A note on maximum likelihood estimation of the initial number of susceptibles in the general stochastic epidemic model

2009 ◽  
Vol 79 (18) ◽  
pp. 1972-1976 ◽  
Author(s):  
Theodore Kypraios
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Epidemic Model ◽  
Likelihood Estimation ◽  
Initial Number ◽  
Stochastic Epidemic ◽  
Stochastic Epidemic Model ◽  
General Stochastic

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.

The transition probabilities of the general stochastic epidemic model

1975 ◽  
Vol 12 (3) ◽  
pp. 415-424 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Stochastic Model ◽  
Convenient Method ◽  
Epidemic Model ◽  
Transition Probabilities ◽  
Stochastic Epidemic ◽  
General Stochastic Model ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
General Stochastic

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.

Maximum likelihood estimation for stochastic rth-order reactions. II

1973 ◽  
Vol 10 (02) ◽  
pp. 441-446
Author(s):  
John P. Mullooly
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Waiting Time ◽  
Likelihood Estimation ◽  
Initial Number ◽  
Asymptotically Normal ◽  
Reaction Event ◽  
Wide Range ◽  
Order Case ◽  
Exact Maximum Likelihood

In this paper we consider maximum likelihood estimation of the rate constant for stochastic rth-order reactions based on observation of the level of the system at time t > 0. Conditions are found for which the waiting time until the nth. reaction event is asymptotically normal, as the initial number of molecules and the number of reaction events become large. This distributional result is used to derive an approximate estimator which is shown for the second-order case to be close to the exact maximum likelihood estimate over a wide range of percentage completion.

Maximum likelihood estimation for stochastic rth -order reactions

1972 ◽  
Vol 9 (01) ◽  
pp. 32-42
Author(s):  
John P. Mullooly
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Asymptotic Solution ◽  
Probability Distributions ◽  
Likelihood Estimation ◽  
Order System ◽  
Initial Number ◽  
Direct Evaluation ◽  
Likelihood Equation ◽  
Satisfactory Approximation

In this paper we derive the probability distributions of the number of molecules and of the lifetime of a molecule in a stochastic rth-order system by direct evaluation of probabilities, avoiding the use of differential-difference equations. Maximum likelihood estimation of the rate constant is based on an observation of the level of the system at time t > 0. We find the asymptotic solution of the likelihood equation for a large initial number of molecules. By comparison with the numerical solution of the likelihood equation, the asymptotic estimator is shown to be a satisfactory approximation for second order reactions which are far from completion.

Maximum likelihood estimation for stochastic rth -order reactions

1972 ◽  
Vol 9 (1) ◽  
pp. 32-42 ◽  
Author(s):  
John P. Mullooly
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Asymptotic Solution ◽  
Probability Distributions ◽  
Likelihood Estimation ◽  
Order System ◽  
Initial Number ◽  
Direct Evaluation ◽  
Likelihood Equation ◽  
Satisfactory Approximation

In this paper we derive the probability distributions of the number of molecules and of the lifetime of a molecule in a stochastic rth-order system by direct evaluation of probabilities, avoiding the use of differential-difference equations. Maximum likelihood estimation of the rate constant is based on an observation of the level of the system at time t > 0. We find the asymptotic solution of the likelihood equation for a large initial number of molecules. By comparison with the numerical solution of the likelihood equation, the asymptotic estimator is shown to be a satisfactory approximation for second order reactions which are far from completion.

The rumour process

1982 ◽  
Vol 19 (04) ◽  
pp. 759-766
Author(s):  
Ross Dunstan
Keyword(s):  
Epidemic Model ◽  
Asymptotic Form ◽  
Deterministic Model ◽  
Final Size ◽  
Stochastic Epidemic ◽  
The Mean ◽  
Stochastic Epidemic Model ◽  
General Stochastic

The general stochastic epidemic model is used as a model for the spread of rumours. Recursive expressions are found for the mean of the final size of each generation of hearers. Simple expressions are found for the generation size and the asymptotic form of its final size in the deterministic model. An approximating process is presented.

The transition probabilities of the general stochastic epidemic model

1975 ◽  
Vol 12 (03) ◽  
pp. 415-424 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Stochastic Model ◽  
Convenient Method ◽  
Epidemic Model ◽  
Transition Probabilities ◽  
Stochastic Epidemic ◽  
General Stochastic Model ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
General Stochastic

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.

The estimation of the parameters in epidemics

1975 ◽  
Vol 7 (3) ◽  
pp. 463-463
Author(s):  
Kevin Gough
Keyword(s):  
Epidemic Model ◽  
Stochastic Epidemic ◽  
Stochastic Epidemic Model ◽  
General Stochastic

The General Stochastic Epidemic model is extended to allow for outside infection. The likelihood is derived for small households, and the difficulties for larger populations are discussed.

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