General stochastic epidemic with recovery

1975 ◽  
Vol 12 (1) ◽  
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.

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 transition probabilities of the general stochastic epidemic model

1975 ◽  
Vol 12 (3) ◽  
pp. 415-424 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Stochastic Model ◽  
Convenient Method ◽  
Epidemic Model ◽  
Transition Probabilities ◽  
Stochastic Epidemic ◽  
General Stochastic Model ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
General Stochastic

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.

The transition probabilities of the general stochastic epidemic model

1975 ◽  
Vol 12 (03) ◽  
pp. 415-424 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Stochastic Model ◽  
Convenient Method ◽  
Epidemic Model ◽  
Transition Probabilities ◽  
Stochastic Epidemic ◽  
General Stochastic Model ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
General Stochastic

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.

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 estimation of parameters from population data on the general stochastic epidemic

1971 ◽  
Vol 3 (2) ◽  
pp. 211-214
Author(s):  
Norman T. J. Bailey ◽  
Anthony S. Thomas
Keyword(s):  
Stochastic Model ◽  
Population Data ◽  
Estimation Of Parameters ◽  
Stochastic Epidemic ◽  
Image Position ◽  
General Stochastic

We consider the usual stochastic model of a general epidemic, consisting of n + 1 homogeneously mixing individuals, assuming that initially when t = 0 there is one infective and n susceptibles (Bailey (1957)). At any time t we suppose that there are r susceptibles still uninfected, s infectives in circulation, and u individuals who have been removed (and are dead, isolated, or recovered and immune), where

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.

The estimation of parameters from population data on the general stochastic epidemic

1971 ◽  
Vol 3 (02) ◽  
pp. 211-214
Author(s):  
Norman T. J. Bailey ◽  
Anthony S. Thomas
Keyword(s):  
Stochastic Model ◽  
Population Data ◽  
Estimation Of Parameters ◽  
Stochastic Epidemic ◽  
Image Position ◽  
General Stochastic

We consider the usual stochastic model of a general epidemic, consisting of n + 1 homogeneously mixing individuals, assuming that initially when t = 0 there is one infective and n susceptibles (Bailey (1957)). At any time t we suppose that there are r susceptibles still uninfected, s infectives in circulation, and u individuals who have been removed (and are dead, isolated, or recovered and immune), where

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.

A modification of the general stochastic epidemic motivated by AIDS modelling

1993 ◽  
Vol 25 (1) ◽  
pp. 39-62 ◽  
Author(s):  
Frank Ball ◽  
Philip O'neill
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Deterministic Model ◽  
Time Dependent ◽  
Total Size ◽  
Model Comparisons ◽  
Stochastic Epidemic ◽  
Time Dependent Solution ◽  
General Stochastic

This paper considers a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate βxy/(x + y), where x and y are the numbers of susceptible and infectious individuals, respectively, and β is an infection parameter. This contrasts with the standard general epidemic in which new infections occur at rate βxy. Both the deterministic and stochastic versions of the modified epidemic are analysed. The deterministic model is completely soluble. The time-dependent solution of the stochastic model is derived and the total size distribution is considered. Threshold theorems, analogous to those of Whittle (1955) and Williams (1971) for the general stochastic epidemic, are proved for the stochastic model. Comparisons are made between the modified and general epidemics. The effect of introducing variability in susceptibility into the modified epidemic is studied.

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