Nonlinear stochastic Markov processes and modeling uncertainty in populations
H.Thomas Banks Shuhua Hu
Mathematical Biosciences & Engineering 2012, 9(1): 1-25 doi: 10.3934/mbe.2012.9.1
We consider an alternative approach to the use of nonlinear stochastic Markov processes (which have a Fokker-Planck or Forward Kolmogorov representation for density) in modeling uncertainty in populations. These alternate formulations, which involve imposing probabilistic structures on a family of deterministic dynamical systems, are shown to yield pointwise equivalent population densities. Moreover, these alternate formulations lead to fast efficient calculations in inverse problems as well as in forward simulations. Here we derive a class of stochastic formulations for which such an alternate representation is readily found.
keywords: forward Kolmogorov uncertainty Fokker-Planck probabilistic structures on deterministic systems pointwise equivalence. Nonlinear Markov processes
Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations
H.Thomas Banks Danielle Robbins Karyn L. Sutton
Mathematical Biosciences & Engineering 2013, 10(5&6): 1301-1333 doi: 10.3934/mbe.2013.10.1301
In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.
keywords: Delay equations Fisher information matrix. differentiability with respect to delays sensitivity generalized sensitivity

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