Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogenous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school. Supplementary material for this article is available online.     

Parametric inference for discrete observations of diffusion processes with mixed effects

Stochastic differential equations with mixed effects provide means to model intra-individual and inter-individual variability in repeated experiments leading to longitudinal data. We consider N i.i.d. stochastic processes defined by a stochastic differential equation with linear mixed effects which are discretely observed. We study a parametric framework with distributions leading to explicit approximate likelihood functions and investigate the asymptotic behavior of estimators under the asymptotic framework : the number N of individuals (trajectories) and the number n of observations per individual tend to infinity within a fixed time interval. The estimation method is assessed on simulated data for various models.     

The Million-Variable “March” for Stochastic Combinatorial Optimization

Combinatorial optimization problems have applications in a variety of sciences and engineering. In the presence of data uncertainty, these problems lead to stochastic combinatorial optimization problems which result in very large scale combinatorial optimization problems. In this paper, we report on the solution of some of the largest stochastic combinatorial optimization problems consisting of over a million binary variables. While the methodology is quite general, the specific application with which we conduct our experiments arises in stochastic server location problems. The main observation is that stochastic combinatorial optimization problems are comprised of loosely coupled subsystems. By taking advantage of the loosely coupled structure, we show that decomposition-coordination methods provide highly effective algorithms, and surpass the scalability of even the most efficiently implemented backtracking search algorithms.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension d=1,2,3. Moreover, for these particular examples, known results in the literature have been improved.      

Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a “match” between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set ofsufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach—including a computationally efficient version based on the auxiliary score—achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multiparameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the article. Supplementary materials for this article are available online.     

On Lp -Theory of stochastic partialdifferential equations of divergence form with continuous coefficients

We develop an L_p -theory of stochastic PDEs of divergence form. Under natural assumptions on the coefficients and the data, we show that the solutions belong to some modified stochastic Sobolev spaces. As a consequence of this result and certain embedding theorem, we also show that the solutions are Holder continuous in space and time a.s. for sufficiently large p     

Constrained Stochastic Estimation Algorithms for a Class of Hybrid Stock Market Models

This paper is concerned with a class of hybrid stock market models, in which both the return rate and the volatility depend on a hidden, continuous-time Markov chain with a finite state space. One of the crucial issues is to estimate the generator of the underlying Markov chain. We develop a stochastic optimization procedure for this task, prove its convergence, and establish the rate of convergence. Numerical tests are carried out via simulation as well as using real market data. In addition, we demonstrate how to use the estimated generator in making stock liquidation decisions.     

Persistence and extinction in stochastic non-autonomous logistic systems

This paper studies two widely used stochastic non-autonomous logistic models. For the first system, sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The critical number between weak persistence and extinction is obtained. For the second system, sufficient criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean and stochastic permanence are established. The critical number between weak persistence in the mean and extinction is obtained. It should be pointed out that this research is systematical and complete. In fact, the behaviors of the two models in every coefficient cases are cleared up by the results obtained in this paper.     

Probability distribution of the number of initiated cells of carcinogenesis under prevention

In this paper, I have shown that under some mild conditions, the number of initiated cells in an extended two-stage model of carcinogenesis can be approximated by a diffusion process. By using this approximation, I have derived the probability distribution for the number of initiated cells in terms of Laguerre polynomials under normal prevention conditions. This follows from the fact that many of the dietary components are antioxidants which would neutralize the hydroxyl free radicals and hence, reduce the proliferation rates of initiated cells to interrupt or slow down the promotion stage in carcinogenesis.     

An approximation for computing the throughput of closed assembly-type queueing networks

In this paper, we examine throughput (mean number of completed assemblies per unit time) of closed assembly type queueing networks where machine processing times are drawn from general distributions. The system dynamics are characterized via a set of stochastic difference equations; it is shown that the system state can be modeled by a discrete index Markov chain on a continuous state space. Standard Markovian analysis is employed to derive an approximate expression for system throughput, following discretisation of state space. Four examples of CONWIP (CONstant Work IN Process) systems are given that illustrate the results.     

Stochastic modeling of phytoplankton allelopathy

The study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research due to the global increase of harmful phytoplankton blooms. Ordinary differential equation models for two competing phytoplankton species, when one or both the species liberate toxic substances, are unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic formulation never predicts the sudden localized extinction of certain species. These obstacles of mathematical modeling can be overcome if we include stochastic variability in our modeling approach. In this investigation, we construct stochastic models of allelopathic interactions between two competing phytoplankton species as a continuous time Markov chain model as well as an Itô stochastic differential equation model. Approximate extinction probabilities for both species are obtained analytically for the continuous time Markov chain model. Analytical estimates are validated with the help of numerical simulations.     

Mathematical modeling of the receptor-mediated endocytosis process of targeted therapeutic agents in drug delivery systems

In this paper, we propose mathematical models to describe receptor-mediated endocytosis processes. One is a stochastic differential model for the agent-target binding process. The mean extinction time and a standard variation over time profile are evaluated. The other is an age-structured model for demonstrating endocytosis and lysosome processes. A targeted drug delivery system has a complex process in how it is to deliver drug molecules in terms of administration, transportation in blood and across membranes to intracellular space, and inhibition to microtubule polymerization. In particular, receptor-mediated endocytosis of targeted therapeutic agents, such as antibody drug conjugates or ligand-targeted liposome encapsulated nanoparticles, is a key step in understanding the drug delivery mechanism. We discuss stochastic quasi steady state approximation when agent-target complex does not appreciably vary compared with the free agents. This reduces the number of the systems and the parameters; however, an initial time phase cannot be captured. In addition, we discuss the strengths and weaknesses when the age-structured model induces the reduced model compared with the full model that considers endocytosis and lysosome processes. If the total mean retention time until payload release in intracellular space is known, then the age-structured model with the Erlang distribution may fairly predict data of the released payload over time profile with far fewer parameters; however, induced compartments lose their physical meaning and describe only a delay.     

Integrated inventory and inspection policies for stochastic demand

In this paper, we develop integrated inventory inspection models with and without replacement of nonconforming items. Inspection policies include no inspection, sampling inspection, and 100% inspection. We consider a buyer who places an order from a supplier when his inventory level drops to a certain point, due to demand which is stochastic in nature. When a lot is received, the buyer uses some type of inspection policy. The fraction nonconforming is assumed to be a random variable following a beta distribution. The order quantity, reorder point and the inspection policy are decision variables. In the inspection policy involving determining sampling plan parameters, constraints on the buyer and manufacturer risks is set in order to obtain a fair plan for both parties. A solution procedure for determining the operating policies for inventory and inspection consisting of order quantity, sample size, and acceptance number is proposed. Numerical examples are presented to conduct a sensitivity analysis for important model parameters and to illustrate important issues about the developed models.     

Large deviations for diffusion processes in duals of nuclear spaces

Motivated by applications to neurophysiological problems, various authors have studied diffusion processes in duals of countably Hilbertian nuclear spaces governed by stochastic differential equations. In these models the diffusion coefficients describe the random stimuli received by spatially extended neurons. In this paper we present a large deviation principle for such processes when the diffusion terms tend to zero in terms of a small parameter. The lower bounds are established by making use of the Girsanov formula in abstract Wiener space. The upper bounds are obtained by Gaussian approximation of the diffusion processes and by taking advantage of the nuclear structure of the state space to pass from compact sets to closed sets.This research was partially supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620-92-J-0154 and the Army Research Office Grant No. DAAL03-92-G-0008.     

A note on wavelet density deconvolution for weakly dependent data

In this paper we investigate the performance of a linear wavelet-type deconvolution estimator for weakly dependent data. We show that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough. The results are applied to a discretely observed continuous-time stochastic volatility model.     

Ensuring service levels in routing problems with time windows and stochastic travel times

In the stochastic variant of the vehicle routing problem with time windows, known as the SVRPTW, travel times are assumed to be stochastic. In our chance-constrained approach to the problem, restrictions are placed on the probability that individual time window constraints are violated, while the objective remains based on traditional routing costs. In this paper, we propose a way to offer this probability, or service level, for all customers. Our approach carefully considers how to compute the start-service time and arrival time distributions for each customer. These distributions are used to create a feasibility check that can be “plugged” into any algorithm for the VRPTW and thus be used to solve large problems fairly quickly. Our computational experiments show how the solutions change for some well-known data sets across different levels of customer service, two travel time distributions, and several parameter settings.     

GPU-Powered Shotgun Stochastic Search for Dirichlet Process Mixtures of Gaussian Graphical Models

Gaussian graphical models (GGMs) are popular for modeling high-dimensional multivariate data with sparse conditional dependencies. A mixture of GGMs extends this model to the more realistic scenario where observations come from a heterogenous population composed of a small number of homogeneous subgroups. In this article, we present a novel stochastic search algorithm for finding the posterior mode of high-dimensional Dirichlet process mixtures of decomposable GGMs. Further, we investigate how to harness the massive thread-parallelization capabilities of graphical processing units to accelerate computation. The computational advantages of our algorithms are demonstrated with various simulated data examples in which we compare our stochastic search with a Markov chain Monte Carlo (MCMC) algorithm in moderate dimensional data examples. These experiments show that our stochastic search largely outperforms the MCMC algorithm in terms of computing-times and in terms of the quality of the posterior mode discovered. Finally, we analyze a gene expression dataset in which MCMC algorithms are too slow to be practically useful.     

Expected present value of total dividends in a delayed claims risk model under stochastic interest rates

In this paper, a compound binomial risk model with a constant dividend barrier under stochastic interest rates is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are K_n distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends.