10,146 research outputs found
Structured populations with distributed recruitment: from PDE to delay formulation
In this work first we consider a physiologically structured population model
with a distributed recruitment process. That is, our model allows newly
recruited individuals to enter the population at all possible individual
states, in principle. The model can be naturally formulated as a first order
partial integro-differential equation, and it has been studied extensively. In
particular, it is well-posed on the biologically relevant state space of
Lebesgue integrable functions. We also formulate a delayed integral equation
(renewal equation) for the distributed birth rate of the population. We aim to
illustrate the connection between the partial integro-differential and the
delayed integral equation formulation of the model utilising a recent spectral
theoretic result. In particular, we consider the equivalence of the steady
state problems in the two different formulations, which then leads us to
characterise irreducibility of the semigroup governing the linear partial
integro-differential equation. Furthermore, using the method of
characteristics, we investigate the connection between the time dependent
problems. In particular, we prove that any (non-negative) solution of the
delayed integral equation determines a (non-negative) solution of the partial
differential equation and vice versa. The results obtained for the particular
distributed states at birth model then lead us to present some very general
results, which establish the equivalence between a general class of partial
differential and delay equation, modelling physiologically structured
populations.Comment: 28 pages, to appear in Mathematical Methods in the Applied Science
Dangerous connections: on binding site models of infectious disease dynamics
We formulate models for the spread of infection on networks that are amenable
to analysis in the large population limit. We distinguish three different
levels: (1) binding sites, (2) individuals, and (3) the population. In the
tradition of Physiologically Structured Population Models, the formulation
starts on the individual level. Influences from the `outside world' on an
individual are captured by environmental variables. These environmental
variables are population level quantities. A key characteristic of the network
models is that individuals can be decomposed into a number of conditionally
independent components: each individual has a fixed number of `binding sites'
for partners. The Markov chain dynamics of binding sites are described by only
a few equations. In particular, individual-level probabilities are obtained
from binding-site-level probabilities by combinatorics while population-level
quantities are obtained by averaging over individuals in the population. Thus
we are able to characterize population-level epidemiological quantities, such
as , , the final size, and the endemic equilibrium, in terms of the
corresponding variables
Variable renewal rate and growth properties of cell populations in colon crypts
A nonlinear mathematical model is used to investigate the time evolution of the cell populations in colon crypts (stem, semidifferentiated and fully differentiated cells). To mimic pathological alteration of the biochemical pathways leading to abnormal proliferative activity of the population of semidifferentiated cells their renewal rate is assumed to be dependent on the population size. Then, the effects of such perturbation on the population dynamics are investigated theoretically. Using both theoretical methods and numerical simulations it is shown that the increase in the renewal rate of semidifferentiated cells strongly impacts the dynamical behavior of the cell populations
A mathematical model of systemic inhibition of angiogenesis in metastatic development
We present a mathematical model describing the time development of a
population of tumors subject to mutual angiogenic inhibitory signaling. Based
on biophysical derivations, it describes organism-scale population dynamics
under the influence of three processes: birth (dissemination of secondary
tumors), growth and inhibition (through angiogenesis). The resulting model is a
nonlinear partial differential transport equation with nonlocal boundary
condition. The nonlinearity stands in the velocity through a nonlocal quantity
of the model (the total metastatic volume). The asymptotic behavior of the
model is numerically investigated and reveals interesting dynamics ranging from
convergence to a steady state to bounded non-periodic or periodic behaviors,
possibly with complex repeated patterns. Numerical simulations are performed
with the intent to theoretically study the relative impact of potentiation or
impairment of each process of the birth/growth/inhibition balance. Biological
insights on possible implications for the phenomenon of "cancer without
disease" are also discussed
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