Bulletin of Mathematical Biology

Mirkin, Boris;Roberts, Fred S.

doi: 10.1007/BF02460669pmid: 8318927

Abstract In recent years, methods of consensus, developed for the solution of problems in the social sciences, have become widely used in molecular biology. Westudy a method of consensus originally due to Watermanet al. (Waterman, Galas and Arratia. 1984. Pattern recognition in several sequences: consensus and alignment.Bull. math. Biol. 46, 515–527) which is used to identify patterns or features in a molecular sequence where a pattern can vary in position within a given window. We show that some well-known consensus methods of the social sciences, the median and the mean, are special cases of this method for certain choices of the parameters used in it and give a precise account of the parameters for which these special cases arise. We also show that the specific parameters used in the method of Watermanet al. make their method equivalent to the median procedure which is widely used in the social sciences.

Beretta, E.;Solimano, F.

doi: 10.1007/BF02460670pmid: 8391359

Abstract The problem of optimal dosage is studied for the administration of ddCyd using erythrocytes as carriers and bioreactors. The volume of erythrocytes and the initial amount of drug to be loaded have to be determined in such a way that the duration of the therapeutic effect is maximized without exceeding the toxic threshold. It is found that the optimal control is unique and it is at the upper vertex of the set of the admissible controls. A more general case is also briefly discussed.

Donnelly, Christl;Leisenring, Wendy;Kanki, Phyllis;Awerbuch, Tamara;Sandberg, Sonja

doi: 10.1007/BF02460671pmid: 8318928

Abstract To explore the biological similarities and differences between the HIV-1 and HIV-2 viruses, we model the probability of male-to-female transmission of either HIV virus as a function of the number of sexual partners, the prevalence of the viruses and the infectivity per contact. Using maximum likelihood estimation theory and data from a prospective study of registered female prostitutes in Dakar, Senegal, we estimate and compare the infectivities of HIV-1 and HIV-2. Graphical goodness-of-fit methods are used to show that our model fits the data well. We find that in male-to-female transmission HIV-1 is significantly more infectious than HIV-2. This findings is consistent with other data from laboratory and epidemiologic studies comparing the biology of HIV-1 and HIV-2.

De Boer, Rob J.;Perelson, Alan S.;Kevrekidis, Ioannis G.

doi: 10.1007/BF02460672pmid: 8318929

Abstract We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation, B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior. After appropriate nondimensinalization, the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-tye equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.

De Boer, Rob J.;Perelson, Alan S.;Kevrekidis, Ioannis G.

doi: 10.1007/BF02460673pmid: 8318930

Abstract Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.

Freedman, H. I.;Mathsen, R. M.

doi: 10.1007/BF02460674pmid: N/A

Abstract Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, i.e. that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Although local extinction certainly is a common occurrence in nature, it cannot be modeled by systems which are ratio-dependent near the axes.

Jeka, John J.;Kelso, J. A. S.;Kiemel, Tim

doi: 10.1007/BF02460675pmid: 8318931

Abstract A relative phase model of four coupled oscillators is used to interpret experiments on the coordination between rhythmically moving human limbs. The pairwise coupling functions in the model are motivated by experiments on two-limb coordination. Stable patterns of coordination between the limbs are represented by fixed points in relative phase coordinates. Four invariant circles exist in the model, each containing two patterns of coordination seen experimentally. The direction of switches between two four-limb patterns on the same circle can be understood in terms of two-limb coordination. Transitions between patterns in the human four-limb system are theoretically interpreted as bifurcations in a nonlinear dynamical system.

Allen, Linda J. S.;Hannigan, Mary K.;Strauss, Monty J.

doi: 10.1007/BF02460676pmid: N/A

Abstract The apple twig borer (Amphicerus bicaudatus) is an insect pest of the grape vine, causing considerable damage to the grape vine in early spring. A simple difference equation model is formulated and analysed for this plant-herbivore system based on two control strategies, cane removal and pesticide application. The system has two equilibria, one where the pest is present and one where the pest is absent. Regions are found in parameter space for global stability of the equilibria and in the absence of global stability it is shown that there exist periodic or quasiperiodic solutions.

Reed, Michael C.

doi: 10.1007/BF02460677pmid: N/A