On the size distribution for some epidemic models

1980 ◽  
Vol 17 (04) ◽  
pp. 912-921 ◽  
Author(s):  
Ray Watson
Keyword(s):  
Size Distribution ◽  
Epidemic Model ◽  
Time Scale ◽  
Epidemic Models ◽  
Random Time ◽  
Scale Transformation ◽  
Asymptotic Approximations

We consider the standard epidemic model and several extensions of this model, including Downton's carrier-borne epidemic model. A random time-scale transformation is used to obtain equations for the size distribution and to derive asymptotic approximations for the size distribution for each of the models

On the size distribution for some epidemic models

1980 ◽  
Vol 17 (4) ◽  
pp. 912-921 ◽  
Author(s):  
Ray Watson
Keyword(s):  
Size Distribution ◽  
Epidemic Model ◽  
Time Scale ◽  
Epidemic Models ◽  
Random Time ◽  
Scale Transformation ◽  
Asymptotic Approximations

We consider the standard epidemic model and several extensions of this model, including Downton's carrier-borne epidemic model. A random time-scale transformation is used to obtain equations for the size distribution and to derive asymptotic approximations for the size distribution for each of the models

A useful random time-scale transformation for the standard epidemic model

1980 ◽  
Vol 17 (2) ◽  
pp. 324-332 ◽  
Author(s):  
Ray Watson
Keyword(s):  
Size Distribution ◽  
Epidemic Model ◽  
Time Scale ◽  
Random Time ◽  
Scale Transformation ◽  
Threshold Parameter ◽  
Asymptotic Results ◽  
Simple Derivation ◽  
Major Outbreak

In this paper it is shown that a random time-scale transformation leads to a simple derivation of some asymptotic results describing the progress of a major outbreak in the standard epidemic model. These results find application in approximation of the size distribution and in estimation of the threshold parameter.

A useful random time-scale transformation for the standard epidemic model

1980 ◽  
Vol 17 (02) ◽  
pp. 324-332 ◽  
Author(s):  
Ray Watson
Keyword(s):  
Size Distribution ◽  
Epidemic Model ◽  
Time Scale ◽  
Random Time ◽  
Scale Transformation ◽  
Threshold Parameter ◽  
Asymptotic Results ◽  
Simple Derivation ◽  
Major Outbreak

In this paper it is shown that a random time-scale transformation leads to a simple derivation of some asymptotic results describing the progress of a major outbreak in the standard epidemic model. These results find application in approximation of the size distribution and in estimation of the threshold parameter.

A bivariate counting process

1993 ◽  
Vol 30 (2) ◽  
pp. 353-364 ◽  
Author(s):  
Ray Watson ◽  
Paul Yip
Keyword(s):  
Epidemic Model ◽  
Time Scale ◽  
Counting Process ◽  
Transition Probabilities ◽  
Random Time ◽  
Scale Transformation ◽  
Asymptotic Results ◽  
Martingale Methods ◽  
Conflict Model ◽  
Population Processes

We consider a bivariate Markov counting process with transition probabilities having a particular structure, which includes a number of useful population processes. Using a suitable random time-scale transformation, we derive some probability statements about the process and some asymptotic results. These asymptotic results are also derived using martingale methods. Further, it is shown that these methods and results can be used for inference on the rate parameters for the process. The general epidemic model and the square law conflict model are used as illustrative examples.

A bivariate counting process

1993 ◽  
Vol 30 (02) ◽  
pp. 353-364
Author(s):  
Ray Watson ◽  
Paul Yip
Keyword(s):  
Epidemic Model ◽  
Time Scale ◽  
Counting Process ◽  
Transition Probabilities ◽  
Random Time ◽  
Scale Transformation ◽  
Asymptotic Results ◽  
Martingale Methods ◽  
Conflict Model ◽  
Population Processes

We consider a bivariate Markov counting process with transition probabilities having a particular structure, which includes a number of useful population processes. Using a suitable random time-scale transformation, we derive some probability statements about the process and some asymptotic results. These asymptotic results are also derived using martingale methods. Further, it is shown that these methods and results can be used for inference on the rate parameters for the process. The general epidemic model and the square law conflict model are used as illustrative examples.

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.

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

2007 ◽  
Vol 44 (1) ◽  
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.

An Alternative Test Using Algal Cultures for Testing Chemicals for Toxicity

1993 ◽  
Vol 21 (2) ◽  
pp. 196-201
Author(s):  
Søren Achim Nielsen ◽  
Thomas Hougaard
Keyword(s):  
Size Distribution ◽  
Mitotic Activity ◽  
Time Scale ◽  
Test System ◽  
Short Time Scale ◽  
Toxic Substances ◽  
Alternative Test ◽  
Short Time ◽  
Algal Cultures ◽  
Test Cultures

An alternative test is presented, in which algal cultures are used for testing toxic substances. This test system is based on variations in the size distribution of cells in test cultures as a measurement of growth. Thus, inhibition of mitotic activity is used as a measurement for toxic effects. The test can be performed on a short time-scale and is very sensitive to even weak toxic doses.

VARIABILITY FOR CARRIER-BORNE EPIDEMICS AND REED–FROST MODELS INCORPORATING UNCERTAINTIES AND DEPENDENCIES FROM SUSCEPTIBLES AND INFECTIVES

2010 ◽  
Vol 24 (2) ◽  
pp. 303-328 ◽  
Author(s):  
Eva María Ortega ◽  
Laureano F. Escudero
Keyword(s):  
Risk Assessment ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Random Environments ◽  
Random Parameters ◽  
Infection Rates ◽  
Disease Propagation ◽  
Increasing Concave Order ◽  
Special Cases ◽  
Frost Model

This article provides analytical results on which are the implications of the statistical dependencies among certain random parameters on the variability of the number of susceptibles of the carrier-borne epidemic model with heterogeneous populations and of the number of infectives under the Reed–Frost model with random infection rates. We consider dependencies among the random infection rates, among the random infectious times, and among random initial susceptibles of several carrier-borne epidemic models. We obtain conditions for the variability ordering between the number of susceptibles for carrier-borne epidemics under two different random environments, at any time-scale value. These results are extended to multivariate comparisons of the random vectors of populations in the strata. We also obtain conditions for the increasing concave order between the number of infectives in the Reed–Frost model under two different random environments, for any generation. Variability bounds are obtained for different epidemic models from modeling dependencies for a range of special cases that are useful for risk assessment of disease propagation.

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