Individuals in communities in which different strains of pathogen are circulating can acquire resistance by accumulating immunity to each strain. After considering susceptibility, models of infection and immunity are defined for vector-borne diseases such as malaria and trypanosomiasis. For these models the prevalence of infection, the number of infections per individual, and the mean duration of infection, increase rapidly in young individuals, but decrease in older individuals as immunity is acquired to the various strains of pathogen; the mean interval between successive infections lengthens with age. The bivariate Poisson distribution is shown to be a close approximation to some stochastic processes. The models explain observed cross-sectional patterns of age prevalence, and longitudinal patterns in which individuals typically continue to become infected as they age, albeit with decreasing frequency. In these models the time spent infected depends on parasite diversity, as well as the inoculation and recovery rates. It is shown that control measures can cause an increase in the number of infections and the prevalence of infection in older individuals, and in the average prevalence in the community, even when strain-specific immunity is life-long.

During the course of a day an individual typically mixes with different groups of individuals. Epidemic models incorporating population structure with individuals being able to infect different groups of individuals have received extensive attention in the literature. However, almost exclusively the models assume that individuals are able to simultaneously infect members of all groups, whereas in reality individuals will typically only be able to infect members of any group they currently reside in. In this paper we develop a model where individuals move between a community and their household during the course of the day, only infecting within their current group. By defining a novel branching process approximation with an explicit expression for the probability generating function of the offspring distribution, we are able to derive the probability of a major epidemic outbreak.

In Bingham and Doney (1988) the authors presented the applied probability community with a question which is very simply stated, yet is extremely difficult to solve: what is the distribution of the quadrant occupation time of planar Brownian motion? In this paper we study an alternate formulation of this long-standing open problem: let X(t), Y(t) t≥0, be standard Brownian motions starting at x, y, respectively. Find the distribution of the total time T=Leb{t∈[0,1]: X(t)×Y(t)>0}, when x=y=0, i.e. the occupation time of the union of the first and third quadrants. If two adjacent quadrants are used, the problem becomes much easier and the distribution of T follows the arcsine law.

We study a network of fluid queues in which exogenous arrivals are modulated by a continuous-time Markov chain. Service rates in each queue are proportional to the queue size, and the network is assumed to be irreducible. The queue levels satisfy a linear, vector-valued differential equation. We obtain joint moments of the queue sizes recursively, and deduce the Laplace transform of the queue sizes in the stationary regime.

The contact process is an interacting particle system which models a spatially restricted infection. In the basic contact process the infection can only spread to an uninfected neighbour, but the diffusive contact process allows an infected individual to move to an uninfected site. If the infection rate is too low, the process will die out. If the individual can move (or diffuse), the disease can spread with a lower infection rate. An idea of the relationship between these rates is obtained by obtaining rigorous lower bounds for the critical infection rate for various values of the diffusion rate. In this paper we also improve the lower bound for the critical infection rate for the basic contact process from 1.539 to 1.5517.

We introduce a new model for the infection of one or more subjects by a single agent, and calculate the probability of infection after a fixed length of time. We model the agent and subjects as random walkers on a complete graph of N sites, jumping with equal rates from site to site. When one of the walkers is at the same site as the agent for a length of time τ, we assume that the infection probability is given by an exponential law with parameter γ, i.e. q(τ) = 1 - e-γ τ . We introduce the boundary condition that all walkers return to their initial site (‘home’) at the end of a fixed period T. We also assume that the incubation period is longer than T, so that there is no immediate propagation of the infection. In this model, we find that for short periods T, i.e. such that γ T ≪ 1 and T ≪ 1, the infection probability is remarkably small and behaves like T 3. On the other hand, for large T, the probability tends to 1 (as might be expected) exponentially. However, the dominant exponential rate is given approximately by 2γ/[(2+γ)N] and is therefore small for large N.

Two related individuals are identical by descent at a genetic locus if they share the same gene copy at that locus due to inheritance from a recent common ancestor. We consider idealized continuous identity by descent (IBD) data in which IBD status is known continuously along chromosomes. IBD data contains information about the relationship between the two individuals, and about the underlying crossover processes. We present a Monte Carlo method for calculating probabilities for IBD data. The method is not restricted to Haldane's Poisson process model of crossing-over but may be used with other models including the chi-square, Kosambi renewal and Sturt models. Results of a simulation study demonstrate that IBD data can be used to distinguish between alternative models for the crossover process.

Let x be a non-negative, non-decreasing, right-continuous function of t defined on the interval [O, T) and let A = x(T–). Define the function y on [O, A] by Then y is non-decreasing and left-continuous and If t is a time variable it is natural to call y the passage time function.

In the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until time t the bacterial burst has not taken place, its occurence between tand t + Δt is a random event with probability f(· | t)Δt + o(Δt), where f is a non-negative and non-decreasing function of the number X(t) of vegetative phages and of Z(t), the number of mature bacteriophages at time t. More specifically it is assumed that f = b(t)X(t) + c(t)Z(t) with b(t), c(t) ≦ 0. Here X(t) denotes the survivors in a linear birth and death process and Z(t) the number of deaths until time t. The joint distribution of XT and ZT, the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution of ZT is then fitted to some observed data of Delbrück [2].

A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.

This article discusses certain aspects of determining heritability estimates from the genetic analyses of data. First, a brief review is given of the major statistical techniques for fitting models to quantitative family data. Then, by reference to a general schematic framework, the basic models that have been proposed for such data will be described. Some new environmental models are also developed. Finally, by reference to some data sets, the interactive approach to data analysis is proposed, advocating the determination of adequacy of fit of all reasonable models to available data.

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in $\mathbb{R}^d$ of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

A model for the transition probability matrices (t.p.m.'s) of finite discrete Markov chains is suggested which may help those who wish to use a larger number of states than would seem reasonable with the data available in the current estimation situation. The model is especially useful in that a finite t.p.m. of arbitrary size can be specified by as few as two parameters. An example of the model's estimation and use is presented, showing it in a fair light in comparison with the conventional method of t.p.m. specification.

Nash (1980) demonstrated that index policies are optimal for a class of generalised bandit problem. A transform of the index concerned has many of the attributes of the Gittins index. The transformed index is positive-valued, with maximal values yielding optimal actions. It may be characterised as the value of a restart problem and is hence computable via dynamic programming methodologies. The transformed index can also be used in procedures for policy evaluation.

In this paper the probability distribution of the discrete busy period of the 〈M, 1, GI/∞, 1〉 queue and some of its stability properties are given.

We consider a Markov-modulated Brownian motion (MMBM) with phase-dependent termination rates, i.e. while in a phase i the process terminates with a constant hazard rate ri ≥ 0. For such a process, we determine the matrix of expected local times (at zero) before termination and hence the resolvent. The results are applied to some recent questions arising in the framework of insurance risk. We further provide expressions for the resolvent and the local times at zero of an MMBM reflected at its infimum.

Slotted ALOHA with a finite number of users, each with infinite buffer, is considered. For stationary, metrically transitive, non-independent input, the problem of existence of stationary queue length distributions is solved. Results are obtained for zero and arbitrary (finite a.s.) initial conditions. Continuity, in probability, of queue lengths with respect to input sequence is proved.

The spacings have been widely applied to many situations, such as for example the coverage problem of the circle or the line, under the hypothesis of uniformity. The paper examines the asymptotic behaviour of the maximal spacings under non-uniformity, assuming that the existing density f has a minimum point m with f(m) = 0. The results are applied to the coverage of the circle by randomly placed arcs.

The constructive solution to the problem of representing a strictly stationary Markov chain ζ with a countable infinity of r-sequences (i1, i2, · ··, ir), r > 1, as states by a class of directed weighted circuits is given. Associating the chain ζ with its dual chain η having reversed states and the same transition law, a connection with physical laws that govern diffusion of electrical current through a directed planar network with r-series-connected nodes is shown.