1
|
Awasthi A, Minin VM, Huang J, Chow D, Xu J. Fitting a stochastic model of intensive care occupancy to noisy hospitalization time series during the COVID-19 pandemic. Stat Med 2023; 42:5189-5206. [PMID: 37705508 DOI: 10.1002/sim.9907] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2022] [Revised: 07/28/2023] [Accepted: 09/01/2023] [Indexed: 09/15/2023]
Abstract
Intensive care occupancy is an important indicator of health care stress that has been used to guide policy decisions during the COVID-19 pandemic. Toward reliable decision-making as a pandemic progresses, estimating the rates at which patients are admitted to and discharged from hospitals and intensive care units (ICUs) is crucial. Since individual-level hospital data are rarely available to modelers in each geographic locality of interest, it is important to develop tools for inferring these rates from publicly available daily numbers of hospital and ICU beds occupied. We develop such an estimation approach based on an immigration-death process that models fluctuations of ICU occupancy. Our flexible framework allows for immigration and death rates to depend on covariates, such as hospital bed occupancy and daily SARS-CoV-2 test positivity rate, which may drive changes in hospital ICU operations. We demonstrate via simulation studies that the proposed method performs well on noisy time series data and apply our statistical framework to hospitalization data from the University of California, Irvine (UCI) Health and Orange County, California. By introducing a likelihood-based framework where immigration and death rates can vary with covariates, we find, through rigorous model selection, that hospitalization and positivity rates are crucial covariates for modeling ICU stay dynamics and validate our per-patient ICU stay estimates using anonymized patient-level UCI hospital data.
Collapse
Affiliation(s)
- Achal Awasthi
- Department of Biostatistics and Bioinformatics, Duke University, Durham, North Carolina, USA
| | - Volodymyr M Minin
- Department of Statistics, University of California, Irvine, Irvine, California, USA
| | - Jenny Huang
- Department of Statistical Science, Duke University, Durham, North Carolina, USA
| | - Daniel Chow
- School of Medicine, University of California, Irvine, Irvine, California, USA
| | - Jason Xu
- Department of Biostatistics and Bioinformatics, Duke University, Durham, North Carolina, USA
- Department of Statistical Science, Duke University, Durham, North Carolina, USA
| |
Collapse
|
2
|
Gao P. The solutions with recurrence property for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations. STOCH DYNAM 2019. [DOI: 10.1142/s0219493719500059] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Stochastic periodic type solution is a powerful tool for studying qualitative analysis of stochastic dynamical systems. In this paper, we will establish the bounded solutions, stationary solutions, periodic solutions, almost periodic solutions, almost automorphic solutions for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations under suitable conditions. The main novelty of this paper is dealing with cubic nonlinear terms and the quintic nonlinear terms which are not Lipschitz. We overcome this difficulty by the semigroup approach, stochastic analysis techniques, energy estimate method and refined inequality technique.
Collapse
Affiliation(s)
- Peng Gao
- School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, P. R. China
| |
Collapse
|
3
|
Crawford FW, Ho LST, Suchard MA. Computational methods for birth-death processes. WILEY INTERDISCIPLINARY REVIEWS. COMPUTATIONAL STATISTICS 2018; 10:e1423. [PMID: 29942419 PMCID: PMC6014701 DOI: 10.1002/wics.1423] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
Many important stochastic counting models can be written as general birth-death processes (BDPs). BDPs are continuous-time Markov chains on the non-negative integers in which only jumps to adjacent states are allowed. BDPs can be used to easily parameterize a rich variety of probability distributions on the non-negative integers, and straightforward conditions guarantee that these distributions are proper. BDPs also provide a mechanistic interpretation - birth and death of actual particles or organisms - that has proven useful in evolution, ecology, physics, and chemistry. Although the theoretical properties of general BDPs are well understood, traditionally statistical work on BDPs has been limited to the simple linear (Kendall) process. Aside from a few simple cases, it remains impossible to find analytic expressions for the likelihood of a discretely-observed BDP, and computational difficulties have hindered development of tools for statistical inference. But the gap between BDP theory and practical methods for estimation has narrowed in recent years. There are now robust methods for evaluating likelihoods for realizations of BDPs: finite-time transition, first passage, equilibrium probabilities, and distributions of summary statistics that arise commonly in applications. Recent work has also exploited the connection between continuously- and discretely-observed BDPs to derive EM algorithms for maximum likelihood estimation. Likelihood-based inference for previously intractable BDPs is much easier than previously thought and regression approaches analogous to Poisson regression are straightforward to derive. In this review, we outline the basic mathematical theory for BDPs and demonstrate new tools for statistical inference using data from BDPs.
Collapse
Affiliation(s)
- Forrest W Crawford
- Departments of Biostatistics, Ecology & Evolutionary Biology, and School of Management, Yale University
| | - Lam Si Tung Ho
- Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada
| | - Marc A Suchard
- Departments of Biomathematics, Biostatistics and Human Genetics, University of California, Los Angeles
| |
Collapse
|
4
|
Ho LST, Xu J, Crawford FW, Minin VN, Suchard MA. Birth/birth-death processes and their computable transition probabilities with biological applications. J Math Biol 2018; 76:911-944. [PMID: 28741177 PMCID: PMC5783825 DOI: 10.1007/s00285-017-1160-3] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2016] [Revised: 04/04/2017] [Indexed: 01/20/2023]
Abstract
Birth-death processes track the size of a univariate population, but many biological systems involve interaction between populations, necessitating models for two or more populations simultaneously. A lack of efficient methods for evaluating finite-time transition probabilities of bivariate processes, however, has restricted statistical inference in these models. Researchers rely on computationally expensive methods such as matrix exponentiation or Monte Carlo approximation, restricting likelihood-based inference to small systems, or indirect methods such as approximate Bayesian computation. In this paper, we introduce the birth/birth-death process, a tractable bivariate extension of the birth-death process, where rates are allowed to be nonlinear. We develop an efficient algorithm to calculate its transition probabilities using a continued fraction representation of their Laplace transforms. Next, we identify several exemplary models arising in molecular epidemiology, macro-parasite evolution, and infectious disease modeling that fall within this class, and demonstrate advantages of our proposed method over existing approaches to inference in these models. Notably, the ubiquitous stochastic susceptible-infectious-removed (SIR) model falls within this class, and we emphasize that computable transition probabilities newly enable direct inference of parameters in the SIR model. We also propose a very fast method for approximating the transition probabilities under the SIR model via a novel branching process simplification, and compare it to the continued fraction representation method with application to the 17th century plague in Eyam. Although the two methods produce similar maximum a posteriori estimates, the branching process approximation fails to capture the correlation structure in the joint posterior distribution.
Collapse
Affiliation(s)
- Lam Si Tung Ho
- Department of Biostatistics, University of California, Los Angeles, Los Angeles, CA, USA.
| | - Jason Xu
- Department of Biomathematics, University of California, Los Angeles, Los Angeles, CA, USA
| | | | - Vladimir N Minin
- Departments of Statistics and Biology, University of Washington, Seattle, WA, USA
| | - Marc A Suchard
- Departments of Biomathematics, Biostatistics and Human Genetics, University of California, Los Angeles, Los Angeles, WA, USA
| |
Collapse
|
5
|
Xu J, Guttorp P, Kato-Maeda M, Minin VN. Likelihood-based inference for discretely observed birth-death-shift processes, with applications to evolution of mobile genetic elements. Biometrics 2015; 71:1009-21. [PMID: 26148963 DOI: 10.1111/biom.12352] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2015] [Revised: 05/01/2015] [Accepted: 05/01/2015] [Indexed: 11/28/2022]
Abstract
Continuous-time birth-death-shift (BDS) processes are frequently used in stochastic modeling, with many applications in ecology and epidemiology. In particular, such processes can model evolutionary dynamics of transposable elements-important genetic markers in molecular epidemiology. Estimation of the effects of individual covariates on the birth, death, and shift rates of the process can be accomplished by analyzing patient data, but inferring these rates in a discretely and unevenly observed setting presents computational challenges. We propose a multi-type branching process approximation to BDS processes and develop a corresponding expectation maximization algorithm, where we use spectral techniques to reduce calculation of expected sufficient statistics to low-dimensional integration. These techniques yield an efficient and robust optimization routine for inferring the rates of the BDS process, and apply broadly to multi-type branching processes whose rates can depend on many covariates. After rigorously testing our methodology in simulation studies, we apply our method to study intrapatient time evolution of IS6110 transposable element, a genetic marker frequently used during estimation of epidemiological clusters of Mycobacterium tuberculosis infections.
Collapse
Affiliation(s)
- Jason Xu
- Department of Statistics, University of Washington, Seattle, WA, U.S.A
| | - Peter Guttorp
- Department of Statistics, University of Washington, Seattle, WA, U.S.A
| | - Midori Kato-Maeda
- School of Medicine, University of California, San Francisco, CA, U.S.A
| | - Vladimir N Minin
- Department of Statistics, University of Washington, Seattle, WA, U.S.A.,Department of Biology, University of Washington, Seattle, WA, U.S.A
| |
Collapse
|
6
|
Xu J, Minin VN. Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing. UNCERTAINTY IN ARTIFICIAL INTELLIGENCE : PROCEEDINGS OF THE ... CONFERENCE. CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE 2015; 2015:952-961. [PMID: 26949377 PMCID: PMC4775097] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Branching processes are a class of continuous-time Markov chains (CTMCs) with ubiquitous applications. A general difficulty in statistical inference under partially observed CTMC models arises in computing transition probabilities when the discrete state space is large or uncountable. Classical methods such as matrix exponentiation are infeasible for large or countably infinite state spaces, and sampling-based alternatives are computationally intensive, requiring integration over all possible hidden events. Recent work has successfully applied generating function techniques to computing transition probabilities for linear multi-type branching processes. While these techniques often require significantly fewer computations than matrix exponentiation, they also become prohibitive in applications with large populations. We propose a compressed sensing framework that significantly accelerates the generating function method, decreasing computational cost up to a logarithmic factor by only assuming the probability mass of transitions is sparse. We demonstrate accurate and efficient transition probability computations in branching process models for blood cell formation and evolution of self-replicating transposable elements in bacterial genomes.
Collapse
Affiliation(s)
- Jason Xu
- Department of Statistics, University of Washington, Seattle, WA 98195
| | - Vladimir N Minin
- Departments of Statistics and Biology, University of Washington, Seattle, WA 98195
| |
Collapse
|
7
|
Abstract
Birth-death processes (BDPs) are continuous-time Markov chains that track the number of "particles" in a system over time. While widely used in population biology, genetics and ecology, statistical inference of the instantaneous particle birth and death rates remains largely limited to restrictive linear BDPs in which per-particle birth and death rates are constant. Researchers often observe the number of particles at discrete times, necessitating data augmentation procedures such as expectation-maximization (EM) to find maximum likelihood estimates. For BDPs on finite state-spaces, there are powerful matrix methods for computing the conditional expectations needed for the E-step of the EM algorithm. For BDPs on infinite state-spaces, closed-form solutions for the E-step are available for some linear models, but most previous work has resorted to time-consuming simulation. Remarkably, we show that the E-step conditional expectations can be expressed as convolutions of computable transition probabilities for any general BDP with arbitrary rates. This important observation, along with a convenient continued fraction representation of the Laplace transforms of the transition probabilities, allows for novel and efficient computation of the conditional expectations for all BDPs, eliminating the need for truncation of the state-space or costly simulation. We use this insight to derive EM algorithms that yield maximum likelihood estimation for general BDPs characterized by various rate models, including generalized linear models. We show that our Laplace convolution technique outperforms competing methods when they are available and demonstrate a technique to accelerate EM algorithm convergence. We validate our approach using synthetic data and then apply our methods to cancer cell growth and estimation of mutation parameters in microsatellite evolution.
Collapse
Affiliation(s)
- Forrest W Crawford
- Department of Biostatistics, Yale University, 60 College Street, Box 208034, New Haven, CT 06510 USA
| | - Vladimir N Minin
- Department of Statistics, University of Washington, Padelford Hall C-315, Box 354322, Seattle, WA 98195-4322 USA
| | - Marc A Suchard
- Department of Biomathematics, University of California Los Angeles, 6558 Gonda Building, Los Angeles, CA 90095-1766 USA ; Department of Biostatistics, University of California Los Angeles, 6558 Gonda Building, Los Angeles, CA 90095-1766 USA ; Department of Human Genetics, University of California Los Angeles, 6558 Gonda Building, Los Angeles, CA 90095-1766 USA
| |
Collapse
|
8
|
Doss CR, Suchard MA, Holmes I, Kato-Maeda M, Minin VN. Fitting Birth-Death Processes to Panel Data with Applications to Bacterial DNA Fingerprinting. Ann Appl Stat 2013; 7:2315-2335. [PMID: 26702330 DOI: 10.1214/13-aoas673] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Abstract
Continuous-time linear birth-death-immigration (BDI) processes are frequently used in ecology and epidemiology to model stochastic dynamics of the population of interest. In clinical settings, multiple birth-death processes can describe disease trajectories of individual patients, allowing for estimation of the effects of individual covariates on the birth and death rates of the process. Such estimation is usually accomplished by analyzing patient data collected at unevenly spaced time points, referred to as panel data in the biostatistics literature. Fitting linear BDI processes to panel data is a nontrivial optimization problem because birth and death rates can be functions of many parameters related to the covariates of interest. We propose a novel expectation-maximization (EM) algorithm for fitting linear BDI models with covariates to panel data. We derive a closed-form expression for the joint generating function of some of the BDI process statistics and use this generating function to reduce the E-step of the EM algorithm, as well as calculation of the Fisher information, to one-dimensional integration. This analytical technique yields a computationally efficient and robust optimization algorithm that we implemented in an open-source R package. We apply our method to DNA fingerprinting of Mycobacterium tuberculosis, the causative agent of tuberculosis, to study intrapatient time evolution of IS6110 copy number, a genetic marker frequently used during estimation of epidemiological clusters of Mycobacterium tuberculosis infections. Our analysis reveals previously undocumented differences in IS6110 birth-death rates among three major lineages of Mycobacterium tuberculosis, which has important implications for epidemiologists that use IS6110 for DNA fingerprinting of Mycobacterium tuberculosis.
Collapse
|