Canada's provincial COVID-19 pandemic modelling efforts: A review of mathematical models and their impacts on the responses.

SETTING: Mathematical modelling played an important role in the public health response to COVID-19 in Canada. Variability in epidemic trajectories, modelling approaches, and data infrastructure across provinces provides a unique opportunity to understand the factors that shaped modelling strategies. INTERVENTION: Provinces implemented stringent pandemic interventions to mitigate SARS-CoV-2 transmission, considering evidence from epidemic models. This study aimed to summarize provincial COVID-19 modelling efforts. We identified modelling teams working with provincial decision-makers, through referrals and membership in Canadian modelling networks. Information on models, data sources, and knowledge translation were abstracted using standardized instruments. OUTCOMES: We obtained information from six provinces. For provinces with sustained community transmission, initial modelling efforts focused on projecting epidemic trajectories and healthcare demands, and evaluating impacts of proposed interventions. In provinces with low community transmission, models emphasized quantifying importation risks. Most of the models were compartmental and deterministic, with projection horizons of a few weeks. Models were updated regularly or replaced by new ones, adapting to changing local epidemic dynamics, pathogen characteristics, vaccines, and requests from public health. Surveillance datasets for cases, hospitalizations and deaths, and serological studies were the main data sources for model calibration. Access to data for modelling and the structure for knowledge translation differed markedly between provinces. IMPLICATION: Provincial modelling efforts during the COVID-19 pandemic were tailored to local contexts and modulated by available resources. Strengthening Canadian modelling capacity, developing and sustaining collaborations between modellers and governments, and ensuring earlier access to linked and timely surveillance data could help improve pandemic preparedness.

RéSUMé: CONTEXTE: La modélisation mathématique a joué un rôle de premier plan dans les ripostes sanitaires à la COVID-19 au Canada. Les différentes trajectoires épidémiques provinciales, leurs approches de modélisation et infrastructures de données représentent une occasion unique de comprendre les facteurs qui ont influencé les stratégies de modélisation provinciales. INTERVENTION: Les provinces ont mis en place des mesures de santé publique strictes afin d'atténuer la transmission du SRAS-CoV-2 en tenant compte des données probantes provenant des modèles épidémiques. Notre étude vise à décrire et résumer les efforts provinciaux de modélisation de la COVID-19. Nous avons identifié les équipes de modélisation travaillant avec les décideurs provinciaux parmi les réseaux Canadiens de modélisation et par référence. Les informations sur les modèles, leurs sources de données et les approches de mobilisation des connaissances ont été obtenues à l'aide d'instruments standardisés. RéSULTATS: Nous avons colligé les informations provenant de six provinces. Pour les provinces qui ont eu de la transmission communautaire soutenue, les efforts de modélisation initiaux se sont concentrés sur la projection des trajectoires épidémiques et des demandes de soins de santé et sur l'évaluation des impacts des interventions proposées. Dans les provinces où la transmission communautaire a été faible, les modèles visaient à quantifier les risques d'importation. La plupart des équipes ont développé des modèles à compartiments déterministes avec des horizons de projection de quelques semaines. Les modèles ont été régulièrement mis à jour ou remplacés par de nouveaux, s'adaptant aux dynamiques locales, à l'arrivée de nouveaux variants, aux vaccins et aux demandes des autorités de santé publique. Les données de surveillance des cas, des hospitalisations et des décès, ainsi que les études sérologiques, ont constitué les principales sources de données pour calibrer les modèles. L'accès aux données pour la modélisation et la structure de mobilisation des connaissances différaient considérablement d'une province à l'autre. IMPLICATION: Les efforts de modélisation provinciaux pendant la pandémie de la COVID-19 ont été adaptés aux contextes locaux et modulés par les ressources disponibles. Le renforcement de la capacité canadienne de modélisation, le développement et le maintien de collaborations entre les modélisateurs et les gouvernements, ainsi qu'un accès rapide et opportun aux données de surveillance individuelles et liées pourraient contribuer à améliorer la préparation aux futures pandémies.

Fitting Epidemic Models to Data: A Tutorial in Memory of Fred Brauer.

Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.

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.

Modelling the impacts of male alternative reproductive tactics on population dynamics.

Observations of male alternative reproductive tactics (ARTs) in a variety of species have stimulated the development of mathematical models that can account for the evolution and stable coexistence of multiple male phenotypes. However, little attention has been given to the population dynamic consequences of ARTs. We present a population model that takes account of the existence of two male ARTs (guarders and sneakers), assuming that tactic frequencies are environmentally determined and tactic reproductive success depends on the densities of both types. The presence of sneakers typically increases overall population density. However, if sneakers comprise a sufficiently large proportion of the population-or, equivalently, if guarders are sufficiently rare-then it is possible for the total population to crash to extinction (in this extreme regime, there is also an Allee effect, i.e. a threshold density below which the population will go extinct). We apply the model to the example of the invasive round goby (Neogobius melanostomus). We argue that ARTs can dramatically influence population dynamics and suggest that considering such phenotypic plasticity in population models is potentially important, especially for species of conservation or commercial importance.

Protocol for a living evidence synthesis on variants of concern and COVID-19 vaccine effectiveness.

BACKGROUND: It is evident that COVID-19 will remain a public health concern in the coming years, largely driven by variants of concern (VOC). It is critical to continuously monitor vaccine effectiveness as new variants emerge and new vaccines and/or boosters are developed. Systematic surveillance of the scientific evidence base is necessary to inform public health action and identify key uncertainties. Evidence syntheses may also be used to populate models to fill in research gaps and help to prepare for future public health crises. This protocol outlines the rationale and methods for a living evidence synthesis of the effectiveness of COVID-19 vaccines in reducing the morbidity and mortality associated with, and transmission of, VOC of SARS-CoV-2. METHODS: Living evidence syntheses of vaccine effectiveness will be carried out over one year for (1) a range of potential outcomes in the index individual associated with VOC (pathogenesis); and (2) transmission of VOC. The literature search will be conducted up to May 2023. Observational and database-linkage primary studies will be included, as well as RCTs. Information sources include electronic databases (MEDLINE; Embase; Cochrane, L*OVE; the CNKI and Wangfang platforms), pre-print servers (medRxiv, BiorXiv), and online repositories of grey literature. Title and abstract and full-text screening will be performed by two reviewers using a liberal accelerated method. Data extraction and risk of bias assessment will be completed by one reviewer with verification of the assessment by a second reviewer. Results from included studies will be pooled via random effects meta-analysis when appropriate, or otherwise summarized narratively. DISCUSSION: Evidence generated from our living evidence synthesis will be used to inform policy making, modelling, and prioritization of future research on the effectiveness of COVID-19 vaccines against VOC.

Face masking and COVID-19: potential effects of variolation on transmission dynamics.

Face masks do not completely prevent transmission of respiratory infections, but masked individuals are likely to inhale fewer infectious particles. If smaller infectious doses tend to yield milder infections, yet ultimately induce similar levels of immunity, then masking could reduce the prevalence of severe disease even if the total number of infections is unaffected. It has been suggested that this effect of masking is analogous to the pre-vaccination practice of variolation for smallpox, whereby susceptible individuals were intentionally infected with small doses of live virus (and often acquired immunity without severe disease). We present a simple epidemiological model in which mask-induced variolation causes milder infections, potentially with lower transmission rate and/or different duration. We derive relationships between the effectiveness of mask-induced variolation and important epidemiological metrics (the basic reproduction number and initial epidemic growth rate, and the peak prevalence, attack rate and equilibrium prevalence of severe infections). We illustrate our results using parameter estimates for the original SARS-CoV-2 wild-type virus, as well as the Alpha, Delta and Omicron variants. Our results suggest that if variolation is a genuine side-effect of masking, then the importance of face masks as a tool for reducing healthcare burdens from COVID-19 may be under-appreciated.

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.

From science to politics: COVID-19 information fatigue on YouTube.

OBJECTIVE: The COVID-19 pandemic is the first pandemic where social media platforms relayed information on a large scale, enabling an "infodemic" of conflicting information which undermined the global response to the pandemic. Understanding how the information circulated and evolved on social media platforms is essential for planning future public health campaigns. This study investigated what types of themes about COVID-19 were most viewed on YouTube during the first 8 months of the pandemic, and how COVID-19 themes progressed over this period. METHODS: We analyzed top-viewed YouTube COVID-19-related videos in English from December 1, 2019 to August 16, 2020 with an open inductive content analysis. We coded 536 videos associated with 1.1 billion views across the study period. East Asian countries were the first to report the virus, while most of the top-viewed videos in English were from the US. Videos from straight news outlets dominated the top-viewed videos throughout the outbreak, and public health authorities contributed the fewest. Although straight news was the dominant COVID-19 video source with various types of themes, its viewership per video was similar to that for entertainment news and YouTubers after March. RESULTS: We found, first, that collective public attention to the COVID-19 pandemic on YouTube peaked around March 2020, before the outbreak peaked, and flattened afterwards despite a spike in worldwide cases. Second, more videos focused on prevention early on, but videos with political themes increased through time. Third, regarding prevention and control measures, masking received much less attention than lockdown and social distancing in the study period. CONCLUSION: Our study suggests that a transition of focus from science to politics on social media intensified the COVID-19 infodemic and may have weakened mitigation measures during the first waves of the COVID-19 pandemic. It is recommended that authorities should consider co-operating with reputable social media influencers to promote health campaigns and improve health literacy. In addition, given high levels of globalization of social platforms and polarization of users, tailoring communication towards different digital communities is likely to be essential.

Adjuvanted trivalent influenza vaccine versus quadrivalent inactivated influenza vaccine in Hutterite Children: A randomized clinical trial.

BACKGROUND: Children play an important role in the transmission of influenza. The best choice of vaccine to achieve both direct and indirect protection is uncertain. The objective of the study was to test whether vaccinating children with MF59 adjuvanted trivalent influenza vaccine (aTIV) can reduce influenza in children and their extended households compared to inactivated quadrivalent vaccine (QIV). METHODS: We conducted a cluster randomized trial in 42 Hutterite colonies in Alberta and Saskatchewan. Colonies were randomized such that children were assigned in a blinded manner to receive aTIV (0.25 ml of pediatric aTIV for ages 6 months to < 36 months or 0.5 ml for ages ≥ 36 months to 6 years) or 0.5 ml of QIV. Participants included 424 children aged 6 months to 6 years who received the study vaccine and 1246 family cluster members who did not receive the study vaccine. The primary outcome was confirmed influenza A and B infection using a real-time reverse transcriptase polymerase chain reaction (RT-PCR) assay. An intent to treat analysis was used. Data were collected from January 2017 to June 2019. RESULTS: The mean percentage of children who received study vaccine was 62% for aTIV colonies and 74% for QIV colonies. There were 66 (3.4%) with RT-PCR confirmed influenza A and B in the aTIV colonies (children and family clusters) versus 93 (4.4%) in the QIV colonies, hazard ratio (HR) 0.78 (95 %CI 0.36-1.71). Of these, 48 (2.5%) in the aTIV colonies and 76 (3.6%) in the QIV colonies had influenza A, HR 0.69, (95 %CI 0.29-1.66) while 18 (0.9%) and 17 (0.8%) in the aTIV versus QIV colonies respectively had influenza B, HR 1.22, (95 %CI 0.20-7.41). In children who received study vaccine, there were 5 Influenza A infections in the aTIV colonies (1.1%) compared to 30 (5.8%) in the QIV colonies, relative efficacy of 80%, HR 0.20, (95 %CI 0.06-0.66). Adverse events were significantly more common among children who received aTIV. No serious vaccine adverse events were reported. CONCLUSION: Vaccinating children with aTIV compared to QIV resulted in similar community RT-PCR confirmed influenza illness and led to significant protection against influenza A in children.

The origins and potential future of SARS-CoV-2 variants of concern in the evolving COVID-19 pandemic.

One year into the global COVID-19 pandemic, the focus of attention has shifted to the emergence and spread of SARS-CoV-2 variants of concern (VOCs). After nearly a year of the pandemic with little evolutionary change affecting human health, several variants have now been shown to have substantial detrimental effects on transmission and severity of the virus. Public health officials, medical practitioners, scientists, and the broader community have since been scrambling to understand what these variants mean for diagnosis, treatment, and the control of the pandemic through nonpharmaceutical interventions and vaccines. Here we explore the evolutionary processes that are involved in the emergence of new variants, what we can expect in terms of the future emergence of VOCs, and what we can do to minimise their impact.

On inferring evolutionary stability in finite populations using infinite population models.

Models of evolution by natural selection often make the simplifying assumption that populations are infinitely large. In this infinite population limit, rare mutations that are selected against always go extinct, whereas in finite populations they can persist and even reach fixation. Nevertheless, for mutations of arbitrarily small phenotypic effect, it is widely believed that in sufficiently large populations, if selection opposes the invasion of rare mutants, then it also opposes their fixation. Here, we identify circumstances under which infinite-population models do or do not accurately predict evolutionary outcomes in large, finite populations. We show that there is no population size above which considering only invasion generally suffices: for any finite population size, there are situations in which selection opposes the invasion of mutations of arbitrarily small effect, but favours their fixation. This is not an unlikely limiting case; it can occur when fitness is a smooth function of the evolving trait, and when the selection process is biologically sensible. Nevertheless, there are circumstances under which opposition of invasion does imply opposition of fixation: in fact, for the [Formula: see text]-player snowdrift game (a common model of cooperation) we identify sufficient conditions under which selection against rare mutants of small effect precludes their fixation-in sufficiently large populations-for any selection process. We also find conditions under which-no matter how large the population-the trait that fixes depends on the selection process, which is important because any particular selection process is only an approximation of reality.

Transient disease dynamics across ecological scales.

Analyses of transient dynamics are critical to understanding infectious disease transmission and persistence. Identifying and predicting transients across scales, from within-host to community-level patterns, plays an important role in combating ongoing epidemics and mitigating the risk of future outbreaks. Moreover, greater emphases on non-asymptotic processes will enable timely evaluations of wildlife and human diseases and lead to improved surveillance efforts, preventive responses, and intervention strategies. Here, we explore the contributions of transient analyses in recent models spanning the fields of epidemiology, movement ecology, and parasitology. In addition to their roles in predicting epidemic patterns and endemic outbreaks, we explore transients in the contexts of pathogen transmission, resistance, and avoidance at various scales of the ecological hierarchy. Examples illustrate how (i) transient movement dynamics at the individual host level can modify opportunities for transmission events over time; (ii) within-host energetic processes often lead to transient dynamics in immunity, pathogen load, and transmission potential; (iii) transient connectivity between discrete populations in response to environmental factors and outbreak dynamics can affect disease spread across spatial networks; and (iv) increasing species richness in a community can provide transient protection to individuals against infection. Ultimately, we suggest that transient analyses offer deeper insights and raise new, interdisciplinary questions for disease research, consequently broadening the applications of dynamical models for outbreak preparedness and management. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s12080-021-00514-w.

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.

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.

Patterns of smallpox mortality in London, England, over three centuries.

Smallpox is unique among infectious diseases in the degree to which it devastated human populations, its long history of control interventions, and the fact that it has been successfully eradicated. Mortality from smallpox in London, England was carefully documented, weekly, for nearly 300 years, providing a rare and valuable source for the study of ecology and evolution of infectious disease. We describe and analyze smallpox mortality in London from 1664 to 1930. We digitized the weekly records published in the London Bills of Mortality (LBoM) and the Registrar General's Weekly Returns (RGWRs). We annotated the resulting time series with a sequence of historical events that might have influenced smallpox dynamics in London. We present a spectral analysis that reveals how periodicities in reported smallpox mortality changed over decades and centuries; many of these changes in epidemic patterns are correlated with changes in control interventions and public health policies. We also examine how the seasonality of reported smallpox mortality changed from the 17th to 20th centuries in London.

Estimating epidemic coupling between populations from the time to invasion.

Identifying the mechanisms by which diseases spread among populations is important for understanding and forecasting patterns of epidemics and pandemics. Estimating transmission coupling among populations is challenging because transmission events are difficult to observe in practice, and connectivity among populations is often obscured by local disease dynamics. We consider the common situation in which an epidemic is seeded in one population and later spreads to a second population. We present a method for estimating transmission coupling between the two populations, assuming they can be modelled as susceptible-infected-removed (SIR) systems. We show that the strength of coupling between the two populations can be estimated from the time taken for the disease to invade the second population. Confidence in the estimate is low if only a single invasion event has been observed, but is substantially improved if numerous independent invasion events are observed. Our analysis of this simplest, idealized scenario represents a first step toward developing and verifying methods for estimating epidemic coupling among populations in an ever-more-connected global human population.

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.

Fast estimation of time-varying infectious disease transmission rates.

Compartmental epidemic models have been used extensively to study the historical spread of infectious diseases and to inform strategies for future control. A critical parameter of any such model is the transmission rate. Temporal variation in the transmission rate has a profound influence on disease spread. For this reason, estimation of time-varying transmission rates is an important step in identifying mechanisms that underlie patterns in observed disease incidence and mortality. Here, we present and test fast methods for reconstructing transmission rates from time series of reported incidence. Using simulated data, we quantify the sensitivity of these methods to parameters of the data-generating process and to mis-specification of input parameters by the user. We show that sensitivity to the user's estimate of the initial number of susceptible individuals-considered to be a major limitation of similar methods-can be eliminated by an efficient, "peak-to-peak" iterative technique, which we propose. The method of transmission rate estimation that we advocate is extremely fast, for even the longest infectious disease time series that exist. It can be used independently or as a fast way to obtain better starting conditions for computationally expensive methods, such as iterated filtering and generalized profiling.