Stochastic process

About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.

On methods for studying stochastic disease dynamics

Abstract: Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.

Fluid limit of the continuous-time random walk with general Lévy jump distribution functions.

Abstract: The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions psi(t) and general jump distribution functions eta(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times psi approximately t(-(1+beta)) and algebraic decaying jump distributions eta approximately x(-(1+alpha)) corresponding to Levy stable processes, the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Levy stochastic processes in the Levy-Khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Levy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda(-alpha/beta), where 1/lambda is the truncation length scale. The asymptotic behavior of the propagator (Green's function) of the truncated fractional equation exhibits a transition from algebraic decay for t >tauc.

Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability

Abstract: A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length of shortest path through a random sample, the length of a rectilinear Steiner tree spanned by a sample, and the length of a minimal matching. Also, a uniform convergence theorem is proved which is needed in Karp's probabilistic algorithm for the traveling salesman problem.

Stochastic Finite-Element Analysis of Seismic Soil-Structure Interaction

Abstract: A procedure is presented for the probabilistic analysis of the seismic soil-structure interaction problem. The procedure accounts for uncertainty in both the free-field input motion as well as in local site conditions, and structural parameters. Uncertain parameters are modeled using a probabilistic framework as stochastic processes. The site amplification effects are accounted for via a randomized relationship between the soil shear modulus and damping on the one hand, and the shear strain of the subgrade on the other hand, as well as by modeling the shear modulus at low strain level as randomly fluctuating with depth. The various random processes are represented by their respective Karhunen-Loeve expansions, and the solution processes, consisting of the accelerations and generalized forces in the structure, are represented by their coordinates with respect to the polynomial chaos basis. These coordinates are then evaluated by a combination of weighted residuals and stratified sampling schemes. The expansion can be used to carry out very efficiently, extensive Monte Carlo simulations. The procedure is applied to the seismic analysis of a nuclear reactor facility.

Transcriptional Bursting in Gene Expression: Analytical Results for General Stochastic Models.

Abstract: Gene expression in individual cells is highly variable and sporadic, often resulting in the synthesis of mRNAs and proteins in bursts. Such bursting has important consequences for cell-fate decisions in diverse processes ranging from HIV-1 viral infections to stem-cell differentiation. It is generally assumed that bursts are geometrically distributed and that they arrive according to a Poisson process. On the other hand, recent single-cell experiments provide evidence for complex burst arrival processes, highlighting the need for analysis of more general stochastic models. To address this issue, we invoke a mapping between general stochastic models of gene expression and systems studied in queueing theory to derive exact analytical expressions for the moments associated with mRNA/protein steady-state distributions. These results are then used to derive noise signatures, i.e. explicit conditions based entirely on experimentally measurable quantities, that determine if the burst distributions deviate from the geometric distribution or if burst arrival deviates from a Poisson process. For non-Poisson arrivals, we develop approaches for accurate estimation of burst parameters. The proposed approaches can lead to new insights into transcriptional bursting based on measurements of steady-state mRNA/protein distributions.