Quantifying infection transmission in a stochastic SVIS model with infection reintroduction when vaccine is partially effective

Abstract

This communication is framed within the area of epidemic modelling and studies infectious disease dynamics in a stochastic Markovian approach. We consider a constant size population where individuals are homogeneous and uniformly mixed. Prior the start of the epidemic, a percentage of the population was immunized preventively to an infectious disease with an available vaccine that fails with a certain probability. The underlying mathematical model is the stochastic SVIS model with infection reintroduction and imperfect vaccine. The evolution of the infectious disease, at each time point t, is represented in terms of the bidimensional CTMC, X = {(V (t), I(t)), t ! 0}, where the random variables V (t) and I(t) count the number of vaccinated and infected individuals at time t, respectively. The basic reproduction number, R0, is probably the most well-known descriptor of disease transmission and plays a privileged role in epidemiology. It is used to determine the herd immunity threshold or the vaccine coverage required to control the spread of a disease when a vaccine offers a complete protection. Due to repeated contacts between the marked infective and previously infected individuals, R0 overestimates the average number of secondary infections and leads to high immunization coverage. In this sense, we propose alternatives exact measures to R0 to quantify the potential transmission of an infectious disease. Specifically, we describe the exact and population reproduction numbers, Re0 and Rp, in a post-vaccination context. For both random variables, we derive theoretical schemes involving their mass probability and generating functions, and moments distributions. We complement theoretical and algorithmic results with several numerical examples.