The soil moisture dynamics under seasonally fixed conditions are studied at a point. The water balance is described through the representation of rainfall as a marked Poisson process, which in turn produces an infiltration into the soil dependent on the existing level of soil moisture. The losses from the soil are due to evapotranspiration and leakage, which are also considered dependent on the existing soil moisture. The steady-state probability distributions for soil moisture are then analytically obtained. The analysis of the distribution allows for the assessment of the role of climate, soil and vegetation on soil moisture dynamics. Further hydrologic insight is obtained by studying the various components of an average water balance. The realistic representation of the processes acting at a site and the analytical tractability of the model make it well suited for further analyses that consider the spatial aspect of soil moisture dynamics.

Despite some notable successes in the control of infectious diseases, transmissible pathogens still pose an enormous threat to human and animal health. The ecological and evolutionary dynamics of infections play out on a wide range of interconnected temporal, organizational and spatial scales, which even within a single pathogen often span hours to months, cellular to ecosystem levels, and local to pandemic spread. Some pathogens are directly transmitted between individuals of a single species, while others circulate among multiple hosts, need arthropod vectors, or can survive in environmental reservoirs. Many factors, including increasing antimicrobial resistance, increased human connectivity, and dynamic human behavior, raise prevention and control from formerly national to international issues. In the face of this complexity, mathematical models offer essential tools for synthesizing information to understand epidemiological patterns, and for developing the quantitative evidence base for decision-making in global health.

Stochastic models are discussed for the variation of rainfall intensity at a fixed point in space. First, models are analysed in which storm events arise in a Poisson process, each such event being associated with a period of rainfall of random duration and constant but random intensity. Total rainfall intensity is formed by adding the contributions from all storm events. Then similar but more complex models are studied in which storms arise in a Poisson process, each storm giving rise to a cluster of rain cells and each cell being associated with a random period of rain. The main properties of these models are determined analytically. Analysis of some hourly rainfall data from Denver, Colorado shows the clustered models to be much the more satisfactory.

[1] We consider the problem of simulating sequences of daily rainfall at a network of sites in such a way as to reproduce a variety of properties realistically over a range of spatial scales. The properties of interest will vary between applications but typically will include some measures of ''extreme'' rainfall in addition to means, variances, proportions of wet days, and autocorrelation structure. Our approach is to fit a generalized linear model (GLM) to rain gauge data and, with appropriate incorporation of intersite dependence structure, to use the GLM to generate simulated sequences. We illustrate the methodology using a data set from southern England and show that the GLM is able to reproduce many properties at spatial scales ranging from a single site to 2000 km 2 (the limit of the available data).Citation: Yang, C., R. E. Chandler, V. S. Isham, and H. S. Wheater (2005), Spatial-temporal rainfall simulation using generalized linear models, Water Resour. Res., 41, W11415,

A stochastic model for rainfall at a single site is studied in which storms arrive in a Poisson process, each storm consisting of a cluster of a random number of rain cells, each cell having random duration and depth. A model studied in an earlier paper is extended to provide a better fit to empirical experience, the extension being based on the attachment of a single random variable to each storm to achieve in particular some correlation between the durations of the cells within a single storm. The properties of the new model are developed, its fitting to two sets of empirical data is described and the examination of adequacy of fit is studied in some detail via properties not used in the fitting procedure. Finally a theoretical study is made of short-term prediction from the model.