@article{Minayev:2009:J-R-Soc-Interface:19158010,
  abstract = {We develop mathematical models of the transmission and evolution of multi-strain pathogens that incorporate strain extinction and the stochastic generation of new strains via mutation. The dynamics resulting from these models is then examined with the applied aim of understanding the mechanisms underpinning the evolution and dynamics of rapidly mutating pathogens, such as human influenza viruses. Our approach, while analytically relatively simple, gives results that are qualitatively similar to those obtained from much more complex individually based simulation models. We examine strain dynamics as a function of cross-immunity and key transmission parameters, and show that introducing strain extinction and modelling mutation as a stochastic process significantly changes the model dynamics, leading to lower strain diversity, reduced infection prevalence and shorter strain lifetimes. Finally, we incorporate transient strain-transcending immunity in the model and demonstrate that it reduces strain diversity further, giving patterns of sequential strain replacement similar to that seen in human influenza A viruses.},
  added-at = {2009-08-05T19:10:59.000+0200},
  author = {Minayev and Ferguson, N},
  biburl = {http://www.bibsonomy.org/bibtex/24a799026e607dbc68b8f86d5e89ac3c9/ebo},
  description = {Incorporating demographic stochasticity into multi...[J R Soc Interface. 2009] - PubMed Result},
  doi = {10.1098/rsif.2008.0467},
  interhash = {e2762d0a95357cbfca38d8ddd4de8f72},
  intrahash = {4a799026e607dbc68b8f86d5e89ac3c9},
  journal = {Journal of the Royal Society Interface},
  keywords = {evolution immunity imported influenza stochastic winmong},
  month = Jan,
  pmid = {19158010},
  timestamp = {2009-08-05T19:10:59.000+0200},
  title = {Incorporating demographic stochasticity into multi-strain epidemic models: application to influenza A},
  url = {http://www.ncbi.nlm.nih.gov/pubmed/19158010?ordinalpos=1&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_DiscoveryPanel.Pubmed_Discovery_RA&linkpos=1&log$=relatedarticles&logdbfrom=pubmed},
  year = 2009
}

@misc{Moschopoulos2009,
  abstract = {  The number of extant individuals within a lineage, as exemplified by counts
of species numbers across genera in a higher taxonomic category, is known to be
a highly skewed distribution. Because the sublineages (such as genera in a
clade) themselves follow a random birth process, deriving the distribution of
lineage sizes involves averaging the solutions to a birth and death process
over the distribution of time intervals separating the origin of the lineages.
In this paper, we show that the resulting distributions can be represented by
hypergeometric functions of the second kind. We also provide approximations of
these distributions up to the second order, and compare these results to the
asymptotic distributions and numerical approximations used in previous studies.
For two limiting cases, one with a relatively high rate of lineage origin, one
with a low rate, the cumulative probability densities and percentiles are
compared to show that the approximations are robust over a wide rane of
parameters. It is proposed that the probability density distributions of
lineage size may have a number of relevant applications to biological problems
such as the coalescence of genetic lineages and in predicting the number of
species in living and extinct higher taxa, as these systems are special
instances of the underlying process analyzed in this paper.
},
  added-at = {2009-07-20T22:39:12.000+0200},
  author = {Moschopoulos, Panagis and Shpak, Max},
  biburl = {http://www.bibsonomy.org/bibtex/2d90bcd22e8c01e07a69873eb971b1a68/ebo},
  description = {[0901.1066] Taxon Size Distribution in a Time Homogeneous Birth and Death Process},
  interhash = {6c6f817241d444f52c70dacf9633fee0},
  intrahash = {d90bcd22e8c01e07a69873eb971b1a68},
  keywords = {imported stochastic},
  note = {cite arxiv:0901.1066
},
  timestamp = {2009-07-20T22:39:12.000+0200},
  title = {Taxon Size Distribution in a Time Homogeneous Birth and Death Process},
  url = {http://arxiv.org/abs/0901.1066},
  year = 2009
}

@article{MR0221600,
  added-at = {2009-07-10T19:23:40.000+0200},
  author = {Athreya, Krishna Balasundaram},
  biburl = {http://www.bibsonomy.org/bibtex/2affdb48572e7bc346cce1a1c600c2b68/ebo},
  description = {MR: Publications results for "MR Number=(221600)"},
  fjournal = {Annals of Mathematical Statistics},
  interhash = {92160d7648f9380c6c6e3bdcafb6b6a7},
  intrahash = {affdb48572e7bc346cce1a1c600c2b68},
  issn = {0003-4851},
  journal = {Annals of Mathematical Statistics},
  keywords = {branching imported stochastic},
  mrclass = {60.67},
  mrnumber = {MR0221600 (36 \#4652)},
  mrreviewer = {F. L. Spitzer},
  pages = {347--357},
  timestamp = {2009-07-10T19:23:40.000+0200},
  title = {Some results on multitype continuous time {M}arkov branching
              processes},
  volume = 39,
  year = 1968
}

@article{kendall1956deterministic,
  added-at = {2009-07-10T02:17:19.000+0200},
  author = {Kendall, D.G.},
  biburl = {http://www.bibsonomy.org/bibtex/2768b177a9276e81345f0e7124e0e2919/ebo},
  booktitle = {Proc. 3rd Berkeley Symp. Math. Statist. Prob},
  interhash = {a1e6b85df175ece04d69a1d1e2b764f3},
  intrahash = {768b177a9276e81345f0e7124e0e2919},
  keywords = {epi imported stochastic},
  pages = {149--165},
  timestamp = {2009-07-10T02:17:19.000+0200},
  title = {{Deterministic and stochastic epidemics in closed populations}},
  volume = 4,
  year = 1956
}

@misc{Yakovlev2009,
  abstract = {  This paper considers the relative frequencies of distinct types of
individuals in multitype branching processes. We prove that the frequencies are
asymptotically multivariate normal when the initial number of ancestors is
large and the time of observation is fixed. The result is valid for any
branching process with a finite number of types; the only assumption required
is that of independent individual evolutions. The problem under consideration
is motivated by applications in the area of cell biology. Specifically, the
reported limiting results are of advantage in cell kinetics studies where the
relative frequencies but not the absolute cell counts are accessible to
measurement. Relevant statistical applications are discussed in the context of
asymptotic maximum likelihood inference for multitype branching processes.
},
  added-at = {2009-07-09T00:05:35.000+0200},
  author = {Yakovlev, Andrei Y. and Yanev, Nikolay M.},
  biburl = {http://www.bibsonomy.org/bibtex/22e1ed1de4ab36e19ccea06c8634111d6/ebo},
  description = {[0902.4773] Relative frequencies in multitype branching processes},
  interhash = {b2e1afdc21649b00811aaf27cb265790},
  intrahash = {2e1ed1de4ab36e19ccea06c8634111d6},
  keywords = {branching imported stochastic},
  note = {cite arxiv:0902.4773
Comment: Published in at http://dx.doi.org/10.1214/08-AAP539 the Annals of
  Applied Probability (http://www.imstat.org/aap/) by the Institute of
  Mathematical Statistics (http://www.imstat.org)},
  timestamp = {2009-07-09T00:05:35.000+0200},
  title = {Relative frequencies in multitype branching processes},
  url = {http://arxiv.org/abs/0902.4773},
  year = 2009
}

@article{charles1969applications,
  abstract = {Without Abstract
ER  -},
  added-at = {2009-07-08T23:39:42.000+0200},
  author = {Mode, Charles},
  biburl = {http://www.bibsonomy.org/bibtex/28be83f74960d3cdf9fdfb37b660f269b/ebo},
  description = {SpringerLink - Journal Article},
  interhash = {09242d3510c4d5706a762e59ce51d6f8},
  intrahash = {8be83f74960d3cdf9fdfb37b660f269b},
  journal = {Bulletin of Mathematical Biology},
  keywords = {branching imported stochastic},
  month = {#sep#},
  number = 3,
  pages = {575--589},
  timestamp = {2009-07-08T23:39:42.000+0200},
  title = {Applications of generalized multi-type age-dependent branching processes in population genetics},
  url = {http://dx.doi.org/10.1007/BF02476639},
  volume = 31,
  year = 1969
}

@article{olle1981convergence,
  abstract = {Convergence in probability of Malthus normed supercritical general branching processes (i.e. Crump-Mode-Jagers branching processes) counted with a general characteristic are established, provided the latter satisfies mild regularity conditions. If the Laplace transform of the reproduction point process evaluated in the Malthusian parameter has a finite ‘x log x-moment’ convergence in probability of the empirical age distribution and more generally of the ratio of two differently counted versions of the process also follow.
ER  -},
  added-at = {2009-07-03T04:33:39.000+0200},
  author = {Nerman, Olle},
  biburl = {http://www.bibsonomy.org/bibtex/25188727be38bd3bc0cca61f86acfaa88/ebo},
  description = {SpringerLink - Journal Article},
  interhash = {3272da18b777448bd2f4dc3447e24edc},
  intrahash = {5188727be38bd3bc0cca61f86acfaa88},
  journal = {Probability Theory and Related Fields},
  keywords = {branching imported stochastic},
  month = {#sep#},
  number = 3,
  pages = {365--395},
  timestamp = {2009-07-03T04:33:39.000+0200},
  title = {On the convergence of supercritical general (C-M-J) branching processes},
  url = {http://dx.doi.org/10.1007/BF00534830},
  volume = 57,
  year = 1981
}

@article{Ball19951,
  abstract = {This paper is concerned with the approximation of early stages of epidemic processes by branching processes. A general model for an epidemic in a closed, homogeneously mixing population is presented. A construction of a sequence of such epidemics, indexed by the initial number of susceptibles N, from the limiting branching process is described. Strong convergence of the epidemic processes to the branching process is shown when the latter goes extinct. When the branching process does not go extinct, necessary and sufficient conditions on the sequence (tN) for strong convergence over the time interval [0, tN] are provided. Convergence of a wide variety of functionals of the epidemic process to corresponding functionals of the branching process is shown, and bounds are provided on the total variation distance for given N. The theory is illustrated by reference to the general stochastic epidemic. Generalisations to, for example, multipopulation epidemics are described briefly.},
  added-at = {2009-07-03T02:50:54.000+0200},
  author = {Ball, Frank and Donnelly, Peter},
  biburl = {http://www.bibsonomy.org/bibtex/27a5dc6b7374c8394bc7653e64b8e790a/ebo},
  description = {ScienceDirect - Stochastic Processes and their Applications : Strong approximations for epidemic models},
  doi = {DOI: 10.1016/0304-4149(94)00034-Q},
  interhash = {36de3db49d73f0d006d554ffcd4b86e0},
  intrahash = {7a5dc6b7374c8394bc7653e64b8e790a},
  issn = {0304-4149},
  journal = {Stochastic Processes and their Applications},
  keywords = {epi stochastic},
  number = 1,
  pages = {1 - 21},
  timestamp = {2009-07-03T02:50:54.000+0200},
  title = {Strong approximations for epidemic models},
  url = {http://www.sciencedirect.com/science/article/B6V1B-3YYTDND-1/2/490700f625cb3bfa0ab13484de175d06},
  volume = 55,
  year = 1995
}