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.

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.

The Early and Final States of an Epidemic in a Large Heterogeneous Population by a Small Initial Number of Infectives

1994 ◽  
Vol 26 (03) ◽  
pp. 671-689
Author(s):  
Steven M. Butler
Keyword(s):  
Random Number ◽  
Death Process ◽  
Time Interval ◽  
Final States ◽  
Birth And Death Process ◽  
Initial Number ◽  
Threshold Behavior ◽  
Final State ◽  
Epidemic Process ◽  
Susceptibility To Infection

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.

The threshold behaviour of epidemic models

1983 ◽  
Vol 20 (2) ◽  
pp. 227-241 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Almost Sure Convergence ◽  
Death Process ◽  
Time Interval ◽  
Birth And Death Process ◽  
Initial Number ◽  
Finite Time Interval ◽  
Threshold Behaviour ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

The threshold behaviour of epidemic models

1983 ◽  
Vol 20 (02) ◽  
pp. 227-241 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Almost Sure Convergence ◽  
Death Process ◽  
Time Interval ◽  
Birth And Death Process ◽  
Initial Number ◽  
Finite Time Interval ◽  
Threshold Behaviour ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

Strong approximations for some open population epidemic models

1996 ◽  
Vol 33 (2) ◽  
pp. 448-457 ◽  
Author(s):  
Philip O'Neill
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Probability Space ◽  
Epidemic Models ◽  
Death Process ◽  
Threshold Analysis ◽  
Birth And Death Process ◽  
Initial Number ◽  
Open Population ◽  
Time Density

This paper considers a class of epidemic models in which susceptibles may enter or leave the population according to a general continuous time density dependent Markov chain. A sequence of such epidemics indexed by N, the initial number of susceptibles, is constructed on the same probability space as a time-inhomogeneous birth-and-death process. A coupling argument is then used to demonstrate the strong convergence of the sequence of infectives to the birth-and-death process. This result is used to provide a threshold analysis of the epidemic model in question.

Strong approximations for some open population epidemic models

1996 ◽  
Vol 33 (02) ◽  
pp. 448-457 ◽  
Author(s):  
Philip O'Neill
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Probability Space ◽  
Epidemic Models ◽  
Death Process ◽  
Threshold Analysis ◽  
Birth And Death Process ◽  
Initial Number ◽  
Open Population ◽  
Time Density

This paper considers a class of epidemic models in which susceptibles may enter or leave the population according to a general continuous time density dependent Markov chain. A sequence of such epidemics indexed by N, the initial number of susceptibles, is constructed on the same probability space as a time-inhomogeneous birth-and-death process. A coupling argument is then used to demonstrate the strong convergence of the sequence of infectives to the birth-and-death process. This result is used to provide a threshold analysis of the epidemic model in question.

The Early and Final States of an Epidemic in a Large Heterogeneous Population by a Small Initial Number of Infectives

1994 ◽  
Vol 26 (3) ◽  
pp. 671-689 ◽  
Author(s):  
Steven M. Butler
Keyword(s):  
Random Number ◽  
Death Process ◽  
Time Interval ◽  
Final States ◽  
Birth And Death Process ◽  
Initial Number ◽  
Threshold Behavior ◽  
Final State ◽  
Epidemic Process ◽  
Susceptibility To Infection

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.

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