, Epidemiology, and Evolution. Viruses 2020, 12, 1257. [Google Scholar] [CrossRef]
- Trejos, D.Y.; Valverde, J.C.; Venturino, E. Dynamics of infectious diseases: A review of the main biological aspects and their mathematical translation. Appl. Math. Nonlinear Sci. 2022, 7, 1–26. [Google Scholar] [CrossRef]
- Peter, O.J.; Oguntolu, F.A.; Ojo, M.M.; Oyeniyi, A.O.; Jan, R.; Khan, I. Fractional order mathematical model of monkeypox transmission dynamics. Phys. Scr. 2022, 97, 084005. [Google Scholar] [CrossRef]
- Peter, O.J.; Kumar, S.; Kumari, N.; Oguntolu, F.A.; Oshinubi, K.; Musa, R. Transmission dynamics of Monkeypox virus: A mathematical modelling approach. Model. Earth Syst. Environ. 2021, 8, 3423–3434. [Google Scholar] [CrossRef]
- Ali, A.K. Dynamics of Sofic Shifts. 3C Tecnol. Glosas de Innov. Apl. a la Pyme 2022, 11, 13–23. [Google Scholar] [CrossRef]
- Peter, O.J.; Viriyapong, R.; Oguntolu, F.A.; Yosyingyong, P.; Edogbanya, H.O.; Ajisope, M.O. Stability and optimal control analysis of an SCIR epidemic model. J. Math. Comput. Sci. 2020, 10, 2722–2753. [Google Scholar] [CrossRef]
- Ojo, M.M.; Gbadamosi, B.; Olukayode, A.; Oluwaseun, O.R. Sensitivity Analysis of Dengue Model with Saturated Incidence Rate. Oalib 2018, 5, 1–17. [Google Scholar] [CrossRef]
- Ayoola, T.A.; Edogbanya, H.O.; Peter, O.J.; Oguntolu, F.A.; Oshinubi, K.; Olaosebikan, M.L. Modelling and optimal control analysis of typhoid fever. J. Math. Comput. Sci. 2021, 11, 6666–6682. [Google Scholar] [CrossRef]
- Umar, M.; Sabir, Z.; Raja, M.; Shoaib, M.; Gupta, M.; Sánchez, Y. A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics. Symmetry 2020, 12, 1628. [Google Scholar] [CrossRef]
- Sabir, Z. Stochastic numerical investigations for nonlinear three-species food chain system. Int. J. Biomath. 2021, 15, 2250005. [Google Scholar] [CrossRef]
- Umar, M.; Sabir, Z.; Raja, M.A.Z.; Aguilar, J.G.; Amin, F.; Shoaib, M. Neuro-swarm intelligent computing paradigm for nonlinear HIV infection model with CD4+ T-cells. Math. Comput. Simul. 2021, 188, 241–253. [Google Scholar] [CrossRef]
- Sabir, Z.; Guirao, J.L.; Saeed, T. Solving a novel designed second order nonlinear Lane–Emden delay differential model using the heuristic techniques. Appl. Soft Comput. 2021, 102, 107105. [Google Scholar] [CrossRef]
- Guirao, J.L.; Sabir, Z.; Saeed, T. Design and Numerical Solutions of a Novel Third-Order Nonlinear Emden–Fowler Delay Differential Model. Math. Probl. Eng. 2020, 2020, 1–9. [Google Scholar] [CrossRef]
- Singh, N. Applications of Fixed Point Theorems to Solutions of Operator Equations in Banach Spaces. In 3C TIC Cuadernos de Desarrollo Aplicados a las TIC; 3ciencias: Alcoy, Spain, 2022; Volume 11, pp. 72–79. [Google Scholar] [CrossRef]
- Andrei, N. Scaled conjugate gradient algorithms for unconstrained optimization. Comput. Optim. Appl. 2007, 38, 401–416. [Google Scholar] [CrossRef]
- Yuan, G.; Li, T.; Hu, W. A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems. Appl. Numer. Math. 2019, 147, 129–141. [Google Scholar] [CrossRef]
- Abubakar, A.B.; Kumam, P.; Malik, M.; Ibrahim, A.H. A hybrid conjugate gradient based approach for solving unconstrained optimization and motion control problems. Math. Comput. Simul. 2021, 201, 640–657. [Google Scholar] [CrossRef]
- Vargas, J.O.; Batista, A.C.; Batista, L.S.; Adriano, R. On the computational complexity of the conjugate-gradient method for solving inverse scattering problems. J. Electromagn. Waves Appl. 2021, 35, 2323–2334. [Google Scholar] [CrossRef]
- Abubakar, A.B.; Kumam, P.; Ibrahim, A.H.; Chaipunya, P.; Rano, S.A. New hybrid three-term spectral-conjugate gradient method for finding solutions of nonlinear monotone operator equations with applications. Math. Comput. Simul. 2021, 201, 670–683. [Google Scholar] [CrossRef]
- Awwal, A.M.; Sulaiman, I.M.; Malik, M.; Mamat, M.; Kumam, P.; Sitthithakerngkiet, K. A Spectral RMIL+ Conjugate Gradient Method for Unconstrained Optimization With Applications in Portfolio Selection and Motion Control. IEEE Access 2021, 9, 75398–75414. [Google Scholar] [CrossRef]
- Chen, Q.; Sabir, Z.; Raja, M.A.Z.; Gao, W.; Baskonus, H.M. A fractional study based on the economic and environmental mathematical model. Alex. Eng. J. 2023, 65, 761–770. [Google Scholar] [CrossRef]
- Sada, S.O. Improving the predictive accuracy of artificial neural network (ANN) approach in a mild steel turning operation. Int. J. Adv. Manuf. Technol. 2021, 112, 2389–2398. [Google Scholar] [CrossRef]
- Khodadadian, A.; Parvizi, M.; Teshnehlab, M.; Heitzinger, C. Rational Design of Field-Effect Sensors Using Partial Differ-ential Equations, Bayesian Inversion, and Artificial Neural Networks. Sensors 2022, 22, 4785. [Google Scholar] [CrossRef]
- Sun, C.; Li, H. Algebraic Formulation and Application of Multi-Input Single-Output Hierarchical Fuzzy Systems with Correction Factors. IEEE Trans. Fuzzy Syst. 2022; Early Access. [Google Scholar] [CrossRef]
- Sun, C.; Li, H. Parallel fuzzy relation matrix factorization towards algebraic formulation, universal approximation and in-terpretability of MIMO hierarchical fuzzy systems. Fuzzy Sets Syst. 2022, 450, 68–86. [Google Scholar] [CrossRef]
- Durur, H.; Yokuş, A. Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation. Appl. Math. Nonlinear Sci. 2020, 6, 381–386. [Google Scholar] [CrossRef]
- Sulaiman, T.A.; Bulut, H.; Baskonus, H.M. On the exact solutions to some system of complex nonlinear models. Appl. Math. Nonlinear Sci. 2020, 6, 29–42. [Google Scholar] [CrossRef]
- Gençoğlu, M.T.; Agarwal, P. Use of Quantum Differential Equations in Sonic Processes. Appl. Math. Nonlinear Sci. 2020, 6, 21–28. [Google Scholar] [CrossRef]
- Aghili, A. Complete Solution for the Time Fractional Diffusion Problem with Mixed Boundary Conditions by Operational Method. Appl. Math. Nonlinear Sci. 2020, 6, 9–20. [Google Scholar] [CrossRef] [Green Version]
- Rahaman, H.; Hasan, M.K.; Ali, A.; Alam, M.S. Implicit Methods for Numerical Solution of Singular Initial Value Problems. Appl. Math. Nonlinear Sci. 2020, 6, 1–8. [Google Scholar] [CrossRef]
- İlhan, E.; Kıymaz, I.O. A generalization of truncated M-fractional derivative and applications to fractional differential equations. Appl. Math. Nonlinear Sci. 2020, 5, 171–188. [Google Scholar]
Index | Settings |
---|---|
Hidden neurons in the first layer | 12 |
Hidden neurons in the second layer | 20 |
Fitness goal (MSE) | 0 |
Maximum Mu values | 108 |
Maximum Epochs | 850 |
Minimum values of the gradient | 10−7 |
Increasing Mu performances | 10 |
Decreeing Mu measures | 0.2 |
Training statics | 0.15 |
Testing data | 0.72 |
Validation data | 0.13 |
Samples selection | Random |
Generation of dataset | Adam method |
Adam method execution and stoppage standards | Default |
Case | MSE | Performance | Gradient | Epoch | ||
---|---|---|---|---|---|---|
Train | Validation | Test | ||||
1 | 2.41 × 10−6 | 4.00 × 10−6 | 8.62 × 10−6 | 2.14 × 10−6 | 9.08 × 10−6 | 74 |
2 | 2.69 × 10−6 | 1.41 × 10−5 | 1.49 × 10−5 | 2.34 × 10−6 | 6.19 × 10−6 | 61 |
3 | 3.14 × 10−6 | 1.06 × 10−5 | 1.90 × 10−5 | 2.46 × 10−6 | 7.57 × 10−6 | 38 |
Absolute Error | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |
6 × 10−3 | 1 × 10−3 | 1 × 10−3 | 2 × 10−3 | 8 × 10−3 | 3 × 10−3 | 8 × 10−3 | 3 × 10−5 | 8 × 10−4 | 3 × 10−3 | 3 × 10−3 | |
1 × 10−2 | 1 × 10−3 | 1 × 10−3 | 7 × 10−6 | 3 × 10−3 | 9 × 10−3 | 3 × 10−3 | 3 × 10−3 | 2 × 10−3 | 1 × 10−2 | 2 × 10−3 | |
1 × 10−3 | 6 × 10−3 | 1 × 10−3 | 9 × 10−3 | 2 × 10−4 | 1 × 10−3 | 8 × 10−4 | 3 × 10−3 | 3 × 10−4 | 8 × 10−3 | 2 × 10−3 | |
2 × 10−5 | 2 × 10−5 | 2 × 10−5 | 1 × 10−5 | 1 × 10−5 | 1 × 10−5 | 1 × 10−5 | 1 × 10−5 | 1 × 10−5 | 6 × 10−6 | 9 × 10−6 | |
3 × 10−5 | 3 × 10−5 | 3 × 10−5 | 1 × 10−5 | 7 × 10−6 | 5 × 10−6 | 3 × 10−6 | 2 × 10−7 | 1 × 10−5 | 1 × 10−5 | 1 × 10−5 | |
3 × 10−5 | 3 × 10−5 | 2 × 10−5 | 1 × 10−5 | 3 × 10−5 | 2 × 10−5 | 4 × 10−5 | 1 × 10−5 | 4 × 10−6 | 3 × 10−5 | 2 × 10−5 | |
1 × 10−3 | 1 × 10−3 | 1 × 10−3 | 8 × 10−4 | 8 × 10−4 | 8 × 10−4 | 6 × 10−4 | 3 × 10−4 | 1 × 10−4 | 5 × 10−4 | 4 × 10−4 | |
9 × 10−4 | 1 × 10−3 | 2 × 10−3 | 9 × 10−4 | 6 × 10−4 | 3 × 10−4 | 4 × 10−4 | 6 × 10−4 | 6 × 10−4 | 1 × 10−3 | 4 × 10−4 | |
1 × 10−3 | 7 × 10−4 | 4 × 10−5 | 1 × 10−3 | 5 × 10−4 | 1 × 10−3 | 4 × 10−4 | 5 × 10−5 | 3 × 10−4 | 1 × 10−4 | 8 × 10−5 | |
9 × 10−4 | 1 × 10−3 | 9 × 10−4 | 8 × 10−4 | 6 × 10−4 | 6 × 10−4 | 5 × 10−4 | 1 × 10−4 | 2 × 10−4 | 1 × 10−4 | 4 × 10−5 | |
4 × 10−5 | 3 × 10−4 | 2 × 10−4 | 1 × 10−4 | 1 × 10−4 | 2 × 10−4 | 3 × 10−4 | 5 × 10−4 | 2 × 10−4 | 1 × 10−4 | 9 × 10−4 | |
6 × 10−4 | 2 × 10−4 | 1 × 10−4 | 2 × 10−4 | 6 × 10−5 | 4 × 10−4 | 2 × 10−4 | 4 × 10−4 | 4 × 10−4 | 1 × 10−3 | 8 × 10−4 | |
6 × 10−5 | 1 × 10−4 | 1 × 10−4 | 2 × 10−4 | 2 × 10−4 | 3 × 10−4 | 3 × 10−4 | 2 × 10−4 | 4 × 10−4 | 4 × 10−4 | 4 × 10−4 | |
2 × 10−7 | 2 × 10−4 | 4 × 10−5 | 4 × 10−5 | 9 × 10−5 | 5 × 10−5 | 9 × 10−5 | 1 × 10−4 | 2 × 10−4 | 3 × 10−5 | 1 × 10−4 | |
5 × 10−4 | 4 × 10−4 | 4 × 10−4 | 6 × 10−4 | 5 × 10−4 | 5 × 10−4 | 2 × 10−4 | 5 × 10−4 | 4 × 10−4 | 3 × 10−4 | 4 × 10−4 | |
3 × 10−3 | 2 × 10−3 | 2 × 10−3 | 2 × 10−3 | 9 × 10−3 | 1 × 10−3 | 7 × 10−3 | 9 × 10−6 | 1 × 10−3 | 2 × 10−3 | 4 × 10−3 | |
1 × 10−2 | 8 × 10−4 | 8 × 10−5 | 9 × 10−4 | 2 × 10−3 | 9 × 10−3 | 4 × 10−3 | 5 × 10−4 | 1 × 10−3 | 1 × 10−2 | 3 × 10−3 | |
5 × 10−3 | 1 × 10−2 | 8 × 10−5 | 1 × 10−2 | 2 × 10−3 | 3 × 10−3 | 5 × 10−4 | 2 × 10−3 | 7 × 10−5 | 1 × 10−2 | 2 × 10−3 | |
2 × 10−3 | 2 × 10−3 | 5 × 10−4 | 9 × 10−5 | 1 × 10−3 | 6 × 10−4 | 1 × 10−3 | 5 × 10−4 | 4 × 10−4 | 1 × 10−3 | 4 × 10−4 | |
4 × 10−3 | 9 × 10−4 | 2 × 10−3 | 7 × 10−4 | 1 × 10−3 | 1 × 10−3 | 1 × 10−3 | 2 × 10−3 | 1 × 10−3 | 5 × 10−3 | 2 × 10−3 | |
4 × 10−4 | 8 × 10−4 | 2 × 10−3 | 5 × 10−4 | 2 × 10−3 | 5 × 10−5 | 4 × 10−4 | 4 × 10−3 | 9 × 10−4 | 3 × 10−3 | 1 × 10−3 | |
1 × 10−3 | 1 × 10−4 | 6 × 10−4 | 1 × 10−3 | 6 × 10−4 | 2 × 10−3 | 2 × 10−3 | 1 × 10−4 | 4 × 10−4 | 3 × 10−3 | 1 × 10−3 | |
2 × 10−3 | 1 × 10−3 | 3 × 10−3 | 2 × 10−3 | 2 × 10−3 | 2 × 10−3 | 2 × 10−3 | 3 × 10−3 | 7 × 10−4 | 8 × 10−3 | 2 × 10−3 | |
2 × 10−3 | 3 × 10−4 | 1 × 10−4 | 3 × 10−3 | 2 × 10−3 | 2 × 10−4 | 3 × 10−3 | 6 × 10−4 | 1 × 10−3 | 8 × 10−3 | 1 × 10−3 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Sabir, Z.; Said, S.B.; Guirao, J.L.G. A Radial Basis Scale Conjugate Gradient Deep Neural Network for the Monkeypox Transmission System. Mathematics 2023, 11, 975. https://doi.org/10.3390/math11040975
Sabir Z, Said SB, Guirao JLG. A Radial Basis Scale Conjugate Gradient Deep Neural Network for the Monkeypox Transmission System. Mathematics. 2023; 11(4):975. https://doi.org/10.3390/math11040975
Chicago/Turabian StyleSabir, Zulqurnain, Salem Ben Said, and Juan L. G. Guirao. 2023. "A Radial Basis Scale Conjugate Gradient Deep Neural Network for the Monkeypox Transmission System" Mathematics 11, no. 4: 975. https://doi.org/10.3390/math11040975
APA StyleSabir, Z., Said, S. B., & Guirao, J. L. G. (2023). A Radial Basis Scale Conjugate Gradient Deep Neural Network for the Monkeypox Transmission System. Mathematics, 11(4), 975. https://doi.org/10.3390/math11040975