Bayesian Inference of State-Level COVID-19 Basic Reproduction Numbers across the United States

Although many persons in the United States have acquired immunity to COVID-19, either through vaccination or infection with SARS-CoV-2, COVID-19 will pose an ongoing threat to non-immune persons so long as disease transmission continues. We can estimate when sustained disease transmission will end in a population by calculating the population-specific basic reproduction number , the expected number of secondary cases generated by an infected person in the absence of any interventions. The value of relates to a herd immunity threshold (HIT), which is given by . When the immune fraction of a population exceeds this threshold, sustained disease transmission becomes exponentially unlikely (barring mutations allowing SARS-CoV-2 to escape immunity). Here, we report state-level estimates obtained using Bayesian inference. Maximum a posteriori estimates range from 7.1 for New Jersey to 2.3 for Wyoming, indicating that disease transmission varies considerably across states and that reaching herd immunity will be more difficult in some states than others. estimates were obtained from compartmental models via the next-generation matrix approach after each model was parameterized using regional daily confirmed case reports of COVID-19 from 21 January 2020 to 21 June 2020. Our estimates characterize the infectiousness of ancestral strains, but they can be used to determine HITs for a distinct, currently dominant circulating strain, such as SARS-CoV-2 variant Delta (lineage B.1.617.2), if the relative infectiousness of the strain can be ascertained. On the basis of Delta-adjusted HITs, vaccination data, and seroprevalence survey data, we found that no state had achieved herd immunity as of 20 September 2021.

ANALYSIS OF CALCULATION METHODS, OBTAINING ESTIMATES AND ENSURING THE ACCURACY OF SIMULATION OF CONTROL OBJECTS

Systems of linear algebraic equations are a mathematical apparatus that is widely used in solving a significant number of problems in the practical application of mathematics and engineering. The analysis of errors in solving linear algebraic equations and the sensitivity function of nonlinear systems using the method of equivalent excitations, which, in turn, makes it possible to make informed decisions on the choice and development of methods for studying the accuracy of computing devices. Methods for constructing estimates «from below» of the distribution functions of the fatal error of the numerical solution of systems of linear algebraic equations are also presented, in particular, a posteriori estimates of the effectiveness of the methods under study are analyzed.

Richardson-Kalitkin method in abstract description

An abstract description of the RichardsonKalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem Ρ{{\Rho}} is considered, the solution of which results in a real number uu. To solve this problem, a numerical method is used, that is, the set Hℝ{H\subset \mathbb{R}} and the mapping uh:Hℝ{u_h:H\to\mathbb{R}} are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set HH and uh{u_h} can be expanded in a convergent series in powers of h:uh=u+c1hk+...{h:u_h=u+c_1h^k+...}. In this very general situation, the RichardsonKalitkin method is formulated for obtaining estimates for uu and cc from two values of uh{u_h}. The question of using a larger number of uh{u_h} values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the RichardsonKalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.

Bayesian Inference of State-Level COVID-19 Basic Reproduction Numbers across the United States

Although many persons in the United States have acquired immunity to COVID-19, either through vaccination or infection with SARS-CoV-2, COVID-19 will pose an ongoing threat to non-immune persons so long as disease transmission continues. We can estimate when sustained disease transmission will end in a population by calculating the population-specific basic reproduction number R_0, the expected number of secondary cases generated by an infected person in the absence of any interventions. The value of R_0 relates to a herd immunity threshold (HIT), which is given by 1-1/R_0. When the immune fraction of a population exceeds this threshold, sustained disease transmission becomes exponentially unlikely (barring mutations allowing SARS-CoV-2 to escape immunity). Here, we report state-level R_0 estimates obtained using Bayesian inference. Maximum a posteriori estimates range from 7.1 for New Jersey to 2.3 for Wyoming, indicating that disease transmission varies considerably across states and that reaching herd immunity will be more difficult in some states than others. R_0 estimates were obtained from compartmental models via the next-generation matrix approach after each model was parameterized using regional daily confirmed case reports of COVID-19 from 21-January-2020 to 21-June-2020. Our R_0 estimates characterize infectiousness of ancestral strains, but they can be used to determine HITs for a distinct, currently dominant circulating strain, such as SARS-CoV-2 variant Delta (lineage B.1.617.2), if the relative infectiousness of the strain can be ascertained. On the basis of Delta-adjusted HITs, vaccination data, and seroprevalence survey data, we find that no state has achieved herd immunity as of 20-September-2021.

Theoretical Analysis of the Relationship Between the Boundary Problem and Irregular Standard Errors in Diagnostic Classification Models

In diagnostic classification models, parameter estimation sometimes provides estimates that stick to the boundaries of the parameter space, which is called the boundary problem, and it may lead to extreme values of standard errors. However, the relationship between the boundary problem and irregular standard errors has not been analytically explored. In addition, prior research has not shown how maximum a posteriori estimates avoid the boundary problem and affect the standard errors of estimates. To analyze these relationships, the expectation-maximization algorithm for maximum a posteriori estimates and a complete data Fisher information matrix are explicitly derived for a mixture formulation of saturated diagnostic classification models. Theoretical considerations show that the emptiness of attribute mastery patterns causes both the boundary problem and the inaccurate standard error estimates. Furthermore, unfortunate boundary problem without emptiness causes shorter standard errors. A simulation study shows that the maximum a posteriori method prevents boundary problems and improves standard error estimates more than maximum likelihood estimates do. The effect is sometimes radical, and the results show that the maximum a posteriori method is more appropriate than the maximum likelihood method.

Score-Based Measurement Invariance Checks for Bayesian Maximum-a-Posteriori Estimates in Item Response Theory

A family of score-based tests has been proposed in the past years for assessing the invariance of model parameters in several models of item response theory. These tests were originally developed in a maximum likelihood framework. This study aims to extend the theoretical framework of these tests to Bayesian maximum-a-posteriori estimates and to multiple group IRT models. We propose two families of statistical tests, which are based on a) an approximation using a pooled variance method, or b) a simulation-based approach based on asymptotic results. The resulting tests were evaluated by a simulation study, which investigated their sensitivity against differential item functioning with respect to a categorical or continuous person covariate in the two- and three-parametric logistic models. Whereas the method based on pooled variance was found to be practically useful with maximum likelihood as well as maximum-a-posteriori estimates, the simulation-based approach was found to require large sample sizes to lead to satisfactory results.