Sequence length bounds for resolving a deep phylogenetic divergence

In evolutionary biology, genetic sequences carry with them a trace of the underlying tree that describes their evolution from a common ancestral sequence. The question of how many sequence sites are required to recover this evolutionary relationship accurately depends on the model of sequence evolution, the substitution rate, divergence times and the method used to infer phylogenetic history. A particularly challenging problem for phylogenetic methods arises when a rapid divergence event occurred in the distant past. We analyse an idealised form of this problem in which the terminal edges of a symmetric four-taxon tree are some factor (λ) times the length of the interior edge. We determine an order λ² lower bound on the growth rate for the sequence length required to resolve the tree (independent of any particular branch length). We also show that this rate of sequence length growth can be achieved by existing methods (including the simple ‘maximum parsimony’ method), and compare these order λ² bounds with an order λ growth rate for a model that describes low-homoplasy evolution. In the final section, we provide a generic bound on the sequence length requirement for a more general class of Markov processes.     

Phylogenetic distances for neighbour dependent substitution processes

We consider models of nucleotidic substitution processes where the rate of substitution at a given site depends on the state of the neighbours of the site. We first estimate the time elapsed between an ancestral sequence at stationarity and a present sequence. Second, assuming that two sequences are issued from a common ancestral sequence at stationarity, we estimate the time since divergence. In the simplest non-trivial case of a Jukes-Cantor model with CpG influence, we provide and justify mathematically consistent estimators in these two settings. We also provide asymptotic confidence intervals, valid for nucleotidic sequences of finite length, and we compute explicit formulas for the estimators and for their confidence intervals. In the general case of an RN model with YpR influence, we extend these results under a proviso, namely that the equation defining the estimator has a unique solution.     

A new look at the critical community size for childhood infections

Quasi-stationarity and time to extinction are studied for the classic endemic model. Attention is restricted to the transition region in parameter space where the quasi-stationary distribution is non-normal. A new approximation of the marginal distribution of infected individuals in quasi-stationarity is presented. It leads to a simple explicit expression for an approximation of the critical community size in terms of model parameters.     

Moment closure and the stochastic logistic model

The quasi-stationary distribution of the stochastic logistic model is studied in the parameter region where its body is approximately normal. Improved asymptotic approximations of its first three cumulants are derived. It is shown that the same results can be derived with the aid of the moment closure method. This indicates that the moment closure method leads to expressions for the cumulants that are asymptotic approximations of the cumulants of the quasi-stationary distribution.     

Type II functional response for continuous, physiologically structured models

The goal of this work is to formulate a general Holling-type functional, or behavioral, response for continuous physiologically structured populations, where both the predator and the prey have physiological densities and certain rules apply to their interactions. The physiological variable can be, for example, a development stage, weight, age, or a characteristic length. The model leads to a Fredholm integral equation for the functional response, and, when inserted into population balance laws, it produces a coupled system of partial differential-integral equations for the two species, with a nonlocal integral term that arises from rules of interaction in the functional response. The general model is, typically, analytically intractable, but specialization to a structured prey-unstructured predator model leads to some analytic results that reveal interesting and unexpected dynamics caused by the presence of size-dependent handling times in the functional response. In this case, steady-states are shown to exist over long times, similar to the stable age-structure solutions for the McKendick-von Foerster model with exponential growth rates determined by the Euler-Lotka equation. But, for type II responses, there are early transient oscillations in the number of births that bifurcate in a few generations into either the decaying or growing steady-state. The bifurcation parameter is the initial level of prey. This special case is applied to a problem of the biological control of a structured pest population (e.g., aphids) by a predator (e.g., lady beetles).     

Stability analysis and optimal vaccination of an SIR epidemic model

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.     

On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment

In this paper we extend the deterministic model for the epidemics induced by virulent phages on bacteria in marine environment introduced by Beretta and Kuang [Math. Biosci. 149 (1998) 57], allowing random fluctuations around the positive equilibrium. The stochastic stability properties of the model are investigated both analytically and numerically suggesting that the deterministic model is robust with respect to stochastic perturbations.     

Encoding phylogenetic trees in terms of weighted quartets

One of the main problems in phylogenetics is to develop systematic methods for constructing evolutionary or phylogenetic trees. For a set of species X, an edge-weighted phylogenetic X-tree or phylogenetic tree is a (graph theoretical) tree with leaf set X and no degree 2 vertices, together with a map assigning a non-negative length to each edge of the tree. Within phylogenetics, several methods have been proposed for constructing such trees that work by trying to piece together quartet trees on X, i.e. phylogenetic trees each having four leaves in X. Hence, it is of interest to characterise when a collection of quartet trees corresponds to a (unique) phylogenetic tree. Recently, Dress and Erdös provided such a characterisation for binary phylogenetic trees, that is, phylogenetic trees all of whose internal vertices have degree 3. Here we provide a new characterisation for arbitrary phylogenetic trees.     

Persistence in reaction diffusion models with weak allee effect

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation. J. S. is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary summer grants, and a grant from Science Council of Heilongjiang Province, China.     

Optimal harvesting and optimal vaccination

Two optimization problems are considered: Harvesting from a structured population with maximal gain subject to the condition of non-extinction, and vaccinating a population with prescribed reduction of the reproduction number of the disease at minimal costs. It is shown that these problems have a similar structure and can be treated by the same mathematical approach. The optimal solutions have a 'two-window' structure: Optimal harvesting and vaccination strategies or policies are concentrated on one or two preferred age classes. The results are first shown for a linear age structure problem and for an epidemic situation at the uninfected state (minimize costs for a given reduction of the reproduction number) and then extended to populations structured by size, to harvesting at Gurtin-MacCamy equilibria and to vaccination at infected equilibria.     

New substitution models for rooting phylogenetic trees

The root of a phylogenetic tree is fundamental to its biological interpretation, but standard substitution models do not provide any information on its position. Here, we describe two recently developed models that relax the usual assumptions of stationarity and reversibility, thereby facilitating root inference without the need for an outgroup. We compare the performance of these models on a classic test case for phylogenetic methods, before considering two highly topical questions in evolutionary biology: the deep structure of the tree of life and the root of the archaeal radiation. We show that all three alignments contain meaningful rooting information that can be harnessed by these new models, thus complementing and extending previous work based on outgroup rooting. In particular, our analyses exclude the root of the tree of life from the eukaryotes or Archaea, placing it on the bacterial stem or within the Bacteria. They also exclude the root of the archaeal radiation from several major clades, consistent with analyses using other rooting methods. Overall, our results demonstrate the utility of non-reversible and non-stationary models for rooting phylogenetic trees, and identify areas where further progress can be made.     

Stepwise mutation likelihood computation by sequential importance sampling in subdivided population models

An importance sampling algorithm for computing the likelihood of a sample of genes at loci under a stepwise mutation model in a subdivided population is developed. This allows maximum likelihood estimation of migration rates between subpopulations. The time to the most recent common ancestor of the sample can also be computed. The technique is illustrated by an analysis of a data set of Australian red fox populations.     

Stochastic search strategy for estimation of maximum likelihood phylogenetic trees

The maximum likelihood (ML) method of phylogenetic tree construction is not as widely used as other tree construction methods (e.g., parsimony, neighbor-joining) because of the prohibitive amount of time required to find the ML tree when the number of sequences under consideration is large. To overcome this difficulty, we propose a stochastic search strategy for estimation of the ML tree that is based on a simulated annealing algorithm. The algorithm works by moving through tree space by way of a "local rearrangement" strategy so that topologies that improve the likelihood are always accepted, whereas those that decrease the likelihood are accepted with a probability that is related to the proportionate decrease in likelihood. Besides greatly reducing the time required to estimate the ML tree, the stochastic search strategy is less likely to become trapped in local optima than are existing algorithms for ML tree estimation. We demonstrate the success of the modified simulated annealing algorithm by comparing it with two existing algorithms (Swofford's PAUP* and Felsenstein's DNAMLK) for several theoretical and real data examples.     

Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls

We investigate a mathematical model of tumor-immune interactions with chemotherapy, and strategies for optimally administering treatment. In this paper we analyze the dynamics of this model, characterize the optimal controls related to drug therapy, and discuss numerical results of the optimal strategies. The form of the model allows us to test and compare various optimal control strategies, including a quadratic control, a linear control, and a state-constraint. We establish the existence of the optimal control, and solve for the control in both the quadratic and linear case. In the linear control case, we show that we cannot rule out the possibility of a singular control. An interesting aspect of this paper is that we provide a graphical representation of regions on which the singular control is optimal.     

A basic limitation on inferring phylogenies by pairwise sequence comparisons

Distance-based approaches in phylogenetics such as Neighbor-Joining are a fast and popular approach for building trees. These methods take pairs of sequences, and from them construct a value that, in expectation, is additive under a stochastic model of site substitution. Most models assume a distribution of rates across sites, often based on a gamma distribution. Provided the (shape) parameter of this distribution is known, the method can correctly reconstruct the tree. However, if the shape parameter is not known then we show that topologically different trees, with different shape parameters and associated positive branch lengths, can lead to exactly matching distributions on pairwise site patterns between all pairs of taxa. Thus, one could not distinguish between the two trees using pairs of sequences without some prior knowledge of the shape parameter. More surprisingly, this can happen for any choice of distinct shape parameters on the two trees, and thus the result is not peculiar to a particular or contrived selection of the shape parameters. On a positive note, we point out known conditions where identifiability can be restored (namely, when the branch lengths are clocklike, or if methods such as maximum likelihood are used).     

Mathematical modeling and qualitative analysis of insulin therapies

Several insulin therapies are widely in clinical use with the basic strategy that mimics insulin secretion in a normal glucose-insulin endocrine metabolic regulatory system. In this paper, we model the insulin therapies using a delay differential equation model. We study the dynamics of the model both qualitatively and quantitatively. The analytical results show the existence and uniqueness of a stable periodic solution that corresponds to ultradian insulin secretion oscillations. Numerically we simulate the insulin administration based on our model. The numerical simulation results are in agreement with findings of clinical studies.     

A stochastic model for prostate-specific antigen levels

We introduce a continuous stochastic model for the prostate-specific antigen (PSA) levels following radiotherapy and derive solutions for the associated partial differential (Kolmogorov-Chapman) equation. The solutions describe the evolution of the time-dependent density for PSA levels which take into account an absorbing condition along the boundary and various initial conditions. We include implications for single-dose and multi-dose radiation treatment regimens and discuss parameter estimation and sensitivity issues.     

Function-valued adaptive dynamics and the calculus of variations

Adaptive dynamics has been widely used to study the evolution of scalar-valued, and occasionally vector-valued, strategies in ecologically realistic models. In many ecological situations, however, evolving strategies are best described as function-valued, and thus infinite-dimensional, traits. So far, such evolution has only been studied sporadically, mostly based on quantitative genetics models with limited ecological realism. In this article we show how to apply the calculus of variations to find evolutionarily singular strategies of function-valued adaptive dynamics: such a strategy has to satisfy Euler's equation with environmental feedback. We also demonstrate how second-order derivatives can be used to investigate whether or not a function-valued singular strategy is evolutionarily stable. We illustrate our approach by presenting several worked examples.     

Statistics for phylogenetic trees

This paper poses the problem of estimating and validating phylogenetic trees in statistical terms. The problem is hard enough to warrant several tacks: we reason by analogy to rounding real numbers, and dealing with ranking data. These are both cases where, as in phylogeny the parameters of interest are not real numbers. Then we pose the problem in geometrical terms, using distances and measures on a natural space of trees. We do not solve the problems of inference on tree space, but suggest some coherent ways of tackling them.