We present a stochastic epidemic model to study the effect of various preventive measures, such as uniform reduction of contacts and transmission, vaccination, isolation, screening and contact tracing, on a disease outbreak in a homogeneously mixing community. The model is based on an infectivity process, which we define through stochastic contact and infectiousness processes, so that each individual has an independent infectivity profile. In particular, we monitor variations of the reproduction number and of the distribution of generation times. We show that some interventions, i.e. uniform reduction and vaccination, affect the former while leaving the latter unchanged, whereas other interventions, i.e. isolation, screening and contact tracing, affect both quantities. We provide a theoretical analysis of the variation of these quantities, and we show that, in practice, the variation of the generation time distribution can be significant and that it can cause biases in the estimation of reproduction numbers. The framework, because of its general nature, captures the properties of many infectious diseases, but particular emphasis is on COVID-19, for which numerical results are provided.
The coupled Wright–Fisher diffusion is a multi-dimensional Wright–Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright–Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright–Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright–Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.
Recent breakthroughs in the reconstruction of ancestral recombination graphs (ARGs) have meant that inference of genome-wide genealogies from large sequencing datasets is now possible. This development has also opened several new directions where research is urgently needed: improving our understanding of which aspects of genealogies are (and which are not) faithfully captured by reconstruction tools, as well as the development of inference methods that directly use ARGs as inputs. By deriving the distribution of branch duration under the SMC' model, defined as the genomic interval spanned by a branch of the ARG until it is disrupted by recombination, we benchmark the quality of ARGs produced by several reconstruction tools: ARGweaver, Relate, tsinfer and tsdate, and ARG-Needle. Further, we develop an accurate and powerful ARG-based test for the presence of chromosomal inversions, based on the idea that suppression of recombination in individuals heterozygous for an inversion has a detectable impact on branch duration.
The results in this paper provide new information on asymptotic properties of classical models: the neutral Kingman coalescent under a general finite-alleles, parentdependent mutation mechanism, and its generalisation, the ancestral selection graph. Several relevant quantities related to these fundamental models are not explicitly known when mutations are parent dependent. Examples include the probability that a sample taken from a population has a certain type configuration, and the transition probabilities of their block counting jump chains. In this paper, asymptotic results are derived for these quantities, as the sample size goes to infinity. It is shown that the sampling probabilities decay polynomially in the sample size with multiplying constant depending on the stationary density of the Wright-Fisher diffusion and that the transition probabilities converge to the limit of frequencies of types in the sample.
The Kingman coalescent is an important and well studied process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, weak convergence is proved for a sequence of Markov chains consisting of two components related to the Kingman coalescent, as the size of the initial configuration, the sample size, grows to infinity. The first component is the normalised jump chain of the block counting processes of the Kingman coalescent with a finite number of d genetic types. The second component is a d 2 -dimensional process counting the number of mutations between types occurring in the Kingman coalescent. Time is scaled by the sample size. The limiting process consists of a deterministic d-dimensional component describing the limit of the jump chain and d 2 independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. The weak convergence result is first proved, using a generator approach, in the setting of parent independent mutations. A change of measure argument is used to extend the weak convergence result to include parent dependent mutations.
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