title: Kinetic theory of age-structured stochastic birth-death processes.
abstract: Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using, e.g., the Bellman-Harris equation do not resolve a population's age structure and are unable to quantify population-size dependencies. Conversely, current theories that include size-dependent population dynamics (e.g., mathematical models that include carrying capacity such as the logistic equation) cannot be easily extended to take into account age-dependent birth and death rates. In this paper, we present a systematic derivation of a new, fully stochastic kinetic theory for interacting age-structured populations. By defining multiparticle probability density functions, we derive a hierarchy of kinetic equations for the stochastic evolution of an aging population undergoing birth and death. We show that the fully stochastic age-dependent birth-death process precludes factorization of the corresponding probability densities, which then must be solved by using a Bogoliubov--Born--Green--Kirkwood--Yvon-like hierarchy. Explicit solutions are derived in three limits: no birth, no death, and steady state. These are then compared with their corresponding mean-field results. Our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age- and population-dependent stochastic dynamics applicable to the study of demography, stem cell dynamics, and disease evolution.
authors: School of Computing Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom.|The Genome Analysis Centre, Norwich Research Park, Norwich NR4 7UH, United Kingdom.
authors: Chris D
authors: CD
authors: Greenman
authors: Departments of Biomathematics and Mathematics, UCLA, Los Angeles, California 90095-1766, USA.
authors: Tom
authors: T
authors: Chou
mesh_terms: D000375:Aging; D003198:Computer Simulation; D003643:Death; D007700:Kinetics; D008954:Models, Biological; D036801:Parturition; D011157:Population Dynamics; D013269:Stochastic Processes