Optimal linear stability condition for scalar differential equations with distributed delay

Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean delay, the linear equation with distributed delay is asymptotically stable if the associated differential equation with a discrete delay is asymptotically stable. We illustrate this criterion on a compartment model of hematopoietic cell dynamics to obtain sufficient conditions for stability

A review on local asymptotic stability analysis for mathematical models of hematopoiesis with delay and delay-dependent coefficients

International audienceStability analysis of mathematical models of hematopoiesis (blood cell production process), described by differential equations with delay, needs to locate eigenvalues of characteristic equations that are usually exponential polynomial functions with delay-dependent coefficients. It is then more complicated than for ordinary differential equations to determine conditions for all roots to have negative real parts. We present, on three models of increasing complexity, the tools and method that can be used, with their advantages and their limitations. The method consists in the reduction of the problem to the localization of roots of a real function, these roots giving critical values of the delay for which stability possibly switches

Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases

Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to carry out explanation on some blood diseases, characterized by oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is analyzed. The existence of a Hopf bifurcation for a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors. This stresses the localization of periodic hematological diseases in the feedback loop

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

International audienceA nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the trivial steady state. The study of eigenvalues of a second degree exponential polynomial characteristic equation allows to conclude to the existence of stability switches for the unique positive steady state. A numerical analysis of the role of each parameter on the appearance of stability switches completes this analysis

A review on local asymptotic stability analysis for mathematical models of hematopoiesis with delay and delay-dependent coefficients

International audienceStability analysis of mathematical models of hematopoiesis (blood cell production process), described by differential equations with delay, needs to locate eigenvalues of characteristic equations that are usually exponential polynomial functions with delay-dependent coefficients. It is then more complicated than for ordinary differential equations to determine conditions for all roots to have negative real parts. We present, on three models of increasing complexity, the tools and method that can be used, with their advantages and their limitations. The method consists in the reduction of the problem to the localization of roots of a real function, these roots giving critical values of the delay for which stability possibly switches

Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics

We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay. We also show that the distributed delay can destabilize the entire system. In particularly, it is shown that Hopf bifurcations can occur

Un mod\`ele non-lin\'eaire de prolif\'eration cellulaire : extinction des cellules et invariance

This paper analyses a nonlinear age-maturity structured system which arises as a model of the blood cellular production in the bone marrow. The resulting model is a nonlinear first-order partial differential equation in which there is a distributed temporal delay and a retardation in the maturation variable. We prove that uniqueness of solutions depends only on small maturity cells (stem cells) and we give a result of invariance