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Our purpose of this paper is to investigate positive solutions of the elliptic equation with regional fractional Laplacian $$ ( - \Delta )_{B_1}^s u +u= h(x,u) \quad {\rm in} \ \, B_1,\qquad u\in C_0(B_1), $$ where $( - \Delta )_{B_1}^s$ with $s\in(0,\frac12]$ is the regional fractional Laplacian and $h$ is the nonlinearity. Ordinarily, positive solutions vanishing at the boundary are not anticipated to be derived for the equations with regional fractional Laplacian of order $s\in(0,\frac12]$. Positive solutions are obtained when the nonlinearity assumes the following two models: $h(x,t)=f(x)$ or $h(x,t)=h_1(x)\, t^p+ \epsilon h_2(x)$, where $p>1$, $\epsilon>0$ small and $f, h_1, h_2$ are H\"older continuous, radially symmetric and decreasing functions under suitable conditions., Comment: 30

Statistical methods are developed for analysis of clinical and virus genetics data from phase 3 randomized, placebo-controlled trials of vaccines against novel coronavirus COVID-19. Vaccine efficacy (VE) of a vaccine to prevent COVID-19 caused by one of finitely many genetic strains of SARS-CoV-2 may vary by strain. The problem of assessing differential VE by viral genetics can be formulated under a competing risks model where the endpoint is virologically confirmed COVID-19 and the cause-of-failure is the infecting SARS-CoV-2 genotype. Strain-specific VE is defined as one minus the cause-specific hazard ratio (vaccine/placebo). For the COVID-19 VE trials, the time to COVID-19 is right-censored, and a substantial percentage of failure cases are missing the infecting virus genotype. We develop estimation and hypothesis testing procedures for strain-specific VE when the failure time is subject to right censoring and the cause-of-failure is subject to missingness, focusing on $J \ge 2$ discrete categorical unordered or ordered virus genotypes. The stratified Cox proportional hazards model is used to relate the cause-specific outcomes to explanatory variables. The inverse probability weighted complete-case (IPW) estimator and the augmented inverse probability weighted complete-case (AIPW) estimator are investigated. Hypothesis tests are developed to assess whether the vaccine provides at least a specified level of efficacy against some viral genotypes and whether VE varies across genotypes, adjusting for covariates. The finite-sample properties of the proposed tests are studied through simulations and are shown to have good performances. In preparation for the real data analyses, the developed methods are applied to a pseudo dataset mimicking the Moderna COVE trial.

For time-to-event data with finitely many competing risks, the proportional hazards model has been a popular tool for relating the cause-specific outcomes to covariates [Prentice et al. Biometrics 34 (1978) 541--554]. This article studies an extension of this approach to allow a continuum of competing risks, in which the cause of failure is replaced by a continuous mark only observed at the failure time. We develop inference for the proportional hazards model in which the regression parameters depend nonparametrically on the mark and the baseline hazard depends nonparametrically on both time and mark. This work is motivated by the need to assess HIV vaccine efficacy, while taking into account the genetic divergence of infecting HIV viruses in trial participants from the HIV strain that is contained in the vaccine, and adjusting for covariate effects. Mark-specific vaccine efficacy is expressed in terms of one of the regression functions in the mark-specific proportional hazards model. The new approach is evaluated in simulations and applied to the first HIV vaccine efficacy trial., Comment: Published in at http://dx.doi.org/10.1214/07-AOS554 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)