A threshold theorem for the Reed-Frost chain-binomial epidemic

1983 ◽  
Vol 20 (1) ◽  
pp. 153-157 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Population Size ◽  
Binomial Model ◽  
Susceptible Individual ◽  
Stochastic Epidemic ◽  
General Stochastic

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.

A threshold theorem for the Reed-Frost chain-binomial epidemic

1983 ◽  
Vol 20 (01) ◽  
pp. 153-157 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Population Size ◽  
Binomial Model ◽  
Susceptible Individual ◽  
Stochastic Epidemic ◽  
General Stochastic

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.

Stochastic approximation for the general epidemic

1973 ◽  
Vol 10 (02) ◽  
pp. 263-276 ◽  
Author(s):  
Donald Ludwig
Keyword(s):  
Exact Solution ◽  
Population Size ◽  
Stochastic Approximation ◽  
Asymptotic Approximation ◽  
Asymptotic Approximations ◽  
System Of Equations ◽  
Numerical Comparisons ◽  
Stochastic Epidemic ◽  
Simple Equations ◽  
General Stochastic

A system of equations is introduced whose solutions are remarkably close to corresponding solutions of the “general stochastic epidemic”. If N is the population size, then there are approximately equations to be solved for the general stochastic epidemic while the number of equations in the approximating system is proportional to N. An asymptotic approximation to the general stochastic epidemic is also introduced. Numerical comparisons of the stochastic and asymptotic approximations with the exact solution are presented.

Stochastic approximation for the general epidemic

1973 ◽  
Vol 10 (2) ◽  
pp. 263-276 ◽  
Author(s):  
Donald Ludwig
Keyword(s):  
Exact Solution ◽  
Population Size ◽  
Stochastic Approximation ◽  
Asymptotic Approximation ◽  
Asymptotic Approximations ◽  
System Of Equations ◽  
Numerical Comparisons ◽  
Stochastic Epidemic ◽  
General Stochastic

A system of equations is introduced whose solutions are remarkably close to corresponding solutions of the “general stochastic epidemic”. If N is the population size, then there are approximately equations to be solved for the general stochastic epidemic while the number of equations in the approximating system is proportional to N. An asymptotic approximation to the general stochastic epidemic is also introduced. Numerical comparisons of the stochastic and asymptotic approximations with the exact solution are presented.

The cost of a general stochastic epidemic

1972 ◽  
Vol 9 (02) ◽  
pp. 257-269 ◽  
Author(s):  
J. Gani ◽  
D. Jerwood
Keyword(s):  
Recursive Equation ◽  
Stieltjes Transform ◽  
Stochastic Epidemic ◽  
The Mean ◽  
The Cost ◽  
Mean Cost ◽  
General Stochastic

This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis ) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis . Using the results obtained, bounds are found for the mean cost of the epidemic.

Asymptotic analysis of the general stochastic epidemic with variable infectious periods

2001 ◽  
Vol 38 (01) ◽  
pp. 18-35 ◽  
Author(s):  
A. N. Startsev
Keyword(s):  
Asymptotic Analysis ◽  
Asymptotic Behaviour ◽  
Size Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Boundary Crossing ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.

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.

Two theorems on solutions of differential-difference equations and applications to epidemic theory

1967 ◽  
Vol 4 (02) ◽  
pp. 271-280 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Differential Equation ◽  
Difference Equations ◽  
Total Population ◽  
Original System ◽  
Initial Distribution ◽  
Fixed Number ◽  
Total Population Size ◽  
Stochastic Epidemic ◽  
Iterative Solutions ◽  
General Stochastic

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.

General stochastic epidemic with recovery

1975 ◽  
Vol 12 (01) ◽  
pp. 29-38 ◽  
Author(s):  
L. Billard
Keyword(s):  
Stochastic Model ◽  
Finite Number ◽  
Epidemic Model ◽  
The State ◽  
Stochastic Epidemic ◽  
General Stochastic

The general epidemic model does not provide for an infective recovering and thence being susceptible to further infection in the course of the epidemic. By considering the case in which recovery can occur once, we show how the state probabilities can be found for the stochastic model. This is readily extended to allow recovery up to a finite number of times.

An optimal isolation policy for an epidemic

1973 ◽  
Vol 10 (02) ◽  
pp. 247-262 ◽  
Author(s):  
Andris Abakuks
Keyword(s):  
Optimal Policy ◽  
Future Cost ◽  
Numerical Examples ◽  
Stochastic Epidemic ◽  
General Stochastic

Policies of isolating infectives in the general stochastic epidemic are considered. With costs assigned to the infection and isolation of individuals, an optimal policy is found, which at any stage minimises the expected future cost. An optimal policy is also found for the general deterministic epidemic and the two policies are compared. Finally, some numerical examples are provided.

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