This work analyzes the spread of an epidemiological disease with a stochastic approach. In the analysis, the number of individuals that each infected member of the population can infect is modeled as a random variable and the number of infected individuals over time is modeled as a stochastic branching process. The focus of the work is to characterize the influence of the probabilistic model of the random variable that models contagion between individuals on the spread of the disease and the probability of extinction, and to analyze the influence of mass vaccination in controlling the spread of a disease. The comparison is based on histograms and sample statistics of the number of infected individuals over time, such as mean and variance. Statistical models for the chapter dealing with a vaccine free population are calculated using Monte Carlo simulations for 3 different families of random variables: binomial, geometric-1 and geometric-0. For each of the 3 families, 21 different distributions were selected and, for each distribution, 4000 simulations of the branching process were computed. Statistical models for a partially vaccinated population were calculated using Monte Carlo simulations for one family of random variable: the binomial. For it, 21 different distributions were selected and, for each of them, 6 different percentages of the vaccinated population were chosen. For each of them, 4 different vaccine efficacy were stipulated. In total, 2.2 million simulations were performed, featuring a big data problem.