Back
 JAMP  Vol.7 No.1 , January 2019
Applications of Dynamic-Equilibrium Continuous Markov Stochastic Processes to Elements of Survival Analysis
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.
Cite this paper: Mamontov, E. and Taib, Z. (2019) Applications of Dynamic-Equilibrium Continuous Markov Stochastic Processes to Elements of Survival Analysis. Journal of Applied Mathematics and Physics, 7, 55-71. doi: 10.4236/jamp.2019.71006.
References

[1]   Mamontov, E. (2008) Dynamic-Equilibrium Solutions of Ordinary Differential Equations and Their Role in Applied Problems. Applied Mathematics Letters, 21, 320-325.
https://doi.org/10.1016/j.aml.2007.02.031

[2]   Mamontov, Y.V. and Willander, M. (2001) High-Dimensional Nonlinear Diffusion Stochastic Processes. World Scientific, River Edge, NJ.
https://doi.org/10.1142/4494

[3]   Arnold, L. (1974) Stochastic Differential Equations: Theory and Applications. John Wiley Sons, New York.

[4]   Il’in, A.M. and Has’minskii, R.Z. (1965) Asymptotic Behavior of Solutions of Parabolic Equations and an Ergodic Property of Nonhomogeneous Diffusion Processes. American Mathematical Society Translations: Series 2, 49, 241-268.

[5]   Mamontov, E. (2005) Nonstationary Invariant Distributions and the Hydrodynamic-Style Generalization of the Kolmogorov-Forward/Fokker-Planck Equation. Applied Mathematics Letters, 18, 976-982.
https://doi.org/10.1016/j.aml.2004.06.027

[6]   Has’minskii, R.Z. (1980) Stochastic Stability of Differential Equations. Sijthoff Noordhoff, Alphen aan den Rijn.
https://doi.org/10.1007/978-94-009-9121-7

[7]   https://en.wikipedia.org/wiki/Survival_analysis

[8]   Bellomo, N., Mamontov, E. and Willander, M. (2003) The Generalized Kinetic Modelling of a Multicomponent “Real-Life’’ Fluid by Means of a Single Distribution Function. Mathematical and Computer Modelling, 38, 637-659.
https://doi.org/10.1016/S0895-7177(03)90033-1

[9]   Willander, M., Mamontov, E. and Chiragwandi, Z. (2004) Modelling Living Fluids with the Subdivision into the Components in Terms of Probability Distributions. Mathematical Models and Methods in Applied Sciences, 14, 1495-1520.
https://doi.org/10.1142/S0218202504003702

[10]   Mamontov, E. (2009) Ordinary Differential Equation System for Population of Individuals and the Corresponding Probabilistic Model. Mathematical and Computer Modelling, 49, 1551-1562.
https://doi.org/10.1016/j.mcm.2008.09.010

[11]   Yashin, A.I., et al. (2007) Stochastic Model for Analysis of Longitudinal Data on Aging and Mortality. Mathematical Biosciences, 208, 538-551.
https://doi.org/10.1016/j.mbs.2006.11.006

[12]   https://en.wikipedia.org/wiki/Dependent_and_independent_variables#Statistics_synonyms

[13]   Ortega, J.M. and Rheinboldt, W.C. (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

[14]   Yashin, A.I., et al. (2012) The Quadratic Hazard Model for Analyzing Longitudinal Data on Aging, Health, and the Life Span. Physics of Life Reviews, 9, 177-188.
https://doi.org/10.1016/j.plrev.2012.05.002

[15]   Arbeev, K.G., et al. (2014) Joint Analyses of Longitudinal and Time-to-Event Data in Research on Aging: Implications for Predicting Health and Survival. Frontiers in Public Health, 2, 228/1-228/12.
https://doi.org/10.3389/fpubh.2014.00228

[16]   Struthers, C.A. and McLeish, D.L. (2011) A Particular Diffusion Model for Incomplete Longitudinal Data: Application to the Multicenter AIDS Cohort Study. Biostatistics, 12, 493-505.
https://doi.org/10.1093/biostatistics/kxq079

[17]   https://en.wikipedia.org/wiki/Airway_obstruction

[18]   https://en.wikipedia.org/wiki/Obstructive_lung_disease

[19]   Decramer, M., Janssens, W. and Miravitlles, M. (2012) Chronic Obstructive Pulmonary Disease. The Lancet, 379, 1341-1351.
https://doi.org/10.1016/S0140-6736(11)60968-9

[20]   Hogg, J.C. and Timens, W. (2009) The Pathology of Chronic Obstructive Pulmonary Disease. Annual Review of Pathology, 4, 435-459.
https://doi.org/10.1146/annurev.pathol.4.110807.092145

[21]   Hogg, J.C., Paré, P.D. and Hackett, T.-L. (2017) The Contribution of Small Airway Obstruction to the Pathogenesis of Chronic Obstructive Pulmonary Disease. Physiological Reviews, 97, 529-552.
https://doi.org/10.1152/physrev.00025.2015

[22]   Hogg, J.C., et al. (2004) The Nature of Small-Airway Obstruction in Chronic Obstructive Pulmonary Disease. The New England Journal of Medicine, 350, 2645-2653.
https://doi.org/10.1056/NEJMoa032158

[23]   Koo, H.K., et al. (2018) Small Airways Disease in Mild and Moderate Chronic Obstructive Pulmonary Disease: A Cross-Sectional Study. The Lancet Respiratory Medicine, 6, 591-602.
https://doi.org/10.1016/S2213-2600(18)30196-6

[24]   Thiboutot, J., et al. (2018) Current Advances in COPD Imaging. Academic Radiology.
https://doi.org/10.1016/j.acra.2018.05.023

[25]   https://en.wikipedia.org/wiki/Membership_function_(mathematics)

[26]   https://en.wikipedia.org/wiki/Chronic_obstructive_pulmonary_disease#Severity

[27]   Liu, D., et al. (2015) Prediction of Short Term Re-Exacerbation in Patients with Acute Exacerbation of Chronic Obstructive Pulmonary Disease. International Journal of COPD, 10, 1265-1273.

 
 
Top