  Complex systems can fail through different routes, often progressing through
a series of (rate-limiting) steps and modified by environmental exposures. The
onset of disease, cancer in particular, is no different. Multi-stage models
provide a simple but very general mathematical framework for studying the
failure of complex systems, or equivalently, the onset of disease. They include
the Armitage Doll multi-stage cancer model as a particular case, and have
potential to provide new insights into how failures and disease, arise and
progress. A method described by E.T. Jaynes is developed to provide an
analytical solution for a large class of these models, and highlights
connections between the convolution of Laplace transforms, sums of random
variables, and Schwinger/Feynman parameterisations. Examples include: exact
solutions to the Armitage-Doll model, the sum of Gamma-distributed variables
with integer-valued shape parameters, a clonal-growth cancer model, and a model
for cascading disasters. Applications and limitations of the approach are
discussed in the context of recent cancer research. The model is sufficiently
general to be used in many contexts, such as engineering, project management,
disease progression, and disaster risk for example, allowing the estimation of
failure rates in complex systems and projects. The intended result is a
mathematical toolkit for applying multi-stage models to the study of failure
rates in complex systems and to the onset of disease, cancer in particular.
