In this paper the fractional order epidemic model of a non-fatal disease in a population which is assumed to have a constant size over the period of the epidemic is considered. Laplace-adomian decomposition method (for short L-ADM) is used to compute an analytical solution of the system of nonlinear fractional differential equations governing the problem. The results are compared with the results obtained by the classical Runge-Kutta method in the case of integer-order derivatives.