Eigenanalysis for the Kolmogorov backward equation for the neutral multi-allelic model

In a previous paper [9], we have given an algorithm for obtaining the time dependent solution of all polynomial moments of gene frequencies in neutral models. The recurrence formula for the moment generating functions obtained in [9] gives information about the eigenvalues and the eigenfunctions in neutral models. In this article, we solve the eigenvalue problem for the Kolmogorov backward equation for the case of neutral alleles under mutation pressure.     

A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence

The iterated birth and death Markov process is defined as an n-fold iteration of a birth and death Markov process describing kinetics of certain population combined with random killing of individuals in the population at moments tau 1,...,tau n with given survival probabilities s1,...,sn. A long-standing problem of computing the distribution of the number of clonogenic tumor cells surviving an arbitrary fractionated radiation schedule is solved within the framework of iterated birth and death Markov process. It is shown that, for any initial population size iota, the distribution of the size N of the population at moment t > or = tau n is generalized negative binomial, and an explicit computationally feasible formula for the latter is found. It is shown that if i --> infinity and sn --> 0 so that the product iota s1...sn tends to a finite positive limit, the distribution of random variable N converges to a probability distribution, which for t = tau n turns out to be Poisson. In the latter case, an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained.     

Neighbourhood control policies and the spread of infectious diseases

We present a model of a control programme for a disease outbreak in a population of livestock holdings. Control is achieved by culling infectious holdings when they are discovered and by the pre-emptive culling of livestock on holdings deemed to be at enhanced risk of infection. Because the pre-emptive control programme cannot directly identify exposed holdings, its implementation will result in the removal of both infected and uninfected holdings. This leads to a fundamental trade-off: increased levels of control produce a greater reduction in transmission by removing more exposed holdings, but increase the number of uninfected holdings culled. We derive an expression for the total number of holdings culled during the course of an outbreak and demonstrate that there is an optimal control policy, which minimizes this loss. Using a metapopulation model to incorporate local clustering of infection, we examine a neighbourhood control programme in a locally spreading outbreak. We find that there is an optimal level of control, which increases with increasing basic reproduction ratio, R(0); moreover, implementation of control may be optimal even when R(0) < 1. The total loss to the population is relatively insensitive to the level of control as it increases beyond the optimal level, suggesting that over-control is a safer policy than under-control.     

Risk of Foot-and-Mouth Disease spread due to sole occupancy authorities and linked cattle holdings

Livestock movements in Great Britain are well recorded, have been extensively analysed with respect to their role in disease spread, and have been used in real time to advise governments on the control of infectious diseases. Typically, livestock holdings are treated as distinct entities that must observe movement standstills upon receipt of livestock, and must report livestock movements. However, there are currently two dispensations that can exempt holdings from either observing standstills or reporting movements, namely the Sole Occupancy Authority (SOA) and Cattle Tracing System (CTS) Links, respectively. In this report we have used a combination of data analyses and computational modelling to investigate the usage and potential impact of such linked holdings on the size of a Foot-and-Mouth Disease (FMD) epidemic. Our analyses show that although SOAs are abundant, their dynamics appear relatively stagnant. The number of CTS Links is also abundant, and increasing rapidly. Although most linked holdings are only involved in a single CTS Link, some holdings are involved in numerous links that can be amalgamated to form "CTS Chains" which can be both large and geographically dispersed. Our model predicts that under a worst case scenario of "one infected - all infected", SOAs do pose a risk of increasing the size (in terms of number of infected holdings) of a FMD epidemic, but this increase is mainly due to intra-SOA infection spread events. Furthermore, although SOAs do increase the geographic spread of an epidemic, this increase is predominantly local. Whereas, CTS Chains pose a risk of increasing both the size and the geographical spread of the disease substantially, under a worse case scenario. Our results highlight the need for further investigations into whether CTS Chains are transmission chains, and also investigations into intra-SOA movements and livestock distributions due to the lack of current data.     

Iterated birth and death process as a model of radiation cell survival

The iterated birth and death process is defined as an n-fold iteration of a stochastic process consisting of the combination of instantaneous random killing of individuals in a certain population with a given survival probability s with a Markov birth and death process describing subsequent population dynamics. A long standing problem of computing the distribution of the number of clonogenic tumor cells surviving a fractionated radiation schedule consisting of n equal doses separated by equal time intervals tau is solved within the framework of iterated birth and death processes. For any initial tumor size i, an explicit formula for the distribution of the number M of surviving clonogens at moment tau after the end of treatment is found. It is shown that if i-->infinity and s-->0 so that is(n) tends to a finite positive limit, the distribution of random variable M converges to a probability distribution, and a formula for the latter is obtained. This result generalizes the classical theorem about the Poisson limit of a sequence of binomial distributions. The exact and limiting distributions are also found for the number of surviving clonogens immediately after the nth exposure. In this case, the limiting distribution turns out to be a Poisson distribution.     

Multiple Decrement Analysis with Dependent Risks

In a competing risks problem where a well-defined population is exposed simultaneously to several causes of death, interest has centered on the estimation of the probability of death from a given cause when one or more other causes have been eliminated. A basic component of all available procedures for estimating these probabilities is the assumption that the several causes of death act independently—an unrealistic assumption in the context of human and animal populations. This article considers the estimation of these probabilities assuming the existence ofinterdependencies among the various causes of death. A general formula is derived based on a given set of crude probabilities of death as well as the characteristics of the joint distribution of random variables indicating death from the various causes. This formula identifies alternative assumptions, less restrictive than that of independent risks, which may he used for estimation purposes.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Some model based considerations on observing generation times for communicable diseases

The generation time of an infectious disease is usually defined as the time from the moment one person becomes infected until that person infects another person. The concept is similar to “generation gap” in demography, with new infections replacing births in a population. Originally applied to diseases such as measles where at least the first generations are clearly discernible, the concept has recently been extended to other diseases, such as influenza, where time order of infections is usually much less apparent.By formulating the relevant statistical questions within a simple yet basic mathematical model for infection spread, it is possible to derive theoretical properties of observations in various situations e.g. in “isolation”, in households, or during large outbreaks. In each case, it is shown that the sampling distribution of observations depends on a number of factors, usually not considered in the literature and that must be taken into account in order to achieve unbiased inference about the generation time distribution. Some implications of these findings for statistical inference methods in epidemic spread models are discussed.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

Efficient prediction of nucleic acid binding function from low-resolution protein structures

Structural genomics projects as well as ab initio protein structure prediction methods provide structures of proteins with no sequence or fold similarity to proteins with known functions. These are often low-resolution structures that may only include the positions of C alpha atoms. We present a fast and efficient method to predict DNA-binding proteins from just the amino acid sequences and low-resolution, C alpha-only protein models. The method uses the relative proportions of certain amino acids in the protein sequence, the asymmetry of the spatial distribution of certain other amino acids as well as the dipole moment of the molecule. These quantities are used in a linear formula, with coefficients derived from logistic regression performed on a training set, and DNA-binding is predicted based on whether the result is above a certain threshold. We show that the method is insensitive to errors in the atomic coordinates and provides correct predictions even on inaccurate protein models. We demonstrate that the method is capable of predicting proteins with novel binding site motifs and structures solved in an unbound state. The accuracy of our method is close to another, published method that uses all-atom structures, time-consuming calculations and information on conserved residues.     

New biomolecular tools for aerobiological monitoring: Identification of major allergenic Poaceae species through fast real‐time PCR

Grasses (Poaceae) are very common plants, which are widespread in all environments and urban areas. Despite their economical importance, they can represent a problem to humans due to their abundant production of allergenic pollen. Detailed information about the pollen season for these species is needed in order to plan adequate therapies and to warn allergic people about the risks they take in certain areas at certain moments. Moreover, precise identification of the causative species and their allergens is necessary when the patient is treated with allergen‐specific immunotherapy. The intrafamily morphological similarity of grass pollen grains makes it impossible to distinguish which particular species is present in the atmosphere at a given moment. This study aimed at developing new biomolecular tools to analyze aerobiological samples and identifying major allergenic Poaceae taxa at subfamily or species level, exploiting fast real‐time PCR. Protocols were tested for DNA extraction from pollen sampled with volumetric and gravimetric methods. A fragment of the matK plastidial gene was amplified and sequenced in Poaceae species known to have high allergological impact. Species‐ and subfamily‐specific primer–probe systems were designed and tested in fast real‐time PCRs to evaluate the presence of these taxa in aerobiological pollen samples. Species‐specific systems were obtained for four of five studied species. A primer–probe set was also proposed for the detection of Pooideae (a grass subfamily that includes also major cereal grains) in aerobiological samples, as this subfamily includes species carrying both grass allergens from groups 1 and 5. These, among the 11 groups in which grass pollen allergens are classified, are considered responsible for the most frequent and severe symptoms.     

Constructing and counting phylogenetic invariants.

The method of invariants is an approach to the problem of reconstructing the phylogenetic tree of a collection of m taxa using nucleotide sequence data. Models for the respective probabilities of the 4m possible vectors of bases at a given site will have unknown parameters that describe the random mechanism by which substitution occurs along the branches of a putative phylogenetic tree. An invariant is a polynomial in these probabilities that, for a given phylogeny, is zero for all choices of the substitution mechanism parameters. If the invariant is typically non-zero for another phylogenetic tree, then estimates of the invariant can be used as evidence to support one phylogeny over another. Previous work of Evans and Speed showed that, for certain commonly used substitution models, the problem of finding a minimal generating set for the ideal of invariants can be reduced to the linear algebra problem of finding a basis for a certain lattice (that is, a free Z-module). They also conjectured that the cardinality of such a generating set can be computed using a simple "degrees of freedom" formula. We verify this conjecture. Along the way, we explain in detail how the observations of Evans and Speed lead to a simple, computationally feasible algorithm for constructing a minimal generating set.     

Extinction risk of a meta-population: aggregation approach

Aggregation of variables of a complex mathematical model with realistic structure gives a simplified model which is more suitable than the original one when the amount of data for parameter estimation is limited. Here we explore use of a formula derived for a single unstructured population (canonical model) in predicting the extinction time for a population living in multiple habitats. In particular we focus multiple populations each following logistic growth with demographic and environmental stochasticities, and examine how the mean extinction time depends on the migration and environmental correlation. When migration rate and/or environmental correlation are very large or very small, we may express the mean extinction time exactly using the formula with properly modified parameters. When parameters are of intermediate magnitude, we generate a Monte Carlo time series of the population size for the realistic structured model, estimate the "effective parameters" by fitting the time series to the canonical model, and then calculate the mean extinction time using the formula for a single population. The mean extinction time predicted by the formula was close to those obtained from direct computer simulation of structured models. We conclude that the formula for an unstructured single-population model has good approximation capability and can be applicable in estimating the extinction risk of the structured meta-population model for a limited data set.     

A new sampling formula for neutral biodiversity

The neutral model of biodiversity, proposed by Hubbell (The Unified Neutral Theory of Biodiversity and Biogeography, Princeton University Press, Princeton, NJ, 2001) to explain the diversity of functionally equivalent species, has been subject of hot debate in community ecology. Whereas Hubbell studied the model mostly by simulations, recently analytical treatments have yielded expressions of the expected number of species of a particular abundance in a local community with dispersal limitation. Moreover, a formula has been offered for the joint likelihood of observing a given species‐abundance dataset in a local community with dispersal limitation, but this formula is too complicated to allow practical applications. Here, I present a much simplified expression that can be regarded as an enhanced version of the famous Ewens sampling formula. It can be used in maximum likelihood methods for quick estimation of the model parameters, using all information in the data, and for model comparison. I also show how to rapidly generate examples of species‐abundance distributions for a given set of model parameters and how to calculate Simpson's diversity index.     

A threshold theorem for the general epidemic in discrete time

Summary A theorem, analogous to the continuous time Threshold Theorem of Kermack and McKendrick, is proved for a certain discrete time epidemic model. This model, in contrast to its continuous time analogue, leads to some solutions in which the total population of susceptibles may become infected in a finite time.     

Household Epidemics: Modelling Effects of Early Stage Vaccination

A Markovian susceptible → infectious → removed (SIR) epidemic model is considered in a community partitioned into households. A vaccination strategy, which is implemented during the early stages of the disease following the detection of infected individuals is proposed. In this strategy, the detection occurs while an individual is infectious and other susceptible household members are vaccinated without further delay. Expressions are derived for the influence on the reproduction numbers of this vaccination strategy for equal and unequal household sizes. We fit previously estimated parameters from influenza and use household distributions for Sweden and Tanzania census data. The results show that the reproduction number is much higher in Tanzania (6 compared with 2) due to larger households, and that infected individuals have to be detected (and household members vaccinated) after on average 5 days in Sweden and after 3.3 days in Tanzania, a much smaller difference.     

A Markov chain occurring in enzyme kinetics

A certain Markov chain which was encountered by T. L. Hill in the study of the kinetics of a linear array of enzymes is studied. An explicit formula for the steady state probabilities is given and some conjectures raised by T. L. Hill are proved.     

Multiple infection dynamics has pronounced effects on the fitness of RNA viruses

Several factors play a role during the replication and transmission of RNA viruses. First, as a consequence of their enormous mutation rate, complex mixtures of genomes are generated immediately after infection of a new host. Secondly, differences in growth and competition rates drive the selection of certain genetic variants within an infected host. Thirdly, but not less important, a random sampling occurs at the moment of viral infectious passage from an infected to a healthy host. In addition, the availability of hosts also influences the fate of a given viral genotype. When new hosts are scarce, different viral genotypes might infect the same host, adding an extra complexity to the competition among genetic variants. We have employed a two‐fold approach to analyse the role played by each of these factors in the evolution of RNA viruses. First, we have derived a model that takes into account all the preceding factors. This model employs the classic Lotka‐Volterra competition equations but it also incorporates the effect of mutation during RNA replication, the effect of the stochastic sampling at the moment of infectious passage among hosts and, the effect of the type of infection (single, coinfection or superinfection). Secondly, the predictions of the model have been tested in an in vitro evolution experiment. Both theoretical and experimental results show that in infection passages with coinfection viral fitness increased more than in single infections. In contrast, infection passages with superinfection did not differ from the single infection. The coinfection frequency also affected the outcome: the larger the proportion of viruses coinfecting a host, the larger increase in fitness observed.     

Some properties of a simple stochastic epidemic model of SIR type

We investigate the properties of a simple discrete time stochastic epidemic model. The model is Markovian of the SIR type in which the total population is constant and individuals meet a random number of other individuals at each time step. Individuals remain infectious for R time units, after which they become removed or immune. Individual transition probabilities from susceptible to diseased states are given in terms of the binomial distribution. An expression is given for the probability that any individuals beyond those initially infected become diseased. In the model with a finite recovery time R, simulations reveal large variability in both the total number of infected individuals and in the total duration of the epidemic, even when the variability in number of contacts per day is small. In the case of no recovery, R=infinity, a formal diffusion approximation is obtained for the number infected. The mean for the diffusion process can be approximated by a logistic which is more accurate for larger contact rates or faster developing epidemics. For finite R we then proceed mainly by simulation and investigate in the mean the effects of varying the parameters p (the probability of transmission), R, and the number of contacts per day per individual. A scale invariant property is noted for the size of an outbreak in relation to the total population size. Most notable are the existence of maxima in the duration of an epidemic as a function of R and the extremely large differences in the sizes of outbreaks which can occur for small changes in R. These findings have practical applications in controlling the size and duration of epidemics and hence reducing their human and economic costs.