Disease Modelling on Measles Immunity: Theoretical and Numerical Analyses

Although measles vaccine is considered safe and highly effective, cases continue to be reported globally, even in countries, such as Canada, where herd immunity (a form of indirect protection provided by immunized individuals) threshold is reached. Biological processes and social behaviours are fundamental factors in understanding the re-emergence of this childhood disease in highly vaccinated populations. In the past decades, the assumption that vaccine-induced immunity is life long has started to vacillate and many studies show how measles antibodies wane over time. However, the time needed to wane immunity partially, or fully, is still unknown. During this waning stage, immunity can experience a boosting process, if an encounter with the pathogen occurs. However, in absence of virus, immunity can wane until individuals return fully susceptible. In a society where mobility, travel and immigration are a daily routine, infections stages and levels of immunity are important factors to potentially increase or reduce the spread of a virus. In particular, with the assumption that measles-induced immunity is lifelong, immigrants immunity provides an increase of protection in the host country. On the other hand, immunity heterogeneity in a community creates pockets of individuals vulnerable to the infection, and movement of infectious cases might lead to relatively big outbreaks. In this thesis, we investigate how waning immunity, boosting and vaccination processes, immigration and migration affect the achievement of herd immunity and the spread of the infection. We propose different compartmental models described by systems of ordinary and partial differential equations, following, and extending, the Susceptible-Exposed-Infectious-Recovered framework. We employ both deterministic and stochastic models in order to capture those factors which mostly affect the infection dynamics and immunity of individuals as well as to investigate the probability of extinction or outbreak. Since measles vaccine is given at different ages, from 12 months up to 6 years, we also employ age structured models, discrete and continuous, to capture the age groups which mostly experience waning immunity and infection. Meta-population models are also used to investigate the effect of mobility on the spread of measles infection. We derive expressions for the basic and control reproduction numbers as well as performing sensitivity analysis on the model parameters and its outcomes.