A population of susceptible individuals partitioned into m groups and exposed to a contagious disease is considered.

The progress of this simple epidemic is modeled by an m -dimensional stochastic process, whose components are the number of infective individuals in the respective groups at time t. Exact and approximate formulas for the joint and marginal state probabilities are obtained.

We analyze a class of linear birth and death processes X(t) with killing. The generator is of the form λ i = bi + θ, µi = ai, γ i = ci, where γ i is the killing rate. Then P{killed in (t, t + h) | X(t) = i} = γ ih + o(h), h ↓ 0. A variety of explicit results are found, and an example from population genetics is given.

We define induced selection differential and discuss asymptotic distribution theory for this quantity. We also obtain the asymptotic joint distribution of the selection differential and the induced selection differential. These are used as measures of improvement in genetic selection programs. We consider the linear regression model set up in detail to obtain various possible limit laws for the induced selection differential.

The threshold behaviour of the stochastic spatial general epidemic model on a discrete location space is investigated by making use of the general percolation theory of McDiarmid.

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn (i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.

The general stochastic epidemic model is used as a model for the spread of rumours. Recursive expressions are found for the mean of the final size of each generation of hearers. Simple expressions are found for the generation size and the asymptotic form of its final size in the deterministic model. An approximating process is presented.

Population genetics and evolution have long supplied a stimulus to mathematical and statistical research. In this note a brief survey of current theory is given, with special reference to the work of P. A. P. Moran. Finally, an analysis of various biometrical questions is presented in the spirit of Moran's contributions to mathematical genetics.

The invariant probability distribution is found for a class of birth-and-death processes on the integers with phases and one or two boundaries. The invariant vector has a matrix geometric form and is found by solving a non-linear matrix equation and then finding an invariant probability distribution on the boundary states. Levy's concept of watching a Markov process in a subset is used to naturally decouple the computation of distributions on the boundary and interior states.

An unsolved problem in the design of a repetitive screening trial is the determination of a procedure which yields the optimal number of screens for efficient evaluation. An approach to this problem described in this paper is to use lead time properties of cases detected at screening to define a stopping rule. Previous results are generalized by allowing the intervals between screens to be arbitrary, and the probability of detecting preclinical disease to vary among screens.

In each of a large number N of independent cells a breakdown mechanism is under way and proceeds until the first of the cells actually fails. At such a time, in each cell, the situation reverts to some initial state and the mechanism restarts. In this paper we consider those mechanisms for which breakdown may be modelled as the explosion of a pure birth process. Of interest is the distribution of time between failures and the possibility of estimating N and/or model parameters by observing a sequence of failure times. Saddlepoint approximation methods are used in the relevant extreme-value theory analysis for two important cases.

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

Saddlepoint approximations to the probabilities for a general time-homogeneous birth process are derived from the known Laplace transform with respect to time. Numerical computations for the simple (logistic) epidemic show the approximation to have a remarkably low relative error over the useful range of times for populations as small as 5. The merit or otherwise of renormalising the approximate probabilities is discussed. To study the accuracy for moderately large population sizes, a special form of birth rate is used, for which the exact probabilities can be computed directly.

Gene survival in a population which increases without density dependence is considered using a generalization of the Moran model for haploid individuals in which we have differential reproductive rates. It is shown under very mild conditions that ultimate fixation is certain. This provides some mathematical support for the phenomenon of reduced genetic variability in isolated populations established from founder groups.

The kin number problem involves the relationship between sibship sizes and offspring numbers, and also numbers of relatives of other degrees of affinity of a random member of a population, to be called Ego. The problem has been well known to demographers for some time, but results obtained only gave expected numbers. Recently a study of it, based on the Galton-Watson process, was made, with a view to obtaining joint distributions (Waugh (1981)). In the latter study it was assumed that the population was large, and thus some of the results obtained were approximations.

In the present paper exact distributions are obtained, for any size of population. This can be of use in applications, where the population considered may be a small, isolated tribe or other special group. As a theoretical investigation, it replaces some heuristic arguments with limiting properties that are intrinsic to the process and it makes it possible to evaluate the previous approximations.

A new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain.

We give necessary and sufficient conditions for in terms of , where Sn is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.

Results connecting Yaglom's limit and Z1 are obtained. For example, if α > 1, E[Zα1] and converge or diverge together.

A Markov step process Z equipped with a possibly non-denumerable state space X can model a variety of queueing, communication and computer networks. The analysis of such networks can be facilitated if certain traffic flows consist of mutually independent Poisson processes with respective deterministic intensities λi (t). Accordingly, we define the multivariate counting process N = (N1, N2, · ·· Nc) induced by Z; a count in Ni occurs whenever Z jumps from x (χ into a (possibly empty) target set . We study N through the infinitesimal operator à of the augmented Markov process W = (Z, N), and the integral relationship connecting à with the transition operator Tt of W. It is then shown that Ni depends on a non-negative function ri defined on χ; ri (x) may be interpreted as the expected rate of increase in Ni, given that Z is in state x.

A multivariate N is Poisson (i.e., composed of mutually independent Poisson streams Ni) if and only if simultaneous jumps are impossible in a certain sense, and if the conditional expectation E[ri (Z(t) | 𝒩i] = E[ri (Z(t))] for i = 1, 2, ···, c and each t ≧ 0, where 𝒩i is the σ-algebra σ{N(s), s ≦ t}. Necessary and sufficient conditions are also specified that, for each s ≦ t, the variates [Ni(t)-Ni(s)] are mutually independent Poisson distributed; this involves a weakened version of E[ri(Z(v)) | N(v) – N(u)] = E [ri(Z(v))] for i = 1, 2, ···, c and all 0 ≦ u ≦ v.

It is shown that the above criteria are automatically met by the more stringent classical requirement that N(t) and Z(t) be independent for each t ≧ 0.

In this paper, we consider a Bienaymé– Galton–Watson process {Xn ; n ≧ 0; Xn = 1} and develop a recurrence formula for P(Xn = k), k = 1, 2, ···. The problem of obtaining the maximum likelihood estimate of the age of the process when p 0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p 0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.