Failure of Complex Systems, Cascading Disasters, and the Onset of Disease

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. A simple but very general mathematical framework is described for studying the failure of complex systems, or equivalently, the onset of disease. It includes the Armitage-Doll multi-stage cancer model as a particular case, and has potential to provide new insights into how diseases arise and progress. A method described by E.T. Jaynes is developed to provide an analytical solution for the models, and highlights connections between the convolution of Laplace transforms, sums of random samples, and Schwinger/Feynmann 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. The approach 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 the study of failure rates in complex systems and the onset of disease, cancer in particular.