TITLE:
Applications of Dynamic-Equilibrium Continuous Markov Stochastic Processes to Elements of Survival Analysis
AUTHORS:
Eugen Mamontov, Ziad Taib
KEYWORDS:
Non-Homogeneous Continuous Markov Stochastic Process, Invariant Process, Dynamic Equilibrium, Diffusion Stochastic Process, Itô Stochastic Ordinary Differential Equation, Survival Analysis, Hazard Rate, Obstructive Lung Disease
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.7 No.1,
January
14,
2019
ABSTRACT: In this
article, we summarize some results on invariant non-homogeneous and
dynamic-equilibrium (DE) continuous Markov stochastic processes. Moreover, we
discuss a few examples and consider a new application of DE processes to
elements of survival analysis. These elements concern the stochastic
quadratic-hazard-rate model, for which our work 1) generalizes the reading of its It? stochastic ordinary differential equation (ISODE) for the hazard-rate-driving
independent (HRDI) variables, 2) specifies key properties of the hazard-rate
function, and in particular, reveals that the baseline value of the HRDI variables is
the expectation of the DE solution of the ISODE, 3) suggests practical settings for obtaining multi-dimensional probability densities necessary for
consistent and systematic reconstruction of missing data by Gibbs sampling and 4) further develops the corresponding line of modeling. The resulting
advantages are emphasized in connection with the framework of clinical trials
of chronic obstructive pulmonary disease (COPD) where we propose the use of an
endpoint reflecting the narrowing of airways. This endpoint is based on a
fairly compact geometric model that quantifies the course of the obstruction,
shows how it is associated with the hazard rate, and
clarifies why it is life-threatening. The work also suggests a few directions for future research.