We consider two interacting particle systems on $\mathbf{Z}^d$ to model predator-prey and host-parasite interactions. In both models we have two types of $\mathbf{Z}^d$ particles (1 and 2) and each site in can be in one of four states: empty, occupied by a type 1 particle, occupied by a type 2 particle or occupied by two particles (one of each type). Each type gives birth to particles of the same type on nearest neighbor sites. The interaction between the two types of particles occurs only when a site is occupied by one particle of each type. For both models we show that coexistence and noncoexistence are possible in any dimension.

Barley yellow dwarf is a widespread disease that affects small grains and many grass species, as well as wheat, barley and oat. The disease is caused by an aphid transmitted virus. Rochow conducted a study near Ithaca, New York, which showed that a shift in the dominant strain occurred between 1957 and 1976. Motivated by this phenomenon, we develop a model for the competition between different strains of the barley yellow dwarf virus. Our main goal is to understand the phase diagram of the model, that is, to identify parameter values where one strain competitively excludes the other strain and where both strains coexist. Our analysis applies to a number of other systems as well, for example to a model of competition of water flea species studied by Hanski and Ranta and Bengtsson.

We consider epidemics with removal (SIR epidemics) in populations that mix at two levels: global and local. We develop a general modelling framework for such processes, which allows us to analyze the conditions under which a large outbreak is possible, the size of such outbreaks when they can occur and the implications for vaccination strategies, in each case comparing our results with the simpler homogeneous mixing case.

More precisely, we consider models in which each infectious individual i has a global probability $p_G$ for infecting each other individual in the population and a local probability $p_L$, typically much larger, of infecting each other individual among a set of neighbors $\mathscr{N}(i)$. Our main concern is the case where the population is partitioned into local groups or households, but our approach also applies to cases where neighborhoods do not form a partition, for instance, to spatial models with a mixture of local (e.g., nearest-neighbor) and global contacts.

We use a variety of theoretical approaches: a random graph framework for the initial exposition of the simple case where an individual's contacts are independent; branching process approximations for the general threshold result; and an embedding representation for rigorous results on the final size of outbreaks.

From the applied viewpoint the key result is that, compared with the homogeneous mixing model in which individuals make contacts simply with probability $p_G$, the local infectious contacts have an "amplification" effect. The basic reproductive ratio of the epidemic is increased from its individual-to-individual value $R_G$ in the absence of local infections to a group-to-group value $R_* = \mu R_G$, where $\mu$ is the mean size of an outbreak, started by a randomly chosen individual, in which only local infections count. Where the groups are large and the within-group epidemics are above threshold, this amplification can permit an outbreak in the whole population at very low levels of $p_G$, for instance, for $p_G = O(1/Nn)$ in a population of N divided into groups of size n.

The implication of these results for control strategies is that vaccination should be directed preferentially toward reducing $\mu$; we discuss the conditions under which the equalizing strategy, aimed at leaving unvaccinated sets of neighbors of equal sizes, is optimal. We also discuss the estimation of our threshold parameter $R_*$ from data on epidemics among households.

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixed population in which new infections occur at rate $\beta(z)xy/(x + y)$, where x, y and z denote, respectively, the numbers of susceptible, infective and removed individuals. Thus the infection mechanism depends upon the number of removals to date, reflecting behavior change in response to the progress of the epidemic. For a deterministic version of the model, a recurrent solution is obtained when $\beta(z)$ is piecewise constant. Equations for the total size distribution of the stochastic model are derived. Stochastic comparison results are obtained using a coupling method. Strong convergence of a sequence of epidemics to an unusual birth-and-death process is exhibited, and the behavior of the limiting birth-and-death process is considered. An epidemic model featuring sudden behavior change is studied as an example, and a stochastic threshold result analagous to that of Whittle is derived.

This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to $\infty$. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain converges to the appropriate limiting Langevin diffusion process.

The limiting diffusion approximation admits a straightforward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions.

The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed.

In applied probability one is often interested in the asymptotic behavior of a certain quantity. If a regenerative phenomenon can be imbedded, then one has the problem that the event of interest may have occurred but cannot be observed at the renewal points. In this paper an extension to the renewal theorem is proved which shows that the quantity of interest converges. As an illustration an open problem in risk theory is solved.

For evanescent Markov processes with a single transient communicating class, it is often of interest to examine the limiting probabilities that the process resides in the various transient states, conditional on absorption not having taken place. Such distributions are known as quasi-stationary (or limiting-conditional) distributions. In this paper we consider the determination of the quasi-stationary distribution of a general level-independent quasi-birth-and-death process (QBD). This distribution is shown to have a form analogous to the matrix-geometric form possessed by the stationary distribution of a positive recurrent QBD. We provide an algorithm for the explicit computation of the quasi-stationary distribution.

Consider a completely asymmetric Lévy process which has absolutely continuous transition probabilities. We determine the exponential decay parameter $\rho$ and the quasistationary distribution for the transition probabilities of the Lévy process killed as it exits from a finite interval, prove that the killed process is $\rho$-positive and specify the $\rho$-invariant function and measure.

Consider a system into which units having random magnitude enter at arbitrary times and remain "active" (present in the system) for random periods. Suppose units of high magnitude have stochastically greater lifetimes (tend to stay active for longer periods) than units of low magnitude. Of interest is the process ${\mu (t): t \geq 0}$ where $\mu (t)$ denotes the average magnitude of all units active at time t. We give conditions which guarantee the convergence of $\mu (t)$ and we determine the form of the limit. Some related processes are also studied.

Suppose that a process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always at most 2. In a previous article, the authors showed that as $n \to \infty$, with probability tending to 1, the result of this process is a graph with n edges. The number of l-cycles in this graph is shown to be asymptotically Poisson $(1 \geq 3)$, and other aspects of this random graph model are studied.

Burgers turbulence is an accepted formalism for the adhesion model of the large-scale distribution of matter in the universe. The paper uses variational methods to establish evolution of quasi-Voronoi (curved boundaries) tessellation structure of shock fronts for solutions of the inviscid nonhomogeneous Burgers equation in $R^d$ in the presence of random forcing due to a degenerate potential. The mean rate of growth of the quasi-Voronoi cells is calculated and a scaled limit random tessellation structure is found. Time evolution of the probability that a cell contains a ball of a given radius is also determined.

We prove that the solution of a system of random ordinary differential equations $d\mathbf{X}(t)/dt = \mathbf{V}(t, \mathbf{X}(t))$ with diffusive scaling, $\mathbf{X}_{\varepsilon}(t) = \varepsilon \mathbf{X}(t/ \varepsilon^2)$, converges weakly to a Brownian motion when $\varepsilon \downarrow 0$. We assume that $\mathbf{V}(t, \mathbf{x}), t \epsilon R, \mathbf{x} \epsilon R^d$ is a d-dimensional, random, incompressible, stationary Gaussian field which has mean zero and decorrelates in finite time.

We consider the diffusive scaling limit for the transport of a passive scalar in a two-dimensional time-dependent incompressible Gaussian velocity field and in the presence of molecular diffusivity. We prove that homogenization holds in this limiting regime and we derive some simple properties of the effective diffusivity tensor.