We develop an extension to differential equation models of dynamical systems to allow us to analyze probabilistic threshold dynamics that fundamentally and globally change system behavior. We apply our novel modeling approach to two cases of interest: a model of infectious disease modified for malware where a detection event drastically changes dynamics by introducing a new class in competition with the original infection; and the Lanchester model of armed conflict, where the loss of a key capability drastically changes the effectiveness of one of the sides. We derive and demonstrate a step-by-step, repeatable method for applying our novel modeling approach to an arbitrary system, and we compare the resulting differential equations to simulations of the system's random progression. Our work leads to a simple and easily implemented method for analyzing probabilistic threshold dynamics using differential equations.

This article was presented presented at the 81.2 MORSS (Military Operations Research Society Symposia), June 17-20, 2013, Alexandria VA. It was a nominee for the Richard H. Barchi Prize. This work, with a slightly different title, was accepted for publication in Mathematical Biosciences. The article of record as published may be located at http://dx.doi.org/10.1016/j.mbs.2013.10.009.