Many mathematical models are formulated by involving fractional differential equations with Caputo derivative. It is widely known that the analytical solutions are complicated to obtain or there may no exact solution in solving the fractional differential equations with Caputo derivative, especially when involving incommensurate fractional-order model. Hence, numerical methods are needed. In this paper, we are interested in solving the incommensurate fractional-order SIR model that involves Caputo derivative. Fractional Runge-Kutta method was used to solve the fractional-order SIR model with the aid of MAPLE software. Subsequently, the solution of different fractional-order SIR models was obtained by changing the values of Caputo fractional parameters and then compared to the real data of COVID-19 cases. It has been shown that the modified incommensurate fractional-order model achieves better results for fitting the real data compared to the other models. The application of the Caputo derivative to the mathematical problem is reliable and can be utilized for other practical problems.