Analysis and statistical inference of infectious diseases systems, with a focus on transient dynamics and delays

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The objective of my proposed research program is to develop techniques in the mathematical and statistical analysis of infectious disease systems, with a focus on the following commonly observed, but under-examined phenomena: 1. Long-lasting transient dynamics that occur after sudden changes in the system. An example of this is the "honeymoon period," a long period of low incidence after the initiation of mass vaccination programs. This could last for decades and may be followed by a resurgence of the disease. The study of honeymoon periods and other slow transient dynamics are still in its early stages, because the focus of much study in mathematical ecology and general complex systems is on faster dynamics (e.g., epidemics) and small deviations from equilibria. 2. Complex dynamics resulting from the inclusion of feedback delays in disease systems, such as when members of a population change their contact rates in response to a disease outbreak, or when public health measures to reduce transmission are applied. These measures take time to be implemented depending on the urgency of the situation. As a result, the delays are "state-dependent," because they depend on the state of the outbreak. The study of state-dependent delay systems is new and challenging, but has many applications in many fields of science and engineering. This research will employ an integrated approach that spans analysis of both deterministic systems and stochastic processes, development of mechanistic model systems for ecological processes, practical considerations for epidemiology, and design of efficient statistical inference techniques for fitting model parameters to data. All analysis will begin with the formulation of a deterministic skeleton of the dynamics written as systems of ordinary, partial and delay differential equations. This way the basic features of the models can be distinguished from the equally important effects of stochasticity such as periodic fluctuations and resonance. Linearization techniques, numerical bifurcation methods, Lyapunov and pseudo-potential functions will be employed. The effects of demographic and environmental stochasticity will be considered. The stochastic models will be fit to data using modern statistical inference techniques, such as iterated filtering. This research program will also develop methods to estimate missing or incomplete covariates (such as vaccine coverage), efficiently consider multiple data sets, correctly propagate uncertainty in estimates and predictions, and fit models to general stochastic systems, including those with time delay. This research program will provide a foundation upon which many different types of models can be built, analyzed and challenged with data. With its multi-faceted approach, this program may be applied to phenomena and models of complex systems even beyond those in disease ecology.