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Dynamic model of human mortality for analyses of longitudinal data on aging, health, and longevity
Igor Akushevich, Senior Research Scientist 
Konstantin G Arbeev, Senior Research Scientist 
Alexander Kulminski, Senior Research Scientist 
Svetlana V Ukraintseva , Senior Research Scientist 
*Anatoliy I. Yashin, Scientific Director of the Center for Population Health and Aging 

Keywords: Health model, physiological state, random stopping, comorbidity risk, combining data

In this paper, we investigate a new mathematical model describing age-trajectories of physiological state, changes in health status, and mortality in a cohort of individuals participating in a longitudinal study. The model contains jumping and continuous components. The jumping component, representing changes in health/well-being status, is described by a finite state continuous time (non-Markov) random process with transition intensities depending on current values of the continuous component. The continuous component, representing gradual aging-related changes in physiological state, is described in terms of (non-Markov)stochastic diffusion type differential equations, whose coefficients depend on the current value of the jumping component. Both components can be stopped at a random time T describing individuals’ life span (age at death). The transition from “alive” to “dead” is characterized by mortality rate. We assume that the elements of the matrix of transition intensities for the jumping process (morbidity and co-morbidity risks functions), as well as mortality rate depend on values of both components. They are quadratic forms with respect to the continuous component and an arbitrary function of the health state. The statistical analyses of longitudinal data depend on the observational plan, which determines the likelihood function of the data. Several situations are considered. In the first one, the values of continuous components are observed in a series of subsequent examinations. In the second, these data are enriched by information about ages of disease onsets, ages of co-morbidity occurrence, or recovery transition. In the third, longitudinal data on aging-related changes in physiological state are combined with the health status data. We show how such data can be analyzed using the Gaussian approximation of the conditional distribution of continuous component processes given the health state. Applications to joint analyses of data collected using different observational plans are discussed.