An HBV model with diffusion and time delay

In this paper, a hepatitis B virus (HBV) model with spatial diffusion and saturation response of the infection rate is investigated, in which the intracellular incubation period is modelled by a discrete time delay. By analyzing the corresponding characteristic equations, the local stability of an infected steady state and an uninfected steady state is discussed. By comparison arguments, it is proved that if the basic reproductive number is less than unity, the uninfected steady state is globally asymptotically stable. If the basic reproductive number is greater than unity, by successively modifying the coupled lower-upper solution pairs, sufficient conditions are obtained for the global stability of the infected steady state. Numerical simulations are carried out to illustrate the main results.     

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.     

On the Population Dynamics of the Malaria Vector

A deterministic differential equation model for the population dynamics of the human malaria vector is derived and studied. Conditions for the existence and stability of a non-zero steady state vector population density are derived. These reveal that a threshold parameter, the vectorial basic reproduction number, exist and the vector can established itself in the community if and only if this parameter exceeds unity. When a non-zero steady state population density exists, it can be stable but it can also be driven to instability via a Hopf Bifurcation to periodic solutions, as a parameter is varied in parameter space. By considering a special case, an asymptotic perturbation analysis is used to derive the amplitude of the oscillating solutions for the full non-linear system. The present modelling exercise and results show that it is possible to study the population dynamics of disease vectors, and hence oscillatory behaviour as it is often observed in most indirectly transmitted infectious diseases of humans, without recourse to external seasonal forcing.     

Viral dynamics model with CTL immune response incorporating antiretroviral therapy

We present two HIV models that include the CTL immune response, antiretroviral therapy and a full logistic growth term for uninfected $\text{ CD4}^+$ T-cells. The difference between the two models lies in the inclusion or omission of a loss term in the free virus equation. We obtain critical conditions for the existence of one, two or three steady states, and analyze the stability of these steady states. Through numerical simulation we find substantial differences in the reproduction numbers and the behaviour at the infected steady state between the two models, for certain parameter sets. We explore the effect of varying the combination drug efficacy on model behaviour, and the possibility of reconstituting the CTL immune response through antiretroviral therapy. Furthermore, we employ Latin hypercube sampling to investigate the existence of multiple infected equilibria.     

Stage-structured cannibalism with delay in maturation and harvesting of an adult predator

A three-dimensional stage-structured predator–prey model is proposed and analyzed to study the effect of predation and cannibalism of the organisms at the highest trophic level with non-constant harvesting. Time lag in maturation of the predator is introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a supercritical Hopf bifurcation when the maturation time lag crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.     

Asymmetry in the Presence of Migration Stabilizes Multistrain Disease Outbreaks

We study the effect of migration between coupled populations, or patches, on the stability properties of multistrain disease dynamics. The epidemic model used in this work displays a Hopf bifurcation to oscillations in a single, well-mixed population. It is shown numerically that migration between two non-identical patches stabilizes the endemic steady state, delaying the onset of large amplitude outbreaks and reducing the total number of infections. This result is motivated by analyzing generic Hopf bifurcations with different frequencies and with diffusive coupling between them. Stabilization of the steady state is again seen, indicating that our observation in the full multistrain model is based on qualitative characteristics of the dynamics rather than on details of the disease model.     

Analysis of unstable behavior in a mathematical model for erythropoiesis

We consider an age-structured model that describes the regulation of erythropoiesis through the negative feedback loop between erythropoietin and hemoglobin. This model is reduced to a system of two ordinary differential equations with two constant delays for which we show existence of a unique steady state. We determine all instances at which this steady state loses stability via a Hopf bifurcation through a theoretical bifurcation analysis establishing analytical expressions for the scenarios in which they arise. We show examples of supercritical Hopf bifurcations for parameter values estimated according to physiological values for humans found in the literature and present numerical simulations in agreement with the theoretical analysis. We provide a strategy for parameter estimation to match empirical measurements and predict dynamics in experimental settings, and compare existing data on hemoglobin oscillation in rabbits with predictions of our model.     

Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.     

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.     

一个带有ATL过程的CD4~+ T-淋巴细胞感染HTL V-I病毒的年龄结构模型分析

本文中,我们提出并分析了一个HTLⅤ-Ⅰ感染的年龄结构模型,得到了决定该模型未感染平衡点和感染平衡点的存在性和局部渐近稳定性的条件,即当一个活跃染病淋巴细胞在其整个染病期间在一个均为易感T-淋巴细胞的细胞群体中所能传染的细胞的平均数量不超过某一个阈值时,系统仅存在局部渐近稳定的未感染平衡点;当一个活跃染病淋巴细胞在其整个染病期间在一个均为易感T-淋巴细胞的细胞群体中所能传染的细胞的平均数量超过这一阈值时,未感染平衡点不稳定,此时存在局部渐近稳定的感染平衡点.     

A delay reaction-diffusion model of the dynamics of botulinum in fish

A model of interaction between fish and a bacterium (Clostridium botulinum) responsible for avian botulism is introduced, considering diffusion of both fish and bacterium in water. The fish population moves randomly in water. Death fish disintegrate in water, at different locations, causing bacteria to diffuse through water and infect other fish. Existence of uniform steady states is investigated and the linearized stability of the positive uniform steady state is analyzed. A Hopf bifurcation is proved to occur from the uniform steady state when the bifurcation parameter, here the time delay, passes through a critical value and diffusion coefficients satisfy some conditions, that induces time oscillations of the populations. Comments on diffusion-driven instability are provided, and numerical simulations are carried out to illustrate the results.     

具免疫时滞的HIV感染模型动力学性质分析

讨论了一类具免疫时滞的HIV感染模型.分析了未感染平衡点的全局渐近稳定性,给出了感染无免疫平衡点及感染免疫平衡点局部渐近稳定的充分条件.数值模拟结果表明,当易感细胞生成率的取值使得基本再生数满足平衡存在的条件且低于某一临界值时,时滞对平衡点的稳定性没有影响;若大于该临界值,随着时滞增大,稳定性开关发生,平衡点不稳定,出现一系列Hopf分支,最终表现为周期波动模式.     

A delay-differential equation model of HIV infection of CD4(+) T-cells

A.S. Perelson, D.E. Kirschner and R. De Boer (Math. Biosci. 114 (1993) 81) proposed an ODE model of cell-free viral spread of human immunodeficiency virus (HIV) in a well-mixed compartment such as the bloodstream. Their model consists of four components: uninfected healthy CD4(+) T-cells, latently infected CD4(+) T-cells, actively infected CD4(+) T-cells, and free virus. This model has been important in the field of mathematical modeling of HIV infection and many other models have been proposed which take the model of Perelson, Kirschner and De Boer as their inspiration, so to speak (see a recent survey paper by A.S. Perelson and P.W. Nelson (SIAM Rev. 41 (1999) 3-44)). We first simplify their model into one consisting of only three components: the healthy CD4(+) T-cells, infected CD4(+) T-cells, and free virus and discuss the existence and stability of the infected steady state. Then, we introduce a discrete time delay to the model to describe the time between infection of a CD4(+) T-cell and the emission of viral particles on a cellular level (see A.V.M. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May, M.A. Nowak [Proc. Nat. Acad. Sci. USA 93 (1996) 7247]). We study the effect of the time delay on the stability of the endemically infected equilibrium, criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. Numerical simulations are presented to illustrate the results.     

捕食者有无限时滞效应的两种群系统的全局渐近稳定性

本文用Liapunov泛函方法研究捕食者有无限时滞效应的捕食-被捕食系统的平衡状态的稳定性．文章提供了判定系统的平衡状态全局渐近稳定的简单条件,不要求积分核指数衰减.     

一类带有肝炎B病毒感染的数学模型的全局稳定性分析（英文）

本文主要分析了一类具有肝炎B病毒感染且带有治愈率的典型的数学模型（HBV）.通过稳定性分析,得到了该模型的无病平衡点与地方病平衡点全局稳定的充分条件,并且证明了当基本再生数R0〈1, HBV感染消失;当R0〉1,HBV感染持续.     

Dynamic stability of steady states and static stabilization in unbranched metabolic pathways

The paper is concerned with the conditions of dynamic (asymptotic) stability of steady states in unbranched metabolic pathways. The stationary flux in such pathways is generally determined by the concentration of the end product due to the effector action of this product on the reactions proceeding in its synthetic pathway. The delay in feedback circuits causes violation of dynamic stability at large static stabilization factors. A method permitting analytic estimation of the critical stabilization factor is suggested. Sufficient and necessary conditions for asymptotic stability of the steady state in the general case of the pathway with a single feedback loop have been established. Mechanisms for maintenance of the steady state asymptotic stability at large static stabilization factors are studied. It has been shown that the range of dynamic stability can be widened greatly, if the pathway contains one or two reactions (but not more) of relatively small effective rate constants. Short strong negative feedback is also found to extend considerably the range of dynamic stability of the pathway. The feedback is more effective if it acts on the reaction with small effective rate constant.     

Analysis of continuous cultures of recombinant methylotrophs

Static and dynamic characteristics of continuous cultures of recombinant methylotrophs, which are designed to improve the selectivity of plasmid-bearing cells and the plasmid stability, are investigated in detail. Operational regions in which coexistence (survival of plasmid-bearing and plasmid-free cells) operation is feasible have been identified in the entire space of kinetic parameters and operating variables. The stability characteristics of each steady state are examined. The existence of oscillatory states around the coexistence steady state is investigated using the dynamic (Hopf) bifurcation analysis. For proper startup of the continuous culture operation, it is critical to identify the sets of initial conditions, if any, which lead to transients that ultimately result in washout of plasmid-bearing cells and avoid such conditions. For the numerical illustrations presented, the coexistence steady state happens to be locally stable over much of its region of existence, particular for the operating conditions corresponding to maximum productivity.     

A kinetic model of CD4+ lymphocytes with the human immunodeficiency virus (HIV).

This report describes a kinetic model of in vitro cytopathology involving interactions of human immunodeficiency virus (HIV) with CD4+ helper T lymphocytes. The model uses nonlinearly coupled, ordinary differential equations to simulate the dynamics of infected and uninfected cells and free virions. It is assumed that resting cells are more readily infected than activated cells, but once infected, only activated cells produce more virus. Resting cells can be activated by some appropriate stimulus (e.g. phytohemagglutinin, soluble antigen). The model predicts that the initial inoculum of virus is taken up by resting cells and without stimulation the system comes to a steady state of two populations, namely infected and uninfected cells. Stimulation of this system produces two additional populations, namely infected and uninfected activated cells which, along with the previous populations, exhibit cyclic behavior of growth, viral expression/release, and death. Additional stimuli enhance or diminish the cyclic behavior depending upon their occurrence in time. These simulations suggest a similar dynamics in human HIV infection and may explain a major factor responsible for the widely varying depletion rate of (CD4+) helper T cells in AIDS patients.     

Delay-induced model for tumor-immune interaction and control of malignant tumor growth

The paper deals with the qualitative analysis of the solutions of a system of delay differential equations describing the interaction between tumor and immune cells. The asymptotic stability of the possible steady states is showed and the occurrence of limit cycle of the system around the interior equilibrium is proved by the application of Hopf bifurcation theorem by using the delay as a bifurcation parameter. The length of the delay parameter for preserving stability of the system is also estimated, which gives the idea about the mode of action for controlling oscillations in malignant tumor cell growth. The theoretical and numerical outcomes have been supported through experimental results from literatures. This approach gives new insight of modeling tumor-immune interactions and provides significant control strategies to overcome the large oscillations in tumor cells.     

Time-delayed SIS epidemic model with population awareness

This paper analyses the dynamics of infectious disease with a concurrent spread of disease awareness. The model includes local awareness due to contacts with aware individuals, as well as global awareness due to reported cases of infection and awareness campaigns. We investigate the effects of time delay in response of unaware individuals to available information on the epidemic dynamics by establishing conditions for the Hopf bifurcation of the endemic steady state of the model. Analytical results are supported by numerical bifurcation analysis and simulations.