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Global Asymptotic Stability for a Periodic Delay Hematopoiesis Model with Impulses
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.
Supplementary resources (2)
... 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. ...
... 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. In [13,35] authors studied the periodic impulsive version of (1.2). It is worthy to notice that while it is more realistic to consider a model with combined effects, periodicity and impulsive actions, the literature on this subject is still very scarce. ...
... It is worthy to notice that while it is more realistic to consider a model with combined effects, periodicity and impulsive actions, the literature on this subject is still very scarce. In [13], authors established new criteria for the global asymptotic stability of a positive periodic solution and in [35], by the continuation theorem of coincidence degree, authors proved the existence and gave sufficient conditions for the global attractivity of the unique positive periodic solution. ...
We establish and prove a fixed point theorem from which some sufficient conditions on the existence of positive almost periodic solutions for a model of hematopoiesis with oscillatory circulation loss rate are deduced. Some particular assumption under the nonlinearity of the equation has been previously considered by authors as fundamental for the study of almost periodic solutions of the model. The aim of this paper is to establish results without such assumption. Some examples are given to illustrate our results.
... Modeling the dynamics of real phenomena with differential equations is a potential branch in mathematics [6,19]. The following well-known model of ...
... . Let ( , ) be the set of all continuous functions We denote by ( 0 , ) a solution of the admissible initial value problem (4) with 0 ( 0 , ) = ∈ + and 0 ∈ . 6 Studies in Engineering and Exact Sciences, Curitiba, v.5, n.1, p. 01-28, 2024 ...
In applied mathematics, many practical problems involving heat flow, species interaction, microbiology, neural networks, and more are associated with delay differential equations. To elucidate regulation and control mechanisms in physiological systems, Mackey and Glass introduced new mathematical models as a representation of hematopoiesis (blood-cell formation) in their influential and highly referenced 1977 paper. The presence of multiple delays, rather than a single delay, can lead to a new range of dynamics: an equation that was stable with one delay may become unstable when the delays are not equal, resulting in sustained oscillations. The dynamics of the non-autonomous Mackey-Glass model with two variable delays have not been thoroughly explored yet, which is presented as an open problem by Berezansky and Braverman. This paper focuses on the existence, uniqueness, and exponential stability of positive periodic solutions of a generalized class of the Hematopoiesis Model with two variable delays. Initially, using properties of cones and a fixed point theorem for a concave and increasing operator, we establish conditions ensuring the existence of a unique positive periodic solution for the model with almost periodic coefficients and delays. Subsequently, we employ a suitable Lyapunov functional method and differential inequality techniques to derive sufficient conditions guaranteeing the exponential stability of the unique positive periodic solution of the system. This approach covers a broad range of models that were previously studied. Additionally, an illustrative example with a numerical simulation is provided to validate the efficiency and reliability of the theoretical results. Our approach can be applied to the case of positive almost periodic solutions and positive pseudo almost periodic solutions of the model under consideration.
... Impulses maybe generated in evolutionary system due to transient changes in state experienced at certain moments. In the hematopoietic process, these instantaneous and abrupt changes are caused by external interference such as received radiation, medication or other forms of blood cell pressure [23,24]. The blood cell model driven by impulse effects allows discrete rapid changes in cell number and is more accurate and effective in capturing external disturbances [25]. ...
... From that point of view, models of blood cells production have been investigated by various studies. Existing studies on continuous models of blood cell dynamics conducted with impulsive effects can be referred to [23,[35][36][37][38]. By contrast, relatively few studies on discrete dynamical models have considered the impulse terms. ...
In this paper, we investigate the multiplicity of positive periodic solutions of a discrete blood cell production model with impulse effects. This model is described by periodic coefficients and time delays, as well as nonlinear feedback with exponential terms. By employing the Krasnosel'skii fixed point theorem, we establish a sufficient condition for the existence of at least two positive periodic solutions. To this end, we construct solution transformation between an impulsive delay difference equation and the corresponding nonimpulsive delay difference equation. Aditionally, a solution representation of the positive periodic solution of the blood cell production model is presented. Moreover, a numerical example and its simulations are given to illustrate the main result.
... However, some dynamical systems exhibit rapid changes at specific points in their evolution, which serves to distinguish them from others. The weather, the availability of resources, the use of drugs or radiation in the treatment of hematological illnesses, food supplies, pharmacological factors, mating behaviors, and other seasonal influences are the main causes of this. Figure 1 illustrates how those events are based on the so-called IDE 18 . Impulsive hematopoiesis model is an extension of the traditional hematopoiesis model that incorporates the concept of rapid bursts or "impulses" of blood cell production in response to certain stimuli or conditions. ...
In this paper, the problem of oscillating solutions for an impulsive hematopoiesis model with positive and negative coefficients is investigated. There are several evolutionary processes, which frequently encounter dramatic shifts at specific times and are sensitive to short-term perturbations. As a result, we construct several oscillation criteria that are either brand-new or enhance many of recent findings in the literature. We also give illustrations of how impulsiveness affects the oscillating solutions of the hematopoiesis model.
... In the hematopoietic system, such instantaneous disturbance results in sudden changes in blood cell number [21,22]. There are some research works that investigated the dynamics of blood cells in the disturbed environment [23][24][25]. These studies are well conducted from continuous perspective. ...
The present article is devoted to the positive periodic solution for impulsive discrete hematopoiesis model. This model is described by a first-order nonlinear difference equation with multiple delays and impulses. The existence result is obtained by applying the Krasnosel’skii fixed point theorem. A sufficient condition that guarantees that there is at least one positive periodic solution is established. Moreover, illustrative example and numerical simulation analysis are provided to show the existence of positive periodic solutions and the effect of impulses on positive periodic solutions.
... Constant discrete delays are mathematically convenient, but their use is not always justified by the physical problem being studied. The same is true for delay differential equations with impulses, of which there numerous applications in biology and control [16,21,30,36,43]. Understanding the solution sets of impulsive systems with state-dependent delay therefore has practical implications. ...
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.
... 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. ...
In silico modeling is a powerful strategy to investigate the biological events occurring at tissue, cellular and subcellular level during bone fracture healing. However, most current models do not consider the impact of the inflammatory response on the later stages of bone repair. Indeed, as initiator of the healing process, this early phase can alter the regenerative outcome: if the inflammatory response is too strongly down- or upregulated, the fracture can result in a non-union. This review covers the fundamental information on fracture healing, in silico modeling and experimental validation. It starts with a description of the biology of fracture healing, paying particular attention to the inflammatory phase and its cellular and subcellular components. We then discuss the current state-of-the-art regarding in silico models of the immune response in different tissues as well as the bone regeneration process at the later stages of fracture healing. Combining the aforementioned biological and computational state-of-the-art, continuous, discrete and hybrid modeling technologies are discussed in light of their suitability to capture adequately the multiscale course of the inflammatory phase and its overall role in the healing outcome. Both in the establishment of models as in their validation step, experimental data is required. Hence, this review provides an overview of the different in vitro and in vivo set-ups that can be used to quantify cell- and tissue-scale properties and provide necessary input for model credibility assessment. In conclusion, this review aims to provide hands-on guidance for scientists interested in building in silico models as an additional tool to investigate the critical role of the inflammatory phase in bone regeneration.
... Constant discrete delays are mathematically convenient, but their use is not always justified by the physical problem being studied. The same is true for delay differential equations with impulses, of which there numerous applications in biology and control [16,21,30,36,43]. Understanding the solution sets of impulsive systems with state-dependent delay therefore has practical implications. ...
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
In the present manuscript, we mainly deal with a first-order differential equation with iterative terms where we revisit a leukopoiesis model with a harvesting strategy by assuming that the production and harvesting terms include iterative terms. We prove the existence of at least one positive periodic solution via the Krasnoselskii fixed point theorem together with some properties of an obtained Green’s function. Furthermore, by employing the Banach contraction principal we discuss the existence of one and only one positive periodic solution that depends continuously on parameters. We furnish two illustrative examples to demonstrate the effectiveness of the derived results which are new and complement some previous studies.
In the present paper, we mainly focus on an iterative erythropoiesis model with a nonlinear harvesting term involving a time-varying delay. Under certain conditions and by the Schauder fixed point theorem, we show that the existence of positive periodic solutions can be guaranteed and based on the Banach fixed point theorem, we establish the existence as well as the continuous dependence on parameters theorems of the unique solution. Moreover, two examples are provided to validate the correctness of our outcomes. Our theoretical findings extend and improve several related ones in the existing literature.
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.
The paper is concerned with a broad family of scalar periodic delay differential equations with linear impulses, for which the existence of a positive periodic solution is estab- lished under very general conditions. The proofs rely on fixed point arguments, employing either the Schauder theorem or Krasnoselskii fixed point theorem in cones. The results are illustrated with applications to an impulsive hematopoiesis model or generalized Nicholson’s equations, among other selected examples from mathematical biology. The method presented here turns out to be very powerful, in the sense that the derived theorems largely generalize and improve other results in recent literature, even for the situation without impulses.
We develop a comprehensive mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo a Hopf bifurcation, and use this information to investigate both normal and pathological platelet production, specifically cyclic thrombocytopenia. Carefully estimating model parameters from laboratory and clinical data, we then argue that a subset of parameters are involved in the genesis of cyclic thrombocytopenia based on clinical information. We provide excellent model fits to the existing data for both platelet counts and thrombopoietin levels by changing six parameters that have physiological correlates. Our results indicate that the primary change in cyclic thrombocytopenia is a major interference with or destruction of the thrombopoietin receptor with secondary changes in other processes, including immune-mediated destruction of platelets and megakaryocyte deficiency and failure in platelet production. This study makes a major contribution to the understanding of the origin of cyclic thrombopoietin as well as significantly extending the modeling of thrombopoiesis.
We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the context of recent literature.
We establish several existence and uniqueness results about positive almost periodic solutions for a class of hematopoiesis model. We give the proof of two properties relative to the composition of almost periodic functions and then use fixed point theory in an appropriate cone of the Banach space of almost periodic functions.
For a family of single-species delayed population models, a new global stability condition is found. This condition is sharp and can be applied in both monotone and nonmonotone cases. Moreover, the consideration of variable or distributed delays is allowed. We illustrate our approach on the Mackey-Glass equations and the Lasota-Wazewska model.
For a family of differential equations with infinitive delay and impulses, we establish conditions for the existence of global solutions and for the global asymptotic and global exponential stabilities of an equilibrium point. The results are used to give stability criteria for a very broad family of impulsive neural network models with both unbounded distributed delays and bounded time-varying discrete delays. Most of the impulsive neural network models with delay recently studied are included in the general framework presented here.
For a family of single-species delayed population models, a new global stability condition is found. This condition is sharp and can be applied in both monotone and nonmonotone cases. Moreover, the consideration of variable or distributed delays is allowed. We illustrate our approach on the Mackey-Glass equations and the Lasota-Wazewska model.
Hematopoiesis is a complex biological process that leads to the production and reg-ulation 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 regu-lation between blood cells and growth factors. This stresses the localization of periodic hematological diseases in the feedback loop.
In this paper we shall consider the nonlinear delay differential equation
p¢(t) = \fracb(t)1 + pn (t - mw) - d(t)p(t),p'(t) = \frac{{\beta (t)}}{{1 + p^n (t - m\omega )}} - \delta (t)p(t),
wherem is a positive integer, β(t) and δ(t) are positive periodic functions of period ω. In the nondelay case we shall show that (*) has a unique positive periodic
solution
[`(p)](t)\bar p(t)
, and show that
[`(p)](t)\bar p(t)
is a global attractor all other positive solutions. In the delay case we shall present sufficient conditions for the oscillation
of all positive solutions of (*) about
[`(p)](t)\bar p(t)
, and establish sufficient conditions for the global attractivity of
[`(p)](t)\bar p(t)
. Our results extend and improve the well known results in the autonomous case.
We consider the delay differential equations
P¢(t) = \fracb0 qn [P(t - t)]j qn + [P(t - t)]n - dP(t), j = 0,1,P'(t) = \frac{{\beta _0 \theta ^n [P(t - \tau )]^j }}{{\theta ^n + [P(t - \tau )]^n }} - \delta P(t),{\rm{ }}j = 0,1, which were proposed by Mackey and Glass as a model of blood cell production. We suggest new conditions sufficient for the
positive equilibrium of the equation considered to be a global attractor. In contrast to the Lasota-Wazewska model, we establish
the existence of the number δj = δj(n, τ) > 0 such that the equilibrium of the equation under consideration is a global attractor for all δ ε (0, δj] independently of β0 and θ.
We consider the nonlinear impulsive delay hematopoiesis model p ' (t)=β(t) 1+p n (t-mω)-γ(t)p(t),t≠t k , p(t k + )=(1-b k )p(t k ),k∈ℕ={1,2,⋯}, where n,m∈ℕ, β(t), γ(t) and ∏ 0<t k <t (1+b k ) are positive periodic functions of period ω>0. We prove that the solutions are bounded and persistent. The persistence implies the survival of the mature cells for a long term. By employing the continuation theorem of coincidence degree, we prove the existence of a positive periodic solution p ¯(t). We establish some sufficient conditions for the global attractivity of p ¯(t). These conditions imply the absence of any disease in the mammal. Moreover, we obtain some sufficient conditions for the oscillation of all positive solutions about the positive periodic solution p ¯(t). These conditions lead to the prevalence of mature cells around the periodic solution. Our results extend and improve some known results in the literature for the autonomous model without impulse. An example is presented to illustrate the main results.
By employing the contraction mapping principle and applying Gronwall-Bellman's inequality, sufficient conditions are established to prove the existence and exponential stability of positive almost periodic solution for nonlinear impulsive delay model of hematopoiesis.
First-order nonlinear differential-delay equations describing physiological control systems are studied. The equations display
a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic
or "chaotic" solutions. These results are discussed in relation to dynamical respiratory and hematopoietic 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).
We review the basic characteristics of four periodic hematological disorders (periodic auto-immune hemolytic anemia, cyclical thrombocytopenia, cyclical neutropenia and periodic chronic myelogenous leukemia) and examine the role that mathematical modeling and numerical simulations have played in our understanding of the origin of these diseases and in the regulation of hematopoiesis.
For the generalized periodic model of hematopoiesis
with , sufficient conditions for the global attractivity of its positive periodic solution are given, improving previous results in the literature. The effectiveness of the present criterion is illustrated by a numerical example.
link for the paper: https://authors.elsevier.com/a/1Ye-f3BGwf3wmZ
A discrete model describing the increase and decrease of blood cells is considered in this paper. This hematopoiesis model is a discretization of a delay differential equation with unimodal production function whose coefficients and delays are periodic discrete functions with ω-period. This paper is concerned with the existence of positive ω-periodic solutions. Our results are proved by using the well-known continuation theorem of coincidence degree theory. The existence range of the positive ω-periodic solutions is also clarified. A concrete example and its simulation are also given to illustrate our result. Finally, we examine how positive numbers and coefficients making up our model influence the upper and lower limits of blood cell counts.
If the Mackey–Glass equationx˙(t)=r(t)[ax(h(t))1+xν(g(t))−x(t)]
with a>1 and ν>0 incorporates not one but two variable delays, some new phenomena arise: there may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain a and ν disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.
This paper is concerned with almost periodicity of impulsive Hematopoiesis model with infinite delay. By employing the decreasing operator fixed point theorem, we obtain suffcient conditions for the existence of unique almost periodic positive solution. In addition, the exponential stability is derived by Liapunov functional. © 2015 International Scientific Research Publications. All rights reserved.
This paper deals with a nonautonomous almost periodic Hematopoiesis model. By applying the decreasing operator fixed point and Gronwall inequality, sufficient conditions are established for the existence and exponential stability of a unique positive almost periodic solution. Our results improve and generalize the past literature.
Theorems for the existence of periodic solutions for diverse models of population dynamics are obtained as corollaries of a few basic theorems, thus unifying the analysis of a broad class of scalar models in a single setting. The latter mechanism allows to obtain existence conditions for a broad class of nonlinear, non-autonomous models and models with state-dependent delays. The technique fulfills multiple roles: it can be used to expand on well-known results as well as to shorten existing proofs. We provide some examples which illustrate the applicability of our results.
For the blood cell production model with a unimodal (hump) feedback function dy/dt = -gamma y(t) + beta theta(n)y(t - tau)/theta(n) + y(n) (t -tau ) we review the known results and investigate generalizations of this equation. Permanence, oscillation and stability of the positive equilibrium are studied for non- autonomous equations, including equations with a distributed delay. In addition, a linear control is introduced, and possibilities to stabilize an otherwise unstable positive equilibrium are explored.
For the blood cell production modeldxdt=-αx(t)+F0θnθn+xn(t-τ),we review known results. Permanence of solutions, oscillation and stability of a positive equilibrium or a positive periodic solution are analyzed for non-autonomous generalizations of this equation, with both concentrated and distributed delays.
The periodicity of an impulsive delay Lasota–Wazewska model is discussed. Sufficient and necessary condition for the existence of a positive periodic solution is established. Sufficient conditions for its global attractivity are also obtained via the methods of Lyapunov function and comparison.
LetP(t) denote the density of mature cells in blood circulation. Mackey and Glass (1977) have proposed the following equations:
[(P)\dot](t) = \fracb0 qn qn + [P(t - t)]n - gP(t)\dot P(t) = \frac{{\beta _0 \theta ^n }}{{\theta ^n + [P(t - \tau )]^n }} - \gamma P(t)
and
[(P)\dot](t) = \fracb0 qn P(t - t)qn + [P(t - t)]n - gP(t)\dot P(t) = \frac{{\beta _0 \theta ^n P(t - \tau )}}{{\theta ^n + [P(t - \tau )]^n }} - \gamma P(t)
as models of hematopoiesis. We obtain sufficient and also necessary and sufficient conditions for all positive solutions to oscillate about their respective positive steady states. We also obtain sufficient conditions for the positive equilibrium to be a global attractor.
Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=–x+D(x
t
), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.
In this paper, the stability for the scalar impulsive delay differential equationwhere delay arguments may be bounded or unbounded is investigated. Some new stability theorems are established which improve and extend several known results in the literature.
In this paper, we consider the generalized model of hematopoiesisx′ (t) = - a (t) x (t) + underover(∑, i = 1, m) frac(bi (t), 1 + xn (t - τi (t))) . By using a fixed point theorem, some criteria are established for the existence of the unique positive ω-periodic solution over(x, ̃) of the above equation. In particular, we not only give the conclusion of convergence of xk to over(x, ̃), where {xk} is a successive sequence, but also show that over(x, ̃) is a global attractor of all other positive solutions.
Sufficient conditions are obtained for the existence and global attractivity of a unique positive periodic solution (x) over bar (t) of [GRAPHICS] where n > 1, a and b are continuous positive periodic function. Also, some sufficient conditions are established for oscillation of all positive solutions of (*) about (x) over tilde (t). For the proof of existence and uniqueness of (x) over tilde (t), the method used here is better than contraction mapping principle. (c) 2006 Elsevier Inc. All rights reserved.
An age-structured model for erythropoiesis is extended to include the active destruction of the oldest mature cells and possible control by apoptosis. The former condition, which is applicable to other population models where the predator satiates, becomes a constant flux boundary condition and results in a moving boundary condition. The method of characteristics reduces the age-structured model to a system of threshold type differential delay equations. Under certain assumptions, this model can be reduced to a system of delay differential equations with a state dependent delay in an uncoupled differential equation for the moving boundary condition. Analysis of the characteristic equation for the linearized model demonstrates the existence of a Hopf bifurcation when the destruction rate of erythrocytes is modified. 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 typical of a blood donation. Numerical studies for a rabbit with an induced auto-immune hemolytic anemia are performed and compared with experimental data.
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.
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