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.

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 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.

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

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 Threshold behaviour and final outcome of an epidemic on a random network with household structure

2009 ◽  
Vol 41 (3) ◽  
pp. 765-796 ◽  
Author(s):  
Frank Ball ◽  
David Sirl ◽  
Pieter Trapman
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Random Network ◽  
Household Structure ◽  
Simulation Studies ◽  
Threshold Parameter ◽  
Threshold Behaviour ◽  
Asymptotic Results ◽  
Social Contacts ◽  
Major Outbreak

In this paper we consider a stochastic SIR (susceptible→infective→removed) epidemic model in which individuals may make infectious contacts in two ways, both within ‘households’ (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly sized finite populations. The extension to unequal-sized households is discussed briefly.

scholarly journals Threshold behaviour and final outcome of an epidemic on a random network with household structure

2009 ◽  
Vol 41 (03) ◽  
pp. 765-796 ◽  
Author(s):  
Frank Ball ◽  
David Sirl ◽  
Pieter Trapman
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Random Network ◽  
Household Structure ◽  
Simulation Studies ◽  
Threshold Parameter ◽  
Threshold Behaviour ◽  
Asymptotic Results ◽  
Social Contacts ◽  
Major Outbreak

In this paper we consider a stochastic SIR (susceptible→infective→removed) epidemic model in which individuals may make infectious contacts in two ways, both within ‘households’ (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly sized finite populations. The extension to unequal-sized households is discussed briefly.

scholarly journals Final size distribution of stochastic susceptible infected recovered (SIR) epidemic model

2021 ◽  
Author(s):  
Jane Lola Mahasmara ◽  
Respatiwulan ◽  
Yuliana Susanti
Keyword(s):  
Size Distribution ◽  
Epidemic Model ◽  
Sir Epidemic Model ◽  
Final Size ◽  
Sir Epidemic

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.

scholarly journals A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Aba Oud ◽  
Aatif Ali ◽  
Hussam Alrabaiah ◽  
Saif Ullah ◽  
Muhammad Altaf Khan ◽  
...  
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Model Parameters ◽  
Disease Dynamics ◽  
Threshold Parameter ◽  
Fractional Model ◽  
Infected People ◽  
Fractional Models ◽  
Disease Epidemics ◽  
The Impact

AbstractCOVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained $\mathcal{R}_{0} \approx 1.50$ R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams–Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

Sign in / Sign up

Export Citation Format

Share Document