This research investigates the effectiveness of using Padé approximation in improving the accuracy of analytical solutions for a mathematical model describing the transmission dynamics of the African Swine Fever (ASF) virus. The ASF model consists of a Susceptible–Latent–Infectious–Susceptible–Contaminated (SLI–SC) compartmental framework that incorporates the role of contaminated human vectors. An analytical solution for the model is first constructed using the classical Adomian Decomposition Method (ADM) up to the eighth-order approximation. A comparison between the classical ADM solution and the Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) numerical method reveals that, although ADM provides reliable approximations near the initial conditions, its accuracy deteriorates over larger time intervals due to slow convergence and a limited radius of convergence limitations. The key novelty of this study is the use of Padé approximations of orders of [4,4] and [3,5] to rationalize the ADM series solution. In contrast with traditional methods that solely use truncated series expansions, the hybrid ADM–Padé framework improves convergence behavior and extends the validity of the analytical solution over a wider time domain. By converting polynomial series into a rational form, the method reduces the divergence and instability that are common in long-term nonlinear system simulations. A systematic comparison among the classical ADM, the hybrid ADM–Padé technique, and the RKF45 numerical method shows that the ADM–Padé results closely agree with RKF45 solutions across an extended interval. Improved accuracy, stability and a wider convergence region were made available by Pade acceleration and further demonstrated via graphical simulations created in Maple. Overall, this work presents a strong and solid semi-analytical approach for solving nonlinear differential equation models in mathematical epidemiology. The hybrid technique improves solution accuracy for larger domains while guaranteeing quicker convergence. Consequently, ADM–Padé can be considered a robust, precise mathematical technique to solve epidemiological models arising in mathematical biology.