Norman C. Severo

Generalizations of some stochastic epidemic models

Abstract In a population of size bounded by N , we let p rs ( t ) be the probability at time t or r susceptibles and s infectives. We assume that the probability for a new infective in the time interval ( t , t + dt ) is β r 1− b s a dt , where we call β the infection rate, a the infection power, and b the safety-in-numbers power. Furthermore, we assume that the probability that any infective is removed in ( t , t + dt ) is γ s 1+ c dt , where r is the removal rate and c the removal power. This article presents the solutions for p rs ( t ) for the cases (i) ϱ≡γ/β=0 and (ii) ϱ≠0.

Some properties of an extended simple stochastic epidemic model involving two additional parameters

Abstract In a fixed population of size N, we assume that the probability for a new infective in the time interval (t, t + dt) is βr1−b(N − r)a dt, where r is the number of susceptibles at time t, N - r is the number of infectives at time t, β is the infection rate, a is the infection power, and b is the safety-in-numbers power. If a = 1 and b = 0, then this probability would determine the familiar simple stochastic epidemic model. This article investigates the effect of the parameters a and b on the state probabilities and other derived quantities of the model.

COMPUTATIONAL AND ESTIMATION PROCEDURES IN MULTIDIMENSIONAL RIGHT-SHIFIt PROCESSES AND SOME APPLICATIONS

A right-shift process is a Markov process with multidimensional finite state space on which the infinitesimal transition movement is a shifting of one unit from one coordinate to some other to its right. A multidimensional right-shift process consists of v ? 1 concurrent and dependent right-shift processes. In this paper applications of multidimensional right-shift processes to some wellknown examples from epidemic theory, queueing theory and the Beetle probblem due to Lucien LeCam are discussed. A transformation which orders the Kolmogorov forward equations into a triangular array is provided and some computational procedures for solving the resulting system of equations are presented. One of these procedures is concerned with the problem of evaluating a given transition probability function rather than obtaining the solution to the complete system of forward equations. This particular procedure is applied to the problem of estimating the parameters of a multidimensional rightshift process which is observed at only a finite number of fixed timepoints.

DISTRIBUTIONS OF BALLOT PROBLEM RANDOM VARIABLES

Suppose that in a ballot candidate A scores a votes and candidate B scores b votes, and that all the possible voting records are equally probable. Corresponding to the first r votes, let a, and fir be the numbers of votes registered for A and B, respectively. Let p be an arbitrary positive real number. Denote by 6(a, b, p)[6*(a, b, p)] the number of values of r for which the inequality arl_ Pfir,[rr > Pfir], r = 1, - - - , a + b, holds. Heretofore the probability distributions of 6 and 6* have been derived for only a restricted set of values of a, b, and p, although, as pointed out here, they are obtainable for all values of (a, b, p) by using a result of Takaics (1964). In this paper we present a derivation of the distribution of 6[6*] whose development, for any (a, b, p), leads to both necessary and sufficient

The distribution of the 'finitely many times' in the strong law of large numbers

Consider independent, identically distributed random variables X1, X2,... with common mean [mu]. Let Sn = [Sigma]1nX1. and for [var epsilon] > 0 let N([var epsilon]) count the number of values of n for which n-1Sn - [mu] > [var epsilon]. This note discusses the distribution of N([var epsilon]), thereby elucidating the fluctuations associated with the Strong Law of Large Numbers.

Applications of Statistics in Post office Automation

The National Bureau of Standards is involved in designing and developing equipments and systems for the improved sorting of mail. As part of this project numerous statistical studies-mostly of a sampling typehave been conducted. These studies were designed to obtain research and development information which is not collected by the Post Office as part of its routine data collecting activities. The data from these studies have proved useful as a basis for making administrative decisions, (7) for developing efficient mechanized schemes of sorting mail (4) and for costing these schemes. (1) In this paper we present a thumbnail sketch of the letter sorting mechanization problem, a description of the statistical problems encountered with some results of corresponding studies, and some of the uses to which the data have been put.

Transition probabilities for a modified general stochastic epidemic model

We present a method for finding the transition probabilities of a modified general stochastic epidemic model containing two infection rates. The method uses interarrival times between events, following Billard, Lacayo and Langberg (1979), and a path counting technique of Kryscio (1975).

Distribution of Total Service Time for a Fixed Observation Interval

Abstract For a queue with many servers and constant holding time, the distribution of the total service time for a fixed observation interval during a period of statistical equilibrium has been derived. The derivation, while peculiar to the special case where holding time is assumed constant, involves elementary probabilistic arguments. An application is given in which the distribution of the total service time for a fixed observation interval is used to solve a problem in plumbing design.