I've been reading a bit about the use of hazard functions to help describe the aging process. I thought this was an interesting idea because reliability theory (normally used in engineering to describe technical devices) could potentially be an extra step back from a purely phenomenological view of aging. Hazard functions are defined as the relative rate of decline of the reliability function. Different system arrangements generate different hazard functions: a system composed of elements connected in series would have a different failure rate than one composed of parallel elements. These functions are nice because they let you model systems with built-in redundancy as parallel system elements.
These redundant systems are interesting: you end up with a system constructed of static (non-aging) elements displaying an aging-like behavior (increase in failure as a power function of age that asymptotically approaches an upper limit, much like the late-life leveling-off of mortality rate for real organisms). In addition, although this kind of parallel system structure is sensitive to different levels of initial damage, the failure rates approach the same upper limit regardless of the initial level of redundancy. So you can model a biological system with distributed redundancy as serially-connected blocks of redundant parallel elements.
Incorporating repair capacity into a system could be done in a variety of ways, but I thought the simplest conceptual way was to designate discrete system elements as 'repair modules.' So mathematically, I thought of the repair rate as a term subtracted from the hazard function. For a system with a fixed number of repair modules, I think this would increase their sensitivity to initial damage load, since the initial damage can knock out repair modules in addition to reducing block redundancy. Repair capacity should also enhance the cooperativity of the system. I came up with expressions for probability distributions for simple system configurations containing varying numbers of repair modules per block. I have not however actually used a computer to calculate the failure kinetics for these systems yet, so...more later!
