In this paper we developed a stochastic model of measles transmission dynamics with double dose vaccination. The total population in this model was sub-divided in to five compartments, namely Susceptible , Infected Vaccinated first dose Vaccinated second dose and Recovered First the model was developed by deterministic approach and then transformed into stochastic one, which is known to play a significant role by providing additional degree of realism compared to the deterministic approach. The analysis of the model was done in both approaches. The qualitative behavior of the model, like conditions for positivity of solutions, invariant region of the solution, the existence of equilibrium points of the model and their stability, and also sensitivity analysis of the model were analyzed. We showed that in both deterministic and stochastic cases if the basic reproduction number is less than 1 or greater than 1 the disease free equilibrium point is stable or unstable respectively, so that the disease dies out or persists within the population. Numerical simulations were carried out using MATLAB to support our analytical solutions. These simulations show that how double dose vaccination affect the dynamics of human population.