Abstract
Stochastic Differential Equations (SDE) are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e.g., oncogenesis) can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a high-level behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discrete-state stochastic models, the analysis of SDE and other continuous-state models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, non-identically distributed samples, and Girsanov's theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of non-identically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur.
Highlights
The dynamics of biological systems are largely driven by stochastic processes and subject to random external perturbations
The goal of this paper is to introduce an algorithm for examining such rare behaviors in Stochastic Differential Equation (SDE) models
Stochastic differential equation models A stochastic differential equation (SDE) [26,27] is a differential equation in which some of the terms evolve according to Brownian Motion [28]
Summary
The dynamics of biological systems are largely driven by stochastic processes and subject to random external perturbations. The consequences of such random processes are often investigated through the development and analysis of stochastic models (e.g., [1,2,3,4]). The validation and analysis of stochastic models can be very challenging [5,6], especially when the model is intended to investigate rare, but biologically significant behaviors (e.g., oncogenesis). The goal of this paper is to introduce an algorithm for examining such rare behaviors in Stochastic Differential Equation (SDE) models. The algorithm takes as input the SDE model, M , and a high-
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