  The purpose of time series analysis via mechanistic models is to reconcile
the known or hypothesized structure of a dynamical system with observations
collected over time. We develop a framework for constructing nonlinear
mechanistic models and carrying out inference. Our framework permits the
consideration of implicit dynamic models, meaning statistical models for
stochastic dynamical systems which are specified by a simulation algorithm to
generate sample paths. Inference procedures that operate on implicit models are
said to have the plug-and-play property. Our work builds on recently developed
plug-and-play inference methodology for partially observed Markov models. We
introduce a class of implicitly specified Markov chains with stochastic
transition rates, and we demonstrate its applicability to open problems in
statistical inference for biological systems. As one example, these models are
shown to give a fresh perspective on measles transmission dynamics. As a second
example, we present a mechanistic analysis of cholera incidence data, involving
interaction between two competing strains of the pathogen Vibrio cholerae.
