Poisson approximation for some epidemic models

1990 ◽  
Vol 27 (3) ◽  
pp. 479-490 ◽  
Author(s):  
Frank Ball ◽  
A. D. Barbour
Keyword(s):  
Limit Theorem ◽  
Directed Graph ◽  
Poisson Approximation ◽  
Epidemic Models ◽  
Final Size ◽  
Stochastic Epidemic ◽  
Order Of Magnitude ◽  
Poisson Limit ◽  
General Stochastic

The Daniels' Poisson limit theorem for the final size of a severe general stochastic epidemic is extended to the Martin-Löf epidemic, and an order of magnitude for the error in the approximation is also given. The argument consists largely of showing that the number of survivors of a severe epidemic is essentially the same as the number of isolated vertices in a random directed graph. Poisson approximation for the latter quantity is proved using the Stein–Chen method and a suitable coupling.

Poisson approximation for some epidemic models

1990 ◽  
Vol 27 (03) ◽  
pp. 479-490 ◽  
Author(s):  
Frank Ball ◽  
A. D. Barbour
Keyword(s):  
Limit Theorem ◽  
Directed Graph ◽  
Poisson Approximation ◽  
Epidemic Models ◽  
Final Size ◽  
Stochastic Epidemic ◽  
Order Of Magnitude ◽  
Poisson Limit ◽  
General Stochastic

The Daniels' Poisson limit theorem for the final size of a severe general stochastic epidemic is extended to the Martin-Löf epidemic, and an order of magnitude for the error in the approximation is also given. The argument consists largely of showing that the number of survivors of a severe epidemic is essentially the same as the number of isolated vertices in a random directed graph. Poisson approximation for the latter quantity is proved using the Stein–Chen method and a suitable coupling.

Compound Poisson limits for household epidemics

2005 ◽  
Vol 42 (2) ◽  
pp. 334-345 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Limit Theorem ◽  
Small Groups ◽  
Sufficient Conditions ◽  
Poisson Approximation ◽  
Current Work ◽  
Compound Poisson ◽  
Household Model ◽  
Stochastic Epidemic ◽  
Type Construction ◽  
Poisson Limit

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

Compound Poisson limits for household epidemics

2005 ◽  
Vol 42 (02) ◽  
pp. 334-345 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Limit Theorem ◽  
Small Groups ◽  
Sufficient Conditions ◽  
Poisson Approximation ◽  
Current Work ◽  
Compound Poisson ◽  
Household Model ◽  
Stochastic Epidemic ◽  
Type Construction ◽  
Poisson Limit

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

scholarly journals Poisson Approximation in a Poisson Limit Theorem Inspired by Coupon Collecting

2009 ◽  
Vol 46 (02) ◽  
pp. 585-592
Author(s):  
Anna Pósfai
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Random Variables ◽  
Poisson Approximation ◽  
Poisson Law ◽  
Coupon Collector’S Problem ◽  
Poisson Limit ◽  
Coupon Collector's Problem ◽  
Error Order

In this paper we refine a Poisson limit theorem of Gnedenko and Kolmogorov (1954): we determine the error order of a Poisson approximation for sums of asymptotically negligible integer-valued random variables that converge in distribution to the Poisson law. As an application of our results, we investigate the case of the coupon collector's problem when the distribution of the collector's waiting time is asymptotically Poisson.

scholarly journals Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

2007 ◽  
Vol 44 (01) ◽  
pp. 41-57 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Size Distribution ◽  
Random Graph ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Final Size ◽  
Sir Epidemic ◽  
Novel Approach ◽  
Stochastic Epidemic ◽  
Sir Epidemic Models ◽  
Variable Susceptibility

The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.

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.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (1) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n2). The general case is discussed in terms of operator semigroups.

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