A Branching Process Approximation of the Final Size of a Multitype Collective Reed-Frost Model

2013 ◽  
Vol 2 (1) ◽  
pp. 47-59 ◽  
Author(s):  
A. Eseghira ◽  
A. Kissamia ◽  
H. El Maroufyb ◽  
T. Ziadc
Keyword(s):  
Branching Process ◽  
Final Size ◽  
Frost Model ◽  
Process Approximation

The asymptotic final size distribution of multitype chain-binomial epidemic processes

1999 ◽  
Vol 31 (01) ◽  
pp. 220-234 ◽  
Author(s):  
Mikael Andersson
Keyword(s):  
Branching Process ◽  
Counting Process ◽  
Contact Structure ◽  
Finite Population ◽  
Convergence Result ◽  
System Of Equations ◽  
Final Size ◽  
Linear System Of Equations ◽  
Non Linear System ◽  
Process Approximation

A multitype chain-binomial epidemic process is defined for a closed finite population by sampling a simple multidimensional counting process at certain points. The final size of the epidemic is then characterized, given the counting process, as the smallest root of a non-linear system of equations. By letting the population grow, this characterization is used, in combination with a branching process approximation and a weak convergence result for the counting process, to derive the asymptotic distribution of the final size. This is done for processes with an irreducible contact structure both when the initial infection increases at the same rate as the population and when it stays fixed.

scholarly journals An epidemic model with exposure-dependent severities

2005 ◽  
Vol 42 (04) ◽  
pp. 932-949 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Branching Process ◽  
Large Population ◽  
Final Outcome ◽  
Model Parameters ◽  
Final Size ◽  
Strong Law ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Infectious State ◽  
Process Approximation

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

The asymptotic final size distribution of multitype chain-binomial epidemic processes

1999 ◽  
Vol 31 (1) ◽  
pp. 220-234 ◽  
Author(s):  
Mikael Andersson
Keyword(s):  
Branching Process ◽  
Counting Process ◽  
Contact Structure ◽  
Finite Population ◽  
Convergence Result ◽  
System Of Equations ◽  
Final Size ◽  
Linear System Of Equations ◽  
Non Linear System ◽  
Process Approximation

A multitype chain-binomial epidemic process is defined for a closed finite population by sampling a simple multidimensional counting process at certain points. The final size of the epidemic is then characterized, given the counting process, as the smallest root of a non-linear system of equations. By letting the population grow, this characterization is used, in combination with a branching process approximation and a weak convergence result for the counting process, to derive the asymptotic distribution of the final size. This is done for processes with an irreducible contact structure both when the initial infection increases at the same rate as the population and when it stays fixed.

scholarly journals An epidemic model with exposure-dependent severities

2005 ◽  
Vol 42 (4) ◽  
pp. 932-949 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Branching Process ◽  
Large Population ◽  
Final Outcome ◽  
Model Parameters ◽  
Final Size ◽  
Strong Law ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Infectious State ◽  
Process Approximation

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

scholarly journals From damage models to SIR epidemics and cascading failures

2009 ◽  
Vol 41 (1) ◽  
pp. 247-269 ◽  
Author(s):  
Maude Gathy ◽  
Claude Lefèvre
Keyword(s):  
Branching Process ◽  
Damage Model ◽  
Exact Distribution ◽  
Partial Sums ◽  
Mathematical Tool ◽  
Close Approximation ◽  
Final Size ◽  
Damage Models ◽  
Starting Point ◽  
Made In

This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

An unusual stochastic order relation with some applications in sampling and epidemic theory

1993 ◽  
Vol 25 (1) ◽  
pp. 63-81 ◽  
Author(s):  
Claude Lefevre ◽  
Philippe Picard
Keyword(s):  
Stochastic Order ◽  
Infection Intensity ◽  
Order Relation ◽  
Final Size ◽  
Factorial Moments ◽  
Comparison Results ◽  
Global Cost ◽  
Frost Model ◽  
Sampling Procedures ◽  
Total Damage

One expects, intuitively, that the total damage caused by an epidemic increases, in a certain sense, with the infection intensity exerted by the infectives during their lifelength. The original object of the present work is to make precise in which probabilistic terms such a statement does indeed hold true, when the spread of the disease is described by a collective Reed–Frost model and the global cost is represented by the final size and severity. Surprisingly, this problem leads us to introduce an order relation for -valued random variables, unusual in the literature, based on the descending factorial moments. Further applications of the ordering occur when comparing certain sampling procedures through the number of un-sampled individuals. In particular, it is used to reinforce slightly comparison results obtained earlier for two such samplings.

scholarly journals Nonstationarity and randomization in the Reed-Frost epidemic model

2005 ◽  
Vol 42 (4) ◽  
pp. 950-963 ◽  
Author(s):  
Claude Lefèvre ◽  
Philippe Picard
Keyword(s):  
Wide Class ◽  
Epidemic Model ◽  
Exact Distribution ◽  
Unified Approach ◽  
Final Size ◽  
Final Number ◽  
Nonstationary Model ◽  
Frost Model ◽  
Resistance To Infection

The purpose of this paper is to determine the exact distribution of the final size of an epidemic for a wide class of models of susceptible–infective–removed type. First, a nonstationary version of the classical Reed–Frost model is constructed that allows us to incorporate, in particular, random levels of resistance to infection in the susceptibles. Then, a randomized version of this nonstationary model is considered in order to take into account random levels of infectiousness in the infectives. It is shown that, in both cases, the distribution of the final number of infected individuals can be obtained in terms of Abel–Gontcharoff polynomials. The new methodology followed also provides a unified approach to a number of recent works in the literature.

scholarly journals Epidemics on Random Graphs with Tunable Clustering

2008 ◽  
Vol 45 (03) ◽  
pp. 743-756 ◽  
Author(s):  
Tom Britton ◽  
Maria Deijfen ◽  
Andreas N. Lagerås ◽  
Mathias Lindholm
Keyword(s):  
Random Graphs ◽  
Branching Process ◽  
Intersection Graph ◽  
Epidemic Threshold ◽  
Random Intersection Graph ◽  
Large Outbreak ◽  
Process Approximation

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

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