A note on the expected number of survivors in supercritical carrier-borne epidemics

We provide a formal proof of a conclusion due to Abakuks (1974) which states that the expected number of survivors in Downton's carrier-borne epidemic model approaches the limit (ρ /π)δ as the initial number of susceptibles tends to infinity. Here ρ denotes the relative removal rate for carriers, π denotes the conditional probability that an infected susceptible will become a carrier, δ denotes the Kronecker delta function and denotes the initial number of carriers.

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Simulation of Local Stability Analysis of An Epidemic Model with Constant Removal Rate of the Infective Between 2017-2019

10.7176/jhmn/61-01 ◽

2019 ◽

Keyword(s):

Stability Analysis ◽

Epidemic Model ◽

Local Stability ◽

Removal Rate ◽

Local Stability Analysis

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Discrete Probability Theory

An Introduction to Statistical Mechanics and Thermodynamics ◽

10.1093/oso/9780198853237.003.0003 ◽

2019 ◽

pp. 16-42

Author(s):

Robert H. Swendsen

Keyword(s):

Probability Theory ◽

Gaussian Distribution ◽

Random Variables ◽

Configurational Entropy ◽

Random Variable ◽

Delta Function ◽

Discrete Probability ◽

Fundamental Assumption ◽

Kronecker Delta ◽

Model Probability

The chapter presents an overview of various interpretations of probability. It introduces a ‘model probability,’ which assumes that all microscopic states that are essentially alike have the same probability in equilibrium. A justification for this fundamental assumption is provided. The basic definitions used in discrete probability theory are introduced, along with examples of their application. One such example, which illustrates how a random variable is derived from other random variables, demonstrates the use of the Kronecker delta function. The chapter further derives the binomial and multinomial distributions, which will be important in the following chapter on the configurational entropy, along with the useful approximation developed by Stirling and its variations. The Gaussian distribution is presented in detail, as it will be very important throughout the book.

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A note on the estimation of the initial number of susceptible individuals in the general epidemic model

Statistics & Probability Letters ◽

10.1016/j.spl.2002.02.001 ◽

2004 ◽

Vol 67 (4) ◽

pp. 321-330 ◽

Cited By ~ 8

Author(s):

Richard M. Huggins ◽

Paul S.F. Yip ◽

Eric H.Y. Lau

Keyword(s):

Epidemic Model ◽

Initial Number

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A note on the cost of carrier-borne, right-shift, epidemic models

Journal of Applied Probability ◽

10.2307/3212520 ◽

1976 ◽

Vol 13 (4) ◽

pp. 652-661 ◽

Cited By ~ 6

Author(s):

Richard J. Kryscio ◽

Roy Saunders

Keyword(s):

Epidemic Models ◽

Sufficient Condition ◽

Initial Number ◽

Expected Number ◽

Expected Cost ◽

Special Cases ◽

The Cost ◽

Special Case

We establish a sufficient condition for which the expected area under the trajectory of the carrier process is directly proportional to the expected number of removed carriers in the class of carrier-borne, right-shift, epidemic models studied by Severo (1969a). This result generalizes the previous work of Downton (1972) and Jerwood (1974) for some special cases of these models. We use the result to compute expected costs in the carrier-borne model due to Downton (1968) when it is unlikely that all the susceptibles will be infected. We conclude by showing that for the special case considered by Weiss (1965) this treatment of the expected cost is reasonable for populations with a large initial number of susceptibles.

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Global analysis of an epidemic model with a constant removal rate

Mathematical and Computer Modelling ◽

10.1016/j.mcm.2006.08.003 ◽

2007 ◽

Vol 45 (7-8) ◽

pp. 834-843 ◽

Cited By ~ 4

Author(s):

Yilei Tang ◽

Weigu Li

Keyword(s):

Epidemic Model ◽

Removal Rate ◽

Global Analysis

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A Simplified Interpolating Moving Least-Squares Method and its Error Estimates

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.101-102.271 ◽

2011 ◽

Vol 101-102 ◽

pp. 271-274

Author(s):

Ju Feng Wang

Keyword(s):

Error Estimate ◽

Least Squares ◽

Shape Function ◽

Least Squares Method ◽

Delta Function ◽

Moving Least Squares ◽

Approximation Function ◽

Kronecker Delta ◽

Moving Least Squares Method ◽

Mls Approximation

A disadvantage of the MLS approximation is that the shape function of this method does not satisfy the property of Kronecker Delta function. Thus developing an interpolating MLS approximation is very important. In this paper, the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas is discussed in detail and a simplified expression of the approximation function of the IMLS method is given. The simpler expression makes it more convenient to use this method. The error estimate of the approximation function also is discussed. And a numerical example is given to confirm the results.

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On the asymptotic distribution of the maximum number of infectives in epidemic models by immigration

Journal of Applied Probability ◽

10.2307/3215141 ◽

1994 ◽

Vol 31 (3) ◽

pp. 606-613 ◽

Cited By ~ 4

Author(s):

V. M. Abramov

Keyword(s):

Asymptotic Distribution ◽

Epidemic Model ◽

Epidemic Models ◽

Death Process ◽

Birth And Death Process ◽

Initial Number ◽

Martingale Approach ◽

Distribution Of The Maximum ◽

Non Linear

This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence.

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On the asymptotic distribution of the maximum number of infectives in epidemic models by immigration

Journal of Applied Probability ◽

10.1017/s0021900200045198 ◽

1994 ◽

Vol 31 (03) ◽

pp. 606-613

Author(s):

V. M. Abramov

Keyword(s):

Asymptotic Distribution ◽

Epidemic Model ◽

Epidemic Models ◽

Death Process ◽

Birth And Death Process ◽

Initial Number ◽

Martingale Approach ◽

Distribution Of The Maximum ◽

Non Linear

This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence.

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An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

Boundary Value Problems ◽

10.1186/s13661-020-01457-7 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Samaneh Soradi-Zeid ◽

Mehdi Mesrizadeh ◽

Thabet Abdeljawad

Keyword(s):

Neumann Boundary ◽

Nonlinear Problems ◽

Delta Function ◽

Collocation Methods ◽

Laplacian Equation ◽

Point Collocation ◽

Kronecker Delta ◽

Radial Point Interpolation ◽

Point Interpolation ◽

Element Free

Abstract This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

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