Showing papers in "Mathematical Medicine and Biology-a Journal of The Ima in 1984"

An age-structured model of pre- and post-vaccination measles transmission.

Abstract: An infection like measles does not spread uniformly in populations from Europe and North America. Of special importance is a pronounced age-dependency in the contact rates, because of increased infection transmission within schools. Therefore an age-structured epidemiologic model is investigated here, which also pays attention to the fact that children are promoted grade-wise into and out of school. Simulation results are contrasted with pre- and post-vaccination measles data from England and Wales and the model is shown to perform better than previous global mass-action models.

Generation of Biological Pattern and Form

Abstract: We propose two models for pattern formation in early embryogenesis. The first deals with patterns in motile mesenchymal cells; the second treats patterns in epithelial sheets. In the mesenchymal model, cells exert tractions which deform the extracellular matrix within which they move. This in turn affects their motion. The model field equations are formulated and analysed, and applied to two widely studied phenomena: skin-organ primordia for feather and scale patterns, and the development of cartilage patterns in limb bone formation. The model for epithelial pattern formation consists of viscoelastic field equations with a calcium-controlled contraction trigger. Preliminary analysis is presented which demonstrates the existence of travelling wave solutions.

Parasitism in Patchy Environments: Inverse Density Dependence can be Stabilizing

Abstract: There are now many examples in the literature where the spatial distribution of per cent parasitism by insect parasitoids is either directly or inversely dependent on host density per patch. While it is well known that direct density dependent relationships can contribute markedly to the stability of a host-parasitoid interaction, inverse relationships have been more-or-less ignored. Using difference equation models, the dynamics of host-parasitoid interactions are described where parasitism per patch varies across the range from direct to inversely density dependent. These models demonstrate for a variety of host distributions that inverse relationships can also strongly promote stability.

Information Gain in Joint Linkage Analysis

Abstract: The advent of DNA sequencing and the prospect of the availability of very large numbers of marker loci have reawakened interest in linkage studies. Statistical analyses of linked genetic loci have previously been based on pairwise analysis of the loci, and/or have assumed a known ordering of them along the chromosome. For the sizes of sample available in human medical genetics, pairwise analysis may be a statistically inefficient and inconclusive procedure. It is shown that three-way data provide very much more information, particularly with regard to the problem of ordering the loci. Correct answers to questions of locus order are of particular importance in current approaches to fetal diagnosis, where counselling is based on genotypes at markers closely linked to a disease locus. Although the three-locus analysis of this paper is only a first step towards joint analysis of multilocus data, the qualitative form of the data is of the joint pattern of cosegregation of loci. The pairwise counts of recombinants and nonrecombinants used for pairwise analysis are both qualitatively and quantitatively less informative. Many practical and theoretical problems of multilocus analysis of data remain to be solved, but the practical importance of doing so is demonstrated.

Matrix-Geometric Methods for the General Stochastic Epidemic

Abstract: This paper outlines a matrix-geometric formulation of the general stochastic epidemic for the case of a generalized infection mechanism. The forward Kolmogorov equations of the system are derived, and the Laplace transforms of the state probabilities obtained recursively. These lead to the probabilities of survivors of the epidemic. The stochastic threshold theorem for the generalized case is stated.

Nonrandom Sampling in Human Genetics: Familial Correlations

Abstract: In the context of human genetics, sampling is often nonrandom in that pedigrees are frequently selected by virtue of their having at least one affected individual. For quantitative traits, it may be that the proband is selected because of a phenotypic value above some predetermined cut-point; this will also be true for diseases defined by a cut-point above which individuals are said to be affected (e.g. diabetes, hypertension, and obesity). Relatively little attention has been given to the implications of these forms of nonrandom sampling. In this paper, we show that the biases introduced by such sampling are sufficient to lead to erroneous model specification and biased parameter estimates in path analysis. Existing methodologies used to correct for such sampling (e.g., elimination of proband or regression techniques) also result in similar bias in estimation and model fitting. Thus, it is possible to make inferences which do not reflect underlying biology, but are artifacts of the sampling design. As a consequence, method-of-moment estimators are developed for means, variances, and correlations under such sampling. Simulation results are used to demonstrate the superiority of this method over alternate strategies.

Behaviour of Some Models of Myelinated Axons

Abstract: In this paper we discuss several approaches to modelling myelinated axons and examine the qualitative behaviour of the models. To facilitate our goal of understanding in detail the differences in mechanisms modelling myelination, we impose at the nodes the simplest nonlinear current-voltage relation which allows the models to possess appropriate threshold behaviour and propagating action potentials (travelling waves). Our type of model is a nonlinear differential-difference system and the resulting travelling wave must satisfy a nonlinear delay-differential equation of mixed type. Another type of model is a diffusion equation coupled nonlinearly to ordinary differential systems whose solutions represent boundary data for the diffusion equation. We give some threshold results and derive a relationship between conduction speed and various model parameters for a few classes of these models.

Evolution of an altruistic trait through group selection as studied by the diffusion equation method.

Abstract: A diffusion model is formulated which incorporates the process of group selection (i.e. interdeme competition) in addition to mutation, migration, individual selection, and random genetic drift. A condition is obtained for group selection to prevail over individual selection in the evolution of an altruistic trait. Let DK = c/(v + v' + m) - 4Nes', where v' and v are mutation rates to and from the altruistic allele (A'), m is the migration rate (assuming Wright's island model), s' is the selective disadvantage of A' with respect to individual selection, Ne is the effective population size of each deme (group) and c is a positive constant such that a deme having A' with frequency x has the advantage c(x - means) relative to the average deme. Then, group selection overrides individual selection if DK greater than 0, while individual selection prevails if DK less than 0.

The master equation for neural interaction.

Abstract: Based on neural interaction equations a random walk model for the stochastic dynamics of a single neuron is introduced. In this model the somatic potential corresponds to a state in the state space and action potentials provide the mechanism causing transitions. Time is made discrete, consisting of small finite increments delta t; assumptions are made about the transitions within such an increment and the associated probabilities are formulated. These quantities depend on delta t and on parameters derived from neural interaction equations. Moreover the model is chosen so that the sequence of somatic potentials is a Markov chain. By appropriately scaling the parameters, in the limit as delta t----0, a master equation for the probability in continuous time is obtained. Depending on the parameters, the master equation describes the evolution of a deterministic, a diffusion, or a discrete process. An interpretation for the diffusion and discrete processes is outlined. The conclusion is that the stochastic equations for neural interaction lead to a master equation representing a diffusion or a discrete process depending on the number, size of synaptic connectivity coefficients, and probability distribution of neural activity. An example is included describing how a master equation may be used to derive properties of the single neuron's output process.

Qualitative Analysis of Reaction-Diffusion Systems Modelling Coupled Unmyelinated Nerve Axons

Abstract: A reaction-diffusion system based on FitzHugh-Nagumo dynamics is used to model the interactions between several neighbouring unmyelinated nerve fibres. A generalization of the idea of contracting blocks is used to establish global existence and stability results.

Some new directions for research in epidemic models.

Abstract: Problems requiring further development of epidemic theory are discussed. One important area is the association between progressive disease and horizontally-transmitted agents, as in, e.g., the current epidemic of acquired immunodeficiency syndrome (AIDS) which has been linked to spread of a virus. Models of disease transmission must be combined with models of disease progression. More work is also needed to explain why the annual incidence of some diseases fluctuates dramatically, often in regular cycles, rather than stabilizing at a predictable level.

A review of semi-parametric models for life histories.

Abstract: The development of various generalizations of Cox's proportional hazards regression model to the study of more complicated life histories is reviewed. Cox's paper provided a framework for the regression modelling of single failure (deaths) in terms of a proportional hazards assumption and subsequent work has extended this approach to the cases of multiple (non-catastrophic) failures, multi-state life histories, non-independent failure times (family studies and matched pairs), and more complicated censoring mechanisms.

Comparison principles in the analysis of reaction-diffusion systems modelling unmyelinated nerve fibres.

Abstract: Comparison principles for systems of reaction-diffusion equations coupled via both the reaction and diffusion terms are considered. The techniques are motivated by models of unmyelinated nerve fibres based on FitzHugh-Nagumo dynamics. Applications to threshold phenomena and ephaptic transmission are included.

A Mathematical Overview of a Computer Simulation Model of Maternity Histories with Illustrative Examples

Abstract: A mathematical overview of a stochastic computer simulation model of maternity histories is provided. Various components of human reproduction are accommodated in the model through distributions of waiting times among live births. Included in these components are distributions of age at first marriage in a cohort of women, waiting times to pregnancy for fecundable women, and the lengths of infecundable periods following live births. Probabilities that pregnancies end in either a live birth, induced abortion, or some other type of outcome are also included. Elements of renewal theory and semi-Markov processes in discrete time were the basic mathematical concepts used in the construction of the model. A brief description of an interactive software package called MATHIST, which may be used to implement the model on a computer, is also included. Four illustrative computer runs with MATHIST, pertinent to the operation of family planning programmes in Africa, are also described and discussed.

A Mathematical Model of Exoprotein Production in Bacteria

Abstract: We present a simple mathematical model for the synthesis of extracellular proteins by a class of bacteria which secrete significant quantities of this exoprotein in late-exponential and stationary phases. This model is the simplest generalization of Michaelis-Men ten kinetics (the Monod model) and agrees well with laboratory experiments in batch culture. The model may serve as a simple prototype for the analysis of certain virulent bacterial infections in vivo, particularly that of Pseudomonas aeruginosa in burn wounds.

A mathematical model for a thermal clearance probe.

Abstract: We introduce a new mathematical model for a thermal clearance probe for the measurement of skin blood flow. It is concluded that the technique is more sensitive to differences in the thermal conductivity of the skin than to differences in blood flow. The depth of measurement is also considered.

Some Population and Epidemic Models Revisited

Abstract: Three problems of population and epidemic models formulated between ten and thirty years ago are reconsidered. In each case, a modified approach to the problem leads to its solution. For the two-sex population model, the solution of a Riccati equation results in an expression for the generating function of the process. The fully stochastic, as against the previously studied semistochastic, model of population growth with random catastrophes yields to hard analysis. Finally a generalized form of the general stochastic epidemic is solved using matrix geometric methods.