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Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), the etiological agent of the Coronavirus Disease 2019 (COVID-19) disease, has moved rapidly around the globe, infecting millions and killing hundreds of thousands. The basic reproduction number, which has been widely used—appropriately and less appropriately—to characterize the transmissibility of the virus, hides the fact that transmission is stochastic, often dominated by a small number of individuals, and heavily influenced by superspreading events (SSEs). The distinct transmission features of SARS-CoV-2, e.g., high stochasticity under low prevalence (as compared to other pathogens, such as influenza), and the central role played by SSEs on transmission dynamics cannot be overlooked. Many explosive SSEs have occurred in indoor settings, stoking the pandemic and shaping its spread, such as long-term care facilities, prisons, meat-packing plants, produce processing facilities, fish factories, cruise ships, family gatherings, parties, and nightclubs. These SSEs demonstrate the urgent need to understand routes of transmission, while posing an opportunity to effectively contain outbreaks with targeted interventions to eliminate SSEs. Here, we describe the different types of SSEs, how they influence transmission, empirical evidence for their role in the COVID-19 pandemic, and give recommendations for control of SARS-CoV-2.

The Poisson distribution is a discrete probability model, widely used in science and engineering to describe various natural and man-made phenomena. It possesses an important feature, namely being inherently asymmetric, but as its parameter becomes large, the distribution becomes approximately symmetric. To broaden its use, multiple extensions and variations have been developed. Determining whether a data set follows a Poisson distribution involves hypothesis testing at a chosen significance level. When sampling from a Poisson distribution, confidence intervals provide an estimated range instead of a single value. Due to the discrete nature of the Poisson distribution, confidence intervals cannot be derived from a simple formula, and are therefore computed using specialized algorithms. In this paper, three alternatives are given and discussed.

Poisson regression is a popular tool for modeling count data and is applied in a vast array of applications from the social to the physical sciences and beyond. Real data, however, are often over- or under-dispersed and, thus, not conducive to Poisson regression. We propose a regression model based on the Conway—Maxwell-Poisson (COM-Poisson) distribution to address this problem. The COM-Poisson regression generalizes the well-known Poisson and logistic regression models, and is suitable for fitting count data with a wide range of dispersion levels. With a GLM approach that takes advantage of exponential family properties, we discuss model estimation, inference, diagnostics, and interpretation, and present a test for determining the need for a COM-Poisson regression over a standard Poisson regression. We compare the COM-Poisson to several alternatives and illustrate its advantages and usefulness using three data sets with varying dispersion.

Studying the positive definiteness of the covariance matrix of discrete samples helps to determine whether the dimensionality of the samples can be reduced, which is beneficial for optimizing the number of samples and designing optimal plans for sampling surveys. This paper aims to provide a method to determine the variable numbers of the sample subjecting to Poisson distribution. Methods . It is based on the theory of I -linear combination and its properties which are the author’s previous studying results. Results . study shows the covariance matrix of multi-Poisson distribution is positively defined and the probability of the sample covariance matrix of multi-poisson distribution is about 1 when the sample capacity is very large. Conclusion . The dimension size of the sample data matrix of multi-poisson distribution can be reduced when the sample capacity n is no more than the dimension size p .

We consider some non-uniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs–Markov–Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball $B(x,r)$ converges to a Poisson distribution as the radius $r\\rightarrow 0$ and after suitable normalization.

In this note, the dispersion of -Poisson distribution is studied. It is specifically shown that this distribution is over-dispersed when and under-dispersed when .

This article aims to gain implicit relations among different subclasses of harmonic family of analytic functions containing with some description for Poisson distribution series to be in the new subclasses of harmonic convex functions and harmonic starlike, ( α ) and U W * ( α ) respectively, in the plane.

We develop a statistical model for the analysis and forecasting of football match results which assumes a bivariate Poisson distribution with intensity coefficients that change stochastically over time. The dynamic model is a novelty in the statistical time series analysis of match results in team sports. Our treatment is based on state space and importance sampling methods which are computationally efficient. The out-of-sample performance of our methodology is verified in a betting strategy that is applied to the match outcomes from the 2010–2011 and 2011–2012 seasons of the English football Premier League. We show that our statistical modelling framework can produce a significant positive return over the bookmaker's odds.

Many countries with weaker health systems are struggling to put together a coherent strategy against the COVID-19 epidemic. We explored COVID-19 control strategies that could offer the greatest benefit in resource limited settings. Using an age-structured SEIR model, we explored the effects of COVID-19 control interventions-a lockdown, physical distancing measures, and active case finding (testing and isolation, contact tracing and quarantine)-implemented individually and in combination to control a hypothetical COVID-19 epidemic in Kathmandu (population 2.6 million), Nepal. A month-long lockdown will delay peak demand for hospital beds by 36 days, as compared to a base scenario of no intervention (peak demand at 108 days (IQR 97-119); a 2 month long lockdown will delay it by 74 days, without any difference in annual mortality, or healthcare demand volume. Year-long physical distancing measures will reduce peak demand to 36% (IQR 23%-46%) and annual morality to 67% (IQR 48%-77%) of base scenario. Following a month long lockdown with ongoing physical distancing measures and an active case finding intervention that detects 5% of the daily infection burden could reduce projected morality and peak demand by more than 99%. Limited resource settings are best served by a combination of early and aggressive case finding with ongoing physical distancing measures to control the COVID-19 epidemic. A lockdown may be helpful until combination interventions can be put in place but is unlikely to reduce annual mortality or healthcare demand.

In this article, new properties of the Poisson distribution of order k with parameter λ are found. Based on them, the modes of the Poisson distributions of order k=3 and 4 are derived for λ in (0,1). They are 0, 3, 5, and 0, 4, 7, 8, respectively, for λ in specified subintervals of (0, 1). In addition, using Mathematica, computational results for the modes of the Poisson distributions of order k=2,3, and 4 are presented for λ in specified subintervals of (0,2).