Studying the effect of vaccination in epidemic models with stochastic transmission

Résumé

Mathematical epidemic models are frequently used in biology for analyzing transmission dynamics of infectious diseases and assessing control measures to interrupt their expansion. In order to select and develop properly the above mathematical models, it is necessary to take into account the particularities of an epidemic process as type of disease, mode of transmission and population characteristics. In this thesis we focus on infectious diseases with stochastic transmission including vaccination as a control measure to stop the spread of the pathogen. To that end, we consider constant and moderate size populations where individuals are homogeneously mixed. We assume that characteristics related to the transmission/recovery of the infectious disease present a common probabilistic behavior for individuals in the population. To assure herd immunity protection, we consider that a percentage of the population is protected against the disease by a vaccine, prior to the start of the outbreak. The administered vaccine is imperfect in the sense that some individuals, who have been previously vaccinated, failed to increase antibody levels and, in consequence, they could be infected. Pathogenic transmission occurs by direct contact with infected individuals. As population is not isolated, disease spreads from direct contacts with infected individuals inside or outside the population. In this context, we describe the compartmental stochastic Susceptible-Vaccinated-Infected-Susceptible (SVIS) y Susceptible-Vaccinated-Infected-Recovered (SVIR) models, considering an external source of infection and imperfect vaccine. We represent the evolution of an epidemic process in terms of multidimensional continuous-time Markov chains and we organize state spaces in terms of levels and sub-levels. Such organization will permit us to simplify the study and to analyze the underlying Markov chains as quasi-birth-and-death (QBD) processes. Under the above model hypothesis, we study the effect of vaccination in the expansion of an epidemic, taking into account different possibilities in the selection of model parameters regarding the transmission of the infectious disease. To attain this objective, we analyze the stationary probabilistic behavior of several random variables related to reproduction numbers, incidence measures and time measures, in a post-vaccination context, by applying specific techniques of stochastic processes. Overall, we show that vaccination plays a fundamental role in the control of an infectious disease. In general, we observe that large vaccine coverage produce less severe epidemics in terms of incidence and speed of transmission of the infectious disease. Vaccine effectiveness also plays an important role in the transmission of the pathogen. Less effective vaccines could produce faster loss of herd immunity and more incidence of the infectious disease than others more effective. The external source of infections also plays a special role to study the long-term behavior of the epidemic. This hypothesis implies that, for both models, the epidemic process does not end when there are not infected individuals within the population, in contradistinction to the traditional stochastic SIS and SIR models. In that sense, although infection clears, there is an eventual reintroduction of the infectious disease in the population. Consequently, epidemic processes are larger in time and produce more incidence of infectious cases.