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

2004 ◽  
Vol 67 (4) ◽  
pp. 321-330 ◽  
Author(s):  
Richard M. Huggins ◽  
Paul S.F. Yip ◽  
Eric H.Y. Lau
Keyword(s):  
Epidemic Model ◽  
Initial Number

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

1979 ◽  
Vol 16 (3) ◽  
pp. 646-650 ◽  
Author(s):  
Roy Saunders ◽  
Claude Lefèvre ◽  
Richard J. Kryscio
Keyword(s):  
Conditional Probability ◽  
Epidemic Model ◽  
Removal Rate ◽  
Delta Function ◽  
Formal Proof ◽  
Initial Number ◽  
Expected Number ◽  
Kronecker Delta

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.

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

1994 ◽  
Vol 31 (3) ◽  
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.

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

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.

Strong Convergence of Stochastic Epidemics

1994 ◽  
Vol 26 (3) ◽  
pp. 629-655 ◽  
Author(s):  
Frank Ball ◽  
Philip O'Neill
Keyword(s):  
Strong Convergence ◽  
Epidemic Model ◽  
Death Process ◽  
Birth And Death Process ◽  
Initial Number ◽  
Positive Real ◽  
Immigration And Emigration

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate f(x, y) and removals occur at rate g(x, y), where x and y are the numbers of susceptible and infective individuals, respectively, and f and g are arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptibles n, are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process as n tends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

Strong Convergence of Stochastic Epidemics

1994 ◽  
Vol 26 (03) ◽  
pp. 629-655 ◽  
Author(s):  
Frank Ball ◽  
Philip O'Neill
Keyword(s):  
Strong Convergence ◽  
Epidemic Model ◽  
Death Process ◽  
Birth And Death Process ◽  
Initial Number ◽  
Positive Real ◽  
Immigration And Emigration

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate f(x, y) and removals occur at rate g(x, y), where x and y are the numbers of susceptible and infective individuals, respectively, and f and g are arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptibles n, are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process as n tends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

scholarly journals Improved Estimation of the Initial Number of Susceptible Individuals in the General Stochastic Epidemic Model Using Penalized Likelihood

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Changhyuck Oh
Keyword(s):  
Epidemic Model ◽  
Likelihood Function ◽  
Hessian Matrix ◽  
Short Note ◽  
Penalized Likelihood ◽  
Initial Number ◽  
Susceptible Population ◽  
Stochastic Epidemic ◽  
Stochastic Epidemic Model ◽  
General Stochastic

The initial size of a completely susceptible population in a group of individuals plays a key role in drawing inferences for epidemic models. However, this can be difficult to obtain in practice because, in any population, there might be individuals who may not transmit the disease during the epidemic. This short note describes how to improve the maximum likelihood estimators of the infection rate and the initial number of susceptible individuals and provides their approximate Hessian matrix for the general stochastic epidemic model by using the concept of the penalized likelihood function. The simulations of major epidemics show significant improvements in performance in averages and coverage ratios for the suggested estimator of the initial number in comparison to existing methods. We applied the proposed method to the Abakaliki smallpox data.

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

1979 ◽  
Vol 16 (03) ◽  
pp. 646-650 ◽  
Author(s):  
Roy Saunders ◽  
Claude Lefèvre ◽  
Richard J. Kryscio
Keyword(s):  
Conditional Probability ◽  
Epidemic Model ◽  
Removal Rate ◽  
Delta Function ◽  
Formal Proof ◽  
Initial Number ◽  
Expected Number ◽  
Kronecker Delta

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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