In this paper, our main purpose is to present an analytical solution for measles spread model with three doses of vaccination using Caputo–Fabrizio fractional derivative (CFFD). The presented solution is based on Laplace transform with Adomian decomposition method (LADM), which is an effective technique to obtain a solution for such type of problems. Our solution involves nonlinear differential equations of fractional order (FODEs) with non-singular kernel. Also, we provide analysis to verify the existence of a solution to the considered model using fixed point theory. Numerical results are presented to verify the model building analysis, which proved to be efficient in solving such problems.