A Comparison of Different Approaches to Find the Probability Distribution of Further Generations in a Branching Process

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 generations 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 evaluate 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 (runtime 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 calculation 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 generation can be achieved independently from the others. This is so-called a local property.