A modification of the general stochastic epidemic motivated by AIDS modelling

1993 ◽  
Vol 25 (01) ◽  
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.

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.

Asymptotic analysis of the general stochastic epidemic with variable infectious periods

2001 ◽  
Vol 38 (01) ◽  
pp. 18-35 ◽  
Author(s):  
A. N. Startsev
Keyword(s):  
Asymptotic Analysis ◽  
Asymptotic Behaviour ◽  
Size Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Boundary Crossing ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.

A note on the total size distribution of epidemic models

1986 ◽  
Vol 23 (03) ◽  
pp. 832-836 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Size Distribution ◽  
Epidemic Models ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

A simple coupling argument is used to obtain a new proof of a result of Daniels (1967) concerning the total size distribution of the general stochastic epidemic. The proof admits a straightforward generalisation to multipopulation epidemics and indicates that similar results are unlikely to be available for epidemics with non-exponential infectious periods.

A note on the total size distribution of epidemic models

1986 ◽  
Vol 23 (3) ◽  
pp. 832-836 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Size Distribution ◽  
Epidemic Models ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

A simple coupling argument is used to obtain a new proof of a result of Daniels (1967) concerning the total size distribution of the general stochastic epidemic. The proof admits a straightforward generalisation to multipopulation epidemics and indicates that similar results are unlikely to be available for epidemics with non-exponential infectious periods.

Asymptotic analysis of the general stochastic epidemic with variable infectious periods

2001 ◽  
Vol 38 (1) ◽  
pp. 18-35 ◽  
Author(s):  
A. N. Startsev
Keyword(s):  
Asymptotic Analysis ◽  
Asymptotic Behaviour ◽  
Size Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Boundary Crossing ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.

A Dynamic Generalization of Zipf's Rank-Size Rule

1982 ◽  
Vol 14 (11) ◽  
pp. 1449-1467 ◽  
Author(s):  
B Roehner ◽  
K E Wiese
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Urban Growth ◽  
Simple Form ◽  
Deterministic Model ◽  
City Size ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
City Size Distribution ◽  
Zero Growth

A dynamic deterministic model of urban growth is proposed, which in its most simple form yields Zipf's law for city-size distribution, and in its general form may account for distributions that deviate strongly from Zipf's law. The qualitative consequences of the model are examined, and a corresponding stochastic model is introduced, which permits, in particular, the study of zero-growth situations.

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.

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 TOTAL SIZE OF A GENERAL STOCHASTIC EPIDEMIC

Biometrika ◽  
1953 ◽  
Vol 40 (1-2) ◽  
pp. 177-185 ◽  
Author(s):  
NORMAN T. J. BAILEY
Keyword(s):  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

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

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