A novel third order integrator using polynomial–exponential–trigonometric basis functions for solving the Omicron virus model: South Africa as a case study

This study introduces and analyzes a third-order integrator method that combines polynomial–exponential terms with specific trigonometric basis functions, referred to as the Novel Third-Order Integrator (NTOI) method. The proposed method was assessed for its properties, including accuracy, consistency, and stability, and applied to study the dynamic behavior of the Omicron virus model, focusing on South Africa as a case study. The simulation effectively captured the transmission and recovery dynamics of the virus across different population segments, demonstrating the ability of the integrator to model complex transmission patterns and disease progression. Key findings highlighted a substantial decrease in the number of susceptible individuals, periodic fluctuations in the exposed and asymptomatic groups, and a significant increase in symptomatic and infected cases in the population. These trends are influenced by seasonal variations and the implementation of specific, public health measures. The integrator’s accuracy in replicating the complex behavior of the Omicron variant provides deep insights into the disease trajectory and supports the formulation of region-specific public health strategies. In terms of numerical performance, the NTOI method demonstrated superior efficiency and precision compared to the classical third-order Runge–Kutta (RK3) scheme. Quantitatively, the proposed integrator achieved a reduction of approximately 45% in the truncation error and a 30% improvement in computational efficiency, while maintaining numerical stability throughout the simulation. The difference norm between the NTOI and RK3 solutions remained within 1.0 × 1 0 − 3 , confirming that the proposed scheme not only preserves the accuracy of traditional algorithms but also enhances the convergence and stability of complex nonlinear systems.