A mathematical model for drug administration by using the phagocytosis of red blood cells

 A mathematical model for the delivery of drug directly to the macrophages by using the phagocytosis of senescent red blood cells is proposed. The model is based on the following assumption: At time t=0 a preassigned red blood cell population n(0, a)=φ(a), a>0, loaded by the drug, is injected in the blood circulation. Among the cells of that population only those with an age a≧ā (ā=120 days) will be phagocytosed by macrophages. Of course, the lifetime of the drug must be higher than ā. Within the red blood cells it cannot be metabolized, neither can it diffuse through their membranes. The emphasis of the paper is on the mathematical properties and on the formulation of the control problem. Received 15 December 1994; received in revised form 20 July 1995     

A mathematical model of the stress induced during avascular tumour growth

In this paper a mathematical model is developed to describe the effect of nonuniform growth on the mechanical stress experienced by cells within an avascular tumour. The constitutive law combines the stress-strain relation of linear elasticity with a growth term that is derived by analogy with thermal expansion. To accommodate the continuous nature of the growth process, the law relates the rate of change of the stress tensor to the rate of change of the strain (rather than relating the stress to the strain directly). By studying three model problems which differ in detail, certain characteristic features are identified. First, cells near the tumour boundary, where nutrient levels and cell proliferation rates are high, are under compression. By contrast, cells towards the centre of the tumour, where nutrient levels are low and cell death dominant, are under tension. The implications of these results and possible model developments are also discussed. Received: 15 November 1999 / Published online: 5 May 2000     

Properties of solutions for a chemotaxis system

 We investigate mathematically the system of equations proposed by Chaplain and Stuart [2], to describe the chemotactic response of endothelial cells under the angiogenesis stimulus. In particular, we characterize the steady state endothelial cell density function, and give conditions on the chemotactic parameter k and cell proliferation parameter b that ensure that migration/ proliferation either does or does not occur in steady state. The time dependent problem is also treated. Received 12 September 1995; received in revised form 6 August 1996     

The periodically forced Droop model for phytoplankton growth in a chemostat

 It is proved that the periodically forced Droop model for phytoplankton growth in a chemostat has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the sub-threshold condition holds, the phytoplankton population is washed out of the chemostat. If the super-threshold condition holds, then there is a unique periodic solution, having the same period as the forcing, characterized by the presence of the phytoplankton population, to which all solutions approach asymptotically. Furthermore, this result holds for a general class of models with monotone growth rate and monotone uptake rate, the latter possibly depending on the cell quota. Received 10 October 1995; received in revised form 26 March 1996     

Competitive exclusion in a vector-host model for the dengue fever

 We study a system of differential equations that models the population dynamics of an SIR vector transmitted disease with two pathogen strains. This model arose from our study of the population dynamics of dengue fever. The dengue virus presents four serotypes each induces host immunity but only certain degree of cross-immunity to heterologous serotypes. Our model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. Dengue is in the Americas an epidemic disease and our model reproduces this kind of dynamics. We consider two viral strains and temporary cross-immunity. Our analysis shows the existence of an unstable endemic state (‘saddle’ point) that produces a long transient behavior where both dengue serotypes cocirculate. Conditions for asymptotic stability of equilibria are discussed supported by numerical simulations. We argue that the existence of competitive exclusion in this system is product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates. Received 4 December 1995; received in revised form 5 March 1996     

A non-local model for a swarm

 This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles. Received: 17 September 1997 / Revised version: 17 March 1998     

A mathematical model to describe the dynamic response of a spruce tree to the wind

 A mathematical, computer-based, dynamic sway model of a Sitka spruce (Picea sitchensis) tree was developed and tested against measurements of the movement of a tree within a forest. The model tree was divided into segments each with a stiffness, mass and damping parameter. Equations were formulated to describe the response of every segment which together form a system of coupled differential equations. These were solved with the aid of matrices and from the resulting modes, the transfer function of the tree was found and used to calculate the movement of the tree in the wind. Comparison of the modelled movement of a tree in response to the measured wind speed above a forest canopy gave good agreement with the measured movement of the top of the tree but less satisfactory agreement close to the base. The comparison also pointed to the complexity of tree response to the wind and inadequacies in the model. In particular, the branches need to be treated as coupled cantilevers attached to the stem rather than simply as masses lumped together. Received: 18 February 1997 / Accepted: 16 December 1997     

A weakly nonlinear analysis of a model of avascular solid tumour growth

 In this paper we study a mathematical model that describes the growth of an avascular solid tumour. Our analysis concentrates on the stability of steady, radially-symmetric model solutions with respect to perturbations taken from the class of spherical harmonics. Using weakly nonlinear analysis, previous results are extended to show how the amplitudes of the asymmetric modes interact. Attention focuses on a special case for which the model equations simplify. Analysis of the simplified model equations leads to the identification of a two-parameter family of asymmetric steady solutions, the dimensions of whose stable and unstable manifolds depend on the system parameters. The asymmetric steady solutions limit the basin of attraction of the radially-symmetric steady state when it is linearly stable. On the basis of these numerical and analytical results we postulate the existence of fully nonlinear steady solutions which are stable with respect to time-dependent perturbations. Received: 25 October 1998 / Revised version: 20 June 1998     

Analysis of a model for macroparasitic infection with variable aggregation and clumped infections

 A model for macroparasitic infection with variable aggregation is considered. The starting point is an immigration-and-death process for parasites within a host, as in [3]; it is assumed however that infections will normally occur with several larvae at the same time. Starting from here, a four-dimensional, where free-living larvae are explicitly considered, and a three-dimensional model are obtained with same methods used in [26]. The equilibria of these models are found, their stability is discussed, as well as some qualitative features. It has been found that the assumption of “clumped” infections may have dramatic effects on the aggregation exhibited by these models. Infections with several larvae at the same time also increases the stability of the endemic equilibria of these models, and makes the occurrence of subcritical bifurcations (and consequently multiple equilibria) slightly more likely. The results of the low-dimensional model have also been compared to numerical simulations of the infinite system that describes the immigration-and-death process. It appears that the results of the systems are, by and large, in close correspondence, except for a parameter region where the four-dimensional model exhibits unusual properties, such as the occurrence of multiple disease-free equilibria, that do not appear to be shared by the infinite system. Received 28 October 1996; in revised form 11 April 1997     

A first model for the oxidized active form of the active site in galactose oxidase: a free-radical copper complex

 Copper(II) complexes derived from the tripodal ligand bis(3′-t–butyl-2′-hydroxybenzyl)(2-pyridylmethyl)amine (LH₂) have been studied in order to mimic the redox active site of the free radical-containing copper metalloenzyme galactose oxidase. In non-coordinating solvents such as dichloromethane, only an EPR-silent dimeric complex was obtained (L₂Cu₂). The crystal structure of L₂Cu₂ revealed a "butterfly" design of the [Cu(μOR)₂Cu] unit, which is not flattened and leads to a short Cu–Cu distance, the t–butyl groups being localized on the same side of the [Cu(μOR)₂Cu] unit. The dimeric structure was broken down by acetonitrile or by alcohols, leading quantitatively to a brown mononuclear copper(II) complex. UV-visible and EPR data indicated the coordination of the solvent in these mononuclear complexes. Electrochemical as well as chemical (silver acetate) one-electron oxidation of acetonitrile solutions of the monomeric complex led to a yellow-green solution. Based on EPR, UV-visible and resonance Raman spectroscopy, the one-electron oxidation product was identified as a cupric phenoxyl radical system. It slowly decomposes into a product where the ligand has been substituted (dimerization) in the para position of the hydroxyl group, for one of the phenolic groups. The data for the one-electron oxidized species provides strong evidence for a free-radical copper (II) complex. Received: 19 July 1996 / Accepted: 16 October 1996     

Multiparametric bifurcations for a model in epidemiology

 In the present paper we make a bifurcation analysis of an SIRS epidemiological model depending on all parameters. In particular we are interested in codimension-2 bifurcations. Received 8 April 1994; received in revised form 29 June 1995     

Identifiability of a stochastic model for cell debris in flow cytometry

 One of the most important problems in recovering DNA distribution from flow cytometric DNA measurements is the presence of background noise. In this paper, we analyse a probabilistic model recently proposed for background debris distribution and based on a specific probabilistic mechanism for the DNA fragmentation process of the cell nucleus. In particular, we carry out some sufficient conditions to uniquely identify the original DNA distribution from the flow cytometric data. Received: 15 June 1997 / Revised version: 18 November 1997     

Travelling waves and pattern formation in a model for fungal development

 Under a variety of conditions, the hyphal density within the expanding outer edge of growing fungal mycelia can be spatially heterogeneous or nearly uniform. We conduct an analysis of a system of reaction-diffusion equations used to model the growth of fungal mycelia and the subsequent development of macroscopic patterns produced by differing hyphal and hence biomass densities. Both local and global results are obtained using analytical and numerical techniques. The emphasis is on qualitative results, including the effects of changes in parameter values on the structure of the solution set. Received 22 November 1995; received in revised form 17 May 1996     

Parabolic bursting revisited

 Many excitable membrane systems display bursting oscillations, in which the membrane potential switches periodically between an active phase of rapid spiking and a silent phase of slow, quasi steady-state behavior. A burster is called parabolic when the spike frequency is lower both at the beginning and end of the active phase. We show that classes of voltage-gated conductance equations can be reduced to the mathematical mechanism previously analyzed by Ermentrout and Kopell in [7]. The reduction uses a series of coordinate changes and shows that the mechanism in [7] applies more generally than previously believed. The key hypothesis for the more general theory is that a certain slow periodic orbit must stay close to a curve of degenerate homoclinic points for the fast system, at least during the active phase. We do not require that the slow system have a periodic orbit when the voltage is held constant. Received 28 March 1995; received in revised form 20 October 1995     

Aggregation and collapse in a mechanical model of fungal tip growth

 Interconnected hyphal tubes form the mycelia of a fungal colony. The growth of the colony results from the elongation and branching of these single hyphae. The material being incorporated into the extending hyphal wall is supplied by vesicles which are formed further back in the hyphal tip. Such wall-destined vesicles appear conspicuously concentrated in the interior of the hypha, just before the hyphal apex, in the form of an apical body or Spitzenk?rper. The cytoskeleton of the hyphal tube has been implicated in the organisation of the Spitzenk?rper and the transport of vesicles, but as yet there is no postulated mechanism for this. We propose a mechanism by which forces generated by the cytoskeleton are responsible for biasing the movement of vesicles. A mathematical model is derived where the cytoskeleton is described as a viscoelastic fluid. Viscoelastic forces are coupled to the conservation equation governing the vesicle dynamics, by weighting the diffusion of vesicles via the strain tensor. The model displays collapse and aggregation patterns in one and two dimensions. These are interpreted in terms of the formation of the Spitzenk?rper and the initiation of apical branching. Received: 16 September 1996 / Revised version: 20 July 1998     

Homoclinic chaos in a model of natural selection

 We modify a simple mathematical model for natural selection originally formulated by Robert M. May in 1983 by permitting one homozygote to have a larger selective advantage when rare than the other, and show that the new model exhibits dynamical chaos. We determine an open region of parameter space associated with homoclinic points, and prove that there are infinite sequences of period-doubling bifurcations along selected paths through parameter space. We also discuss the possibility of chaos arising from imbalance in the homozygote fitnesses in more realistic biological situations, beyond the constraints of the model. Received 3 February 1995; received in revised form 1 November 1995     

Analysis of a mathematical model for the growth of tumors under the action of external inhibitors

 In this paper we make rigorous analysis to a mathematical model for the growth of nonnecrotic tumors under the action of external inhibitors. By external inhibitor we mean an inhibitor that is either developed from the immune system of the body or administered by medical treatment to distinguish with that secreted by tumor itself. The model modifies a similar model proposed by H. M. Byrne and M. A. J. Chaplain. After simply establishing the well-posedness of the model, we discuss the asymptotic behavior of its solutions by rigorous analysis. The result shows that an evolutionary tumor will finally disappear, or converge to a stationary state (dormant state), or expand unboundedly, depending on which of the four disjoint regions Δ ₁ , ..., Δ ₄ the parameter vector (A ₁ ,A ₂ ) belongs to, how large the scaled apoptosis number ˜σ is, and how large the initial radius R ₀ of the tumor is. Finally, we discuss some biological implications of the result, which reveals how a tumor varies when inhibitor supply is increased and nutrient supply is reduced. Received: 6 June 2000 / Revised version: 7 November 2001 / Published online: 8 May 2002     

Parameter domains for instability of uniform states in systems with many delays

 We present a computational method for determining regions in parameter space corresponding to linear instability of a spatially uniform steady state solution of any system of two coupled reaction-diffusion equations containing up to four delay terms. At each point in parameter space the required stability properties of the linearised system are found using mainly the Principle of the Argument. The method is first developed for perturbations of a particular wavenumber, and then generalised to allow arbitrary perturbations. Each delay term in the system may be of either a fixed or a distributed type, and spatio-temporal delays are also allowed. Received 19 September 1995; received in revised form 4 September 1996     

Convergence in almost periodic Fisher and Kolmogorov models

 We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed. Received: 11 November 1996 / Revised version: 8 January 1998