An approximation technique is developed for studying the deterministic spread of a simple epidemic which avoids a difficulty inherent in the diffusion approximation. Travelling wave solutions are found and calculated for various contact distributions. The solution developing from an isolated infective is also briefly discussed.

The susceptible→exposed→infectious→susceptible (SEIS) model is well known in mathematical epidemiology as a model of infection in which there is a latent period between the moment of infection and the onset of infectiousness. The compartment model is well studied, but the corresponding particle system has so far received no attention. For the particle system model in one spatial dimension, we give upper and lower bounds on the critical values, prove convergence of critical values in the limit of small and large latent time, and identify a limiting process to which the SEIS model converges in the limit of large latent time.

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

Items arrive at a processing plant at a Poisson rate λ. At time T, all items are dispatched from the system. An intermediate dispatch time is to be chosen to minimize the total wait of all items. It is shown that if the dispatch time must be chosen at time 0 then T/2 not only minimizes the expected total wait but it also maximizes the probability that the total wait is less than a for every a > 0. If the intermediate dispatch time is allowed to be a (random) stopping time, then it is shown that the policy which dispatches at time t iff N(t) > λ(T – t) is optimal, where N(t) denotes the number of items present at time t. The distribution of the optimal dispatch time and the optimal expected total wait are determined. A generalization to the case of a non-homogeneous Poisson process, a time lag, and batch arrivals is given. Finally, the case where the process goes on indefinitely and any number of dispatches are allowed (at a cost K per dispatch) is considered.

In a class of population-size-dependent Galton-Watson processes where extinction does not occur with probability 1 we describe the rate of decay of qi (the probability that the process starting from i ancestors will become extinct) as the number i of ancestors increases.

The results are related to the asymptotic behavior of the Green's function of the critical Galton-Watson process with immigration.

A dam model is proposed for which it is assumed that the conditional distributions of the inputs, given the past, are known only to lie in some class M. For selected M, bounds are derived on various quantities of interest such as the mean time to first emptiness. The case of normal inputs is treated in greater detail and a release rule is discussed. The techniques used are similar to those used in the theory of gambling as developed by Dubins and Savage (1965).

For positive recurrent nearest-neighbour, semi-homogeneous random walks on the lattice {0, 1, 2, …} X {0, 1, 2, …} the bivariate generating function of the stationary distribution is analysed for the case where one-step transitions to the north, north-east and east at interior points of the state space all have zero probability. It is shown that this generating function can be represented by meromorphic functions. The construction of this representation is exposed for a variety of one-step transition vectors at the boundary points of the state space.

The optimal strategy for a class of non-Markov decision processes is characterised and has the property that changes of action may occur between successive transitions of the process. Results are given which enable the optimal strategy to be computed iteratively.

Stochastic models for the origin and extinction of species have been rather neglected in applied probability. As an alternative to modelling speciation and extinction as intrinsically random, I shall describe and show simulations of a rulebased model. This involves mathematical representations of notions such as genetic type of species, environmental niche, fitness of a species in a niche, and adaptation. There are underlying random mechanisms for changes of niche sizes and for disconnection and reconnection of geographical regions, and these ultimately drive the evolution of species.

Other approaches to mathematical modelling of evolution are briefly mentioned.

It is known that for a GI/G/1 queue, if the interarrival time has finite first moment, then the (k + 1)th moment of the service time is finite if the kth moment of the stationary workload is finite. This result is extended to certain tandem and multiple-server queues.

In this paper exchangeable bisexual models with fixed population size and non-overlapping generations are introduced. Each generation consists of N pairs of individuals. The pairs of a generation have altogether 2N children. These individuals form randomly the N pairs of the next generation. The extinction probability of the descendants of a fixed number of pairs of generation 0 is discussed. Under suitable conditions it can be approximately described by the extinction probability of a Galton–Watson process, if the population size is large. Special examples are a bisexual Wright–Fisher model and models with a uniformly bounded number of children of a pair.

We establish heavy-traffic limits for nearly deterministic queues, such as the G/D/n many-server queue. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn /D/1 model, where the Gn interarrival times are the sum of n consecutive interarrival times in the original model, we focus on the Gn /D/1 model and the generalization to Gn /Gn /1, where ‘cyclic thinning’ is applied to both the arrival and service processes. We establish different limits in two cases: (i) when (1 − ρn )√n → β as n → ∞ and (ii) when (1 − ρn )n → β as n → ∞, where ρn is the traffic intensity in model n. The nearly deterministic feature leads to interesting nonstandard scaling.

The cars are considered as points on an infinite road with no intersections. They can overtake each other without any delay and they travel at constant speeds. These are independent identically distributed random variables also independent of the initial positions of the cars. The main purpose of the paper is the study of the asymptotic distribution for the number of overtakings (and/or meetings) in increasing rectangles in the time-road plane. Under the assumption of (weighted) Poisson distributed cars along the time-axis we deduce the asymptotic distribution of the standardized number of overtakings (and/or meetings) for large rectangles in the time-road plane. Lastly we shall indicate an application of the results.

The theory of majorization is applied to investigate the effects of heterogeneity in a class of epidemics. In particular, the heterogeneity of the components of the contact vector of a given susceptible is studied. It is shown that if the heterogeneity can be partially ordered through majorization, then the probability functions of escaping infection can be similarly ordered. The results can be applied to AIDS research on the spread of the HIV virus.

Dr. J. Alan Steppe has kindly pointed out two serious flaws, and some others less serious, in the proof of our paper [2]. However these can all be corrected, so that the result remains valid.

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.

Let X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

In this paper we find conditions under which the epoch times of two nonhomogeneous Poisson processes are ordered in the multivariate dispersive order. Some consequences and examples of this result are given. These results extend a recent result of Brown and Shanthikumar (1998).

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.