Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics (original) (raw)
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Systems & Control Letters, 2017
A nonlinear system with distributed delays describing cell dynamics in hematopoiesis is analyzed-in the time-domain-via a construction of suitable Lyapunov-Krasovskii functionals (LKFs). Two interesting biological situations lead us to re-investigate the stability properties of two meaningful steady states: the 0-equilibrium for unhealthy hematopoiesis and the positive equilibrium for the healthy case. Biologically, convergence to the 0-equilibrium means the extinction of all the generations of blood cells while the positive equilibrium reflects the normal process where blood cells survive. Their analyses are slightly different in the sense that we take advantage of positivity of the system to construct linear functionals to analyze the 0-equilibrium, while we use some quadratic functionals to investigate the stability properties of the positive equilibrium. For both equilibria, we establish the exponential stability of solutions and we provide an estimate of their rates of convergence. Moreover, a robustness analysis is performed when the model is subject to some nonvanishing perturbations. Numerical examples are provided.
Absolute stability of a system with distributed delays modeling cell dynamics in leukemia
In this paper we consider a mathematical model proposed recently by Adimy et al. (2008) for studying the cell dynamics in Acute Myelogenous Leukemia (AML). By using the circle and Popov criteria, we derive absolute stability conditions for this nonlinear system with distributed delays. Connections with the earlier results on stability of the linearized model are also made. The results are illustrated with a numerical example and simulations.
Stability Analysis of a Nonlinear System with Infinite Distributed Delays Describing Cell Dynamics
2018 Annual American Control Conference (ACC), 2018
We want to reconcile some earlier modeling ways of the cell cycle in one common framework. Accordingly, we consider a model that contains a compartment where cells may be quiescent for an unlimited time, along with a proliferating phase in which most of the cells may divide, or die, while few of them may be arrested during their cycle for unlimited time. In fact, the cell-cycle arrest may occur for many reasons (DNA damages detected at some checkpoints, insufficient resources for cell grow, drug infusions). We actually extend some early models involving finite distributed delays (taken from [2], [1]) to the case of infinite distributed delays and time-varying parameters. Our main result relies on the construction of a novel Lyapunov-Krasovskii functional, suitable for the analysis of the origin of the system involving infinite distributed delays and time-varying parameters.
2019 American Control Conference (ACC), 2019
This paper addresses the stability problem of a biological system that describes the proliferation of sick cells in Acute Myeloid Leukemia (AML). AML therapies aim at eradicating malignant cells, reaching a biological status represented by the zero equilibrium point of the age-structured mathematical model describing pathological hematopoeisis. First, the AML stability problem is reformulated into a stability problem of a nonlinear cascaded system. Then based on a positivity property of the system, non quadratic Lyapunov candidates are constructed. Finally, necessary and stability conditions are obtained. These conditions complete and generalize previous results where the main contribution consists in providing necessary and sufficient conditions based on a general model that incorporates fast self renewal. This model is complex but more realistic from a practical pint of view. Further more, unlike previously published works, the proposed conditions do not depend on auxiliary parameters which are biologically ambiguous but depend only on the AML system which makes the results more biologically relevant for AML treatment.
Absolute Stability of a System with Distributed Delays
2010
In this paper we consider a mathematical model proposed recently by Adimy et al. (2008) for studying the cell dynamics in Acute Myelogenous Leukemia (AML). By using the circle and Popov criteria, we derive absolute stability conditions for this nonlinear system with distributed delays. Connections with the earlier results on stability of the linearized model are also made. The results are illustrated with a numerical example and simulations.
2016 IEEE 55th Conference on Decision and Control (CDC), 2016
A new mathematical model that represents the coexistence between normal and leukemic populations of cells is proposed and analyzed. It is composed by a nonlinear time-delay system describing the dynamics of ordinary stem cells, coupled to a differential-difference system governing the dynamics of mutated cells. A Lyapunov-like technique is developed in order to investigate the stability properties of a steady state where healthy cells survive while leukemic ones are eradicated. Exponential stability of solutions is established, estimate of their decay rate is given and a subset of the basin of attraction of the desired steady state is provided.
SIAM Journal on Applied Mathematics, 2007
We study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise. The results established here are applied to examining the local stability of the hematopoietic stem cell (HSC) regulation system in the presence of noise. The stability of the first moment for the solutions of a linear differential delay equation under stochastic perturbation is identical to that of the unperturbed system. However, the stability of the second moment is altered by the perturbation. We obtain, using Laplace transform techniques, necessary and sufficient conditions for the second moment to be bounded. In applying the results to the HSC system, we find that the system stability is sensitive to perturbations in the stem cell differentiation and death rates, but insensitive to perturbations in the proliferation rate.
Mathematical Modelling of Natural Phenomena, 2006
We propose and analyze a mathematical model of hematopoietic stem cell dynamics. This model takes into account a finite number of stages in blood production, characterized by cell maturity levels, which enhance the difference, in the hematopoiesis process, between dividing cells that differentiate (by going to the next stage) and dividing cells that keep the same maturity level (by staying in the same stage). It is described by a system of n nonlinear differential equations with n delays. We study some fundamental properties of the solutions, such as boundedness and positivity, and we investigate the existence of steady states. We determine some conditions for the local asymptotic stability of the trivial steady state, and obtain a sufficient condition for its global asymptotic stability by using a Lyapunov functional. Then we prove the instability of axial steady states. We study the asymptotic behavior of the unique positive steady state and obtain the existence of a stability area depending on all the time delays. We give a numerical illustration of this result for a system of four equations.