On the asymptotic distribution of the size of a stochastic epidemic

1983 ◽  
Vol 20 (2) ◽  
pp. 390-394 ◽  
Author(s):  
Thomas Sellke
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Original Number ◽  
Intuitive Proof

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.

On the asymptotic distribution of the size of a stochastic epidemic

1983 ◽  
Vol 20 (02) ◽  
pp. 390-394 ◽  
Author(s):  
Thomas Sellke
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Original Number ◽  
Intuitive Proof

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.

Bounds on the velocity of spread of infection for a spatially connected epidemic process

1980 ◽  
Vol 17 (3) ◽  
pp. 839-845 ◽  
Author(s):  
M. J. Faddy ◽  
I. H. Slorach
Keyword(s):  
Asymptotic Velocity ◽  
Epidemic Process ◽  
Lower Bounding ◽  
Stochastic Epidemic ◽  
Spread Of Infection

The simple (non-spatial) stochastic epidemic is generalised to allow infected individuals to move forward through a system of spatially connected colonies C1, C2, C3, ·· ·each containing susceptible individuals. Upper and lower bounding processes are considered, to establish bounds on the asymptotic velocity of forward spread of the infection through these spatially connected colonies. These bounds are shown to be asymptotically equivalent under certain conditions, and some simulations reveal other features of the process.

Asymptotic distribution for the coupon — collector's and sampling-tagging problems, when tagging affects catchability

1975 ◽  
Vol 12 (3) ◽  
pp. 625-628 ◽  
Author(s):  
Ester Samuel-Cahn
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Waiting Time ◽  
Limiting Distributions

The coupon-collector's and sampling tagging problems are considered, under the model that each tagged element is γ > 0 times as likely to be caught at the next stage as every untagged element. Let WN(γ, k) and SN(γ, j) denote, respectively, the waiting time until k + 1 distinct elements are obtained, and the number of distinct elements in a sample of size j, when the population size is N. Complete characterisations are obtained for the limiting distributions of WN (γ, k) and SN(γ, j), in terms of the rates at which k and j tend to infinity with N.

Stochastic approximation for the general epidemic

1973 ◽  
Vol 10 (02) ◽  
pp. 263-276 ◽  
Author(s):  
Donald Ludwig
Keyword(s):  
Exact Solution ◽  
Population Size ◽  
Stochastic Approximation ◽  
Asymptotic Approximation ◽  
Asymptotic Approximations ◽  
System Of Equations ◽  
Numerical Comparisons ◽  
Stochastic Epidemic ◽  
Simple Equations ◽  
General Stochastic

A system of equations is introduced whose solutions are remarkably close to corresponding solutions of the “general stochastic epidemic”. If N is the population size, then there are approximately equations to be solved for the general stochastic epidemic while the number of equations in the approximating system is proportional to N. An asymptotic approximation to the general stochastic epidemic is also introduced. Numerical comparisons of the stochastic and asymptotic approximations with the exact solution are presented.

scholarly journals The effect of population size for pathogen transmission on prediction of COVID-19 spread

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Xuqi Zhang ◽  
Haiqi Liu ◽  
Hanning Tang ◽  
Mei Zhang ◽  
Xuedong Yuan ◽  
...  
Keyword(s):  
Public Health ◽  
Dynamic System ◽  
Population Size ◽  
Nonlinear Filtering ◽  
Hidden Markov ◽  
Pathogen Transmission ◽  
Health Interventions ◽  
Public Health Interventions ◽  
Stochastic Epidemic ◽  
Epidemic Dynamic

AbstractExtreme public health interventions play a critical role in mitigating the local and global prevalence and pandemic potential. Here, we use population size for pathogen transmission to measure the intensity of public health interventions, which is a key characteristic variable for nowcasting and forecasting of COVID-19. By formulating a hidden Markov dynamic system and using nonlinear filtering theory, we have developed a stochastic epidemic dynamic model under public health interventions. The model parameters and states are estimated in time from internationally available public data by combining an unscented filter and an interacting multiple model filter. Moreover, we consider the computability of the population size and provide its selection criterion. With applications to COVID-19, we estimate the mean of the effective reproductive number of China and the rest of the globe except China (GEC) to be 2.4626 (95% CI: 2.4142–2.5111) and 3.0979 (95% CI: 3.0968–3.0990), respectively. The prediction results show the effectiveness of the stochastic epidemic dynamic model with nonlinear filtering. The hidden Markov dynamic system with nonlinear filtering can be used to make analysis, nowcasting and forecasting for other contagious diseases in the future since it helps to understand the mechanism of disease transmission and to estimate the population size for pathogen transmission and the number of hidden infections, which is a valid tool for decision-making by policy makers for epidemic control.

A note on perturbation techniques for epidemics

1971 ◽  
Vol 3 (2) ◽  
pp. 214-218
Author(s):  
H. E. Daniels
Keyword(s):  
Difference Equations ◽  
Zero Order ◽  
Similar Type ◽  
Perturbation Techniques ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Explicit Formulae

This note is prompted by the papers of Weiss (this Symposium) and Bailey (1968). Weiss develops a technique for approximation to the moments of an epidemic process by regarding them as expandable in powers of N-1 where N is the size of the population, assumed constant. He first considers the simple stochastic epidemic with no removals and obtains explicit formulae for the terms of order N-1, the zero order terms being the deterministic values. Bailey is concerned with a similar type of approximation and he derives explicit results to the same order. Bailey uses an eigenfunction approach whereas Weiss's method is more direct and perhaps easier to generalise. However, in attempting to extend the method to the case of a closed epidemic with removals Weiss is led to intractable difference equations.

A threshold theorem for the Reed-Frost chain-binomial epidemic

1983 ◽  
Vol 20 (1) ◽  
pp. 153-157 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Population Size ◽  
Binomial Model ◽  
Susceptible Individual ◽  
Stochastic Epidemic ◽  
General Stochastic

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.

Stochastic approximation for the general epidemic

1973 ◽  
Vol 10 (2) ◽  
pp. 263-276 ◽  
Author(s):  
Donald Ludwig
Keyword(s):  
Exact Solution ◽  
Population Size ◽  
Stochastic Approximation ◽  
Asymptotic Approximation ◽  
Asymptotic Approximations ◽  
System Of Equations ◽  
Numerical Comparisons ◽  
Stochastic Epidemic ◽  
General Stochastic

A system of equations is introduced whose solutions are remarkably close to corresponding solutions of the “general stochastic epidemic”. If N is the population size, then there are approximately equations to be solved for the general stochastic epidemic while the number of equations in the approximating system is proportional to N. An asymptotic approximation to the general stochastic epidemic is also introduced. Numerical comparisons of the stochastic and asymptotic approximations with the exact solution are presented.

The age of a Galton-Watson population with a geometric offspring distribution

2002 ◽  
Vol 39 (4) ◽  
pp. 816-828 ◽  
Author(s):  
F. C. Klebaner ◽  
S. Sagitov
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Initial Population ◽  
Asymptotic Results ◽  
The Past ◽  
Offspring Distribution ◽  
Initial Population Size ◽  
Watson Process ◽  
Looking Back

Motivated by the question of the age in a branching population we try to recreate the past by looking back from the currently observed population size. We define a new backward Galton-Watson process and study the case of the geometric offspring distribution with parameter p in detail. The backward process is then the Galton-Watson process with immigration, again with a geometric offspring distribution but with parameter 1-p, and it is also the dual to the original Galton-Watson process. We give the asymptotic distribution of the age when the initial population size is large in supercritical and critical cases. To this end, we give new asymptotic results on the Galton-Watson immigration processes stopped at zero.

Asymptotic distribution for the coupon — collector's and sampling-tagging problems, when tagging affects catchability

1975 ◽  
Vol 12 (03) ◽  
pp. 625-628
Author(s):  
Ester Samuel-Cahn
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Waiting Time ◽  
Limiting Distributions

The coupon-collector's and sampling tagging problems are considered, under the model that each tagged element is γ > 0 times as likely to be caught at the next stage as every untagged element. Let WN (γ, k) and SN (γ, j) denote, respectively, the waiting time until k + 1 distinct elements are obtained, and the number of distinct elements in a sample of size j, when the population size is N. Complete characterisations are obtained for the limiting distributions of WN (γ, k) and SN (γ, j), in terms of the rates at which k and j tend to infinity with N.

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