Differential Equations for Sharp Thresholds

We develop an extension to differential equation models of dynamical systems to allowus to analyze probabilistic threshold dynamics that fundamentally change systembehavior. We apply our novel modeling approach to two cases of interest: a model ofcyber infection, where a detection event drastically changes dynamics, and theLanchester model of armed conflict, where the loss of a key capability drasticallychanges dynamics. We derive and demonstrate a step-by-step, repeatable method forapplying our novel modeling approach to an arbitrary system, and we compare theresulting differential equations to simulations of the system’s random progression. Ourwork leads to a simple and easily implemented method for analyzing probabilisticthreshold dynamics using differential equations.

Papers

Harrison C. Schramm and Nedialko B. Dimitrov. Differential Equation Models for Sharp Threshold Dynamics. Mathematical Biosciences, October 2013, doi:10.1016/j.mbs.2013.10.009

Harrison C. Schramm and Nedialko B. Dimitrov . Differential Equation Models for Sharp Threshold Dynamics. Naval Postgraduate School Technical Report, NPS-OR-12-003, August, 2012

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