Large effects and the infinitesimal model.

The infinitesimal model of quantitative genetics relies on the Central Limit Theorem to stipulate that under additive models of quantitative traits determined by many loci having similar effect size, the difference between an offspring's genetic trait component and the average of their two parents' genetic trait components is Normally distributed and independent of the parents' values. Here, we investigate how the assumption of similar effect sizes affects the model: if, alternatively, the tail of the effect size distribution is polynomial with exponent α<2, then a different Central Limit Theorem implies that sums of effects should be well-approximated by a "stable distribution", for which single large effects are often still important. Empirically, we first find tail exponents between 1 and 2 in effect sizes estimated by genome-wide association studies of many human disease-related traits. We then show that the independence of offspring trait deviations from parental averages in many cases implies the Gaussian aspect of the infinitesimal model, suggesting that non-Gaussian models of trait evolution must explicitly track the underlying genetics, at least for loci of large effect. We also characterize possible limiting trait distributions of the infinitesimal model with infinitely divisible noise distributions, and compare our results to simulations.

The probability of epidemic burnout in the stochastic SIR model with vital dynamics.

We present an approach to computing the probability of epidemic "burnout," i.e., the probability that a newly emergent pathogen will go extinct after a major epidemic. Our analysis is based on the standard stochastic formulation of the Susceptible-Infectious-Removed (SIR) epidemic model including host demography (births and deaths) and corresponds to the standard SIR ordinary differential equations (ODEs) in the infinite population limit. Exploiting a boundary layer approximation to the ODEs and a birth-death process approximation to the stochastic dynamics within the boundary layer, we derive convenient, fully analytical approximations for the burnout probability. We demonstrate-by comparing with computationally demanding individual-based stochastic simulations and with semi-analytical approximations derived previously-that our fully analytical approximations are highly accurate for biologically plausible parameters. We show that the probability of burnout always decreases with increased mean infectious period. However, for typical biological parameters, there is a relevant local minimum in the probability of persistence as a function of the basic reproduction number [Formula: see text]. For the shortest infectious periods, persistence is least likely if [Formula: see text]; for longer infectious periods, the minimum point decreases to [Formula: see text]. For typical acute immunizing infections in human populations of realistic size, our analysis of the SIR model shows that burnout is almost certain in a well-mixed population, implying that susceptible recruitment through births is insufficient on its own to explain disease persistence.

Rates of SARS-CoV-2 transmission between and into California state prisons.

Correctional institutions are a crucial hotspot amplifying SARS-CoV-2 spread and disease disparity in the U.S. In the California state prison system, multiple massive outbreaks have been caused by transmission between prisons. Correctional staff are a likely vector for transmission into the prison system from surrounding communities. We used publicly available data to estimate the magnitude of flows to and between California state prisons, estimating rates of transmission from communities to prison staff and residents, among and between residents and staff within facilities, and between staff and residents of distinct facilities in the state's 34 prisons through March 22, 2021. We use a mechanistic model, the Hawkes process, reflecting the dynamics of SARS-CoV-2 transmission, for joint estimation of transmission rates. Using nested models for hypothesis testing, we compared the results to simplified models (i) without transmission between prisons, and (ii) with no distinction between prison staff and residents. We estimated that transmission between different facilities' staff is a significant cause of disease spread, and that staff are a vector of transmission between resident populations and outside communities. While increased screening and vaccination of correctional staff may help reduce introductions, large-scale decarceration remains crucially needed as more limited measures are not likely to prevent large-scale disease spread.

Assessing the risk of cascading COVID-19 outbreaks from prison-to-prison transfers.

COVID-19 transmission has been widespread across the California prison system, and at least two of these outbreaks were caused by transfer of infected individuals between prisons. Risks of individual prison outbreaks due to introduction of the virus and of widespread transmission within prisons due to poor conditions have been documented. We examine the additional risk potentially posed by transfer between prisons that can lead to large-scale spread of outbreaks across the prison system if the rate of transfer is sufficiently high. We estimated the threshold number of individuals transferred per prison per month to generate supercritical transmission between prisons, a condition that could lead to large-scale spread across the prison system. We obtained numerical estimates from a range of representative quantitative assumptions, and derived the percentage of transfers that must be performed with effective quarantine measures to prevent supercritical transmission given known rates of transfers occurring between California prisons. Our mean estimate of the critical threshold rate of transfers was 27 individuals transferred per prison per month, with standard deviation 26, in the absence of quarantine measures. Available data documents transfers occurring at a rate of 61 transfers per prison per month. At that rate, estimates of the threshold rate of adherence to quarantine precautions had mean 61%, with standard deviation 32%. While the impact of vaccination and possible decarceration measures is unclear, we include estimates of the above quantities given reductions in the probability and extent of outbreaks. We conclude that the risk of supercritical transmission between California prisons has been substantial, requiring quarantine protocols to be followed rigorously to manage this risk. The rate of outbreaks occurring in California prisons suggests that supercritical transmission may have occurred. We stress that the thresholds we estimate here do not define a safe level of transfers, even if supercritical transmission between prisons is avoided, since even low rates of transfer can cause very large outbreaks. We note that risks may persist after vaccination, due for example to variant strains, and in prison systems where widespread vaccination has not occurred. Decarceration remains urgently needed as a public health measure.

Assessing the Risk of Cascading COVID-19 Outbreaks from Prison-to-Prison Transfers.

COVID-19 transmission has been widespread across the California prison system, and at least two of these outbreaks were caused by transfer of infected individuals between prisons. Risks of individual prison outbreaks due to introduction of the virus and of widespread transmission within prisons due to poor conditions have been documented. We examine the additional risk potentially posed by transfer between prisons that can lead to large-scale spread of outbreaks across the prison system if the rate of transfer is sufficiently high. We estimated the threshold number of individuals transferred per prison per month to generate supercritical transmission between prisons, a condition that could lead to large-scale spread across the prison system. We obtained numerical estimates from a range of representative quantitative assumptions, and derived the percentage of transfers that must be performed with effective quarantine measures to prevent supercritical transmission given known rates of transfers occurring between California prisons. Our mean estimate of the critical threshold rate of transfers was 14.38 individuals transferred per prison per month in the absence of quarantine measures. Available data documents transfers occurring at a rate of 60 transfers per prison per month. At that rate, estimates of the threshold rate of adherence to quarantine precautions had mean 76.03%. While the impact of vaccination and possible decarceration measures is unclear, we include estimates of the above quantities given reductions in the probability and extent of outbreaks. We conclude that the risk of supercritical transmission between California prisons has been substantial, requiring quarantine protocols to be followed rigorously to manage this risk. The rate of outbreaks occurring in California prisons suggests that supercritical transmission may have occurred. We stress that the thresholds we estimate here do not define a safe level of transfers, even if supercritical transmission between prisons is avoided, since even low rates of transfer can cause very large outbreaks. We note that risks may persist after vaccination, due for example to variant strains, and in prison systems where widespread vaccination has not occurred. Decarceration remains urgently needed as a public health measure.

Pathogen evolution in finite populations: slow and steady spreads the best.

The theory of life-history evolution provides a powerful framework to understand the evolutionary dynamics of pathogens. It assumes, however, that host populations are large and that one can neglect the effects of demographic stochasticity. Here, we expand the theory to account for the effects of finite population size on the evolution of pathogen virulence. We show that demographic stochasticity introduces additional evolutionary forces that can qualitatively affect the dynamics and the evolutionary outcome. We discuss the importance of the shape of the pathogen fitness landscape on the balance between mutation, selection and genetic drift. This analysis reconciles Adaptive Dynamics with population genetics in finite populations and provides a new theoretical toolbox to study life-history evolution in realistic ecological scenarios.

Invasion probabilities, hitting times, and some fluctuation theory for the stochastic logistic process.

We consider excursions for a class of stochastic processes describing a population of discrete individuals experiencing density-limited growth, such that the population has a finite carrying capacity and behaves qualitatively like the classical logistic model Verhulst (Corresp Math Phys 10:113-121, 1838) when the carrying capacity is large. Being discrete and stochastic, however, our population nonetheless goes extinct in finite time. We present results concerning the maximum of the population prior to extinction in the large population limit, from which we obtain establishment probabilities and upper bounds for the process, as well as estimates for the waiting time to establishment and extinction. As a consequence, we show that conditional upon establishment, the stochastic logistic process will with high probability greatly exceed carrying capacity an arbitrary number of times prior to extinction.

Evolutionary consequences of behavioral diversity.

Iterated games provide a framework to describe social interactions among groups of individuals. This body of work has focused primarily on individuals who face a simple binary choice, such as "cooperate" or "defect." Real individuals, however, can exhibit behavioral diversity, varying their input to a social interaction both qualitatively and quantitatively. Here we explore how access to a greater diversity of behavioral choices impacts the evolution of social dynamics in populations. We show that, in public goods games, some simple strategies that choose between only two possible actions can resist invasion by all multichoice invaders, even while engaging in relatively little punishment. More generally, access to a larger repertoire of behavioral choices results in a more "rugged" fitness landscape, with populations able to stabilize cooperation at multiple levels of investment. As a result, increased behavioral choice facilitates cooperation when returns on investments are low, but it hinders cooperation when returns on investments are high. Finally, we analyze iterated rock-paper-scissors games, the nontransitive payoff structure of which means that unilateral control is difficult to achieve. Despite this, we find that a large proportion of multichoice strategies can invade and resist invasion by single-choice strategies-so that even well-mixed populations will tend to evolve and maintain behavioral diversity.

Environmental robustness and the adaptability of populations.

Recent work has shown that genetic robustness can either facilitate or impede adaptation. But the impact of environmental robustness on adaptation remains unclear. Environmental robustness helps ensure that organisms consistently develop the same phenotype in the face of "environmental noise" during development. Under purifying selection, those genotypes that express the optimal phenotype most reliably will be selectively favored. The resulting reduction in genetic variation tends to slow adaptation when the population is faced with a novel target phenotype. However, environmental noise sometimes induces the expression of an alternative advantageous phenotype, which may speed adaptation by genetic assimilation. Here, we use a population-genetic model to explore how these two opposing effects of environmental noise influence the capacity of a population to adapt. We analyze how the rate of adaptation depends on the frequency of environmental noise, the degree of environmental robustness in the population, the distribution of environmental robustness across genotypes, the population size, and the strength of selection for a newly adaptive phenotype. Over a broad regime, we find that environmental noise can either facilitate or impede adaptation. Our analysis uncovers several surprising insights about the relationship between environmental noise and adaptation, and it provides a general framework for interpreting empirical studies of both genetic and environmental robustness.

Reconciling molecular phylogenies with the fossil record.

Historical patterns of species diversity inferred from phylogenies typically contradict the direct evidence found in the fossil record. According to the fossil record, species frequently go extinct, and many clades experience periods of dramatic diversity loss. However, most analyses of molecular phylogenies fail to identify any periods of declining diversity, and they typically infer low levels of extinction. This striking inconsistency between phylogenies and fossils limits our understanding of macroevolution, and it undermines our confidence in phylogenetic inference. Here, we show that realistic extinction rates and diversity trajectories can be inferred from molecular phylogenies. To make this inference, we derive an analytic expression for the likelihood of a phylogeny that accommodates scenarios of declining diversity, time-variable rates, and incomplete sampling; we show that this likelihood expression reliably detects periods of diversity loss using simulation. We then study the cetaceans (whales, dolphins, and porpoises), a group for which standard phylogenetic inferences are strikingly inconsistent with fossil data. When the cetacean phylogeny is considered as a whole, recently radiating clades, such as the Balaneopteridae, Delphinidae, Phocoenidae, and Ziphiidae, mask the signal of extinctions. However, when isolating these groups, we infer diversity dynamics that are consistent with the fossil record. These results reconcile molecular phylogenies with fossil data, and they suggest that most extant cetaceans arose from four recent radiations, with a few additional species arising from clades that have been in decline over the last ~10 Myr.

Epistasis increases the rate of conditionally neutral substitution in an adapting population.

Kimura observed that the rate of neutral substitution should equal the neutral mutation rate. This classic result is central to our understanding of molecular evolution, and it continues to influence phylogenetics, genomics, and the interpretation of evolution experiments. By demonstrating that neutral mutations substitute at a rate independent of population size and selection at linked sites, Kimura provided an influential justification for the idea of a molecular clock and emphasized the importance of genetic drift in shaping molecular evolution. But when epistasis among sites is common, as numerous empirical studies suggest, do neutral mutations substitute according to Kimura's expectation? Here we study simulated, asexual populations of RNA molecules, and we observe that conditionally neutral mutations--i.e., mutations that do not alter the fitness of the individual in which they arise, but that may alter the fitness effects of subsequent mutations--substitute much more often than expected while a population is adapting. We quantify these effects using a simple population-genetic model that elucidates how the substitution rate at conditionally neutral sites depends on the population size, mutation rate, strength of selection, and prevalence of epistasis. We discuss the implications of these results for our understanding of the molecular clock, and for the interpretation of molecular variation in laboratory and natural populations.

Some consequences of demographic stochasticity in population genetics.

Much of population genetics is based on the diffusion limit of the Wright-Fisher model, which assumes a fixed population size. This assumption is violated in most natural populations, particularly for microbes. Here we study a more realistic model that decouples birth and death events and allows for a stochastically varying population size. Under this model, classical quantities such as the probability of and time before fixation of a mutant allele can differ dramatically from their Wright-Fisher expectations. Moreover, inferences about natural selection based on Wright-Fisher assumptions can yield erroneous and even contradictory conclusions: at small population densities one allele will appear superior, whereas at large densities the other allele will dominate. Consequently, competition assays in laboratory conditions may not reflect the outcome of long-term evolution in the field. These results highlight the importance of incorporating demographic stochasticity into basic models of population genetics.

Mutational robustness can facilitate adaptation.

Robustness seems to be the opposite of evolvability. If phenotypes are robust against mutation, we might expect that a population will have difficulty adapting to an environmental change, as several studies have suggested. However, other studies contend that robust organisms are more adaptable. A quantitative understanding of the relationship between robustness and evolvability will help resolve these conflicting reports and will clarify outstanding problems in molecular and experimental evolution, evolutionary developmental biology and protein engineering. Here we demonstrate, using a general population genetics model, that mutational robustness can either impede or facilitate adaptation, depending on the population size, the mutation rate and the structure of the fitness landscape. In particular, neutral diversity in a robust population can accelerate adaptation as long as the number of phenotypes accessible to an individual by mutation is smaller than the total number of phenotypes in the fitness landscape. These results provide a quantitative resolution to a significant ambiguity in evolutionary theory.

Absorption and fixation times for neutral and quasi-neutral populations with density dependence.

We study a generalisation of Moran's population-genetic model that incorporates density dependence. Rather than assuming fixed population size, we allow the number of individuals to vary stochastically with the same events that change allele number, according to a logistic growth process with density dependent mortality. We analyse the expected time to absorption and fixation in the 'quasi-neutral' case: both types have the same carrying capacity, achieved through a trade-off of birth and death rates. Such types would be competitively neutral in a classical, fixed-population Wright-Fisher model. Nonetheless, we find that absorption times are skewed compared to the Wright-Fisher model. The absorption time is longer than the Wright-Fisher prediction when the initial proportion of the type with higher birth rate is large, and shorter when it is small. By contrast, demographic stochasticity has no effect on the fixation or absorption times of truly neutral alleles in a large population. Our calculations provide the first analytic results on hitting times in a two-allele model, when the population size varies stochastically.

Fixation in haploid populations exhibiting density dependence II: the quasi-neutral case.

We determine fixation probabilities in a model of two competing types with density dependence. The model is defined as a two-dimensional birth-and-death process with density-independent death rates, and birth rates that are a linearly decreasing function of total population density. We treat the 'quasi-neutral case' where both types have the same equilibrium population densities. This condition results in birth rates that are proportional to death rates. This can be viewed as a life history trade-off. The deterministic dynamics possesses a stable manifold of mixtures of the two types. We show that the fixation probability is asymptotically equal to the fixation probability at the point where the deterministic flow intersects this manifold. The deterministic dynamics predicts an increase in the proportion of the type with higher birth rate in growing populations (and a decrease in shrinking populations). Growing (shrinking) populations therefore intersect the manifold at a higher (lower) than initial proportion of this type. On the center manifold, the fixation probability is a quadratic function of initial proportion, with a disadvantage to the type with higher birth rate. This disadvantage arises from the larger fluctuations in population density for this type. These results are asymptotically exact and have relevance for allele fixation, models of species abundance, and epidemiological models.

Fixation in haploid populations exhibiting density dependence I: The non-neutral case.

We extend the one-locus two allele Moran model of fixation in a haploid population to the case where the total size of the population is not fixed. The model is defined as a two-dimensional birth-and-death process for allele number. Changes in allele number occur through density-independent death events and birth events whose per capita rate decreases linearly with the total population density. Uniquely for models of this type, the latter is determined by these same birth-and-death events. This provides a framework for investigating both the effects of fluctuation in total population number through demographic stochasticity, and deterministic density-dependent changes in mean density, on allele fixation. We analyze this model using a combination of asymptotic analytic approximations supported by numerics. We find that for advantageous mutants demographic stochasticity of the resident population does not affect the fixation probability, but that deterministic changes in total density do. In contrast, for deleterious mutants, the fixation probability increases with increasing resident population fluctuation size, but is relatively insensitive to initial density. These phenomena cannot be described by simply using a harmonic mean effective population size.