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The aim of the article is to analyze the dynamics of Rubella disease model with fractal-fractional exponential decay kernel. Different fractal dimensions and fractional-orders are used to investigate various aspects of the model. It is observed that the considered operator is very effective for the proposed model. The existence and uniqueness of the model is obtained using Banach fixed point theorems. The stability of the system is verified by Ulam–Hyers stability analysis. A numerical technique is established based on the Adam–Bashforth scheme and Lagrange piecewise interpolation. The numerical illustrations are graphically displayed, reflecting great behavior of every class in the model. The behavior of infected people is reported to be largely influenced by recovery and transmission rates. This demonstrates that quarantining Rubella-infected patients for a week or two reduces the virus transmission. It is revealed that the fractal dimensions has significant impact on the system’s dynamics.