Delay-dependent Coefficients Research Articles

Oscillations in a patchy environment disease model

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R 0 is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R 0 < 1 . For R 0 > 1 , local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.

Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay

Under the action of growth factors, proliferating and nonproliferating hematopoietic stem cells differentiate and divide, so as to produce blood cells. Growth factors act at different levels in the differentiation process, and we consider their action on the mortality rate (apoptosis) of the proliferating cell population. We propose a mathematical model describing the evolution of a hematopoietic stem cell population under the action of growth factors. It consists of a system of two age-structured evolution equations modelling the dynamics of the stem cell population coupled with a delay differential equation describing the evolution of the growth factor concentration. We first reduce our system of three differential equations to a system of two nonlinear differential equations with two delays and a distributed delay. We investigate some positivity and boundedness properties of the solutions, as well as the existence of steady states. We then analyze the asymptotic stability of the two steady states by studying the characteristic equation with delay-dependent coefficients obtained while linearizing our system. We obtain necessary and sufficient conditions for the global stability of the steady state describing the cell population's dying out, using a Lyapunov function, and we prove the existence of periodic solutions about the other steady state through the existence of a Hopf bifurcation.

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 understand some blood diseases characterized by very long period oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is proposed and analyzed. The existence of a Hopf bifurcation at 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, for reasonable cell cycle durations. These oscillations can be related to observations on some periodic hematological diseases (such as chronic myelogenous leukemia, for example).

Numerical detection of instability regions for delay models with delay-dependent parameters

In this paper we are interested in gaining local stability insights about the interior equilibria of delay models arising in biomathematics. The models share the property that the corresponding characteristic equations involve delay-dependent coefficients. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work harder so that numerical techniques must be used. Most existing methods for studying stability switching of equilibria fail when applied to such a class of delay models. To this aim, an efficient criterion for stability switches was recently introduced in [E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002) 1144–1165] and extended [E. Beretta, Y. Tang, Extension of a geometric stability switch criterion, Funkcial Ekvac 46(3) (2003) 337–361]. We describe how to numerically detect the instability regions of positive equilibria by using such a criterion, considering both discrete and distributed delay models.

Periodic oscillations in leukopoiesis models with two delays

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analysing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.

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

We analyze the asymptotic stability of a nonlinear system of two differential equations with delay, describing the dynamics of blood cell produc- tion. This process takes place in the bone marrow, where stem cells differen- tiate throughout division in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells in the introduction of cells in cycle, we are led 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 popula- tion'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.

A Delay Reaction-Diffusion Model of the Spread of Bacteriophage Infection

This paper is a continuation of recent attempts to understand, via mathematical modeling, the dynamics of marine bacteriophage infections. Previous authors have proposed systems of ordinary differential delay equations with delay dependent coefficients. In this paper we continue these studies in two respects. First, we show that the dynamics is sensitive to the phage mortality function, and in particular to the parameter we use to measure the density dependent phage mortality rate. Second, we incorporate spatial effects by deriving, in one spatial dimension, a delay reaction- diffusion model in which the delay term is rigorously derived by solving a von Foerster equation. Using this model, we formally compute the speed at which the viral infection spreads through the domain and investigate how this speed depends on the system parameters. Numerical simulations suggest that the minimum speed according to linear theory is the asymptotic speed of propagation. − KS(t)P (t), dI dt = −µiI(t )+ KS(t)P (t) − e −µiT KS(t − T )P (t − T ), dP dt = β − µpP (t) − KS(t)P (t )+ be −µiT KS(t − T )P (t − T ).

Numerical Determination of the Instability Region for a Delay Model of Phage–Bacteria Interaction

In this paper we describe an algorithm to determine the instability region of the endemic equilibrium of a delay model for phage–bacteria interaction in open marine environment. The algorithm relies on a new geometric criterion for stability switches introduced by Beretta and Kuang for delay models for which the corresponding characteristic equations have delay-dependent coefficients [6]. The outputs of the algorithm presented in this paper, which can be equivalently applied to similar 2D delay models, show that for this kind of models large delays are usually stabilizing.