Path-properties of the tree-valued Fleming-Viot process

We consider the tree-valued Fleming-Viot process, $(\mathcal X_t)_{t\geq 0}$, with mutation and selection as studied in Depperschmidt, Greven, Pfaffelhuber (2012). This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming-Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent. In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the neutral Fleming-Viot genealogies at fixed time $t$ arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than \ve apart in the limit \ve\to 0. Second, we answer two open questions about the sample paths of the tree-valued Fleming-Viot process. We show that for all $t>0$ almost surely the marked metric measure space $\mathcal X_t$ has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming-Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov's formula giving absolute continuity.Comment: 47 pages, 1 figur

Tree-valued Fleming-Viot dynamics with mutation and selection

The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.Comment: Published in at http://dx.doi.org/10.1214/11-AAP831 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Marked metric measure spaces

A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I. It arises as state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm- spaces and identify a convergence determining algebra of functions, called polynomials.Comment: 15 page

Implementation of a standardized protocol to manage elderly patients with low energy pelvic fractures: can service improvement be expected?

Purpose: The incidence of low energy pelvic fractures (FPFs) in the elderly is increasing. Comorbidities, decreased bone-quality, problematic fracture fixation and poor compliance represent some of their specific difficulties. In the absence of uniform management, a standard operating procedure (SOP) was introduced to our unit, aiming to improve the quality of services provided to these patients. Methods: A cohort study was contacted to test the impact of (1) using a specific clinical algorithm and (2) using different antiosteoporotic drugs. Multivariate regression analysis was used to determine prognostic factors. Study endpoints were the time-to-healing, length-of-stay, return to pre-injury mobility, union status, mortality and complications. Results: A total of 132 elderly patients (≥65 years) admitted during the period 2012–2014 with FPFs were enrolled. High-energy fractures, acetabular fractures, associated trauma affecting mobility, pathological pelvic lesions and operated FPFs were used as exclusion criteria. The majority of included patients were females (108/132; 81.8%), and the mean age was 85.8 years (range 67–108). Use of antiosteoporotics was associated with a shorter time of healing (p = 0.036). Patients treated according to the algorithm showed a significant protection against malunion (p < 0.001). Also, adherence to the algorithm allowed more patients to return to their pre-injury mobility status (p = 0.039). Conclusions: The use of antiosteoporotic medication in elderly patients with fragility pelvic fractures was associated with faster healing, whilst the adherence to a structured clinical pathway led to less malunions and non-unions and return to pre-injury mobility state