Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model

TL;DR

This paper investigates the stability and oscillatory behavior of a delay differential equation model for blood cell production, identifying conditions for stability, population extinction, and periodic blood cell fluctuations linked to diseases.

Contribution

It provides a new analysis of a delay differential equation model, establishing stability criteria and demonstrating the occurrence of Hopf bifurcations leading to periodic solutions.

Findings

Necessary and sufficient condition for global stability of the zero population state.

Existence of Hopf bifurcation at the positive steady state.

Periodic solutions related to cyclic blood diseases.

Abstract

We analyze the asymptotic stability of a nonlinear system of two differential equations with delay describing the dynamics of blood cell production. This process takes place in the bone marrow where stem cells differentiate throughout divisions in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells on the introduction of cells in cycle, we are lead to study a characteristic equation with delay-dependent coefficients. We determine a necessary and sufficient condition for the global stability of the first steady state of our model, which describes the population's dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial positive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature.