A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models

1986 ◽  
Vol 18 (2) ◽  
pp. 289-310 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Probabilistic Approach ◽  
Heterogeneous Population ◽  
Epidemic Models ◽  
Infectious Period ◽  
Unified Approach ◽  
Total Size ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
General Stochastic

We provide a unified probabilistic approach to the distribution of total size and total area under the trajectory of infectives for a general stochastic epidemic with any specified distribution of the infectious period. The key tool is a Wald&s identity for the epidemic process. The generalisation of our results to epidemics spreading amongst a heterogeneous population is straightforward.

A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models

1986 ◽  
Vol 18 (02) ◽  
pp. 289-310 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Probabilistic Approach ◽  
Heterogeneous Population ◽  
Epidemic Models ◽  
Infectious Period ◽  
Unified Approach ◽  
Total Size ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
General Stochastic

We provide a unified probabilistic approach to the distribution of total size and total area under the trajectory of infectives for a general stochastic epidemic with any specified distribution of the infectious period. The key tool is a Wald&s identity for the epidemic process. The generalisation of our results to epidemics spreading amongst a heterogeneous population is straightforward.

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 (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.

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 threshold behaviour of epidemic models

1983 ◽  
Vol 20 (2) ◽  
pp. 227-241 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Almost Sure Convergence ◽  
Death Process ◽  
Time Interval ◽  
Birth And Death Process ◽  
Initial Number ◽  
Finite Time Interval ◽  
Threshold Behaviour ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

The Total Size of a General Stochastic Epidemic

Biometrika ◽  
1953 ◽  
Vol 40 (1/2) ◽  
pp. 177 ◽  
Author(s):  
Norman T. J. Bailey
Keyword(s):  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

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 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.

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.

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