Bulletin of Mathematical Biology

Filipe, J.; Gibson, G.

doi: 10.1006/bulm.2001.0234pmid: 11497160

Analytical methods for predicting and exploring the dynamics of stochastic, spatially interacting populations have proven to have useful application in epidemiology and ecology. An important development has been the increasing interest in spatially explicit models, which require more advanced analytical techniques than the usual mean-field or mass-action approaches. The general principle is the derivation of differential equations describing the evolution of the expected population size and other statistics. As a result of spatial interactions no closed set of equations is obtained. Nevertheless, approximate solutions are possible using closure relations for truncation. Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. This study is made in the context of an SIS model for plant-disease epidemics introduced in Filipe and Gibson (1998, Studying and approximating spatio-temporal models for epidemic spread and control, Phil. Trans. R. Soc. Lond. B 353, 2153–2162) of which the contact process [Harris, T. E. (1974), Contact interactions on a lattice, Ann. Prob. 2, 969] is a special case. The various methods of approximation are derived and explained and their predictions are compared and tested against simulation. The merits and limitations of the various approximations are discussed. A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.

Keener, James

doi: 10.1006/bulm.2001.0235pmid: 11497161

Oscillatory secretion of insulin has been observed in many different experimental preparations. Here we examine a mathematical model for in vitro insulin secretion from pancreatic beta cells in a flow-through reactor. The analysis shows that oscillations result because of an important interplay between flow rate of the reactor and insulin diffusion. In particular, if the ratio of flow rate to volume of the reaction bed is too large, oscillations are eliminated, in contradiction to the conclusions of Maki and Keizer (L. W. Maki and Keizer J. Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells. Bull. Math. Biol., 57 (1995), 569–591). Furthermore, with reasonable numbers for the experimental parameters and the diffusion of insulin, the model equations do not exhibit oscillations.

Oosawa, Fumio

doi: 10.1006/bulm.2001.0236pmid: 11497162

Living cells often generate signals spontaneously in the absence of external stimuli. Those signals play an important role in their tactic behaviors. This paper presents a theoretical treatment on the mechanism of spontaneous signal generation. The mechanism consists of two steps: (1) production of the basic fluctuation of the intracellular electric potential due to the open-closed fluctuation of the gates of ion channels and (2) generation of a spike-like fluctuation of potential depending on the positive shift of the basic fluctuation. The first step is described by an equation of the Langevin type, where the random force is proportional to the circulating ion current across the membrane; the average of the square of the random force is proportional to the rate of free-energy consumption by the current. The second step is described by a rate equation of transition of field-sensitive channel gates which contains the fluctuating electric field in the exponential term. There, the fluctuation has a nonlinear effect. Such a two-step process may work in various kinds of living cells. The presence of circulating ion current in the resting state is a most important key. Some cells may be quiet and some cells may be active to generate spontaneous signals.

Owen, M.; Lewis, M.

doi: 10.1006/bulm.2001.0239pmid: 11497163

Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, 238–251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predator-prey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.

Utzny, C.; Burroughs, N.

doi: 10.1006/bulm.2001.0242pmid: 11497164

The correlation between properties of the T cell memory pool and the two regulatory mechanisms of cell death (apoptosis) and memory entry (differentiation) is investigated mathematically. Apoptosis of T cells occurs at the end of an immune response, removing unwanted activated T cells. T cells escaping apoptosis enter the memory pool composed of T cells specific for previously encountered antigens. We find that the relative efficiencies of these two pathways determine the clonal distribution and the long-term stability of the memory pool by regulating the number of new entries. The main result presented in this paper is that immunological memory of previously encountered pathogens cannot be erased by either severe or repeat infections with a particular pathogen (the diversity of the memory pool is ensured) only if apoptosis and/or memory differentiation are regulated by population dependent processes. Furthermore, vaccination properties are improved significantly by population dependent mechanisms and our mathematical analysis reveals that the T cell population must communicate with other parts of the immune system to ensure optimal performance of immunological memory.

Wilke, Claus

doi: 10.1006/bulm.2001.0244pmid: 11497165

We study the evolution of large but finite asexual populations evolving in fitness landscapes in which all mutations are either neutral or strongly deleterious. We demonstrate that despite the absence of higher fitness genotypes, adaptation takes place as regions with more advantageous distributions of neutral genotypes are discovered. Since these discoveries are typically rare events, the population dynamics can be subdivided into separate epochs, with rapid transitions between them. Within one epoch, the average fitness in the population is approximately constant. The transitions between epochs, however, are generally accompanied by a significant increase in the average fitness. We verify our theoretical considerations with two analytically tractable bitstring models.

Wu, Joseph; Byrne, Helen; Kirn, David; Wein, Lawrence

doi: 10.1006/bulm.2001.0245pmid: 11497166

Replication-competent viruses have shown considerable promise in overcoming the inefficient gene transduction experienced by traditional gene therapy approaches to cancer treatment. The viruses infect tumor cells and replicate inside them, eventually causing lysis. Virus particles released during lysis are then able to infect other tumor cells, and, in this way, continuous rounds of infection and lysis allow the virus to spread throughout the tumor. Motivated by this novel cancer treatment, we formulate and analyse a system of partial differential equations that is essentially a radially-symmetric epidemic model embedded in a Stefan problem. We compare three, alternative virus-injection strategies: a fixed fraction of cells pre-infected with the virus are introduced throughout the entire tumor volume, within the tumor core, or within the tumor rim. For all three injection methods, simple and accurate conditions that predict whether the virus will control the tumor are derived.

Kooi, B.; Hallam, T.; Kelpin, F.; Krohn, C.; Kooijman, S.

doi: 10.1006/bulm.2001.0246pmid: 11497167

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.

DeAngelis, Donald; Cocho, Germinal

doi: 10.1006/bulm.2001.0233pmid: N/A