A kinetic theory for age-structured stochastic birth-death processes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2015
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| author | Greenman, Chris D. Chou, Tom |
| author_facet | Greenman, Chris D. Chou, Tom |
| contents | Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but they are structurally 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 ageing 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 BBGKY-like hierarchy. However, explicit solutions are derived in two simple limits and 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1506_02111 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | A kinetic theory for age-structured stochastic birth-death processes Greenman, Chris D. Chou, Tom Statistical Mechanics Populations and Evolution Quantitative Methods Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but they are structurally 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 ageing 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 BBGKY-like hierarchy. However, explicit solutions are derived in two simple limits and 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. |
| title | A kinetic theory for age-structured stochastic birth-death processes |
| topic | Statistical Mechanics Populations and Evolution Quantitative Methods |
| url | https://arxiv.org/abs/1506.02111 |