Chaos, bifurcation and statistical analysis of rubella disease dynamical system by utilizing different fractional approach
Keywords:
Rubella, Chaos test, Lyapunov function, Transcritical bifurcation, Fractional derivative, Fractal-fractional operator, Mittag Leffler Kernel, Statistical analysis.Abstract
Rubella is a virus cause serious health problems, particularly for expectant mothers and their unborn children. Understanding its dynamical transitions is crucial for elimination strategies. In this paper, we sug gest SVEIR model Rubella model that captures the lifelong immunity after natural infection acquired through vaccination. This work presents the dynamics of Rubella spread using fractal fractional Mittag Leffler operator and Atangana Baleanu Caputo operator. Both of the operators provide reliable solutions in simulation, while the fractal fractional derivative provide better results then ABC operator due to its fractal structure solved through Newton polynomial based numerical approach. A comprehensive mathematical examination of the suggested model is presented, covering existence of bounded and positive solution, convergence and stability. We also prove the existence of transcritical bifurcation at R0 = 1 using center manifold theory, which determines the threshold between disease elimination and endemic persistence. We provide impact of parameters through 3D bar plot that shows transmission rate and vaccination rate are very important in Rubella disease dynamics. The chaos test shows that model experiences bounded behaviour and does not exhibit chaotic oscillations. The sta tistical analysis provides information about symmetric behaviour of the compartments of the model. The work serves as a complete reference for mathematical modeling of rubella control.
References
Rubella. (2026). In Wikipedia. Retrieved June 14, 2026, from https://en.wikipedia.org/wiki/Rubella
World Health Organization. (2026). Rubella – Number of reported cases. Global Health Observatory. Retrieved June 14, 2026, from https://www.who.int/data/gho/data/indicators/indicator-details/GHO/rubella---number-of-reported-cases
Hasnan, N. H., & Puzi, S. M. (n.d.). Numerical solution for rubella transmission with vaccination effect using adaptive Runge–Kutta method.
Ochigbo, J. E., Okorie, C. E., & Ojeka, P. I. (n.d.). Dynamics of measles–rubella co-circulation: A mathematical model incorporating waning immunity, isolation, vertical transmission and adaptive public health response.
Zeb, S., Yatim, S. A. M., Ahmad, A., Kamran, A., & Rafiq, M. (2025). Numerical modelling of SEIR on two-dose vaccination against the rubella virus. Malaysian Journal of Fundamental and Applied Sciences, 21(1), 1577–1601.
Zhang, Y., Zhang, T., & Li, Z. (2025). A rubella epidemic model with seasonality. Journal of Applied Analysis & Computation, 16(3), 1207–1225.
Tilahun, G. T., Tolasa, T. M., & Wole, G. A. (2022). Modeling the dynamics of rubella disease with vertical transmission. Heliyon, 8(11).
Peter, O. J., Kumari, N., Kumar, A., Omede, B. I., Ogunmola, O. P., Balogun, G. B., & Godara, G. (2026). Analysis and dynamics of rubella with control strategies: A data-driven mathematical modeling approach. Boletín de la Sociedad Matemática Mexicana, 32(1), 36.
Amelia, D., & Tasman, H. (2021). Mathematics model of measles and rubella with vaccination. In Journal of Physics: Conference Series (Vol. 1725, No. 1, Article 012017). IOP Publishing.
Jain, K., & Bhattacharjee, A. (2022). Mathematical modeling of rubella disease dynamics with vertical transmission. Global Journal of Pure and Applied Mathematics, 18, 392–400.
Tilahun, G. T., Tolasa, T. M., Asfaw, M. D., & Wole, G. A. (2024). Stochastic and deterministic models for rubella dynamics with two doses of vaccination and vertical transmission. Discrete Dynamics in Nature and Society, 2024(1), 9697951.
Ahmad, W., Butt, A. I. K., Akhtar, N., Rafiq, M., Gohar, M., Idrees, Z., & Ahmad, N. (2024). Developing computationally efficient optimal control strategies to eradicate rubella disease. Physica Scripta, 99(3), 035202.
Duru, E. C., Anyanwu, M. C., & Nwosu, C. N. (2023). Optimal vaccination and treatment strategies for control of rubella with vertical transmission. International Journal of Mathematical Analysis and Modelling, 6(2).
Asri, M., Sidjara, S., Sanusi, W., Side, S., & Pratama, M. I. (2021). Analysis and solution of the SEIRS model for the rubella transmission with vaccination effect using Runge–Kutta method. In Journal of Physics: Conference Series (Vol. 1899, No. 1, Article 012090). IOP Publishing.
Prawoto, B. P., Abadi, & Artiono, R. (2020). Dynamic of re-infection rubella transmission model with vaccination. In AIP Conference Proceedings (Vol. 2264, No. 1, Article 020005). AIP Publishing.
Younus, A., Asif, M., Atta, U., Bashir, T., & Abdeljawad, T. (2023). Applications of fuzzy conformable Laplace transforms for solving fuzzy conformable differential equations. Soft Computing, 27(13), 8583–8597.
Abdeljawad, T., Younus, A., Alqudah, M., & Atta, U. (2023). On fuzzy conformable double Laplace transform with applications to partial differential equations. Computer Modeling in Engineering & Sciences, 134(3), 2163.
Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 73–85.
Losada, J., & Nieto, J. J. (2015). Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 87–92.
Atangana, A., & Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. arXiv. https://arxiv.org/abs/1602.03408
Atangana, A. (2017). Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex systems. Chaos, Solitons & Fractals, 102, 396–406.
Ahmad, W., Nazir, M. A., Rafiq, M., Butt, A. I. K., Ahmad, N., & Hussain, M. (2025). Analyzing optimal control techniques in a nonlinear fractional rubella model with the Atangana–Baleanu derivative. Computers in Biology and Medicine, 197, 110954.
Elivina, E. F., & Kusumawinahyu, W. M. (2023). Dynamics of a fractional-order rubella disease model with vertical transmission and saturated incidence rate. Communications in Mathematical and Biological Neurosciences, 2023.
Atangana, A., & Alkahtani, B. S. T. (2016). Modeling the spread of rubella disease using the concept of local derivative with fractional parameter: Beta-derivative. Complexity, 21(6), 442–451.
Al Qurashi, M. M. (2020). Role of fractal-fractional operators in modeling of rubella epidemic with optimized orders. Open Physics, 18(1), 1111–1120.
Koca, I. (2018). Analysis of rubella disease model with non-local and non-singular fractional derivatives. An International Journal of Optimization and Control, 8(1), 17–25.
Alkahtani, B. S. T. (2023). Dynamical analysis of rubella disease model in the context of fractional piecewise derivative: Simulations with real statistical data. Fractal and Fractional, 7(10), 746.
Owolabi, K. M., Gómez-Aguilar, J. F., & Karaagac, B. (2019). Modelling, analysis and simulations of some chaotic systems using derivative with Mittag–Leffler kernel. Chaos, Solitons & Fractals, 125, 54–63.
Farman, M., Ahmad, A., Atta, U., Nisar, K. S., & Ghaffar, A. (2025). Hyers–Ulam stability and bifurcation control of leptospirosis disease dynamics and preventions: Modeling with singular and non-singular kernels. PLOS ONE, 20(3), e0314095.
Shah, K., Arfan, M., Mahariq, I., Ahmadian, A., Salahshour, S., & Ferrara, M. (2020). Fractal-fractional mathematical model addressing the situation of coronavirus in Pakistan. Results in Physics, 19, 103560.
Diekmann, O., Heesterbeek, J. A. P., & Roberts, M. G. (2010). The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47), 873–885.
Castillo-Chavez, C., Feng, Z., & Huang, W. (2002). On the computation of R 0 and its role in global stability. In Mathematical approaches for emerging and reemerging infectious diseases: An introduction (Vol. 1, p. 229).
Gottwald, G. A., & Melbourne, I. (2004). A new test for chaos in deterministic systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 460(2042), 603–611.
Gottwald, G. A., & Melbourne, I. (2009). On the implementation of the 0–1 test for chaos. SIAM Journal on Applied Dynamical Systems, 8(1), 129–145.
Ahmad, A., Atta, U., Akgül, A., Sattar, S., & Ahmad, M. O. (2025). A study on mathematical modeling and control of leptospirosis transmission dynamics during a hurricane with asymptomatic measures. Modeling Earth Systems and Environment, 11(5), 361.
Atangana, A., & Araz, S. I. (2021). New numerical scheme with Newton polynomial: Theory, methods, and applications. Academic Press.
