This paper considers the appropriateness of certain stochastic models for the representation of disease etiologies. Models for disease as the result of multiple events, and as the endpoint of a network of events, are discussed.

We consider the problem of number and weight distributions for breakage-mechanism branching processes whose break distributions are general. We derive a recursive relation between the expected empirical distribution after (n + 1) breaks and after n breaks, making use of length-biased sampling. Using this relation and the strong law of large numbers, we derive integro-differential equations for the asymptotic expected empirical distribution and its associated weight distribution. The mean of the asymptotic number distribution is derived using the integro-differential equation. We then provide approximate solutions to these equations and the moments of these approximations. Finally we apply these results to the case where the break distribution is a symmetric beta distribution.

A generalization of Bailey's general epidemic model is considered. In this generalized model, it is assumed that the probability of any particular susceptible becoming infected during the small time interval (t, t + Δt) is α(X(t))Δt + o(Δt), for some function a, where X(t) is the proportion of infected individuals in the entire population, the probability that an infected individual is infected for at least a length of time t is F(t), and recovered individuals are permanently immune from further attack. In this paper, central limit theorems are obtained for the proportion of infected individuals and the proportion of susceptibles in the entire population.

A random graph is a collection of n points and n directed arcs: a directed arc goes equiprobably from each point to one of (n – 1) other points. m points are initially ‘infected'. We consider several schemes of epidemic process, e.g. when the infection is delivered according to arc direction. We prove that the probability of infecting all the n points with m = 1 is ∼ e/n, when n → ∞; another result is that m = o(√ n) cannot infect an essential part of the graph (having the size of O(n)). Possible applications of the models to real world phenomena are briefly discussed.

Consider a diploid population of N individuals undergoing random mating and mutation as in the infinite-allele Wright model. Choose a particular gene in generation Q for large Q, and let ft (0 ≦ t ≦ Q) be the frequency of the allelic type of the predecessor of that particular gene (or the gene itself) in generation t. By considering a ‘diffusion approximation' {xt} of {ft}, we find the distribution of the age of an allelic type now known to have frequency p, and of its distribution of frequencies since the allele came into existence. A novelty here is that the process {xt} is not a diffusion, but a process with jumps; it has x = 0 as an inaccessible entrance boundary but periodically jumps to it from the interior of [0, 1]. The formulas obtained are the same as those derived by Maruyama and Kimura, who used a totally different approach.

A general epidemic process is said to be completed whenever the number of susceptibles or the number of infectives reduces to zero. The mean duration time for this process to be completed is derived.

A classical theorem on the convergence to a diffusion of a sequence of branching processes will be proved in the case where the branching processes are modified to allow dependence on population size of the generating function governing reproduction. Some recent semigroup convergence theorems due to Kurtz (1969), (1975) provide the main tools.

The linear birth-and-death process is elaborated to allow the elements of the process to live as members of linear clusters which have the possibility of breaking up. For the supercritical case, expressions, based on an approximation, are derived for the mean numbers of clusters of the various sizes as time → ∞. These expressions compare very well with exact solutions obtained by the method of Runge-Kutta. Exact solutions for the mean values for all time are given for when the death rate is zero.

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

A result like the Kesten-Stigum theorem is obtained for certain martingales associated with the branching random walk. A special case, when a ‘Malthusian parameter’ exists, is considered in greater detail and a link with some known results about the Crump-Mode model for a population is established.

It is known that, in the deterministic model of a one-dimensional spatial linear birth process, a finite initial distribution of individuals develops into an advancing wave with the minimum velocity possible for such waves. In this paper, the spatial covariance density of the process is studied in the vicinity of a point moving outwards with a given velocity. Its behaviour casts doubt on the applicability of the deterministic result to individual realisations of the process.

The characteristic values of certain matrices (underlying multiple allele models in population genetics) are investigated and related to the asymptotic behavior of Pr(at least k distinct alleles in the population at time t).

The sign-changes over the entire real line of a Gaussian process with unit variance and mean zero are observed. The Gaussian process is reconstructed by an interpolator which is optimal in the class of linear interpolators based on that process which is one when the original process is positive and minus one when it is negative. Sufficient conditions for the interpolator to be a convolution of the sign process and a square integrable function are given. Analytical expressions of the interpolation error are derived, and the behaviour of the interpolator is studied by means of simulations.

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

The problem of determining the asymptotic form of the stock vector n(t) in a continuous time Markovian manpower model is solved for asymptotically exponential recruitment functions {R(t)}. A new approach to the limiting behaviour of some manpower systems with given total sizes {N(t)} is then given by means of time-inhomogeneous Markov processes.

A two-dimensional Poisson cluster point process is formulated by the use of a probability generating functional. Moment measures of both the cluster centre and member processes are discussed. An example is provided, and the magnitude frequency law is proved in this case.

This paper compares estimators of u = |P2 — P1| based on partial knowledge about points P1, P2 which are chosen independently from the same probability distribution in the plane. Typical estimators are the conditional expectations E(u | r1, r2), E(u | s), E(u | rM), E(u | θ) where ri is the distance from Pi to the origin 0, s = r1 + r2, rM = Max{r1, r2}, θ = ∠ P10P2. These expectations are the least squares estimators, given the conditioned variables. Simpler linear estimators are also considered.

A rigorous treatment is given for a construction via Markov chains of a binary (0–1) stationary homogeneous Markov random field on Z × Z. The resulting process possesses rather interesting properties. For example, its correlation structure is geometric and it may be easily simulated. Some of the results are rather unintuitive — indeed counter-intuitive — but their demonstration is straightforward involving only the most elementary properties of Markov chains.

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

If events occur in time according to a stochastic process then, under not very restrictive conditions, each realization will appear to come from a Poisson process with its own rate provided that the events in the realization occur at effectively random times. This result is related to de Finetti's theorem on exchangeable events. Particular applications are to the Pólya process describing accidents and the pure birth process.