Stochastic dynamics of immunity in small populations: A general framework

Assessment of immunological status is a powerful tool in the surveillance and control of infectious pathogens in livestock and human populations. The distribution of immunity levels in the population provides information on time and age dependent transmission. A stochastic model is developed for a livestock population which relates the dynamics of the distribution of immunity levels at the population level to those of pathogen transmission. A general model with K immunity level categories is first proposed, taking into account the increase of the immunity level due to an infection or a re-exposure, the decrease of the immunity level with time since infection or exposure, and the effect of immunity level on the susceptibility and the infectivity of individuals. Numerical results are presented in the particular cases with K = 2 and K = 3 immunity level categories. We demonstrate that for a given distribution of the immunity levels at the population level, the model can be used to identify quantities such as most likely periods of time since introduction of infection. We discuss this approach in relation to analysis of serological data.