State-dependent neutral delay equations from population dynamics.

A novel class of state-dependent delay equations is derived from the balance laws of age-structured population dynamics, assuming that birth rates and death rates, as functions of age, are piece-wise constant and that the length of the juvenile phase depends on the total adult population size. The resulting class of equations includes also neutral delay equations. All these equations are very different from the standard delay equations with state-dependent delay since the balance laws require non-linear correction factors. These equations can be written as systems for two variables consisting of an ordinary differential equation (ODE) and a generalized shift, a form suitable for numerical calculations. It is shown that the neutral equation (and the corresponding ODE--shift system) is a limiting case of a system of two standard delay equations.

Quiescence, excitability, and heterogeneity in ecological models.

Introducing quiescent phases into dynamical systems and ecological models tends to stabilize equilibria against the onset of oscillations and also to lower the amplitudes of existing periodic orbits. However, these effects occur when all interacting species go quiescent with the same rates and return to activity with the same rates. On the other hand, if the species differ with respect to these rates, then an equilibrium may even be destabilized. At least in the case of two interacting species this bifurcation phenomenon is closely related to the well-known Turing instability. In particular, for two species it is true that an equilibrium can be destabilized by quiescent phases if and only if it is excitable in the Turing sense. These effects are thoroughly studied and exhibited at the example of classical ecological models and epidemic models. Similar effects occur in delay equations and reaction-diffusion equations. The effect of stabilization against oscillations by quiescent phases can be shown as a special realization of a general principle saying that spatial heterogeneity stabilizes. The results on local stability of stationary points can be extended to periodic orbits. In particular, a geometric argument on the flow along a periodic orbit explains why convex periodic orbits, as observed in numerical simulations, tend to shrink when quiescent phases are introduced.

Pair formation.

A multitype pair formation model for a one-sex population, without separation, with given type distribution of singles, produces a distribution of pairs with the given type distribution as a marginal distribution. The pair distribution can be seen as a nonnegative symmetric matrix. For this matrix representation formulas have been given years ago and have been widely used. The goal of the paper is to understand these formulas in probabilistic terms and give a meaning to their coefficients. Our approach connects the formulas to the problem of completing a substochastic matrix to a stochastic matrix. In this way the coefficients in the representation formula can be interpreted as preferences and insight can be gained into the set of distributions respecting given preferences. In order to put these questions into a wider perspective, the classical two-sex pair formation models are reviewed and embedded into the class of one-sex models, and dynamic models are designed that yield pair distributions as limit elements.

Parameter identification in epidemic models.

Starting from a recent paper of Pollicott, Wang and Weiss we try to obtain improved representation formulas for the estimation of the time-dependent transmission rate of an epidemic in terms of either incidence or prevalence data. Although the formulas are (trivially) mathematically equivalent to previous formulas, the new representations need no additional estimates and they should be more stable numerically. We review the discrete time and the stochastic continuous time approach. We replace the assumption that recovery follows an exponential distribution and get estimates for the transmission rate for constant duration of the infectious phase.

The random walk of Azospirillum brasilense.

The bacterium Azospirillum brasilense has been frequently studied in laboratory experiments. It performs movements in space where long forward and backward runs on a straight line occur simultaneously with slow changes of direction of the line. A model is presented in which a correlated random walk on a line is joined to diffusion on a sphere of directions. For this transport system, a hierarchy of moment approximations is derived, ranging from a hyperbolic system with four dependent variables to a scalar damped wave equation (telegraph equation) and then to a single diffusion equation for particle density. The original parameters are compounded in the diffusion quotient. The effects of these parameters, such as particle speed or turning rate, on the diffusion coefficient are discussed in detail.

Quiescence stabilizes predator-prey relations.

The classical MacArthur Rosenzweig predator-prey system has a stable coexistence point or, if either the prey capacity is large or the predator mortality is low, a stable limit cycle. The question here is how the stability properties of the coexistence point change when the prey or the predator or both can go quiescent. It can be shown that a stable equilibrium stays stable, but an unstable equilibrium may become stable. The exact stability domain is determined. In general, increasing the duration of the quiescent phase of the prey or of the predator widens the stability window. Numerical studies show that limit cycles shrink when quiescent phases are introduced.

Monotone dependence of the spectral bound on the transition rates in linear compartment models.

For linear compartment models or Leslie-type staged population models with quasi-positive matrix the spectral bound of the matrix (the eigenvalue determining stability) is studied in the situation where particles or individuals leave a compartment or stage with some rate and enter another with the same rate. Then the matrix carries the rate with a positive sign in some off-diagonal entry and with a negative sign in the corresponding diagonal entry. Hence the matrix does not depend on the rate in a monotone way. It is shown, however, that the spectral bound is a monotone function of the rate. It is all the time strictly increasing or strictly decreasing or it is constant. A simple algebraic criterion distinguishes between the three cases. The results can be applied to linear systems and to the stability of stationary states in non-linear systems, in particular to models for the transmission of infectious diseases, and in population dynamics.

Optimal harvesting and optimal vaccination.

Two optimization problems are considered: Harvesting from a structured population with maximal gain subject to the condition of non-extinction, and vaccinating a population with prescribed reduction of the reproduction number of the disease at minimal costs. It is shown that these problems have a similar structure and can be treated by the same mathematical approach. The optimal solutions have a 'two-window' structure: Optimal harvesting and vaccination strategies or policies are concentrated on one or two preferred age classes. The results are first shown for a linear age structure problem and for an epidemic situation at the uninfected state (minimize costs for a given reduction of the reproduction number) and then extended to populations structured by size, to harvesting at Gurtin-MacCamy equilibria and to vaccination at infected equilibria.

Insensitivity of cardiovascular function to low power cm-/mm-microwaves.

A previous study failed to disclose an effect of short (15 min) exposure to low level energy microwaves (3 microW/cm2) emitted by a commercially available automobile radar system (77 GHz) for adaptive cruise control (ACC) on cardiovascular function. The present study explored whether a 15 min exposure to higher level energy microwaves of frequencies varying from 5.8 to 110 GHz influences cardiovascular function. To this end heart rate, skin temperature (thermocouple), skin conductance (Ag/AgCl electrodes), systolic and diastolic blood pressure (automatic cuff) were recorded in 50 test persons before, during and after a 15 min exposure to a sequential pattern of microwaves varying from 5.8 to 110 GHz (59.7 microW/cm2). After an equilibration period of 30 min the first group of test persons and after additional 30 min the second group of test persons were exposed. The study has been performed in a strict double blind design. While significant effects on the measured parameters were observed depending on time ("calming" effect), no significant difference was observed between exposure and sham exposure to microwaves. In view of the small scatter of the data the present study rules out physiologically relevant effects of moderate energy (59.7 microW/cm2) microwaves varying from 5.8 to 110 GHz on cardiovascular function.

Influence of low power cm-/mm-microwaves on cardiovascular function.

The present study has been designed to investigate physiological effects of short (15 min) exposure to low level energy microwaves (< 10 microW cm(-2)). To this end heart rate, PQ, QS and ST (electrocardiography), respiration (conductive stretch band around the thorax), skin temperature (thermocouple), skin conductance (Ag/AgCl electrodes), systolic and diastolic blood pressure (automatic cuff) were continuously recorded in a group of 50 test persons before, during and after a 15 min exposure to 3 microW cm(-2) high frequency (77 GHz) microwaves. After an equilibration period of 30 min the first group of test persons and after additional 30 min the second group of test persons were exposed. The study has been performed in a strict double blind design. While significant effects on the measured parameters were observed depending on time ('calming' effect), no significant difference was observed between exposure and sham exposure to microwaves. In view of the small scatter of the data the present study rules out physiologically relevant effects of low level energy on the autonomic nervous system and cardiovascular function.

Cleaving proteins for the immune system.

Proteasomes are enzymes in eukaryotic cells which cut proteins marked for degradation into fragments. In mammals some of these fragments are used by the immune system to detect proteins of foreign, e.g. viral, origin. Hence reproducing, predicting and possibly understanding the cleaving patterns of proteasomes is an interesting theoretical problem and its solution would be beneficial for vaccine design. The equations connecting cut probabilities, fragment frequencies and so-called cut strengths are derived. A simple model for the time course of protein digestion is used to explain the problem of fragment competition and the possible deviation of in vitro fragment frequencies from those that can be expected in vivo. A family of neural network proteasome models for the reproduction and prediction of cleavage patterns is described in detail together with the webtool PAProC. The first model is based on the experimentally observed cleavage pattern, an intermediate model on the distinction between weak and strong cuts, and the most elaborate model uses quantitative data, i.e., fragment frequencies.

PAProC: a prediction algorithm for proteasomal cleavages available on the WWW.

The first version of PAProC (Prediction Algorithm for Proteasomal Cleavages) is now available to the general public. PAProC is a prediction tool for cleavages by human and yeast proteasomes, based on experimental cleavage data. It will be particularly useful for immunologists working on antigen processing and the prediction of major histocompatibility complex class I molecule (MHC I) ligands and cytotoxic T-lymphocyte (CTL) epitopes. Likewise, in cases in which proteasomal protein degradation has been indicated in disease, PAProC can be used to assess the general cleavability of disease-linked proteins. On its web site (http://www.paproc.de), background information and hyperlinks are provided for the user (e.g., to SYFPEITHI, the database for the prediction of MHC I ligands).

Ticks and tick-borne diseases: a vector-host interaction model for the brown ear tick (Rhipicephalus appendiculatus).

An analytical model is derived for the interaction of the brown ear tick (Rhipicephalus appendiculatus) with its hosts. Such models are rare due to the complexity and lack of information on the entire stages of ticks life cycles. Most models are simulations rather than analytical. The vector is categorized into a discrete number of compartments according to its life cycle. The starting model in this article consists of a system of differential equations with constant coefficients. A general model on a stage structured population with unlimited host density is developed. From the characteristic polynomial of the system a sensitivity analysis for the population parameters is carried out in detail. The model is then improved by incorporating host abundance and availability. This is done on the basis of a demand-driven and ratio-dependent functional response model. The improved model adequately represents the dynamics of a stage-structured vector population under conditions of varying host density. The model allows the qualitative evaluation of several management strategies and is expected to guide future research work.

An algorithm for the prediction of proteasomal cleavages.

Proteasomes, major proteolytic sites in eukaryotic cells, play an important part in major histocompatibility class I (MHC I) ligand generation and thus in the regulation of specific immune responses. Their cleavage specificity is of outstanding interest for this process. In order to generalize previously determined cleavage motifs of 20 S proteasomes, we developed network-based model proteasomes trained by an evolutionary algorithm with experimental cleavage data of yeast and human 20 S proteasomes. A window of ten flanking amino acid residues proved sufficient for the model proteasomes to reproduce the experimental results with 98-100 % accuracy. Actual experimental data were reproduced significantly better than randomly selected cleavage sites, suggesting that our model proteasomes were able to extract rules inherent to proteasomal cleavage data. The affinity parameters of the model, which decide for or against cleavage, correspond with the cleavage motifs determined experimentally. The predictive power of the model was verified for unknown (to the program) test conditions: the prediction of cleavage numbers in proteins and the generation of MHC I ligands from short peptides. In summary, our model proteasomes reproduce and predict proteasomal cleavages with high degree of accuracy. They present a promising approach for predicting proteasomal cleavage products in future attempts and, in combination with existing algorithms for MHC I ligand prediction, will be tested to improve cytotoxic T lymphocyte epitope prediction.

Model of plasmid-bearing, plasmid-free competition in the chemostat with nutrient recycling and an inhibitor.

In this paper, we consider competition between plasmid-bearing and plasmid-free organisms with nutrient recycling and an inhibitor in a chemostat-type systems. We discuss the cases where the nutrient is supplied at a constant rate and the nutrient supply is time-dependent. For each case, we obtain criteria for the boundedness of solutions and persistence.

Backward bifurcation in epidemic control.

For a class of epidemiological SIRS models that include public health policies, the stability at the uninfected state and the prevalence at the infected state are investigated. Backward bifurcation from the uninfected state and hysteresis effects are shown to occur for some range of parameters. In such cases, the reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. An explicit expression is given for this quantity. The phenomenon of subcritical bifurcation in epidemic modeling is also discussed in terms of group models, pair formation, and macroparasite infection.

A core group model for disease transmission.

Models for sexually transmitted diseases generally assume that the size of the core group is fixed. Publicly available information on disease prevalence may influence the recruitment of new susceptibles into highly sexually active populations. It is assumed that the recruitment rate into the core population is low while disease prevalence is high, core group members mix only with each other, disease levels outside the core are negligible, and some core group members reduce their risk through the use of a partially effective vaccine or prophylactics. A demographic-epidemic model is formulated in which the combined size of the core and non-core population is constant. A simpler version models the epidemic in an isolated core population of constant size under the influence of educational programs and measures that reduce susceptibility. The threshold condition for an endemic infection is determined. Backward bifurcations, multiple infective stationary states, and hysteresis phenomena can be observed even in the simplified version. Abrupt changes in disease prevalence levels may result from small changes in the disease management parameters and do not occur in the absence of such a program. The general conclusion is that partially effective vaccination or education programs may increase the total number of cases while decreasing the relative frequency of cases in the core group. The study throws some new light on the role of the reproduction number in connection with elimination attempts. It shows that although the reproduction number defines the threshold for the spread of the disease in a susceptible population, it is of limited value when elimination of an existing epidemic is planned.

Deterministic models for the eradication of poliomyelitis: vaccination with the inactivated (IPV) and attenuated (OPV) polio virus vaccine.

Currently two polio vaccines, IPV and OPV, are in use which differ markedly in their epidemiological parameters. A simple epidemiological model in terms of ordinary differential equations is proposed to study the effects of vaccination campaigns using these vaccines. The numbers of interest are the reproduction number of the disease in the presence of vaccination and the critical vaccination coverage necessary to prevent an outbreak. For these numbers explicit representations are determined which can be used in comparing different vaccination strategies.

The discrete Rosenzweig model.

Discrete time versions of the Rosenweig predator-prey model are studied by analytic and numerical methods. The interaction of the Hopf bifurcation leading to periodic orbits and the period-doubling bifurcation is investigated. It is shown that for certain choices of the parameters there is stable coexistence of both species together with a local attractor at which the prey is absent.