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This work is concerned with the transmissibility of a disease, on observation of an outbreak of limited size. When such an outbreak occurs, an accurate estimate of the transmissibility of the responsible pathogen is essential for an appropriate response to future outbreaks. Transmissibility is usually characterised in terms of the reproduction number, R, which is the mean number of new cases of infection produced by a single infectious individual. A subcritical reproduction number (R < 1) guarantees that an outbreak will eventually die out of its own accord. By contrast, a supercritical reproduction number (R > 1) does not guarantee spread of the disease, since even with appreciable transmissibility, an outbreak may become extinct due to stochastic effects associated with a small number of infected individuals. Once the number of infectious individuals is conditioned on extinction, we show that an exact symmetry of the underlying theory ensures two distinct values of R, one larger than unity, the other smaller than unity, for which all outbreak properties are identical, with no signature of difference. Therefore a disease with a subcritical R, or its supercritical counterpart, when conditioned on extinction, have, for a given outbreak, identical individual likelihoods. In the full likelihood, this symmetry is lost, since the individual likelihood for a subcritical R is weighted by an extinction probability of unity, but the individual likelihood of a supercritical R is weighted by a sub-unity extinction probability. However, the inference can still benefit from the underlying symmetry, since it yields a mapping of all supercritical reproduction numbers onto the subcritical domain (R < 1), thereby speeding up evaluation of the likelihood profile. The symmetry holds in the standard situation, where the distribution of secondary cases is Poisson, as well as where this distribution has a negative binomial form and super-spreading can occur., (Copyright © 2019 The Authors. Published by Elsevier Ltd.. All rights reserved.)

Adaptation is a defining property of living systems. It occurs when organisms become better suited to their environment. The phenomena that people find most fascinating about biological systems are, in general, the result of adaptive processes. Examples include the mammalian central nervous system, the flight of birds and insects, photosynthesis, and the human hand. However, despite the centrality of adaptation for biology, there is no generally agreed quantitative way to describe the degree to which an organism is adapted. Here, we address this situation by proposing a quantitative measure of adaptation. We also present results of computer simulations which demonstrate that some changes in parameter values cause mean adaptedness and mean relative fitness to change in opposite directions. This indicates that adaptedness and relative fitness are distinct concepts. We suggest that the measure of adaptedness proposed in this work may help to resolve questions about 'units of selection' and 'major transitions in evolution'., (Copyright © 2018 Elsevier Ltd. All rights reserved.)

In this work we consider fixation of an allele in a population. Fixation is key to understanding the way long-term evolutionary change occurs at the gene and molecular levels. Two basic aspects of fixation are: (i) the chance it occurs and (ii) the way the gene frequency progresses to fixation. We present exact results for both aspects of fixation for the Wright-Fisher model. We give the exact fixation probability for some different schemes of frequency-dependent selection. We also give the corresponding exact stochastic difference equation that generates frequency trajectories which ultimately fix. Exactness of the results means selection need not be weak. There are possible applications of this work to data analysis, modelling, and tests of approximations. The methodology employed illustrates that knowledge of the fixation probability, for all initial frequencies, fully characterises the dynamics of the Wright-Fisher model. The stochastic equations for fixing trajectories allow insight into the way fixation occurs. They provide the alternative picture that fixation is driven by the injection of one carrier of the fixing allele into the population each generation. The stochastic equations allow explicit calculation of some properties of fixing trajectories and their efficient simulation. The results are illustrated and tested with simulations., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

Random processes in biology, in particular random genetic drift, often make it difficult to predict the fate of a particular mutation in a population. Using principles of theoretical population genetics, we present a form of biological control that ensures a focal allele's frequency, at a given locus, achieves a prescribed probability distribution at a given time. This control is in the form of an additional evolutionary force that acts on a population. We provide the mathematical framework that determines the additional force. Our analysis indicates that generally the additional force depends on the frequency of the focal allele, and it may also depend on the time. We argue that translating this additional force into an externally controlled process, which has the possibility of being implemented in a number of different ways corresponding to selection, migration, mutation, or a combination of these, may provide a flexible instrument for targeted change of traits of interest in natural populations. This framework may be applied, or used as an informed form of guidance, in a variety of different biological scenarios including: yield and pesticide optimisation in crop production, biofermentation, the local regulation of human-associated natural populations, such as parasitic animals, or bacterial communities in hospitals., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

In this work we assume that we have some knowledge about the state of a population at two known times, when the dynamics is governed by a Markov chain such as a Wright-Fisher model. Such knowledge could be obtained, for example, from observations made on ancient and contemporary DNA, or during laboratory experiments involving long term evolution. A natural assumption is that the behaviour of the population, between observations, is related to (or constrained by) what was actually observed. The present work shows that this assumption has limited validity. When the time interval between observations is larger than a characteristic value, which is a property of the population under consideration, there is a range of intermediate times where the behaviour of the population has reduced or no dependence on what was observed and an equilibrium-like distribution applies. Thus, for example, if the frequency of an allele is observed at two different times, then for a large enough time interval between observations, the population has reduced or no dependence on the two observed frequencies for a range of intermediate times. Given observations of a population at two times, we provide a general theoretical analysis of the behaviour of the population at all intermediate times, and determine an expression for the characteristic time interval, beyond which the observations do not constrain the population's behaviour over a range of intermediate times. The findings of this work relate to what can be meaningfully inferred about a population at intermediate times, given knowledge of terminal states., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

The Wright-Fisher model is an important model in evolutionary biology and population genetics. It has been applied in numerous analyses of finite populations with discrete generations. It is recognised that real populations can behave, in some key aspects, as though their size that is not the census size, N, but rather a smaller size, namely the effective population size, Ne. However, in the Wright-Fisher model, there is no distinction between the effective and census population sizes. Equivalently, we can say that in this model, Ne coincides with N. The Wright-Fisher model therefore lacks an important aspect of biological realism. Here, we present a method that allows Ne to be directly incorporated into the Wright-Fisher model. The modified model involves matrices whose size is determined by Ne. Thus apart from increased biological realism, the modified model also has reduced computational complexity, particularly so when Ne⪡N. For complex problems, it may be hard or impossible to numerically analyse the most commonly-used approximation of the Wright-Fisher model that incorporates Ne, namely the diffusion approximation. An alternative approach is simulation. However, the simulations need to be sufficiently detailed that they yield an effective size that is different to the census size. Simulations may also be time consuming and have attendant statistical errors. The method presented in this work may then be the only alternative to simulations, when Ne differs from N. We illustrate the straightforward application of the method to some problems involving allele fixation and the determination of the equilibrium site frequency spectrum. We then apply the method to the problem of fixation when three alleles are segregating in a population. This latter problem is significantly more complex than a two allele problem and since the diffusion equation cannot be numerically solved, the only other way Ne can be incorporated into the analysis is by simulation. We have achieved good accuracy in all cases considered. In summary, the present work extends the realism and tractability of an important model of evolutionary biology and population genetics., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

We consider the simplest form of negative frequency-dependent selection in a biallelic haploid population, where the selection coefficient of a mutant allele is a linear function of the allele's frequency, and changes from positive to negative as the frequency is increased. In an effectively infinite population this behaviour leads to a stable polymorphism. We present a theoretical investigation of what occurs in a finite population, where a long-lived polymorphism may be formed, but which fluctuates and ultimately disappears due to random genetic drift. We model the dynamics as a branching process and explicitly take into account differences between the census population size and the effective population size, which play different roles in the dynamics. We characterise the behaviour of the population in terms of three distinct timescales associated with: (i) early loss of mutant alleles, (ii) achievement of the long-lived polymorphism, (iii) disappearance of the polymorphism. Timescales (i) and (iii) depend on the effective population size and are, as a consequence, affected by random genetic drift, while timescale (ii) depends primarily on the census size and is relatively insensitive to genetic drift. Analysis and simulations of the branching process clarify the different influences of the census and effective population sizes. One substantial quantitative difference, between populations where the effective and census population sizes coincide and where they differ, lies in the number of mutant alleles in the long-lived polymorphism. This number is approximately proportional to the census size. Thus assuming the census size equals a much smaller effective population size predicts a much smaller number of mutants in the long-lived polymorphism., (Copyright © 2015 Elsevier Ltd. All rights reserved.)

We consider a population whose size changes stochastically under a branching process, with the added modification that each generation a fixed number of individuals are removed, irrespective of the size of the population. We call removal that is independent of population size 'hard harvesting'. A key feature of hard harvesting occurs if the size of the population is smaller than the fixed number that are harvested. In such a case, the dynamics cannot continue and must terminate. We find that even for populations with a tendency to grow, there is a finite probability of termination. We determine the probability of termination, and given that termination occurs, we characterise the statistical properties of the random time to termination. We determine the impact of hard harvesting on the size of the population, in populations where termination has not occurred., (Copyright © 2015 Elsevier Ltd. All rights reserved.)

The way population size, population structure (with migration), and spatially dependent selection (where there is no globally optimal allele), combine to affect the substitution rate is poorly understood. Here, we consider a two patch model where mutant alleles are beneficial in one patch and deleterious in the other patch. We assume that the spatial average of selection on mutant alleles is zero. We take each patch to maintain a finite number of N adults each generation, hence random genetic drift can independently occur in each patch. We show that the principal way the population size, N, when large, affects the substitution rate, R∞, is through its dependence on two composite parameters. These are the scaled migration rate M (∝ population size × migration rate), and the scaled selection intensity S (∝population size × beneficial effect of a mutant). Any relation between S and M that arises for ecological/evolutionary reasons can strongly influence the way the substitution rate, R∞, depends on the population size, N. In the simplest situation, both M and S are proportional to N, and this is shown to lead to R∞ increasing with N when S is not large. The behaviour, that R∞ increases with N, is not inevitable; a more complex relation between S and M can lead to the opposite or other behaviours. In particular, let us assume that dM/dN is positive, as would occur if the migration rate were constant, S is not large, and S depends on M (i.e., S=S(M)). We then find that if S(M) satisfies S(M)>((1+M)/1+2M)S(0) then the substitution rate, R∞, increases with N, but if S(M)<((1+M)/1+2M)S(0) then R∞ decreases with N., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Forward and backward simulations play an increasing role in population genetics, in particular when inferring the relative importance of evolutionary forces. It is therefore important to develop fast and accurate simulation methods for general population genetics models. Here we present an exact simulation method that generates trajectories of an allele׳s frequency in a finite population, as described by a general Wright-Fisher model. The method generates conditioned trajectories that start from a known frequency at a known time, and which achieve a specific final frequency at a known final time. The simulation method applies irrespective of the smallness of the probability of the transition between the initial and final states, because it is not based on rejection of trajectories. We illustrate the method on several different populations where a Wright-Fisher model (or related) applies, namely (i) a locus with 2 alleles, that is subject to selection and mutation; (ii) a locus with 3 alleles, that is subject to selection; (iii) a locus in a metapopulation consisting of two subpopulations of finite size, that are subject to selection and migration. The simulation method allows the generation of conditioned trajectories that can be used for the purposes of visualisation, the estimation of summary statistics, and the development/testing of new inferential methods. The simulated trajectories provide a very simple approach to estimating quantities that cannot easily be expressed in terms of the transition matrix, and can be applied to finite Markov chains other than the Wright-Fisher model., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Retrotransposons occur in extremely large numbers in many eukaryotic genomes. However, little is known of the factors which affect the distribution of close proximity elements. In this work we investigate the frequency of close facing retrotransposons in a plant species with extremely high numbers of retrotransposons. Molecular observations are compared with predictions of a mathematical model that assumes a uniform probability of retrotransposon insertion into the genome. The mathematical model plays the role of a null hypothesis. We find that compared with the predictions of the model, there is a statistically significant deficit of identical copies of facing retroelements that are close to one another. This suggests that an efficient mechanism exists that removes or limits close facing retroelements., (Copyright (c) 2009 Elsevier Ltd. All rights reserved.)

Phylogenetic comparative methods have been used to study patterns of correlated evolution between biological traits of all kinds, and are increasingly used to identify predictors of the rate of DNA substitution. Substitution rate differs from most traits studied in that it cannot be observed directly, but must be inferred from substitutions accrued over a long period of time. Studying a mathematical model of trait and rate evolution, we show that the special nature of substitution rates can lead to a severe loss of power for standard comparative methods. We further show how strategies designed to maximise power, by increasing the number of data points, can have the opposite effect when substitution rate is involved. We then propose two modifications of the standard methods that can mitigate these problems. First, we show how pre-existing tests for homogeneity of variance can be used to identify and exclude those data from which changes in substitution rate cannot be reliably inferred. Second, we show that power can be increased by comparing substitution rate with the time average of the predictor trait along the history of the lineage, and introduce a maximum likelihood estimator of this quantity.

The nk model of fitness interactions is examined. This model has been used by previous authors to investigate the effects of fitness epistasis on substitution dynamics in molecular evolution, and to make broader claims about the importance of epistasis. To examine these claims, an infinite-allele approximation is introduced. In this limit, it is shown that the nk model is, at an appropriate level of description, formally identical to the non-epistatic House-of-Cards model--a well-studied model in theoretical population genetics. It is further shown that in many parameter regimes, the analytical results obtained from this infinite-allele approximation are very close to results from the full nk model (with a finite number of alleles per locus). The findings presented shed light on a number of previous results.