Article

Global Asymptotic Stability for a Periodic Delay Hematopoiesis Model with Impulses

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Abstract

In this paper, sufficient conditions for the global asymptotic stability of a broad family of periodic impulsive scalar delay differential equations are obtained. These conditions are applied to a periodic hematopoiesis model with multiple time-dependent delays and linear impulses, in order to establish criteria for the global asymptotic stability of a positive periodic solution. The present results are discussed within the context of recent literature. In conclusion, when compared with previous works, not only sharper stability criteria are obtained here, even for models without impulses, but also the usual constraints imposed on the linear impulses are relaxed.

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... It is well known that a periodically varying environment is a foundation for the theory of natural selection. Hence, periodic effects for these type of dynamic populations have received great attention by numerous authors (e.g., [1,4,6,8,13,18,28,32,35,36,38] and references therein). In particular, the authors in [36] developed a new method and employed a generalized Lyapunov functional to establish delay-dependent criteria to ensure the existence and global exponential stability of positive periodic solutions for the model (1.2) with a discontinuous harvesting function. ...
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... DDEs model biological processes that not only depend on the current time, but also on an earlier time; representing e.g. hematopoiesis regulation (Mackey and Glass, 1977;Faria and Oliveira, 2020) or inflammatory responses (Nagaraja et al., 2014). PDEs describe the spatiotemporal evolution of biological entities, using e.g. ...
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We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds
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An age-structured model is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations using the method of characteristics. Under certain assumptions, this model can be reduced to a system of delay differential equations with two delays. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood. The characteristic equation of the two-delay equation is analyzed and shown to exhibit Hopf bifurcations when the destruction rate of erythrocytes is increased. A numerical study for a rabbit with autoimmune hemolytic anemia is performed and compared with experimental data.
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Although all blood cells are derived from hematopoietic stem cells, the regulation of this production system is only partially understood. Negative feedback control mediated by erythropoietin and thrombopoietin regulates erythrocyte and platelet production, respectively, but the regulation of leukocyte levels is less well understood. The local regulatory mechanisms within the hematopoietic stem cells are also not well characterized at this point. Because of their dynamic character, cyclical neutropenia and other periodic hematological disorders offer a rare opportunity to more fully understand the nature of these regulatory processes. We review the salient clinical and laboratory features of cyclical neutropenia (and the less common disorders periodic chronic myelogenous leukemia, periodic auto-immune hemolytic anemia, polycythemia vera, aplastic anemia, and cyclical thrombocytopenia) and the insight into these diseases afforded by mathematical modeling. We argue that the available evidence indicates that the locus of the defect in most of these dynamic diseases is at the stem cell level (auto-immune hemolytic anemia and cyclical thrombocytopenia seem to be the exceptions). Abnormal responses to growth factors or accelerated cell loss through apoptosis may play an important role in the genesis of these disorders. © 1998 by The American Society of Hematology.
A note on stability of impulsive scalar delay differential equations
  • Faria
T. Faria, J.J. Oliveira, A note on stability of impulsive scalar delay differential equations, Electron. J. Qual. Theory Differ. Equ. 2016, Paper No. 69, 14 pp. https://doi.org/10.14232/ejqtde.2016.1.69