A model for the spread of an epidemic

A temporally continuous and spatially discrete stochastic model for the spread of an epidemic within some set of holdings is constructed. A recursion formula is given for the probability that a certain set of holdings is infected at a certain moment. Moreover, under an additional condition (which will always be satisfied in practice) a formula for the expected value and the variance of the moment when a certain holding is infected the first time is given.     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

Nets with distance bias

A distance bias is imposed on the probability of direct connection between every pair of points in a random net. The probability that there exists a path from a given point in the net to another point is now a function of both the axone density and the distance between the points. A recursion formula is derived in terms of which this probability can be computed. The rate of spread of an epidemic where probability of contact depends on the distance between the individuals can also be computed from the recursion formula.     

The organization of time in biological systems

A notion of organization of time similar to the notion of organization of space in architecture has been introduced. The level and pattern of organization of time in biological systems differs from that in physical and chemical systems, which presents an independent problem. Analysis of the problem leads to a new definition of life as a process of renormalization of possibilities described by a Bayes formula. This definition leads to the notion of self-monitoring as a property of every biological system, and of complicated structure of the biological present, including the physical past and physical future. This is naturally followed by determination by far past, and, hence, memory, and determination by future, i.e. preadaptation, surpassing reflection, aim-setting etc. A direct dependence of a number of elements of a complex system on its stability has been demonstrated. The self-monitoring and organization of time can be traced at various levels of biological hierarchy from intracellular to biosphere level.     

An Exact Paired Rank Test

An exact rank test for two dependent samples based on overall mid‐ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data.     

Ancestral covariance and the Bulmer effect

Summary An n-generation pedigree is set up and selection is carried out generation by generation. The influence of this procedure on the covariances in subsequent generations is assessed and, ultimately, Bulmer's general recursion formula for the reduction in genetic variance due to selection is obtained. The results are extended to assess the effect of selection of one or more characters on the genetic covariance matrix of a number of characters. The concept of ancestral regression is also used to provide a different insight into the selection process and to justify some models used in the analysis of assortative mating.     

Dynamics of granzyme B-induced apoptosis: mathematical modeling

Apoptosis is mediated by an intracellular biochemical system that mainly includes proteins (procaspases, caspases, inhibitors, Bcl-2 protein family as well as substances released from mitochondrial intermembrane space). The dynamics of caspase activation and target cleavage in apoptosis induced by granzyme B in a single K562 cell was studied using a mathematical model of the dynamics of granzyme B-induced apoptosis developed in this work. Also the first application of optimization approach to determination of unknown kinetic constants of biochemical apoptotic reactions was presented. The optimization approach involves solving of two problems: direct and inverse. Solving the direct optimization problem, we obtain the initial (baseline) concentrations of procaspases for known kinetic constants through conditional minimization of a cost function based on the principle of minimum protein consumption by the apoptosis system. The inverse optimization problem is aimed at determination of unknown kinetic constants of apoptotic biochemical reactions proceeding from the condition that the optimal concentrations of procaspases resulting from the solution of the direct optimization problem coincide with the observed ones, that is, those determined by biochemical methods. The Multidimensional Index Method was used to perform numerical solution of the inverse optimization problem.     

Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind

In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.     

The graduation of secondary structure elements

A method of graduating (i.e., least-squares fitting) a smooth polynomial curve through long elements of protein secondary structure is described. It uses the Chebyshev polynomials of a discrete (integer) variable with several restraints to prevent artifactual curvatures. A new recursion formula is given which allows the evaluation of the polynomials on rational-number points as well as on the integer points. High-order splines suitable for interpolation between integer points are also discussed. The new method finds applications in graphics and in structural analysis.     

Fitting Circles and Spheres to Coordinate Measuring Machine Data

This work addresses the problem of enclosing given data points between two concentric circles (spheres) of minimum distance whose associated annulus measures the out-of-roundness (OOR) tolerance. The problem arises in analyzing coordinate measuring machine (CMM) data taken against circular (spherical) features of manufactured parts. It also can be interpreted as the “geometric” Chebychev problem of fitting a circle (sphere) to data so as to minimize the maximum distance deviation. A related formulation, the “algebraic” Chebychev formula, determines the equation of a circle (sphere) to minimize the maximum violation of the equation by the data points. In this paper, we describe a linear-programming approach for the algebraic Chebychev formula that determines reference circles (spheres) and related annuluses whose widths are very close to the widths of the true geometric Chebychev annuluses. We also compare the algebraic Chebychev formula against the popular algebraic least-squares solutions for various data sets. In most of these examples, the algebraic and geometric Chebychev solutions coincide, which appears to be the case for most real applications. Such solutions yield concentric circles whose separation is less than that of the corresponding least-squares solution. It is suggested that the linear-programming approach be considered as an alternate solution method for determining OOR annuluses for CMM data sets.     

Analytical solution of the evolution dynamics on a multiplicative-fitness landscape

In an infinite population the frequency distribution of individuals carrying a given number of mutations obeys a set of recursion equations, the equilibrium solution of which describes the mutation-selection balance. Although this solution is well-known in the case of a multiplicative-fitness landscape, where it is assumed that all mutations are deleterious and that each new mutation reduces the fitness of the individual by the same fraction, we are not aware of the existence of an analytical solution for the full dynamics. Using the generating function approach, we present here an explicit analytical solution for the frequency distribution recursion equations valid for all generations and initial conditions.This research was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Proj. No. 99/09644-9. The work of J.F.F. was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).     

A modified method for determination of diffusivities of low molecular substances in non-Newtonian liquids

A modified method for determination of diffusivities of low molecular substances in non-Newtonian liquids described by the power-law model has been proposed. It is based on the dissolution of Geiss body, with a parameter m=1/3 rotating in an infinite fluid. In this case, the solution of the differential equations of motion and mass transfer is available as an analytical formula for calculating the diffusivity coefficient.The method allows the extension of the variety of media and diffusing species. It has been illustrated with dissolving of gypsum in water and five non-Newtonian liquids. The results obtained have been interpreted taking into account the interaction between calcium ions and polymer molecules of the non-Newtonian system, as well as the heterogeneity of the system near to the dissolving surface.     

A Note on Variance Estimation of the Aalen–Johansen Estimator of the Cumulative Incidence Function in Competing Risks,with a View towards Left‐Truncated Data

The Aalen–Johansen estimator is the standard nonparametric estimator of the cumulative incidence function in competing risks. Estimating its variance in small samples has attracted some interest recently, together with a critique of the usual martingale‐based estimators. We show that the preferred estimator equals a Greenwood‐type estimator that has been derived as a recursion formula using counting processes and martingales in a more general multistate framework. We also extend previous simulation studies on estimating the variance of the Aalen–Johansen estimator in small samples to left‐truncated observation schemes, which may conveniently be handled within the counting processes framework. This investigation is motivated by a real data example on spontaneous abortion in pregnancies exposed to coumarin derivatives, where both competing risks and left‐truncation have recently been shown to be crucial methodological issues (Meister and Schaefer (2008), Reproductive Toxicology 26 , 31–35). Multistate‐type software and data are available online to perform the analyses. The Greenwood‐type estimator is recommended for use in practice.     

Analysis of linear determinacy for spread in cooperative models

 The discrete-time recursion system $\u_{n+1}=Q[\u_n]$ with $\u_n(x)$ a vector of population distributions of species and $Q$ an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on $Q$ are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models. Received: 7 August 2000 / Revised version: 5 January 2002 / Published online: 17 July 2002     

Can abstractions be causes?

The Empiricist or Lockean view says natural kinds do not exist objectively in nature but are practical categories reflecting use of words. The Modern, Ostensive view says they do exist, and one can refer to such a kind by ostention and recursion, assuming his designation of it is related causally to the kind itself. However, this leads to a problem: Kinds are abstract repeatables, and it seems impossible that abstractions could have causal force. In defence of the Modern view, I suggest we can think of kinds as — or as like — ecological niches existing in nature, which are causally effective by virtue of the fact that they predictively determine (some) properties of the things that happen to occupy them.     

Gompertzian re-evaluation of the growth patterns of transplantable mammary tumours in sialoadenectomized mice

Abstract
The method of the recursion formula of the Gompertz function (Bassukas & Maurer-Schultze 1988) has been applied to analyse tumour growth data taken from the literature; namely the growth perturbation of transplantable mammary tumours in sialoadenectomized mice with or without subsequent epidermal growth factor substitution (results on two mouse strains, C3H or SHN, have been reported; Inui, Tsubura & Morii 1989). The recursion formula of the Gompertz function fits growth curves to all seven sets of data well ( P > 0.05 for lack of fit test). The growth pattern of the tumours in the unperturbed hosts is Gompertzian and does not change if tumours are transplanted in sialoadenectomized mice, although the starting specific growth rate decreases in C3H mice. However, if sialoadenectomy is carried out after tumour inoculation, a complex alteration of the tumour growth evolves: tumour growth does not simply decelerate but it also shifts from the conventional Gompertzian to an exponential or even 'hyperexpo-nential' growth pattern, i.e. with an accelerating specific growth rate. Some theoretical mechanisms of this alteration, as well as the differences between the present Gompertzian analysis and a previously published Verhulstian analysis of part of the same data (Leith, Harrigan & Michelson 1991), are discussed. It is concluded that the quantitative analysis of tumour growth patterns by the method of the difference equation of the Gompertz function presently applied may substantially contribute to the improvement of the interpretation of perturbations of tumour growth-irrespective of their genesis. In contrast to the application of some a priori fixed growth function, e.g. the Verhulstian one, the present method can quantitatively interprete different growth patterns and their classification on the basis of linear regression analysis.     

Counting coalescent histories.

Given a species tree and a gene tree, a valid coalescent history is a list of the branches of the species tree on which coalescences in the gene tree take place. I develop a recursion for the number of valid coalescent histories that exist for an arbitrary gene tree/species tree pair, when one gene lineage is studied per species. The result is obtained by defining a concept of m-extended coalescent histories, enumerating and counting these histories, and taking the special case of m = 1. As a sum over valid coalescent histories appears in a formula for the probability that a random gene tree evolving along the branches of a fixed species tree has a specified labeled topology, the enumeration of valid coalescent histories can considerably reduce the effort required for evaluating this formula.     

Firing rates of neurons with random excitation and inhibition

The expectation of the interspike interval for a Stein model neuron receiving Poisson excitation and inhibition is determined by solving a differential difference equation with both forward and backward differences. The method of solution relies on an asymptotic expansion at large initial hyperpolarizations. The asymptotic solution is continued to near threshold depolarization whereupon the boundary condition is employed along with recursion relations to obtain the complete solution. The dependency of the mean firing rate on excitation at fixed inhibition and on inhibition at fixed excitation is investigated as well as the threshold dependence at fixed input rates. The results are discussed in relation to those for intracellular current injection and synaptic input to real neurons.