Factorial moments and probabilities for the general stochastic epidemic

1973 ◽  
Vol 10 (2) ◽  
pp. 277-288 ◽  
Author(s):  
L. Billard
Keyword(s):  
Difference Equations ◽  
The State ◽  
Factorial Moments ◽  
Stochastic Epidemic ◽  
The Matrix ◽  
Mean And Variance ◽  
The Mean ◽  
General Stochastic

By an appropriate partitioning of the matrix of coefficients in the system of differential difference equations for the general stochastic epidemic, the nature of the state probabilities is shown to consist of combinations of factorial terms. Further, factorial moments are readily obtained. In particular, the mean and variance of the number of susceptibles are derived.

Factorial moments and probabilities for the general stochastic epidemic

1973 ◽  
Vol 10 (02) ◽  
pp. 277-288 ◽  
Author(s):  
L. Billard
Keyword(s):  
Difference Equations ◽  
The State ◽  
Factorial Moments ◽  
Stochastic Epidemic ◽  
The Matrix ◽  
Mean And Variance ◽  
The Mean ◽  
General Stochastic

By an appropriate partitioning of the matrix of coefficients in the system of differential difference equations for the general stochastic epidemic, the nature of the state probabilities is shown to consist of combinations of factorial terms. Further, factorial moments are readily obtained. In particular, the mean and variance of the number of susceptibles are derived.

The cost of a general stochastic epidemic

1972 ◽  
Vol 9 (02) ◽  
pp. 257-269 ◽  
Author(s):  
J. Gani ◽  
D. Jerwood
Keyword(s):  
Recursive Equation ◽  
Stieltjes Transform ◽  
Stochastic Epidemic ◽  
The Mean ◽  
The Cost ◽  
Mean Cost ◽  
General Stochastic

This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis ) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis . Using the results obtained, bounds are found for the mean cost of the epidemic.

Two theorems on solutions of differential-difference equations and applications to epidemic theory

1967 ◽  
Vol 4 (02) ◽  
pp. 271-280 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Differential Equation ◽  
Difference Equations ◽  
Total Population ◽  
Original System ◽  
Initial Distribution ◽  
Fixed Number ◽  
Total Population Size ◽  
Stochastic Epidemic ◽  
Iterative Solutions ◽  
General Stochastic

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.

General stochastic epidemic with recovery

1975 ◽  
Vol 12 (01) ◽  
pp. 29-38 ◽  
Author(s):  
L. Billard
Keyword(s):  
Stochastic Model ◽  
Finite Number ◽  
Epidemic Model ◽  
The State ◽  
Stochastic Epidemic ◽  
General Stochastic

The general epidemic model does not provide for an infective recovering and thence being susceptible to further infection in the course of the epidemic. By considering the case in which recovery can occur once, we show how the state probabilities can be found for the stochastic model. This is readily extended to allow recovery up to a finite number of times.

The distribution and moments of the number of components of a random function

1990 ◽  
Vol 27 (01) ◽  
pp. 202-207 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Probability Distribution ◽  
Random Function ◽  
Simple Formula ◽  
Computer Calculation ◽  
Factorial Moments ◽  
Number Of Components ◽  
Mean And Variance ◽  
The Mean

A relatively simple formula is presented for the probability distribution of the number K of components of a random function. This formula facilitates the (computer) calculation of the factorial moments of K and yields new expressions for the mean and variance of K.

scholarly journals Prediction of rates of inbreeding in selected populations

1990 ◽  
Vol 55 (1) ◽  
pp. 41-54 ◽  
Author(s):  
Naomi R. Wray ◽  
Robin Thompson
Keyword(s):  
Selective Advantage ◽  
Mass Selection ◽  
Effective Population ◽  
The Matrix ◽  
Mean And Variance ◽  
Unselected Population ◽  
The Mean ◽  
Number Of Individuals ◽  
The Relationship

SummaryA method is presented for the prediction of rate of inbreeding for populations with discrete generations. The matrix of Wright's numerator relationships is partitioned into ‘contribution’ matrices which describe the contribution of the Mendelian sampling of genes of ancestors in a given generation to the relationship between individuals in later generations. These contributions stabilize with time and the value to which they stabilize is shown to be related to the asymptotic rate of inbreeding and therefore also the effective population size, where N is the number of individuals per generation and μr and are the mean and variance of long-term relationships or long-term contributions. These stabilized values are then predicted using a recursive equation via the concept of selective advantage for populations with hierarchical mating structures undergoing mass selection. Account is taken of the change in genetic parameters as a consequence of selection and also the increasing ‘competitiveness’ of contemporaries as selection proceeds. Examples are given and predicted rates of inbreeding are compared to those calculated in simulations. For populations of 20 males and 20, 40, 100 or 200 females the rate of inbreeding was found to increase by as much as 75% over the rate of inbreeding in an unselected population depending on mating ratio, selection intensity and heritability of the selected trait. The prediction presented here estimated the rate of inbreeding usually within 5% of that calculated from simulation.

Two theorems on solutions of differential-difference equations and applications to epidemic theory

1967 ◽  
Vol 4 (2) ◽  
pp. 271-280 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Differential Equation ◽  
Difference Equations ◽  
Total Population ◽  
Original System ◽  
Initial Distribution ◽  
Fixed Number ◽  
Total Population Size ◽  
Stochastic Epidemic ◽  
Iterative Solutions ◽  
General Stochastic

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.

The cost of a general stochastic epidemic

1972 ◽  
Vol 9 (2) ◽  
pp. 257-269 ◽  
Author(s):  
J. Gani ◽  
D. Jerwood
Keyword(s):  
Recursive Equation ◽  
Stieltjes Transform ◽  
Stochastic Epidemic ◽  
The Mean ◽  
The Cost ◽  
Mean Cost ◽  
General Stochastic

This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis. Using the results obtained, bounds are found for the mean cost of the epidemic.

Optimal Distributed Fusion Filtering with the Mean and Variance Information of the State*

2019 ◽  
Author(s):  
Jiabing Sun ◽  
Chengjin Zhang ◽  
Runtao Yang ◽  
Xiaoxia Sun
Keyword(s):  
The State ◽  
Mean And Variance ◽  
The Mean ◽  
Distributed Fusion

The rumour process

1982 ◽  
Vol 19 (04) ◽  
pp. 759-766
Author(s):  
Ross Dunstan
Keyword(s):  
Epidemic Model ◽  
Asymptotic Form ◽  
Deterministic Model ◽  
Final Size ◽  
Stochastic Epidemic ◽  
The Mean ◽  
Stochastic Epidemic Model ◽  
General Stochastic

The general stochastic epidemic model is used as a model for the spread of rumours. Recursive expressions are found for the mean of the final size of each generation of hearers. Simple expressions are found for the generation size and the asymptotic form of its final size in the deterministic model. An approximating process is presented.

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