A law of large numbers approximation for Markov population processes with countably many types

When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since the population size has no natural upper limit, this leads to systems in which there are countably infinitely many possible types of individual. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove a law of large numbers for rather general systems of this kind, together with a rather sharp bound on the rate of convergence in an appropriately chosen weighted $\ell_1$ norm.Comment: revised version in response to referee comments, 34 page

Total variation approximation for quasi-equilibrium distributions

Quasi-stationary distributions, as discussed by Darroch & Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In this paper, we give biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute.Comment: 16 page

Estimation of the transmission dynamics of Theileria equi and Babesia caballi in horses

For the evaluation of the epidemiology of Theileria equi and Babesia caballi in a herd of 510 horses in SW Mongolia, several mathematical models of the transmission dynamics were constructed. Because the field data contain information on the presence of the parasite (determined by PCR) and the presence of antibodies (determined by IFAT), the models cater for maternal protection with antibodies, susceptible animals, infected animals and animals which have eliminated the parasite and also allow for age-dependent infection in susceptible animals. Maximum likelihood estimation procedures were used to estimate the model parameters and a Monte Carlo approach was applied to select the best fitting model. Overall, the results are in line with previous experimental work, and add evidence that the epidemiology of T. equi differs from that of Babesia spp. The presented modelling approach provides a useful tool for the investigation of some vector-borne diseases and the applied model selection procedure avoids asymptotical assumptions that may not be adequate for the analysis of epidemiological field dat

Genetic differentiation in Scottish populations of the pine beauty moth Panolis flammea (Lepidoptera: Noctuidae)

Pine beauty moth, Panolis flammea (Denis & Schiffermüller), is a recent but persistent pest of lodgepole pine plantations in Scotland, but exists naturally at low levels within remnants and plantations of Scots pine. To test whether separate host races occur in lodgepole and Scots pine stands and to examine colonization dynamics, allozyme, randomly amplified polymorphic DNA (RAPD) and mitochondrial variation were screened within a range of Scottish samples. RAPD analysis indicated limited long distance dispersal (FST = 0.099), and significant isolation by distance (P < 0.05); but that colonization between more proximate populations was often variable, from extensive to limited exchange. When compared with material from Germany, Scottish samples were found to be more diverse and significantly differentiated for all markers. For mtDNA, two highly divergent groups of haplotypes were evident, one group contained both German and Scottish samples and the other was predominantly Scottish. No genetic differentiation was evident between P. flammea populations sampled from different hosts, and no diversity bottleneck was observed in the lodgepole group. Indeed, lodgepole stands appear to have been colonized on multiple occasions from Scots pine sources and neighbouring populations on different hosts are close to panmixia.A.J. Lowe, B.J. Hicks, K. Worley, R.A. Ennos, J.D. Morman, G. Stone and A.D. Wat

Smallest small-world network

Efficiency in passage times is an important issue in designing networks, such as transportation or computer networks. The small-world networks have structures that yield high efficiency, while keeping the network highly clustered. We show that among all networks with the small-world structure, the most efficient ones have a single ``center'', from which all shortcuts are connected to uniformly distributed nodes over the network. The networks with several centers and a connected subnetwork of shortcuts are shown to be ``almost'' as efficient. Genetic-algorithm simulations further support our results.Comment: 5 pages, 6 figures, REVTeX

Variations on the Seventh Route to Relativity

As motivated in the full abstract, this paper further investigates Barbour, Foster and O Murchadha (BFO)'s 3-space formulation of GR. This is based on best-matched lapse-eliminated actions and gives rise to several theories including GR and a conformal gravity theory. We study the simplicity postulates assumed in BFO's work and how to weaken them, so as to permit the inclusion of the full set of matter fields known to occur in nature. We study the configuration spaces of gravity-matter systems upon which BFO's formulation leans. In further developments the lapse-eliminated actions used by BFO become impractical and require generalization. We circumvent many of these problems by the equivalent use of lapse-uneliminated actions, which furthermore permit us to interpret BFO's formulation within Kuchar's generally covariant hypersurface framework. This viewpoint provides alternative reasons to BFO's as to why the inclusion of bosonic fields in the 3-space approach gives rise to minimally-coupled scalar fields, electromagnetism and Yang--Mills theory. This viewpoint also permits us to quickly exhibit further GR-matter theories admitted by the 3-space formulation. In particular, we show that the spin-1/2 fermions of the theories of Dirac, Maxwell--Dirac and Yang--Mills--Dirac, all coupled to GR, are admitted by the generalized 3-space formulation we present. Thus all the known fundamental matter fields can be accommodated. This corresponds to being able to pick actions for all these theories which have less kinematics than suggested by the generally covariant hypersurface framework. For all these theories, Wheeler's thin sandwich conjecture may be posed, rendering them timeless in Barbour's sense.Comment: Revtex version; Journal-ref adde

The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications

An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is {\em approximately sparse}, i.e., roughly speaking, its $L_1$-norm is small outside a small set. By building on this result, we obtain the following applications: {\bf Learning Theory.} We design the first computationally efficient learning algorithm for PMDs with respect to the total variation distance. Our algorithm learns an arbitrary $(n, k)$-PMD within variation distance $\epsilon$ using a near-optimal sample size of $\widetilde{O}_k(1/\epsilon^2),$ and runs in time $\widetilde{O}_k(1/\epsilon^2) \cdot \log n.$ Previously, no algorithm with a $\mathrm{poly}(1/\epsilon)$ runtime was known, even for $k=3.$ {\bf Game Theory.} We give the first efficient polynomial-time approximation scheme (EPTAS) for computing Nash equilibria in anonymous games. For normalized anonymous games with $n$ players and $k$ strategies, our algorithm computes a well-supported $\epsilon$-Nash equilibrium in time $n^{O(k^3)} \cdot (k/\epsilon)^{O(k^3\log(k/\epsilon)/\log\log(k/\epsilon))^{k-1}}.$ The best previous algorithm for this problem had running time $n^{(f(k)/\epsilon)^k},$ where $f(k) = \Omega(k^{k^2})$, for any $k>2.$ {\bf Statistics.} We prove a multivariate central limit theorem (CLT) that relates an arbitrary PMD to a discretized multivariate Gaussian with the same mean and covariance, in total variation distance. Our new CLT strengthens the CLT of Valiant and Valiant by completely removing the dependence on $n$ in the error bound.Comment: 68 pages, full version of STOC 2016 pape

Survival of branching random walks in random environment

We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${\mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2\times 2$ random matrices.Comment: 17 pages; to appear in Journal of Theoretical Probabilit