Using early detection data to estimate the date of emergence of an epidemic outbreak.

While the first infection of an emerging disease is often unknown, information on early cases can be used to date it. In the context of the COVID-19 pandemic, previous studies have estimated dates of emergence (e.g., first human SARS-CoV-2 infection, emergence of the Alpha SARS-CoV-2 variant) using mainly genomic data. Another dating attempt used a stochastic population dynamics approach and the date of the first reported case. Here, we extend this approach to use a larger set of early reported cases to estimate the delay from first infection to the Nth case. We first validate our framework by running our model on simulated data. We then apply our model using data on Alpha variant infections in the UK, dating the first Alpha infection at (median) August 21, 2020 (95% interpercentile range across retained simulations (IPR): July 23-September 5, 2020). Next, we apply our model to data on COVID-19 cases with symptom onset before mid-January 2020. We date the first SARS-CoV-2 infection in Wuhan at (median) November 28, 2019 (95% IPR: November 2-December 9, 2019). Our results fall within ranges previously estimated by studies relying on genomic data. Our population dynamics-based modelling framework is generic and flexible, and thus can be applied to estimate the starting time of outbreaks in contexts other than COVID-19.

Drivers and impact of the early silent invasion of SARS-CoV-2 Alpha.

SARS-CoV-2 variants of concern (VOCs) circulated cryptically before being identified as a threat, delaying interventions. Here we studied the drivers of such silent spread and its epidemic impact to inform future response planning. We focused on Alpha spread out of the UK. We integrated spatio-temporal records of international mobility, local epidemic growth and genomic surveillance into a Bayesian framework to reconstruct the first three months after Alpha emergence. We found that silent circulation lasted from days to months and decreased with the logarithm of sequencing coverage. Social restrictions in some countries likely delayed the establishment of local transmission, mitigating the negative consequences of late detection. Revisiting the initial spread of Alpha supports local mitigation at the destination in case of emerging events.

Quantification and modeling of turnover dynamics of de novo transcripts in Drosophila melanogaster.

Most of the transcribed eukaryotic genomes are composed of non-coding transcripts. Among these transcripts, some are newly transcribed when compared to outgroups and are referred to as de novo transcripts. De novo transcripts have been shown to play a major role in genomic innovations. However, little is known about the rates at which de novo transcripts are gained and lost in individuals of the same species. Here, we address this gap and estimate the de novo transcript turnover rate with an evolutionary model. We use DNA long reads and RNA short reads from seven geographically remote samples of inbred individuals of Drosophila melanogaster to detect de novo transcripts that are gained on a short evolutionary time scale. Overall, each sampled individual contains around 2500 unspliced de novo transcripts, with most of them being sample specific. We estimate that around 0.15 transcripts are gained per year, and that each gained transcript is lost at a rate around 5× 10-5 per year. This high turnover of transcripts suggests frequent exploration of new genomic sequences within species. These rate estimates are essential to comprehend the process and timescale of de novo gene birth.

A stochastic analysis of the interplay between antibiotic dose, mode of action, and bacterial competition in the evolution of antibiotic resistance.

The use of an antibiotic may lead to the emergence and spread of bacterial strains resistant to this antibiotic. Experimental and theoretical studies have investigated the drug dose that minimizes the risk of resistance evolution over the course of treatment of an individual, showing that the optimal dose will either be the highest or the lowest drug concentration possible to administer; however, no analytical results exist that help decide between these two extremes. To address this gap, we develop a stochastic mathematical model of bacterial dynamics under antibiotic treatment. We explore various scenarios of density regulation (bacterial density affects cell birth or death rates), and antibiotic modes of action (biostatic or biocidal). We derive analytical results for the survival probability of the resistant subpopulation until the end of treatment, the size of the resistant subpopulation at the end of treatment, the carriage time of the resistant subpopulation until it is replaced by a sensitive one after treatment, and we verify these results with stochastic simulations. We find that the scenario of density regulation and the drug mode of action are important determinants of the survival of a resistant subpopulation. Resistant cells survive best when bacterial competition reduces cell birth and under biocidal antibiotics. Compared to an analogous deterministic model, the population size reached by the resistant type is larger and carriage time is slightly reduced by stochastic loss of resistant cells. Moreover, we obtain an analytical prediction of the antibiotic concentration that maximizes the survival of resistant cells, which may help to decide which drug dosage (not) to administer. Our results are amenable to experimental tests and help link the within and between host scales in epidemiological models.

From individual-based epidemic models to McKendrick-von Foerster PDEs: a guide to modeling and inferring COVID-19 dynamics.

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual's infection age, i.e., the time elapsed since infection, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type i is simply obtained by integrating the probability of being in state i at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (ODEs) to account for disease complexity. We show that such models can always be rewritten in our framework, thus, providing a low-dimensional yet equivalent representation of these complex models. Third, beyond the simplicity of the approach, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.

Revisiting the number of self-incompatibility alleles in finite populations: From old models to new results.

Under gametophytic self-incompatibility (GSI), plants are heterozygous at the self-incompatibility locus (S-locus) and can only be fertilized by pollen with a different allele at that locus. The last century has seen a heated debate about the correct way of modelling the allele diversity in a GSI population that was never formally resolved. Starting from an individual-based model, we derive the deterministic dynamics as proposed by Fisher (The genetical theory of natural selection - A complete, Variorum edition, Oxford University Press, 1958) and compute the stationary S-allele frequency distribution. We find that the stationary distribution proposed by Wright (Evolution, 18, 609, 1964) is close to our theoretical prediction, in line with earlier numerical confirmation. Additionally, we approximate the invasion probability of a new S-allele, which scales inversely with the number of resident S-alleles. Lastly, we use the stationary allele frequency distribution to estimate the population size of a plant population from an empirically obtained allele frequency spectrum, which complements the existing estimator of the number of S-alleles. Our expression of the stationary distribution resolves the long-standing debate about the correct approximation of the number of S-alleles and paves the way for new statistical developments for the estimation of the plant population size based on S-allele frequencies.

The stochastic dynamics of early epidemics: probability of establishment, initial growth rate, and infection cluster size at first detection.

Emerging epidemics and local infection clusters are initially prone to stochastic effects that can substantially impact the early epidemic trajectory. While numerous studies are devoted to the deterministic regime of an established epidemic, mathematical descriptions of the initial phase of epidemic growth are comparatively rarer. Here, we review existing mathematical results on the size of the epidemic over time, and derive new results to elucidate the early dynamics of an infection cluster started by a single infected individual. We show that the initial growth of epidemics that eventually take off is accelerated by stochasticity. As an application, we compute the distribution of the first detection time of an infected individual in an infection cluster depending on testing effort, and estimate that the SARS-CoV-2 variant of concern Alpha detected in September 2020 first appeared in the UK early August 2020. We also compute a minimal testing frequency to detect clusters before they exceed a given threshold size. These results improve our theoretical understanding of early epidemics and will be useful for the study and control of local infectious disease clusters.

Understanding evolutionary and ecological dynamics using a continuum limit.

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual-based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright-Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi-stationary distribution in a finite population.

The Effect of Habitat Choice on Evolutionary Rescue in Subdivided Populations.

AbstractEvolutionary rescue is the process by which a population, in response to an environmental change, successfully avoids extinction through adaptation. In spatially structured environments, dispersal can affect the probability of rescue. Here, we model an environment consisting of patches that degrade one after another, and we investigate the probability of rescue by a mutant adapted to the degraded habitat. We focus on the effects of dispersal and of immigration biases. We identify up to three regions delimiting the effect of dispersal on the probability of evolutionary rescue: (i) starting from low dispersal rates, the probability of rescue increases with dispersal; (ii) at intermediate dispersal rates, it decreases; and (iii) at large dispersal rates, it increases again with dispersal, except if mutants are too counterselected in not-yet-degraded patches. The probability of rescue is generally highest when mutant and wild-type individuals preferentially immigrate into patches that have already undergone environmental change. Additionally, we find that mutants that will eventually rescue the population most likely first appear in nondegraded patches. Overall, our results show that habitat choice, compared with the often-studied unbiased immigration scheme, can substantially alter the dynamics of population survival and adaptation to new environments.

Success of prophylactic antiviral therapy for SARS-CoV-2: Predicted critical efficacies and impact of different drug-specific mechanisms of action.

Repurposed drugs that are safe and immediately available constitute a first line of defense against new viral infections. Despite limited antiviral activity against SARS-CoV-2, several drugs are being tested as medication or as prophylaxis to prevent infection. Using a stochastic model of early phase infection, we evaluate the success of prophylactic treatment with different drug types to prevent viral infection. We find that there exists a critical efficacy that a treatment must reach in order to block viral establishment. Treatment by a combination of drugs reduces the critical efficacy, most effectively by the combination of a drug blocking viral entry into cells and a drug increasing viral clearance. Below the critical efficacy, the risk of infection can nonetheless be reduced. Drugs blocking viral entry into cells or enhancing viral clearance reduce the risk of infection more than drugs that reduce viral production in infected cells. The larger the initial inoculum of infectious virus, the less likely is prevention of an infection. In our model, we find that as long as the viral inoculum is smaller than 10 infectious virus particles, viral infection can be prevented almost certainly with drugs of 90% efficacy (or more). Even when a viral infection cannot be prevented, antivirals delay the time to detectable viral loads. The largest delay of viral infection is achieved by drugs reducing viral production in infected cells. A delay of virus infection flattens the within-host viral dynamic curve, possibly reducing transmission and symptom severity. Thus, antiviral prophylaxis, even with reduced efficacy, could be efficiently used to prevent or alleviate infection in people at high risk.

The search for sexually antagonistic genes: Practical insights from studies of local adaptation and statistical genomics.

Sexually antagonistic (SA) genetic variation-in which alleles favored in one sex are disfavored in the other-is predicted to be common and has been documented in several animal and plant populations, yet we currently know little about its pervasiveness among species or its population genetic basis. Recent applications of genomics in studies of SA genetic variation have highlighted considerable methodological challenges to the identification and characterization of SA genes, raising questions about the feasibility of genomic approaches for inferring SA selection. The related fields of local adaptation and statistical genomics have previously dealt with similar challenges, and lessons from these disciplines can therefore help overcome current difficulties in applying genomics to study SA genetic variation. Here, we integrate theoretical and analytical concepts from local adaptation and statistical genomics research-including F ST and F IS statistics, genome-wide association studies, pedigree analyses, reciprocal transplant studies, and evolve-and-resequence experiments-to evaluate methods for identifying SA genes and genome-wide signals of SA genetic variation. We begin by developing theoretical models for between-sex F ST and F IS, including explicit null distributions for each statistic, and using them to critically evaluate putative multilocus signals of sex-specific selection in previously published datasets. We then highlight new statistics that address some of the limitations of F ST and F IS, along with applications of more direct approaches for characterizing SA genetic variation, which incorporate explicit fitness measurements. We finish by presenting practical guidelines for the validation and evolutionary analysis of candidate SA genes and discussing promising empirical systems for future work.

Years of life lost estimates cannot always be taken at face value: Response to "COVID-19 - exploring the implications of long-term condition type and extent of multimorbidity on years of life lost: a modelling study".

In their recent analysis, Hanlon et al. set out to estimate the years of life lost (YLL) in people who have died with COVID-19 by following and expanding on the WHO standard approach. We welcome this research as an attempt to draw a more accurate picture of the mortality burden of this disease which has been involved in the deaths of more than 300,000 people worldwide as of May 2020. However, we argue that obtained YLL estimates (13 years for men and 11 years for women) are interpreted in a misleading way. Even with the presented efforts to control for the role of multimorbidity in COVID-19 deaths, these estimates cannot be interpreted to imply "how long someone who died from COVID-19 might otherwise have been expected to live". By example we analyze the underlying problem of data selection bias which, in the context of COVID-19, renders such an interpretation of YLL estimates impossible, and outline potential approaches to control for the problem.

Evolution of mating types in finite populations: The precarious advantage of being rare.

Sexually reproducing populations with self-incompatibility bear the cost of limiting potential mates to individuals of a different type. Rare mating types escape this cost since they are unlikely to encounter incompatible partners, leading to the deterministic prediction of continuous invasion by new mutants and an ever-increasing number of types. However, rare types are also at an increased risk of being lost by random drift. Calculating the number of mating types that a population can maintain requires consideration of both the deterministic advantages and the stochastic risks. By comparing the relative importance of selection and drift, we show that a population of size N can maintain a maximum of approximately N1/3 mating types for intermediate population sizes, whereas for large N, we derive a formal estimate. Although the number of mating types in a population is quite stable, the rare-type advantage promotes turnover of types. We derive explicit formulas for both the invasion and turnover probabilities in finite populations.

Invasion and Extinction Dynamics of Mating Types Under Facultative Sexual Reproduction.

In sexually reproducing isogamous species, syngamy between gametes is generally not indiscriminate, but rather restricted to occurring between complementary self-incompatible mating types. A longstanding question regards the evolutionary pressures that control the number of mating types observed in natural populations, which ranges from two to many thousands. Here, we describe a population genetic null model of this reproductive system, and derive expressions for the stationary probability distribution of the number of mating types, the establishment probability of a newly arising mating type, and the mean time to extinction of a resident type. Our results yield that the average rate of sexual reproduction in a population correlates positively with the expected number of mating types observed. We further show that the low number of mating types predicted in the rare-sex regime is primarily driven by low invasion probabilities of new mating type alleles, with established resident alleles being very stable over long evolutionary periods. Moreover, our model naturally exhibits varying selection strength dependent on the number of resident mating types. This results in higher extinction and lower invasion rates for an increasing number of residents.

Disentangling eco-evolutionary effects on trait fixation.

In population genetics, fixation of traits in a demographically changing population under frequency-independent selection has been extensively analysed. In evolutionary game theory, models of fixation have typically focused on fixed population sizes and frequency-dependent selection. A combination of demographic fluctuations with frequency-dependent interactions such as Lotka-Volterra dynamics has received comparatively little attention. We consider a stochastic, competitive Lotka-Volterra model with higher order interactions between two traits. The emerging individual-based model allows for stochastic fluctuations in the frequencies of the two traits and the total population size. We calculate the fixation probability of a trait under differing competition coefficients. This fixation probability resembles, qualitatively, the deterministic evolutionary dynamics. Furthermore, we partially disentangle the selection effects into their ecological and evolutionary components. We find that changing the evolutionary selection strength also changes the population dynamics and vice versa. Thus, a clean separation of the ecological and evolutionary effects is not possible. Instead, our results imply a nested interaction of the evolutionary and ecological effects. The entangled eco-evolutionary processes thus cannot be ignored when determining fixation properties in a co-evolutionary system.

Fixation probabilities in populations under demographic fluctuations.

We study the fixation probability of a mutant type when introduced into a resident population. We implement a stochastic competitive Lotka-Volterra model with two types and intra- and interspecific competition. The model further allows for stochastically varying population sizes. The competition coefficients are interpreted in terms of inverse payoffs emerging from an evolutionary game. Since our study focuses on the impact of the competition values, we assume the same net growth rate for both types. In this general framework, we derive a formula for the fixation probability [Formula: see text] of the mutant type under weak selection. We find that the most important parameter deciding over the invasion success of the mutant is its death rate due to competition with the resident. Furthermore, we compare our approximation to results obtained by implementing population size changes deterministically in order to explore the parameter regime of validity of our method. Finally, we put our formula in the context of classical evolutionary game theory and observe similarities and differences to the results obtained in that constant population size setting.

Limits of noise for autoregulated gene expression.

Gene expression is influenced by extrinsic noise (involving a fluctuating environment of cellular processes) and intrinsic noise (referring to fluctuations within a cell under constant environment). We study the standard model of gene expression including an (in-)active gene, mRNA and protein. Gene expression is regulated in the sense that the protein feeds back and either represses (negative feedback) or enhances (positive feedback) its production at the stage of transcription. While it is well-known that negative (positive) feedback reduces (increases) intrinsic noise, we give a precise result on the resulting fluctuations in protein numbers. The technique we use is an extension of the Langevin approximation and is an application of a central limit theorem under stochastic averaging for Markov jump processes (Kang et al. in Ann Appl Probab 24:721-759, 2014). We find that (under our scaling and in equilibrium), negative feedback leads to a reduction in the Fano factor of at most 2, while the noise under positive feedback is potentially unbounded. The fit with simulations is very good and improves on known approximations.

Classification of phenotypic subpopulations in isogenic bacterial cultures by triple promoter probing at single cell level.

Phenotypic heterogeneity, defined as the unequal behavior of individuals in an isogenic population, is prevalent in microorganisms. It has a significant impact both on industrial bioprocesses and microbial ecology. We introduce a new versatile reporter system designed for simultaneous monitoring of the activities of three different promoters, where each promoter is fused to a dedicated fluorescent reporter gene (cerulean, mCherry, and mVenus). The compact 3.1 kb triple reporter cassette can either be carried on a replicating plasmid or integrated into the genome avoiding artifacts associated with variation in copy number of plasmid-borne reporter constructs. This construct was applied to monitor promoter activities related to quorum sensing (sinI promoter) and biosynthesis of the exopolysaccharide galactoglucan (wgeA promoter) at single cell level in colonies of the symbiotic nitrogen-fixing alpha-proteobacterium Sinorhizobium meliloti growing in a microfluidics system. The T5-promoter served as a constitutive and homogeneously active control promoter indicating cell viability. wgeA promoter activity was heterogeneous over the whole period of colony development, whereas sinI promoter activity passed through a phase of heterogeneity before becoming homogeneous at late stages. Although quorum sensing-dependent regulation is a major factor activating galactoglucan production, activities of both promoters did not correlate at single cell level. We developed a novel mathematical strategy for classification of the gene expression status in cell populations based on the increase in fluorescence over time in each individual. With respect to galactoglucan biosynthesis, cells in the population were classified into non-contributors, weak contributors, and strong contributors.