Global stability of a partial differential equation with distributed delay due to cellular replication

Abstract

In this paper, we investigate a nonlinear partial differential equation, arising from a model of cellular proliferation. This model describes the production of blood cells in the bone marrow. It is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay. Our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population.

Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling

“…Consider now Mackey's model (Figure 1), with the same assumptions as in the previous section, except that the length of the proliferating phase is assumed to be distributed according to a probability density -a probability kernel -denoted by f (a), with a ∈ (0, τ ), where a denotes the time spent by a cell in the proliferating phase (its age) [2,9,33]. In the original Mackey's model, presented in Section 3, the proliferating phase duration was supposed to be constant, the parameter τ representing an average duration of this phase.…”

Modelling Physiological and Pharmacological Control on Cell Proliferation to Optimise Cancer Treatments

“…Other models structured in age and maturity have been proposed [1,184,185], with comparable transport equations. It is also possible to discretise the maturity variable to yield a proliferationquiescence model with communicating compartments for differentiation, with distinction between self-renewal and differentiation rates, as in [3].…”

Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch

“…Since then it has been improved and analyzed by many authors, including Mackey and co-authors [17,18,20,21] and Adimy et al [1,2,3,4]. All these works considered that the nonlinear term, which describes the rate of introduction of nonproliferating cells in the proliferating compartment, depends only on the nonproliferating cell number.…”

“…One question may arise: Is the number of activated cells independent of the number of actually proliferating cells ? Since a majority of hematopoietic stem cells (95%) is known to be nonproliferating (see [1,16]), one may suppose that a control operates on the proportion of proliferating cells, and the total number of hematopoietic stem cells probably plays a role in the introduction of nonproliferating cells in the proliferating compartment. The nature of this role cannot be detailed yet.…”