On an approximation made when analysing stochastic processes

We investigate the effect of a particular mode of approximation by means of four examples of its use; in each case the model approximated is a Markov process with discrete states in continuous time.

Stepping stone models for population growth

A population is composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis. Each colony develops according to a simple birth and death process and migration is allowed between nearest neighbours. An approximate solution is obtained for the probability structure of the population size, and exact results are derived for the process when immigration is introduced into a single colony from outside the system.

A birth, death and migration process by immigration

A finite number of colonies, each subject to a simple birth-death and immigration process is studied under the condition of migration between the colonies.

Kolmogorov's backward equations for the process are solved for some special cases, and a sequence of functions uniformly converging to the p.g.f. of the process is given for the general case. Further, a set of algebraic equations for the extinction probabilities are studied for the process without immigration, and a necessary and sufficient condition that the extinction probability be one is obtained.

Stochastic compartmental models as approximations to more general stochastic systems with the general stochastic epidemic as an example

A general time-dependent stochastic compartmental model is considered, where particles may move between a set of m compartments or out of the system, and new particles may be introduced into the compartments by immigration. A simple argument not relying on generating function techniques is given for the solution of such a system. It is then demonstrated that this class of compartmental models can be used as approximations to more complex stochastic systems by replacing dependence on certain stochastic variables by dependence on corresponding deterministic variables, with some recent examples being discussed. A compartmental approximation to the general stochastic epidemic is then constructed which appears on the basis of some simulations to compare favourably with the true process, particularly after the infection has got established in the population.

A survey of stepping-stone models in population dynamics

A survey is presented of stochastic and deterministic developments in the study of the effects of nearest-neighbour ‘migration’ between spatially separated ‘colonies’. Such processes are of general applicability, and are relevant to any vector processX(t) = (X1(t), · ··,XN(t)) in which the arrival, departure and transfer rates for the states {X(t) = n} may be written in the formαi(ni), βi(ni) andγij(ni,nj), respectively, wheren =(n1,· ··, nN). Whilst the main body of results are described in terms of birth-death, genetic and epidemic situations, the final section examines within colony interaction in the context of spatial predator-prey processes.

Stochastic compartmental models as approximations to more general stochastic systems with the general stochastic epidemic as an example

A general time-dependent stochastic compartmental model is considered, where particles may move between a set of m compartments or out of the system, and new particles may be introduced into the compartments by immigration. A simple argument not relying on generating function techniques is given for the solution of such a system. It is then demonstrated that this class of compartmental models can be used as approximations to more complex stochastic systems by replacing dependence on certain stochastic variables by dependence on corresponding deterministic variables, with some recent examples being discussed. A compartmental approximation to the general stochastic epidemic is then constructed which appears on the basis of some simulations to compare favourably with the true process, particularly after the infection has got established in the population.

Applying the saddlepoint approximation to bivariate stochastic processes

The problem of moment closure is central to the study of multitype stochastic population dynamics since equations for moments up to a given order will generally involve higher-order moments. To obtain a Normal approximation, the standard approach is to replace third- and higher-order moments by zero, which may be severely restrictive on the structure of the p.d.f. The purpose of this paper is therefore to extend the univariate truncated saddlepoint procedure to multivariate scenarios. This has several key advantages: no distributional assumptions are required; it works regardless of the moment order deemed appropriate; and, we obtain an algebraic form for the associated p.d.f. irrespective of whether or not we have complete knowledge of the cumulants. The latter is especially important, since no families of distributions currently exist which embrace all cumulants up to any given order. In general the algorithm converges swiftly to the required p.d.f.; analysis of a severe test case illustrates its current operational limit.

Regularity and reversibility results for birth-death-migration processes

A spatial process is considered in which two general birth-death processes are linked by migration of individuals. We examine conditions for weak symmetry and regularity, and develop necessary and sufficient conditions for recurrence. The results are easily extended to the k-process case.

A carcinogenesis model describing mutational events at the DNA adduct level

A stochastic carcinogenesis model is proposed to describe a sequence of component mutational changes that constitute the G:C-->A:T base substitution. This paper provides the biological basis and mathematical formulation underlying the proposed model. In addition, the paper elaborates on a numerical approach for studying the cumulant functions, survival functions, and hazard functions of the model. Several numerical examples are given of potential applications of the model.

Two variants of the first Nåsell-Hirsch model

Two models are propounded on the general setting of the first Nåsell-Hirsch model for helminthic infections. They differ from it in the two processes of transmission of the infection, which are treated dissymmetrically. In each model only one of these processes will be deterministic, instead of both as in the Nåsell-Hirsch case, the other process being kept fully stochastic. These new models lead to results that show a general agreement with those of Nåsell and Hirsch but also some interesting differences.