Serik Sagitov, Peter Jagers
We establish convergence to the Kingman coalescent for a class of age-structured population models with time-constant population size. Time is discrete with unit called a year. Offspring numbers in a year may depend on mother's age.
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We establish convergence to the Kingman coalescent for the genealogy of a geographically - or otherwise - structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model - random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration…
Multi-type $Ξ$-coalescents from structured population models with bottlenecks
该研究提出带瓶颈的结构化种群个体模型，证明等位基因频率过程收敛到带跳跃的马尔可夫扩散，其谱系对应多类型$Ξ$-合并，数值研究显示模型灵活，可预测与真实数据一致的位点频率谱。
Structured coalescents, coagulation equations and multi-type branching processes
本文研究了结构人口中祖先谱系的凝聚过程，通过分析临界和大样本情形，发现经验测度过程收敛于d维凝聚方程的解，分别由多型分支过程和进入法表示。
The quenched coalescent for structured diploid populations with large migrations and uneven offspring distributions
本文提出了一种新的二倍体结构种群模型，通过淬火尺度极限揭示大规模迁移和不均等后代分布对经典退火理论的影响。
We investigate the infinitely many demes limit of the genealogy of a sample of individuals from a subdivided population subject to sporadic mass extinction events. By exploiting a separation of timescales property of Wright's island model, we show that as the number of demes tends to infinity the limiting form of the genealogy can be described in terms of the alternation of instantaneous 'scattering' phases dominated by local demographic processes, and extended 'collecting' phases dominated by global processes. When extinction and recolonization events are local, this genealogy is given by Kin…
The coalescent in finite populations with arbitrary, fixed structure
本文为具有任意固定空间、年龄、性别和类别结构的有限种群建立了基于映射序列的溯祖过程，并分析了溯祖时间、分支长度、突变及同源相同与状态相同概率，发现二倍体种群中遗传差异度量按性别比例缩放为4r(1-r)。
Conditional sampling distributions (CSDs), sometimes referred to as copying models, underlie numerous practical tools in population genomic analyses. Though an important application that has received much attention is the inference of population structure, the explicit exchange of migrants at specified rates has not hitherto been incorporated into the CSD in a principled framework. Recently, in the case of a single panmictic population, a sequentially Markov CSD has been developed as an accurate, efficient approximation to a principled CSD derived from the diffusion process dual to the coalesc…
Two-type branching processes with immigration, and the structured coalescents
本文研究了两种类型个体在两个岛屿上的分支过程，通过两型Feller扩散模型探讨了频率变化，并推导了不对称两岛频率过程的性质及大种群极限，揭示了平衡选择的新机制。
We identify a new natural coalescent structure, which we call the seed-bank coalescent, that describes the gene genealogy of populations under the influence of a strong seed-bank effect, where "dormant forms" of individuals (such as seeds or spores) may jump a significant number of generations before joining the "active" population. Mathematically, our seed-bank coalescent appears as scaling limit in a Wright-Fisher model with geometric seed-bank age structure if the average time of seed dormancy scales with the order of the total population size $N$. This extends earlier results of Kaj, Krone…
We derive the asymptotic behavior of the total, active and inactive branch lengths of the seed bank coalescent, when the size of the initial sample grows to infinity. Those random variables have important applications for populations evolving under some seed bank effects, such as plants and bacteria, and for some cases of structured populations like metapopulations. The proof relies on the study of the tree at a stopping time corresponding to the first time that a deactivated lineage reactivates. We also give conditional sampling formulas for the random partition and we study the system at the…
In this article, we derive statistical tools to analyze and distinguish the patterns of genetic variability produced by classical and recent population genetic models related to seed banks. In particular, we are concerned with models described by the Kingman coalescent (K), models exhibiting so-called weak seed banks described by a time-changed Kingman coalescent (W), models with so-called strong seed bank described by the seed bank coalescent (S) and the classical two-island model by Wright, described by the structured coalescent (TI). As the presence of a (strong) seed bank should stratify a…
We give constructions of age-structured branching processes without or with immigration as pathwise unique solutions to stochastic integral equations. A necessary and sufficient condition for the ergodicity of the model with immigration is also given.
We examine genetic statistics used in the study of structured populations. In a 1999 paper, Wakeley observed that the coalescent process associated with the finite island model can be decomposed into a scattering phase and a collecting phase. In this paper, we introduce a class of population structure models, which we refer to as G/KC models, that obey such a decomposition. In a large population, large sample limit we derive the distribution of the statistic Fst for all G/KC models under the assumptions of strong or weak mutation. We show that in the large population, large sample limit the is…
Remaining-lifetime age-structured branching processes
本文研究了繁殖律依赖于亲本剩余寿命的年龄结构分支模型，通过随机方程构造了无移民和有移民的测度值过程，并在次临界情形下证明了遍历性条件及逗留时间的极限定理。
Limit theorems for supercritical remaining-lifetime age-structured branching processes
本文研究超临界年龄结构分支模型，给出马尔萨斯归一化随机测度收敛的充要条件，并在特定条件下加强极限理论，证明随机归一化的中心极限定理。
A superprocess limit for an interacting birth-death particle system modelling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to e…
Age-structured stochastic populations under dynamic harvesters' behavior: well-posedness, asymptotic stability and numerically-amenable approximations
本文提出年龄结构种群与收获者反馈的个体模型，证明收敛到耦合随机微分方程，并建立适定性、正性、灭绝边界不可达性及长期行为，附数值实验验证。
Exact phylodynamic likelihood via structured Markov genealogy processes
本文提出了一种基于结构化马尔可夫基因谱过程的方法，通过滤波方程精确计算观察到的基因谱的似然，展示了现有基于凝聚过程和线性生灭过程的系统发育动力学方法是其特殊情形，并提供了用于数值求解滤波方程的算法类，为更广泛的模型提供了统计高效似然基于的系统发育推断的可能性。
We present a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered by Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. We prove that for age distributions with finite mean…
We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. …
We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we present a full kinetic framework for age-structured interacting populations undergoing birth, death and fission processes, in spatially dependent environments. We define the complete probability density for the population-size-age-chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalizati…
We examine the age-structured SIR model, a variant of the classical Susceptible-Infected-Recovered (SIR) model of epidemic propagation, in the context of COVID-19. In doing so, we provide a theoretical basis for the model, perform an empirical validation, and discover the limitations of the model in approximating arbitrary epidemics. We first establish the differential equations defining the age-structured SIR model as the mean-field limits of a continuous-time Markov process that models epidemic spreading on a social network involving random, asynchronous interactions. We then show that, as t…
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