Implementation of Polynomial Identities from the Chain Rule in Probability Generating Functions for Branching Processes

Abstract

In this paper, the Bienaymé-Galton-Watson process is used to model in a stochastic sense the spreading of a disease over time. This is a type of stochastic process, which belongs to the class of branching processes. They can also be employed in other propagation phenomena, such as spam, fake news, innovation adoption and market value. Uncertainty is introduced in the way a single individual interacts with others. This is modeled as discrete random variable. Specifically for this work, it is treated as a Binomial distribution and named contagion. As a consequence, the number of subsequent people infected at a specific time is also a discrete random variable, named generation's size. A fundamental aspect to understand the evolution of the disease in this perspective is to find the mass function of such generations' size. A classical technique to do this is manipulating probability generating functions. This technique works well for early generations. However, it turns very time consuming as the generation number increases, becoming non-feasible for higher generations. The reason is that higher-order derivatives of more complex functions are tackled as the generation number increases. Therefore, a novelty in here is introducing a polynomial identity based on the chain rule approach to relieve the computational costs to perform the necessary derivatives. These two techniques are in here compared in terms of runtime and mathematical implications to find the mass functions for higher generations' size. Results show that the runtime spent for the probability generating function's technique in the absence of polynomial identities is smaller for the early generations. Moreover, analytical expressions relating the random variables of the contagion and the generation's size for different generations are preserved. On the other hand, as the generation number increases, the use of polynomial identities becomes advantageous to decrease the runtime spent and to cover a larger interval from the support of the mass functions.