A comparison of different approaches to find the probability distribution of further generations in a branching process Jo˜aoPedroFreitas 1 , Roberta Lima 1 and Rubens Sampaio 1 Laborat´orio de Vibra¸ c˜ oes, Departamento de Engenharia Mecˆ anica, Pontif´ ıcia Universidade Cat´ olica do Rio de Janeiro, Rua Marquˆ es de S˜ ao Vicente, 255, G´ avea, 22451-900, Rio de Janeiro, RJ, Brazil joaopxf@aluno.puc-rio.br, robertalima@puc-rio.br, rsampaio@puc-rio.br Abstract. In this paper, the spread of a general epidemic over time is modeled as a branching process. It is a stochastic process sorted as an individual-based model, which records population growth over genera- tions with uncertainties to its size. The source of randomness is inherently related to the individual behavior of each member in a population. In this context, the transmissibility of the disease, i.e., the contagion from an infected person to susceptible ones is the root. Therefore, a discrete random variable models the number of infections per infector and rules the branching process. Given the probabilistic model of the contagion, the objective of the paper is to compare three methodologies to evalu- ate the mass functions of further generations of the branching process: probability generating functions (pgf), Markov chains (MC) and Monte Carlo simulations (MCS). The former gives analytical expressions, that can be symbolic computed, to evaluate the probability of an arbitrary number of infected members for a desired generation, whereas MC is a semi-numerical methodology and the latter is indeed a numerical one. The comparison between all of them relies on computational cost (run- time and storage) and limitation of applicability in relation to the mass function of the contagion. One of the characteristics of interest in the analysis is the determination of which methodologies allow the calcula- tion of the mass function of a further generation without computing the mass functions of previous ones. This feature is referred in here as not time-dependent. Another characteristic of interest is the determination of which methodologies allow the computation of just some values of the mass function of a generation, i.e., probabilities related to the same gen- eration can be achieved independently from the others. This is so-called a local property. Keywords: epidemic; branching process; probability generating func- tions; Markov chains; Monte Carlo simulations 1 Introduction The transmissibility of an epidemic is related to how easily a disease can spread from the contagion of an infected person (infector) to susceptible ones (infectees)