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 they are structurally unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov agedependent processes using e.g., the Bellman-Harris equation, do not resolve a population's agestructure 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.

Fitness response relation of a multitype age-structured population dynamics

“…From the same tree, a retrospective path was sampled, and the response coefficients, g * B (τ, x) , µ * B (a, x) and j * B (x, x ′ ) = ∞ 0 dτ ′ j * B (x; τ ′ , x ′ ) were empirically calculated from the path (see right panels in FIGs. [4][5][6]. For all the cases we have tested, the stationary growth rate exactly responds to the changes in the parameters as predicted by Eq.…”

“…The evaluation of the fitness or its proxy, the population growth rate, has been conducted in the context of ordinary or partial differential equations by focusing on the time-slice distribution of the population [3][4][5][6]. In these approaches, the problems are mostly reduced to eigenvalue problems of the differential equations with appropriate boundary conditions, the largest eigenvalue of which corresponds to the stationary population growth rate.…”

“…In Sec. III, by incorporating the birth and death effects, we formulate the MTASP, which is represented by a McKendric equation with the type transition [3][4][5][6]. In Sec.…”

A model for meme popularity growth in social networking systems based on biological principle and human interest dynamics

“…Initially, the two normalized probabilities have relatively low values but they begin to increase after certain time. This initial "silent" phase corresponds to the lag phase that occurs before the accumulation phase for cell growth in microbial ecology [34][35][36][37][38][39][40][41][42][43][44][45][46] , where the growth and mortality rates are expected to be low at the beginning but increase with time. The striking phenomenon is that, for the five data sets arising from diverse social networking contexts, the time evolution of the probabilities exhibits quite similar features, suggesting a universal mechanism underlying the dynamical evolution of meme popularity.…”

“…Basic principles underlying the construction of a universal model for meme popularity dynamics Our first step is to hypothesize the equivalence between meme evolution in OSN systems and microbial cell population growth so as to develop a probabilistic, populationlevel base model. In such a dynamical evolution model of cell population [34][35][36][37][38][39][40][41][42][43][44][45][46] , at any given time a cell can experience one of the three possible events: division (generation), death, and survival. Likewise, memes are the "microscopic" elements of OSN systems.…”