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.

The role of fire in terrestrial vertebrate richness patterns.

Productivity is strongly associated with terrestrial species richness patterns, although the mechanisms underpinning such patterns have long been debated. Despite considerable consumption of primary productivity by fire, its influence on global diversity has received relatively little study. Here we examine the sensitivity of terrestrial vertebrate biodiversity (amphibians, birds and mammals) to fire, while accounting for other drivers. We analyse global data on terrestrial vertebrate richness, net primary productivity, fire occurrence (fraction of productivity consumed) and additional influences unrelated to productivity (i.e., historical phylogenetic and area effects) on species richness. For birds, fire is associated with higher diversity, rivalling the effects of productivity on richness, and for mammals, fire's positive association with diversity is even stronger than productivity; for amphibians, in contrast, there are few clear associations. Our findings suggest an underappreciated role for fire in the generation of animal species richness and the conservation of global biodiversity.

Comparing machine learning approaches to incorporate time-varying covariates in predicting cancer survival time.

The Cox proportional hazards model is commonly used in evaluating risk factors in cancer survival data. The model assumes an additive, linear relationship between the risk factors and the log hazard. However, this assumption may be too simplistic. Further, failure to take time-varying covariates into account, if present, may lower prediction accuracy. In this retrospective, population-based, prognostic study of data from patients diagnosed with cancer from 2008 to 2015 in Ontario, Canada, we applied machine learning-based time-to-event prediction methods and compared their predictive performance in two sets of analyses: (1) yearly-cohort-based time-invariant and (2) fully time-varying covariates analysis. Machine learning-based methods-gradient boosting model (gbm), random survival forest (rsf), elastic net (enet), lasso and ridge-were compared to the traditional Cox proportional hazards (coxph) model and the prior study which used the yearly-cohort-based time-invariant analysis. Using Harrell's C index as our primary measure, we found that using both machine learning techniques and incorporating time-dependent covariates can improve predictive performance. Gradient boosting machine showed the best performance on test data in both time-invariant and time-varying covariates analysis.

Incorporating nonlinearity with generalized functional responses to simulate multiple predator effects.

Predicting the combined effects of predators on shared prey has long been a focus of community ecology, yet quantitative predictions often fail. Failure to account for nonlinearity is one reason for this. Moreover, prey depletion in multiple predator effects (MPE) studies generates biased predictions in applications of common experimental and quantitative frameworks. Here, we explore additional sources of bias stemming from nonlinearities in prey predation risk. We show that in order to avoid bias, predictions about the combined effects of independent predators must account for nonlinear size-dependent risk for prey as well as changes in prey risk driven by nonlinear predator functional responses and depletion. Historical failure to account for biases introduced by well-known nonlinear processes that affect predation risk suggest that we may need to reevaluate the general conclusions that have been drawn about the ubiquity of emergent MPEs over the past three decades.

The importance of the generation interval in investigating dynamics and control of new SARS-CoV-2 variants.

Inferring the relative strength (i.e. the ratio of reproduction numbers) and relative speed (i.e. the difference between growth rates) of new SARS-CoV-2 variants is critical to predicting and controlling the course of the current pandemic. Analyses of new variants have primarily focused on characterizing changes in the proportion of new variants, implicitly or explicitly assuming that the relative speed remains fixed over the course of an invasion. We use a generation-interval-based framework to challenge this assumption and illustrate how relative strength and speed change over time under two idealized interventions: a constant-strength intervention like idealized vaccination or social distancing, which reduces transmission rates by a constant proportion, and a constant-speed intervention like idealized contact tracing, which isolates infected individuals at a constant rate. In general, constant-strength interventions change the relative speed of a new variant, while constant-speed interventions change its relative strength. Differences in the generation-interval distributions between variants can exaggerate these changes and modify the effectiveness of interventions. Finally, neglecting differences in generation-interval distributions can bias estimates of relative strength.

Testing and Isolation Efficacy: Insights from a Simple Epidemic Model.

Testing individuals for pathogens can affect the spread of epidemics. Understanding how individual-level processes of sampling and reporting test results can affect community- or population-level spread is a dynamical modeling question. The effect of testing processes on epidemic dynamics depends on factors underlying implementation, particularly testing intensity and on whom testing is focused. Here, we use a simple model to explore how the individual-level effects of testing might directly impact population-level spread. Our model development was motivated by the COVID-19 epidemic, but has generic epidemiological and testing structures. To the classic SIR framework we have added a per capita testing intensity, and compartment-specific testing weights, which can be adjusted to reflect different testing emphases-surveillance, diagnosis, or control. We derive an analytic expression for the relative reduction in the basic reproductive number due to testing, test-reporting and related isolation behaviours. Intensive testing and fast test reporting are expected to be beneficial at the community level because they can provide a rapid assessment of the situation, identify hot spots, and may enable rapid contact-tracing. Direct effects of fast testing at the individual level are less clear, and may depend on how individuals' behaviour is affected by testing information. Our simple model shows that under some circumstances both increased testing intensity and faster test reporting can reduce the effectiveness of control, and allows us to explore the conditions under which this occurs. Conversely, we find that focusing testing on infected individuals always acts to increase effectiveness of control.

Age-dependence of healthcare interventions for COVID-19 in Ontario, Canada.

BACKGROUND: Patient age is one of the most salient clinical indicators of risk from COVID-19. Age-specific distributions of known SARS-CoV-2 infections and COVID-19-related deaths are available for many regions. Less attention has been given to the age distributions of serious medical interventions administered to COVID-19 patients, which could reveal sources of potential pressure on the healthcare system should SARS-CoV-2 prevalence increase, and could inform mass vaccination strategies. The aim of this study is to quantify the relationship between COVID-19 patient age and serious outcomes of the disease, beyond fatalities alone. METHODS: We analysed 277,555 known SARS-CoV-2 infection records for Ontario, Canada, from 23 January 2020 to 16 February 2021 and estimated the age distributions of hospitalizations, Intensive Care Unit admissions, intubations, and ventilations. We quantified the probability of hospitalization given known SARS-CoV-2 infection, and of survival given COVID-19-related hospitalization. RESULTS: The distribution of hospitalizations peaks with a wide plateau covering ages 60-90, whereas deaths are concentrated in ages 80+. The estimated probability of hospitalization given known infection reaches a maximum of 27.8% at age 80 (95% CI 26.0%-29.7%). The probability of survival given hospitalization is nearly 100% for adults younger than 40, but declines substantially after this age; for example, a hospitalized 54-year-old patient has a 91.7% chance of surviving COVID-19 (95% CI 88.3%-94.4%). CONCLUSIONS: Our study demonstrates a significant need for hospitalization in middle-aged individuals and young seniors. This need is not captured by the distribution of deaths, which is heavily concentrated in very old ages. The probability of survival given hospitalization for COVID-19 is lower than is generally perceived for patients over 40. If acute care capacity is exceeded due to an increase in COVID-19 prevalence, the distribution of deaths could expand toward younger ages. These results suggest that vaccine programs should aim to prevent infection not only in old seniors, but also in young seniors and middle-aged individuals, to protect them from serious illness and to limit stress on the healthcare system.

Forward-looking serial intervals correctly link epidemic growth to reproduction numbers.

The reproduction number R and the growth rate r are critical epidemiological quantities. They are linked by generation intervals, the time between infection and onward transmission. Because generation intervals are difficult to observe, epidemiologists often substitute serial intervals, the time between symptom onset in successive links in a transmission chain. Recent studies suggest that such substitution biases estimates of R based on r. Here we explore how these intervals vary over the course of an epidemic, and the implications for R estimation. Forward-looking serial intervals, measuring time forward from symptom onset of an infector, correctly describe the renewal process of symptomatic cases and therefore reliably link R with r. In contrast, backward-looking intervals, which measure time backward, and intrinsic intervals, which neglect population-level dynamics, give incorrect R estimates. Forward-looking intervals are affected both by epidemic dynamics and by censoring, changing in complex ways over the course of an epidemic. We present a heuristic method for addressing biases that arise from neglecting changes in serial intervals. We apply the method to early (21 January to February 8, 2020) serial interval-based estimates of R for the COVID-19 outbreak in China outside Hubei province; using improperly defined serial intervals in this context biases estimates of initial R by up to a factor of 2.6. This study demonstrates the importance of early contact tracing efforts and provides a framework for reassessing generation intervals, serial intervals, and R estimates for COVID-19.

Modelling song popularity as a contagious process.

Popular songs are often said to be 'contagious', 'infectious' or 'viral'. We find that download count time series for many popular songs resemble infectious disease epidemic curves. This paper suggests infectious disease transmission models could help clarify mechanisms that contribute to the 'spread' of song preferences and how these mechanisms underlie song popularity. We analysed data from MixRadio, comprising song downloads through Nokia cell phones in Great Britain from 2007 to 2014. We compared the ability of the standard susceptible-infectious-recovered (SIR) epidemic model and a phenomenological (spline) model to fit download time series of popular songs. We fitted these same models to simulated epidemic time series generated by the SIR model. Song downloads are captured better by the SIR model, to the same extent that actual SIR simulations are fitted better by the SIR model than by splines. This suggests that the social processes underlying song popularity are similar to those that drive infectious disease transmission. We draw conclusions about song popularity within specific genres based on estimated SIR parameters. In particular, we argue that faster spread of preferences for Electronica songs may reflect stronger connectivity of the 'susceptible community', compared with the larger and broader community that listens to more common genres.

Consequences of nest site selection vary along a tidal gradient.

Parents providing care must sometimes choose between rearing locations that are most favourable for offspring versus those that are most favourable for themselves. Here, we measured how both parental and offspring performance varied in nest sites distributed along an environmental gradient. The plainfin midshipman fish Porichthys notatus nests along a tidal gradient. When ascending from the subtidal to the high intertidal at low tide, both nest temperature and frequency of air exposure increase. We used one lab and two field experiments to investigate how parental nest site choices across tidal elevations are linked to the physiological costs incurred by parents and the developmental benefits accrued by offspring. Under warmer incubation conditions, simulating high intertidal nests, offspring developed faster but had higher mortality rates compared to those incubated in cooler conditions that mimicked subtidal nests. In the field, males in higher intertidal nests were more active caregivers, but their young still died at the fastest rates. Larger males claimed and retained low intertidal nests, where offspring survival and development rates were also highest. Our results suggest that males compete more intensively for nest sites in the low intertidal, where they can raise their young quickly and with lower per-offspring investments. Smaller, less-competitive males forced into higher intertidal sites nest earlier in the season and provide more active parental care, possibly to bolster brood survival under harsh environmental conditions.

Acceleration of plague outbreaks in the second pandemic.

Historical records reveal the temporal patterns of a sequence of plague epidemics in London, United Kingdom, from the 14th to 17th centuries. Analysis of these records shows that later epidemics spread significantly faster ("accelerated"). Between the Black Death of 1348 and the later epidemics that culminated with the Great Plague of 1665, we estimate that the epidemic growth rate increased fourfold. Currently available data do not provide enough information to infer the mode of plague transmission in any given epidemic; nevertheless, order-of-magnitude estimates of epidemic parameters suggest that the observed slow growth rates in the 14th century are inconsistent with direct (pneumonic) transmission. We discuss the potential roles of demographic and ecological factors, such as climate change or human or rat population density, in driving the observed acceleration.

Reconciling early-outbreak estimates of the basic reproductive number and its uncertainty: framework and applications to the novel coronavirus (SARS-CoV-2) outbreak.

A novel coronavirus (SARS-CoV-2) emerged as a global threat in December 2019. As the epidemic progresses, disease modellers continue to focus on estimating the basic reproductive number [Formula: see text]-the average number of secondary cases caused by a primary case in an otherwise susceptible population. The modelling approaches and resulting estimates of [Formula: see text] during the beginning of the outbreak vary widely, despite relying on similar data sources. Here, we present a statistical framework for comparing and combining different estimates of [Formula: see text] across a wide range of models by decomposing the basic reproductive number into three key quantities: the exponential growth rate, the mean generation interval and the generation-interval dispersion. We apply our framework to early estimates of [Formula: see text] for the SARS-CoV-2 outbreak, showing that many [Formula: see text] estimates are overly confident. Our results emphasize the importance of propagating uncertainties in all components of [Formula: see text], including the shape of the generation-interval distribution, in efforts to estimate [Formula: see text] at the outset of an epidemic.

A Note on Observation Processes in Epidemic Models.

Many disease models focus on characterizing the underlying transmission mechanism but make simple, possibly naive assumptions about how infections are reported. In this note, we use a simple deterministic Susceptible-Infected-Removed (SIR) model to compare two common assumptions about disease incidence reports: Individuals can report their infection as soon as they become infected or as soon as they recover. We show that incorrect assumptions about the underlying observation processes can bias estimates of the basic reproduction number and lead to overly narrow confidence intervals.

Patterns of seasonal and pandemic influenza-associated health care and mortality in Ontario, Canada.

BACKGROUND: Mathematical and statistical models are used to project the future time course of infectious disease epidemics and the expected future burden on health care systems and economies. Influenza is a particularly important disease in this context because it causes annual epidemics and occasional pandemics. In order to forecast health care utilization during epidemics-and the effects of hospitalizations and deaths on the contact network and, in turn, on transmission dynamics-modellers must make assumptions about the lengths of time between infection, visiting a physician, being admitted to hospital, leaving hospital, and death. More reliable forecasts could be be made if the distributions of times between these types of events ("delay distributions") were known. METHODS: We estimated delay distributions in the province of Ontario, Canada, between 2006 and 2010. To do so, we used encrypted health insurance numbers to link 1.34 billion health care billing records to 4.27 million hospital inpatient stays. Because the four year period we studied included three typical influenza seasons and the 2009 influenza pandemic, we were able to compare the delay distributions in non-pandemic and pandemic settings. We also estimated conditional probabilities such as the probability of hospitalization within the year if diagnosed with influenza. RESULTS: In non-pandemic [pandemic] years, delay distribution medians (inter-quartile ranges) were: Service to Admission 6.3 days (0.1-17.6 days) [2.4 days (-0.3-13.6 days)], Admission to Discharge 3 days (1.4-5.9 days) [2.6 days (1.2-5.1 days)], Admission to Death 5.3 days (2.1-11 days) [6 days (2.6-13.1 days)]. (Service date is defined as the date, within the year, of the first health care billing that included a diagnostic code for influenza-like-illness.) Among individuals diagnosed with either pneumonia or influenza in a given year, 19% [16%] were hospitalized within the year and 3% [2%] died in hospital. Among all individuals who were hospitalized, 10% [12%] were diagnosed with pneumonia or influenza during the year and 5% [5%] died in hospital. CONCLUSION: Our empirical delay distributions and conditional probabilities should help facilitate more accurate forecasts in the future, including improved predictions of hospital bed demands during influenza outbreaks, and the expected effects of hospitalizations on epidemic dynamics.

Predicting West Nile virus transmission in North American bird communities using phylogenetic mixed effects models and eBird citizen science data.

BACKGROUND: West Nile virus (WNV) is a mosquito-transmitted disease of birds that has caused bird population declines and can spill over into human populations. Previous research has identified bird species that infect a large fraction of the total pool of infected mosquitoes and correlate with human infection risk; however, these analyses cover small spatial regions and cannot be used to predict transmission in bird communities in which these species are rare or absent. Here we present a mechanistic model for WNV transmission that predicts WNV spread (R0) in any bird community in North America by scaling up from the physiological responses of individual birds to transmission at the level of the community. We predict unmeasured bird species' responses to infection using phylogenetic imputation, based on these species' phylogenetic relationships with bird species with measured responses. RESULTS: We focused our analysis on Texas, USA, because it is among the states with the highest total incidence of WNV in humans and is well sampled by birders in the eBird database. Spatio-temporal patterns: WNV transmission is primarily driven by temperature variation across time and space, and secondarily by bird community composition. In Texas, we predicted WNV R0 to be highest in the spring and fall when temperatures maximize the product of mosquito transmission and survival probabilities. In the most favorable months for WNV transmission (April, May, September and October), we predicted R0 to be highest in the "Piney Woods" and "Oak Woods & Prairies" ecoregions of Texas, and lowest in the "High Plains" and "South Texas Brush County" ecoregions. Dilution effect: More abundant bird species are more competent hosts for WNV, and predicted WNV R0 decreases with increasing species richness. Keystone species: We predicted that northern cardinals (Cardinalis cardinalis) are the most important hosts for amplifying WNV and that mourning doves (Zenaida macroura) are the most important sinks of infection across Texas. CONCLUSIONS: Despite some data limitations, we demonstrate the power of phylogenetic imputation in predicting disease transmission in heterogeneous host communities. Our mechanistic modeling framework shows promise both for assisting future analyses on transmission and spillover in heterogeneous multispecies pathogen systems and for improving model transparency by clarifying assumptions, choices and shortcomings in complex ecological analyses.

Statistical modeling of patterns in annual reproductive rates.

Reproduction by individuals is typically recorded as count data (e.g., number of fledglings from a nest or inflorescences on a plant) and commonly modeled using Poisson or negative binomial distributions, which assume that variance is greater than or equal to the mean. However, distributions of reproductive effort are often underdispersed (i.e., variance < mean). When used in hypothesis tests, models that ignore underdispersion will be overly conservative and may fail to detect significant patterns. Here we show that generalized Poisson (GP) and Conway-Maxwell-Poisson (CMP) distributions are better choices for modeling reproductive effort because they can handle both overdispersion and underdispersion; we provide examples of how ecologists can use GP and CMP distributions in generalized linear models (GLMs) and generalized linear mixed models (GLMMs) to quantify patterns in reproduction. Using a new R package, glmmTMB, we construct GLMMs to investigate how rainfall and population density influence the number of fledglings in the warbler Oreothlypis celata and how flowering rate of Heliconia acuminata differs between fragmented and continuous forest. We also demonstrate how to deal with zero-inflation, which occurs when there are more zeros than expected in the distribution, e.g., due to complete reproductive failure by some individuals.

Interactive effects of tree size, crown exposure and logging on drought-induced mortality.

Large trees in the tropics are reportedly more vulnerable to droughts than their smaller neighbours. This pattern is of interest due to what it portends for forest structure, timber production, carbon sequestration and multiple other values given that intensified El Niño Southern Oscillation (ENSO) events are expected to increase the frequency and intensity of droughts in the Amazon region. What remains unclear is what characteristics of large trees render them especially vulnerable to drought-induced mortality and how this vulnerability changes with forest degradation. Using a large-scale, long-term silvicultural experiment in a transitional Amazonian forest in Bolivia, we disentangle the effects of stem diameter, tree height, crown exposure and logging-induced degradation on risks of drought-induced mortality during the 2004/2005 ENSO event. Overall, tree mortality increased in response to drought in both logged and unlogged plots. Tree height was a much stronger predictor of mortality than stem diameter. In unlogged plots, tree height but not crown exposure was positively associated with drought-induced mortality, whereas in logged plots, neither tree height nor crown exposure was associated with drought-induced mortality. Our results suggest that, at the scale of a site, hydraulic factors related to tree height, not air humidity, are a cause of elevated drought-induced mortality of large trees in unlogged plots.This article is part of a discussion meeting issue 'The impact of the 2015/2016 El Niño on the terrestrial tropical carbon cycle: patterns, mechanisms and implications'.