Modeling the propagation of an epidemic in a stochastic SVIS model when a re-vaccination of the susceptible population is considered

Abstract

This presentation is focused on the use of continuous-time Markov chains (CTMC) to model the transmission of contagious diseases that do not confer permanent immunity. Population is not isolated and in consequence, infections can arise from direct contact with infected individuals within the population or from an external source of infection. A proportion of the population receives an imperfect vaccine that fails with a certain probability in the sense that some individuals that have been previously vaccinated to prevent disease could be infected. We describe the evolution of the infectious process in terms of a bi-dimensional CTMC representing the number of vaccinated and infected individuals during the epidemic. The number of immunized individuals decreases over time due to the imperfect vaccine and external source of infection hypothesis, which can lead to the loss of herd immunity. To prevent this, it is established an alarm threshold for the number of protected individuals, which we refer to as the warning level. The viability of a re-vaccination program is evaluated in order to arise vaccine coverage to the initial situation under a Markovian approach. To achieve that objective it is analyzed the size of the susceptible population when the alarm threshold for vaccinated individuals is reached. We also quantify the time until a re-vaccination program can be launched. We provide theoretical and algorithmic results to obtain statistic characteristics for both random variables and also present some numerical results for several infectious disease outbreaks.