This paper designs an object-oriented, continuous-time, full simulation model for addressing a wide range of clinical, procedural, administrative, and financial decisions in health care at a high level of biological, clinical, and administrative detail. The full model has two main parts, which with some simplification can be designated "physiology models" and "models of care processes." The models of care processes, although highly detailed, are mathematically straightforward. However, the mathematics that describes human biology, diseases, and the effects of interventions are more difficult. This paper describes the mathematical formulation and methods for deriving equations, for a variety of different sources of data. Although Archimedes was originally designed for health care applications, the formulation, and equations are general and can be applied to many natural systems.