A branching process approximation to cascading load-dependent system failure

2004 ◽  
Author(s):  
I. Dobson ◽  
B.A. Carreras ◽  
D.E. Newman
Keyword(s):  
Branching Process ◽  
System Failure ◽  
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 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.

Yule process approximation for the skeleton of a branching process

1993 ◽  
Vol 30 (03) ◽  
pp. 725-729 ◽  
Author(s):  
Neil O'Connell
Keyword(s):  
Branching Process ◽  
Birth Process ◽  
Long Period ◽  
Genetic Types ◽  
Yule Process ◽  
Watson Process ◽  
Process Approximation ◽  
Growing Population

The skeleton of a (super-) critical Galton-Watson process with offspring mean 1 + r, r ≧ 0, and finite offspring variance, is considered. When r = 0 it is trivial. If r > 0 is small and the time unit is taken as α /r generations (α > 0) then the skeleton can be approximated by a Yule (linear pure birth) process of rate α. This approximation can be used to study the evolution of genetic types over a long period of time in an exponentially growing population.

scholarly journals The effective founder effect in a spatially expanding population

2014 ◽  
Author(s):  
Benjamin Marco Peter ◽  
Montgomery Slatkin
Keyword(s):  
Founder Effect ◽  
Process Model ◽  
Branching Process ◽  
Model Fit ◽  
Data Set ◽  
Gradual Loss ◽  
Loss Of Diversity ◽  
The Difference ◽  
Expanding Population ◽  
Process Approximation

The gradual loss of diversity associated with range expansions is a well known pattern observed in many species, and can be explained with a serial founder model. We show that under a branching process approximation, this loss in diversity is due to the difference in offspring variance between individuals at and away from the expansion front, which allows us to measure the strength of the founder effect, dependant on an effective founder size. We demonstrate that the predictions from the branching process model fit very well with Wright-Fisher forward simulations and backwards simulations under a modified Kingman coalescent, and further show that estimates of the effective founder size are robust to possibly confounding factors such as migration between subpopulations. We apply our method to a data set ofArabidopsis thaliana, where we find that the founder effect is about three times stronger in the Americas than in Europe, which may be attributed to the more recent, faster expansion.

scholarly journals Introgression of a block of genome under infinitesimal selection

2017 ◽  
Author(s):  
Himani Sachdeva ◽  
Nicholas H. Barton
Keyword(s):  
Genetic Variability ◽  
Infinite Number ◽  
Branching Process ◽  
Directional Selection ◽  
Native Population ◽  
Adaptive Introgression ◽  
Long Run ◽  
Number Of Genes ◽  
Polygenic Selection ◽  
Process Approximation

AbstractAdaptive introgression is common in nature and can be driven by selection acting on multiple, linked genes. We explore the effects of polygenic selection on introgression under the infinitesimal model with linkage. This model assumes that the introgressing block has an effectively infinite number of genes, each with an infinitesimal effect on the trait under selection. The block is assumed to introgress under directional selection within a native population that is genetically homogeneous. We use individual-based simulations and a branching process approximation to compute various statistics of the introgressing block, and explore how these depend on parameters such as the map length and initial trait value associated with the introgressing block, the genetic variability along the block, and the strength of selection. Our results show that the introgression dynamics of a block under infinitesimal selection is qualitatively different from the dynamics of neutral introgression. We also find that in the long run, surviving descendant blocks are likely to have intermediate lengths, and clarify how the length is shaped by the interplay between linkage and infinitesimal selection. Our results suggest that it may be difficult to distinguish introgression of single loci from that of genomic blocks with multiple, tightly linked and weakly selected loci.

scholarly journals Epidemics on Random Graphs with Tunable Clustering

2008 ◽  
Vol 45 (3) ◽  
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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