In two well-known studies [Mathematics and Computers in Simulation 79(2008) 622-633] and [Mathematics and Computers in Simulation 182(2021) 397-410], some nonstandard finite difference (NSFD) schemes for an SIRC epidemic model of influenza A have been proposed. There have been attempts to prove that these NSFD schemes can preserve the positivity of the solutions, the invariance (conservation law) of the total population, equilibrium points and their asymptotic stability of the continuous-time model, for all finite values of the step size. Nevertheless, although the SIRC model possesses two equilibrium points, a unique disease-free equilibrium (DFE) point and a unique disease-endemic equilibrium (DEE) point, only the local asymptotic stability (LAS) of the DFE point has been established theoretically, whereas the LAS of the DEE point has only been confirmed through numerical simulations using some specific parameter sets. In this work, we construct a new class of NSFD schemes for the SIRC epidemic model. The LAS of the equilibrium points of the constructed NSFD schemes is rigorously established from a theoretical perspective and validated through numerical experiments. These NSFD schemes are constructed based on a weighted approximation for linear terms and the renormalization of the denominator function. Thereafter, we give dynamic consistency thresholds that lead to easily-verified conditions, ensuring the NSFD schemes preserve all the qualitative dynamical properties of the continuous-time model, regardless of the values of the step size. In particular, thanks to the simple structure of the constructed NSFD schemes, their LAS can be easily established by the linearization method. Furthermore, they are capable of providing numerical approximations with higher-order accuracy compared to the existing NSFD schemes. Additionally, Richardson’s extrapolation technique can be conveniently applied to increase the accuracy of the constructed NSFD schemes. Consequently, we obtain a new class of dynamically consistent NSFD schemes, which is not only simple but also efficient for numerical simulation of the SIRC model. Also, the constructed NSFD schemes improve those proposed in the two aforementioned studies in terms of both qualitative analysis and computational efficiency. Lastly, numerical experiments are conducted to support the theoretical findings and demonstrate the advantages of the constructed NSFD schemes.