Modeling transient and sustained epidemic dynamics: Cholera, influenza and rubella as case studies

Understanding the underlying mechanisms driving epidemics has public health value for the control of infectious diseases. Mathematical models can be useful to quantify specific aspects of the transmission dynamics of infectious diseases. The present work focuses on the development of techniques to model transient and sustained epidemiological outbreak patterns that are characteristic of diseases such as cholera, influenza and rubella. First, I develop a stochastic model describing the interactions between humans and an aquatic reservoir contaminated with V. cholerae in an effort to reproduce sustained oscillations observed in epidemiological data I show that in a limiting sense, the stochastic path of my system, as observed in simulations, approximates a circular motion modulated by an Ornstein-Uhlenbeck process. My results allow us to explore appropriate parameter regimes where the phenomenon of sustained oscillations can be observed. Secondly, I also examine the transient setting of historical pandemic influenza as observed during the 1918 influenza pandemic and use pandemic data from Geneva, Switzerland. Here, I model the transmission dynamics of two strains of influenza interacting via cross-immunity to simulate two temporal waves of influenza and explore the impact of the basic reproduction number, as a measure of transmissibility associated to each influenza strain, cross-immunity and the timing of the onset of the second influenza variant on the pandemic profile. My results indicate that avoiding a second influenza infection is plausible given sufficient levels of cross-protection are attained via natural infection during an early wave of infection or vaccination campaigns prior to a second wave. Lastly, I examine the patterns of rubella transmission using data from Peru's national epidemiological surveillance system to assess the effects of intervention strategies such as vaccination. For this purpose I use highly refined spatial, temporal and age-specific incidence data of Peru, a geographically diverse country, to quantify spatial-temporal patterns of incidence and transmissibility for rubella during the period 1997-2006. These findings highlight the importance in appropriately disentangling and interpreting the relevant patterns of seasonality and persistence of infectious diseases.