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.

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.

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.

Compound Poisson limit theorems for Markov chains

1997 ◽  
Vol 34 (1) ◽  
pp. 24-34 ◽  
Author(s):  
Shoou-Ren Hsiau
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Markov Chains ◽  
Limit Theorems ◽  
Compound Poisson ◽  
Poisson Limit

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.

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.

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.

Compound Poisson limit theorems for Markov chains

1997 ◽  
Vol 34 (01) ◽  
pp. 24-34
Author(s):  
Shoou-Ren Hsiau
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Markov Chains ◽  
Limit Theorems ◽  
Compound Poisson ◽  
Poisson Limit

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.

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

2009 ◽  
Vol 46 (2) ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document