A DETERMINISTIC METHODOLOGY FOR ESTIMATION OF PARAMETERS IN DYNAMIC MARKOV CHAIN MODELS

Abstract

A method for estimating parameters in dynamic stochastic (Markov Chain) models based on Kurtz's limit theory coupled with inverse problem methods developed for deterministic dynamical systems is proposed and illustrated in the context of disease dynamics. This methodology relies on finding an approximate large-population behavior of an appropriate scaled stochastic system. The approach leads to a deterministic approximation obtained as solutions of rate equations (ordinary differential equations) in terms of the large sample size average over sample paths or trajectories (limits of pure jump Markov processes). Using the resulting deterministic model, we select parameter subset combinations that can be estimated using an ordinary-least-squares (OLS) or generalized-least-squares (GLS) inverse problem formulation with a given data set. The selection is based on two criteria of the sensitivity matrix: the degree of sensitivity measured in the form of its condition number and the degree of uncertainty measured in the form of its parameter selection score. We illustrate the ideas with a stochastic model for the transmission of vancomycin-resistant enterococcus (VRE) in hospitals and VRE surveillance data from an oncology unit.