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Adv Appl Probab 2021 Sep 8;53(3):609-648. Epub 2021 Oct 8.

Texas A&M, College Station.

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [24], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [28]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. Read More

Adv Appl Probab 2014 Sep;46(3):704-718

University of Rochester.

Age-dependent branching processes are increasingly used in analyses of biological data. Despite being central to most statistical procedures, the identifiability of these models has not been studied. In this paper, we partition a family of age-dependent branching processes into equivalence classes over which the distribution of the population size remains identical. Read More

Adv Appl Probab 2012 Jun;44(2):391-407

University of California, Berkeley.

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. Read More

Adv Appl Probab 2012 Jun;44(2):408-428

University of California, Berkeley.

Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. Read More

Adv Appl Probab 2008 Jun;40(2):473-500

University of Oxford.

The diffusion-generator approximation technique developed by De Iorio and Griffiths (2004a) is a very useful method of constructing importance sampling proposal distributions. Being based on general mathematical principles, the method can be applied to various models in population genetics. In this paper we extend the technique to the neutral coalescent model with recombination, thus obtaining novel sampling distributions for the two-locus model. Read More