scholarly journals Pathogen Evolution when Transmission and Virulence are Stochastic

2019 ◽  
Author(s):  
Pooya Aavani ◽  
Sean H. Rice
Keyword(s):  
Opposite Sign ◽  
Random Variables ◽  
Sir Model ◽  
Stochastic Effects ◽  
Pathogen Evolution ◽  
Individual Fitness ◽  
Pathogen Dynamics ◽  
General Stochastic Model ◽  
Special Case ◽  
Vital Parameters

AbstractEvolutionary processes are inherently stochastic, since we can never know with certainty exactly how many descendants an individual will leave, or what the phenotypes of those descendants will be. Despite this, models of pathogen evolution have nearly all been deterministic, treating values such as transmission and virulence as parameters that can be known ahead of time. We present a broadly applicable analytic approach for modeling pathogen evolution in which vital parameters such as transmission and virulence are treated as random variables, rather than as fixed values. Starting from a general stochastic model of evolution, we derive specific equations for the evolution of transmission and virulence, and then apply these to a particular special case; the SIR model of pathogen dynamics. We show that adding stochasticity introduces new directional components to pathogen evolution. In particular, two kinds of covariation between traits emerge as important: covariance across the population (what is usually measured), and covariance between random variables within an individual. We show that these different kinds of trait covariation can be of opposite sign and contribute to evolution in very different ways. In particular, probability covariation between random variables within an individual is sometimes a better way to capture evolutionarily important tradeoffs than is covariation across a population. We further show that stochasticity can influence pathogen evolution through directional stochastic effects, which results from the inevitable covariance between individual fitness and mean population fitness.

scholarly journals A general stochastic model for bivariate episodes driven by a gamma sequence

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Charles K. Amponsah ◽  
Tomasz J. Kozubowski ◽  
Anna K. Panorska
Keyword(s):  
Stochastic Model ◽  
Joint Distribution ◽  
Pareto Distribution ◽  
Integral Transforms ◽  
Bivariate Distribution ◽  
Conditional Distributions ◽  
General Stochastic Model ◽  
Heavy Tailed ◽  
Discrete Pareto Distribution ◽  
Special Case

AbstractWe propose a new stochastic model describing the joint distribution of (X,N), where N is a counting variable while X is the sum of N independent gamma random variables. We present the main properties of this general model, which include marginal and conditional distributions, integral transforms, moments and parameter estimation. We also discuss in more detail a special case where N has a heavy tailed discrete Pareto distribution. An example from finance illustrates the modeling potential of this new mixed bivariate distribution.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

scholarly journals Genetic Diversity in the SIR Model of Pathogen Evolution

PLoS ONE ◽  
2009 ◽  
Vol 4 (3) ◽  
pp. e4876 ◽  
Author(s):  
Isabel Gordo ◽  
M. Gabriela M. Gomes ◽  
Daniel G. Reis ◽  
Paulo R. A. Campos
Keyword(s):  
Genetic Diversity ◽  
Sir Model ◽  
Pathogen Evolution

scholarly journals Improved bounds for the availability and unavailability in a fixed time interval for systems of maintained, interdependent components

1980 ◽  
Vol 12 (01) ◽  
pp. 200-221 ◽  
Author(s):  
B. Natvig
Keyword(s):  
Lower Bound ◽  
System Reliability ◽  
Nuclear Reactors ◽  
Random Variables ◽  
Fixed Time ◽  
Time Interval ◽  
Coherent System ◽  
Exact Results ◽  
Fixed Time Interval ◽  
Special Case

In this paper we arrive at a series of bounds for the availability and unavailability in the time interval I = [t A , t B ] ⊂ [0, ∞), for a coherent system of maintained, interdependent components. These generalize the minimal cut lower bound for the availability in [0, t] given in Esary and Proschan (1970) and also most bounds for the reliability at time t given in Bodin (1970) and Barlow and Proschan (1975). In the latter special case also some new improved bounds are given. The bounds arrived at are of great interest when trying to predict the performance process of the system. In particular, Lewis et al. (1978) have revealed the great need for adequate tools to treat the dependence between the random variables of interest when considering the safety of nuclear reactors. Satyanarayana and Prabhakar (1978) give a rapid algorithm for computing exact system reliability at time t. This can also be used in cases where some simpler assumptions on the dependence between the components are made. It seems, however, impossible to extend their approach to obtain exact results for the cases treated in the present paper.

scholarly journals Upgrading Probability via Fractions of Events

2016 ◽  
Vol 24 (1) ◽  
pp. 29-41 ◽  
Author(s):  
Roman Frič ◽  
Martin Papčo
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Random Variable ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Black And White ◽  
Random Events ◽  
Indicator Functions ◽  
Quantum Phenomena ◽  
Special Case

Abstract The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

scholarly journals NOTES AND PROBLEMS A GENERAL BOUND FOR THE LIMITING DISTRIBUTION OF BREITUNG'S STATISTIC

2008 ◽  
Vol 24 (5) ◽  
pp. 1443-1455 ◽  
Author(s):  
James Davidson ◽  
Jan R. Magnus ◽  
Jan Wiegerinck
Keyword(s):  
General Result ◽  
Random Variables ◽  
Nonparametric Test ◽  
Limiting Distribution ◽  
Residue Theorem ◽  
Prove Theorem ◽  
General Bound ◽  
Result Theorem ◽  
Special Case ◽  
Cauchy’S Residue Theorem

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue theorem.

scholarly journals Cumulants and partition lattices III: Multiply-indexed arrays

1986 ◽  
Vol 40 (2) ◽  
pp. 161-182 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Random Variables ◽  
Unbiased Estimation ◽  
Linear Combinations ◽  
Components Of Variance ◽  
Partition Lattices ◽  
Special Case

AbstractEarlier work of the author exploiting the role of partition lattices and their Mbius functions in the theory of cumulants, k-statistics and their generalisations is extended to multiply-indexed arrays of random variables. The natural generalisations of cumulants and k-statistics to this context are shown to include components of variance and the associated linear combinations of mean-squares which are used to estimate them. Expressions for the generalised cumulants of arrays built up as sums of independent arrays of effects as in anova models are derived in terms of the generalized cumulants of the effects. The special case of degree two, covering the unbiased estimation of components of variance, is discussed in some detail.

A representation for discrete distributions by equiprobable mixtures

1980 ◽  
Vol 17 (01) ◽  
pp. 102-111 ◽  
Author(s):  
Arthur V. Peterson ◽  
Richard A. Kronmal
Keyword(s):  
Random Variables ◽  
Discrete Distribution ◽  
Discrete Distributions ◽  
Important Special Case ◽  
Discrete Random Variables ◽  
Special Case

We obtain a representation of an arbitrary discrete distribution with n mass points by an equiprobable mixture of r distributions, each of which has no more than a (≧2) mass points, where r is the smallest integer greater than or equal to (n – 1)/(a – 1). An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.

A d-DEFORMATION OF FREE GAUSSIAN RANDOM VARIABLES

2005 ◽  
Vol 08 (04) ◽  
pp. 669-680 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Tensor Product ◽  
Hypergeometric Series ◽  
Fock Space ◽  
Recurrence Formula ◽  
Random Variables ◽  
Product Formula ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Gaussian Random Variables ◽  
Special Case

We define a deformation of free creations (and annihilations), given by operators on the full Fock space, acting nontrivially only between the vacuum subspace ℂΩ and the twofold tensor product [Formula: see text]. Then we study the distribution of the deformed free gaussian operators, with the deformation containing also a real parameter d. The recurrence formula for moments is shown, and the Cauchy transform of the distribution measure is computed. This yields the description of the measure: absolutely continuous part and the atomic part. The existence of atoms depends on the parameter d. The special case d =1 is studied with all details, with the formula for moments is given as values of the hypergeometric series. Finally we show the formula for computing the mixed moments of the deformed operators.

Existence and positivity of the limit in processes with a branching structure

1999 ◽  
Vol 36 (1) ◽  
pp. 132-138
Author(s):  
M. P. Quine ◽  
W. Szczotka
Keyword(s):  
Stochastic Process ◽  
Branching Process ◽  
Random Variables ◽  
Random Variable ◽  
Natural Generalization ◽  
Partial Sums ◽  
Urn Model ◽  
Special Case ◽  
Independent And Identically Distributed

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

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