In this study, the Laguerre wavelets exact Parseval frame is introduced and proposed an effective numerical algorithm to get a numerical solution for the system of differential equations based on the Laguerre wavelets exact Parseval frame. This algorithm includes the collocation method and truncated Laguerre wavelet frames. Here, we reduce the system of differential equations into a set of algebraic equations which are having unknown Laguerre wavelet frame coefficients. Some numerical examples are given and compared to the numerical solution by the present method with the Adomian decomposition method. Moreover, the modeling of the spreading of a non-fatal disease in a population, which represents a system of an ordinary differential equation is numerically solved by the proposed technique and compared with the Adomina decomposed method. The obtained results reveal that the present algorithm provides a good approximation than existing methods.