Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

A dynamic model for the p53 stress response networks under ion radiation

Summary. P53 controls the cell cycle arrest and cell apoptosis through interaction with the downstream genes and their signal pathways. To stimulate the investigation into the complicated responses of p53 under the circumstance of ion radiation (IR) in the cellular level, a dynamic model for the p53 stress response networks is proposed. The model can be successfully used to simulate the dynamic processes of generating the double-strand breaks (DSBs) and their repairing, ataxia telangiectasia mutated (ATM) activation, as well as the oscillations occurring in the p53-MDM2 feedback loop.     

Linear stability criteria for population models with periodically perturbed delays

We examine some simple population models that incorporate a time delay which is not a constant but is instead a known periodic function of time. We examine what effect this periodic variation has on the linear stability of the equilibrium states of scalar population models and of a simple predator prey system. The case when the delay differs from a constant by a small amplitude periodic perturbation can be treated analytically by using two-timing methods. Of particular interest is the case when the system is initially marginally stable. The introduction of variation in the delay can then have either a stabilising effect or a destabilizing one, depending on the frequency of the periodic perturbation. The case when the periodic perturbation has large amplitude is studied numerically. If the fluctuation is large enough the effect can be stabilising.     

A new explicit stability criterion for human periodic breathing<--RID="*"--><--ID="*" This research was supported by grants from DRED (96-920).-->

The aim of this paper is to carry out a stability analysis for periodic breathing in humans that incorporates the dynamic characteristics of ventilation control. A simple CO₂ model that takes into account the main elements of the respiratory system, i.e. the lungs and the ventilatory controller with its dynamic properties, is presented. This model results in a three-dimensional non-linear delay differential system for which there exists a unique equilibrium point. Our stability analysis of this equilibrium point leads to the definition of a new explicit stability criterion and to the demonstration of the existence of a Hopf bifurcation. Numerical simulations illustrate the influence of physiological parameters on the stability of ventilation, and particularly the major role of the dynamic characteristics of the respiratory controller. Received: 2 February 1999 / Revised version: 18 June 1999 / Published online: 23 October 2000     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Equality of average and steady-state levels in some nonlinear models of biological oscillations

Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis. This paper is dedicated to the memory of the late Reinhart Heinrich, who was the academic teacher of S.S. and, to a great extent, also of M.M.     

Stability analysis of the partial selfing selection model

We undertake a detailed study of the one-locus two-allele partial selfing selection model. We show that a polymorphic equilibrium can exist only in the cases of overdominance and underdominance and only for a certain range of selfing rates. Furthermore, when it exists, we show that the polymorphic equilibrium is unique. The local stability of the polymorphic equilibrium is investigated and exact analytical conditions are presented. We also carry out an analysis of local stability of the fixation states and then conclude that only overdominance can maintain polymorphism in the population. When the linear local analysis is inconclusive, a quadratic analysis is performed. For some sets of selective values, we demonstrate global convergence. Finally, we compare and discuss results under the partial selfing model and the random mating model.     

Gain-induced oscillations in blood pressure

 “Mayer waves” are long-period (6 to 12 seconds) oscillations in arterial blood pressure, which have been observed and studied for more than 100 years in the cardiovascular system of humans and other mammals. A mathematical model of the human cardiovascular system is presented, incorporating parameters relevant to the onset of Mayer waves. The model is analyzed using methods of Liapunov stability and Hopf bifurcation theory. The analysis shows that increase in the gain of the baroreflex feedback loop controlling venous volume may lead to the onset of oscillations, while changes in the other parameters considered do not affect stability of the equilibrium state. The results agree with clinical observations of Mayer waves in human subjects, both in the period of the oscillations and in the observed age-dependence of Mayer waves. This leads to a proposed explanation of their occurrence, namely that Mayer waves are a gain-induced oscillation. Received: 15 September 1997/Revised version: 15 March 1998     

Electrocoupling of Ion Transporters in Plants: Interaction with Internal Ion Concentrations

There are five major electroenzymes in the plasmalemma of plant cells: a driving electrogenic pump, inward and outward rectifying K⁺ channels, a Cl⁻-2H⁺ symporter, and Cl⁻-channels. It has been demonstrated previously (Gradmann, Blatt & Thiel 1993, J. Membrane Biol. 136:327–332) how voltage-gating of these electroenzymes causes oscillations of the transmembrane voltage (V) at constant substrate concentrations. The purpose of this study is to examine the interaction of the same transporter ensemble with cytoplasmic concentrations of K⁺ and Cl⁻. The former model system has been extended to account for changing internal concentrations. Constant-field theory has been applied to describe the influence of ion concentrations on current-voltage relationships of the active channels. The extended model is investigated using a reference set of model parameters. In this configuration, the system converges to stable slow oscillations with intrinsic changes in cytoplasmic K⁺ and Cl⁻ concentrations. These slow oscillations reflect alternation between a state of salt uptake at steady negative values of V and a state of net salt loss at rapidly oscillating V, the latter being analogous to the previously reported oscillations. By switching off either concentration changes or gating, it is demonstrated that the fast oscillations are mostly due to the gating properties of the Cl⁻ channel, whereas the slow oscillations are controlled by the effect of the Cl⁻ concentration on the current. The sensitivity of output results y (e.g., frequency of oscillations) to changes of the model parameters x (e.g., maximum Cl⁻ conductance) has been investigated for the reference system. Further examples are presented where some larger changes of specific model parameters cause fundamentally different behavior, e.g., convergence towards a stable state of only the fast oscillations without intrinsic concentration changes, or to a steady-state without any oscillations. The main and general result of this study is that the osmotic status of a plant cell is stabilized by the ensemble of familiar electroenzymes through oscillatory interactions with the internal concentrations of the most abundant ions. This convergent behavior of the stand-alone system is an important prerequisite for osmotic regulation by means of other physiological mechanisms, like second messengers and gating modifiers. Received: 23 February/Revised: 16 July 1998     

Mathematical modelling of the intravenous glucose tolerance test

 Several attempts at building a satisfactory model of the glucose-insulin system are recorded in the literature. The minimal model, which is the model currently mostly used in physiological research on the metabolism of glucose, was proposed in the early eighties for the interpretation of the glucose and insulin plasma concentrations following the intravenous glucose tolerance test. It is composed of two parts: the first consists of two differential equations and describes the glucose plasma concentration time-course treating insulin plasma concentration as a known forcing function; the second consists of a single equation and describes the time course of plasma insulin concentration treating glucose plasma concentration as a known forcing function. The two parts are to be separately estimated on the available data. In order to study glucose-insulin homeostasis as a single dynamical system, a unified model would be desirable. To this end, the simple coupling of the original two parts of the minimal model is not appropriate, since it can be shown that, for commonly observed combinations of parameter values, the coupled model would not admit an equilibrium and the concentration of active insulin in the “distant” compartment would be predicted to increase without bounds. For comparison, a simple delay-differential model is introduced, is demonstrated to be globally asymptotically stable around a unique equilibrium point corresponding to the pre-bolus conditions, and is shown to have positive and bounded solutions for all times. The results of fitting the delay-differential model to experimental data from ten healthy volunteers are also shown. It is concluded that a global unified model is both theoretically desirable and practically usable, and that any such model ought to undergo formal analysis to establish its appropriateness and to exclude conflicts with accepted physiological notions. Received: 22 June 1998 / Revised version: 24 February 1999     

Parametric analysis of the ratio-dependent predator–prey model

We present a complete parametric analysis of stability properties and dynamic regimes of an ODE model in which the functional response is a function of the ratio of prey and predator abundances. We show the existence of eight qualitatively different types of system behaviors realized for various parameter values. In particular, there exist areas of coexistence (which may be steady or oscillating), areas in which both populations become extinct, and areas of "conditional coexistence" depending on the initial values. One of the main mathematical features of ratio-dependent models, distinguishing this class from other predator-prey models, is that the Origin is a complicated equilibrium point, whose characteristics crucially determine the main properties of the model. This is the first demonstration of this phenomenon in an ecological model. The model is investigated with methods of the qualitative theory of ODEs and the theory of bifurcations. The biological relevance of the mathematical results is discussed both regarding conservation issues (for which coexistence is desired) and biological control (for which extinction is desired).     

Equilibrium structure and stability in a frequency-dependent,two-population diploid model

We investigate the equilibrium structure for an evolutionary genetic model in discrete time involving two monoecious populations subject to intraspecific and interspecific random pairwise interactions. A characterization for local stability of an equilibrium is found, related to the proximity of this equilibrium with evolutionarily stable strategies (ESS). This extends to a multi-population framework a principle initially proposed for single populations, which states that the mean population strategy at a locally stable equilibrium is as close as possible to an ESS.     

Current Oscillations Under Voltage-Clamp Conditions: An Interplay of Series Resistance and Negative Slope Conductance

Using the patch-clamp technique, we observed profound oscillations of the whole-vacuole outward current across the tonoplast of Mesembryanthemum crystallinum L. (common ice plant). These current oscillations showed a clear voltage dependence and appeared at membrane potentials more positive than 90–100 mV. This paper describes the oscillations in terms of two separate mechanisms. First, the Mesembryanthemum vacuolar membrane shows a negative slope conductance at membrane potentials more positive than 100–120 mV. The fact that the oscillations and the negative slope conductance show a similar threshold potential suggests that (part of) the same mechanism is involved in both phenomena. The second mechanism involved is the voltage drop across the series resistance. As a result, the potential actually experienced by the vacuolar membrane deviates from the command potential defined by the patch-clamp amplifier. This deviation depends in an Ohmic manner on the current magnitude. We suggest that the interplay of the negative slope conductance and the voltage drop across the series resistance can cause a positive feedback which is responsible for the current oscillations. Received: 30 April 1999/Revised: 9 September     

Numerical bifurcation analysis of delay differential equations arising from physiological modeling

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.     

Teratogenicity of 3-hydroxynorvaline in chicken and mouse embryos

Summary. 3-Hydroxynorvaline (HNV; 2-amino-3-hydroxypentanoic acid), a microbial L-threonine analogue, is toxic to mammalian cells and displays antiviral properties. In view of this, we investigated the toxicity and/or potential teratogenicity of HNV in developing chicken and mouse embryos. HNV was administered to chicken embryos (in ovo; dose 75–300 μmole/egg; 48 h post-incubation) and pregnant Hanover NMRI mice (per os; total dose 900–1800 mg/kg body mass; gestation days 7–9). Control animals received sterile saline solutions. Harvested embryos (chicken embryos, 10 days post-incubation; mouse embryos; gestation day 18) were fixed in glutaraldehyde and stereomicroscopically inspected for signs of dysmorphogenesis. Body mass, body and toe length and mortality of chicken embryos, and the body mass and mortality of mouse embryos were recorded. HNV exposure significantly increased the incidence of embryotoxic (growth retardation, toxic mortality) and congenital defects in both chicken and mouse embryos. All the observed effects were dose-dependent. In conclusion, HNV is an embryotoxic and teratogenic compound, which caused significant developmental delay and congenital defects in developing chicken and mouse embryos.     

Periodic solutions in a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

We obtain necessary and sufficient conditions on the existence of a unique positive equilibrium point and a set of sufficient conditions on the existence of periodic solutions for a 3-dimensional system which arises from a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor. Our results improve the corresponding results obtained by Hsu, Luo, and Waltman [1]. Received: 20 November 1997 / Revised version: 12 February 1999 / Published online: 20 December 2000     

Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

Light-dependent bicarbonate uptake and CO2 efflux in the marine microalga Nannochloropsis gaditana

 Inorganic carbon (Ci) uptake and efflux has been investigated in the marine microalga Nannochloropsis gaditana Lubian by monitoring CO₂ fluxes in cell suspensions using mass spectrometry. Addition of H¹³CO₃ ⁻ to cell suspensions in the dark caused a transient increase in the CO₂ concentration in the medium far in excess of the equilibrium CO₂ concentration. The magnitude of this release was dependent on the length of time the cells had been kept in the dark. Once equilibrium between the Ci species had been achieved, a CO₂ efflux was observed after saturating light intensity was applied to the cells. External carbonic anhydrase (CA) was not detected nor does this species demonstrate a capacity to take up CO₂ by active transport. Photosynthetic O₂ evolution and the release CO₂ in the dark depend on HCO₃ ⁻ uptake since both were inhibited by the anion exchange inhibitor, 4,4′-diisothiocyanatostilbene-2,2′-disulfonic acid (DIDS). The bicarbonate uptake mechanism requires light but can also continue for short periods in the dark. Ethoxyzolamide, a CA inhibitor, markedly inhibited CO₂ efflux in the dark, indicating that CO₂ efflux was dependent upon the intracellular dehydration of HCO₃ ⁻. These results indicate that Nannochloropsis possesses a bicarbonate uptake system which causes the accumulation of high intracellular Ci levels and an internal CA which maintains the equilibrium between CO₂ and HCO₃ ⁻ and thus causes a subsequent release of CO₂ to the external medium. Received: 20 September 1999 / Accepted: 25 October 1999