On the construction and topology of multi-type ancestral trees

Branching processes or Galton-Watson processes can be used to model the genealogy of a population of different species, where birth and death events represent speciation and extinction. In the more general context of multi-type branching processes, species are classified under phenotypical traits, and the probability of speciation and extinction is dependent on individual types. Since most accessible biological data concerns surviving species, it becomes necessary to extract information about the shape of genealogical trees from the available knowledge on the standing population, and to devise random models allowing backward reconstruction of ancestry under the rules of a particular branching process. We present two investigations on the topology of ancestral multi-type branching trees, generalizing several known results from the single-type case, and obtaining some new results that can only be formulated in the multi-type setting. In the first part of the thesis, we present a backward construction algorithm for the ancestral tree of a planar embedding of a multi-type Galton-Watson tree assumed to be quasi-stationary, and we derive formulae for the conditional distribution of the time to the most recent common ancestor of two consecutive individuals at the present time, and of two individuals of the same type. We specialize these formulae to multi-type linear-fractional branching processes, and observe some effects of the symmetry of the parameters in the two-type case. In the second part of the thesis, we extend the concepts of cherries and pendant edges from rooted binary trees to the multi-type setting, and compute expressions and asymptotic properties for mean numbers and variances of these structures under the neutral two-type Yule model. We explain how type mutations appear naturally in ancestral trees of multi-type birth-death processes, and show that these ancestral trees are Markovian and behave as pure-birth processes, by giving explicit time-dependent rates. We derive formulae and asymptotic properties for the mean number of cherries and pendant edges of each type in a multi-type pure-birth process with mutations. We show that sometimes it is possible to recover the defining rates of this process from the asymptotic proportion of leaves, cherries and pendant edges of each type

Moment asymptotics for multitype branching random walks in random environment

We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter $1/\rho_{ij}$ for the upper tail of the mean number of $j$ type particles produced by an $i$ type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system

Likelihood-Based Inference for Discretely Observed Birth-Death-Shift Processes, with Applications to Evolution of Mobile Genetic Elements

Continuous-time birth-death-shift (BDS) processes are frequently used in stochastic modeling, with many applications in ecology and epidemiology. In particular, such processes can model evolutionary dynamics of transposable elements - important genetic markers in molecular epidemiology. Estimation of the effects of individual covariates on the birth, death, and shift rates of the process can be accomplished by analyzing patient data, but inferring these rates in a discretely and unevenly observed setting presents computational challenges. We propose a mutli-type branching process approximation to BDS processes and develop a corresponding expectation maximization (EM) algorithm, where we use spectral techniques to reduce calculation of expected sufficient statistics to low dimensional integration. These techniques yield an efficient and robust optimization routine for inferring the rates of the BDS process, and apply more broadly to multi-type branching processes where rates can depend on many covariates. After rigorously testing our methodology in simulation studies, we apply our method to study intrapatient time evolution of IS6110 transposable element, a frequently used element during estimation of epidemiological clusters of Mycobacterium tuberculosis infections.Comment: 31 pages, 7 figures, 1 tabl