Methods for obtaining mass functions in branching processes: a comparative computational framework

In this paper, the spread of an epidemiological disease over time is modeled as a Bienaymé–Galton–Watson process. Therefore, a discrete random variable models the number of infections per infector and rules the branching process. Given this probabilistic model, the main aim is to compare computationally methodologies to get mass functions of further generations’ size: probability generating functions, polynomial identities, Markov chain and Monte Carlo simulation. Comparisons are done in two distinct levels. The first is a local one, in which for the same generation, the analysis are done state by state. The other is a global, which focus on the whole support of each generation. Results show that the first two methodologies are not able to cover the entire support, due runtime issues in the probability generating functions approach and lack of RAM in the polynomial one. On the other hand, they give likelihood functions for Bayesian inference, because their algorithm works with symbolic computation. Moreover, they are the only ones to also perform a local analysis. Despite of the limitation of the random variable modeling the contagion in Markov chain’s technique, it has more advantages in terms of runtime and storage than the previous methods and still provides analytical relations, since it also works with symbolic computation. The further the generation is, Monte Carlo seems to be the only feasible approach, but as a consequence all the features evaluated are then random objects that need a more careful analysis and the requirement to perform Bayesian inference is lost.