Modello epidemiologico stocastico SIS con periodi di recupero a distribuzione Erlang generalizzata

In recent years, mathematical epidemiology research has experienced a remarkable growth, largely driven by the impact of COVID-19. Early contributions to the field date back to the beginning of the twentieth century with compartmental models. Today, mathematical epidemiology is often integrated with artificial intelligence tools to enable more precise simulations using real data, while still providing the structural framework for transmission dynamics. We begin our investigation with the well-known epidemiological SIS model based on Markov chains. The population is compartmentalised into susceptibles, who can contract the disease, and infectious individuals, who can spread it among the population or to newborns. In the standard formulation, recovery occurs in an exponentially distributed time. However, exponential distributions fail to capture the complexity of recovery in real-life. We therefore generalise the model so that the recovery times follow a generalised Erlang distribution, which we develop here for the case of two successive infectious stages, representing a first step towards the broader class of Phase-Type distributions. Inspired by a previous work of Gómez-Corral A. et al., we then quantify the severity of disease spread by tracking the number of secondary infections — both horizontal and vertical — produced by a single infectious individual introduced, at time zero, into an entirely susceptible population.