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Abstract

The dynamics of an infectious disease outbreak are studied under a stochastic framework in a closed population where individuals are homogeneously mixed. A vaccination program is implemented in the community prior to the start of the epidemic. The administered vaccine is imperfect, meaning that vaccinated individuals may still become infected upon contact with infected individuals. The mathematical Susceptible-Vaccinated-Infected-Recovered (SVIR) model describing the evolution of the epidemic is formulated as an absorbing, continuous-time, three-dimensional Markov chain with compartments for vaccinated, susceptible, infected and recovered individuals. The aim of this study is to characterize two fundamental epidemiological quantities: the maximum number of simultaneous infections during an outbreak and an improved measure of the traditional epidemic size. This refined random variable is defined as the sum of two quantities: the total number of infections among both vaccinated and susceptible individuals during an outbreak. This improvement allows for a much better understanding of the effect of vaccination and enables the calculation of measures related to vaccine efficacy. Recursive schemes are derived to compute probability mass functions and moments of both random variables, enabling the calculation of summary statistics that provide meaningful insights into epidemic behavior under different model parameter scenarios. Some numerical experiments are presented to illustrate how imperfect vaccination and the selection of adequate vaccine coverage jointly influence the overall impact and intensity of an epidemic, highlighting the role of theoretical results in understanding these effects.  Based on the random variables analyzed, some methods are presented to optimize epidemic management, including the selection of appropriate vaccine coverage levels, resource allocation and economic cost assessment. Additionally,  an application for estimating vaccine efficacy, under a stochastic approach, is presented.

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