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Structured epidemic models can be formulated as first-order hyperbolic PDEs, where the "spatial" variables represent individual traits, called structures. For models with two structures, we propose a numerical technique to approximate $R_{0}$, which measures the transmissibility of an infectious disease and, rigorously, is defined as the dominant eigenvalue of a next-generation operator. Via bivariate collocation and cubature on tensor grids, the latter is approximated with a finite-dimensional matrix, so that its dominant eigenvalue can easily be computed with standard techniques. We use test examples to investigate experimentally the behavior of the approximation: the convergence order appears to be infinite when the corresponding eigenfunction is smooth, and finite for less regular eigenfunctions. To demonstrate the effectiveness of the technique for more realistic applications, we present a new epidemic model structured by demographic age and immunity, and study the approximation of $R_{0}$ in some particular cases of interest., 24 pages, 5 figures, submitted for publication

Background Predicting the likely size of future SARS-CoV-2 waves is necessary for public health planning. In England, voluntary “plan B” mitigation measures were introduced in December 2021 including increased home working and face coverings in shops but stopped short of restrictions on social contacts. The impact of voluntary risk mitigation behaviours on future SARS-CoV-2 burden is unknown. Methods We developed a rapid online survey of risk mitigation behaviours ahead of the winter 2021 festive period and deployed in two longitudinal cohort studies in the UK (Avon Longitudinal Study of Parents and Children (ALSPAC) and TwinsUK/COVID Symptom Study (CSS) Biobank) in December 2021. Using an individual-based, probabilistic model of COVID-19 transmission between social contacts with SARS-CoV-2 Omicron variant parameters and realistic vaccine coverage in England, we predicted the potential impact of the SARS-CoV-2 Omicron wave in England in terms of the effective reproduction number and cumulative infections, hospital admissions and deaths. Using survey results, we estimated in real-time the impact of voluntary risk mitigation behaviours on the Omicron wave in England, if implemented for the entire epidemic wave. Results Over 95% of survey respondents (NALSPAC = 2686 and NTwins = 6155) reported some risk mitigation behaviours, with vaccination and using home testing kits reported most frequently. Less than half of those respondents reported that their behaviour was due to “plan B”. We estimate that without risk mitigation behaviours, the Omicron variant is consistent with an effective reproduction number between 2.5 and 3.5. Due to the reduced vaccine effectiveness against infection with the Omicron variant, our modelled estimates suggest that between 55% and 60% of the English population could be infected during the current wave, translating into between 12,000 and 46,000 cumulative deaths, depending on assumptions about severity and vaccine effectiveness. The actual number of deaths was 15,208 (26 November 2021–1 March 2022). We estimate that voluntary risk reduction measures could reduce the effective reproduction number to between 1.8 and 2.2 and reduce the cumulative number of deaths by up to 24%. Conclusions Predicting future infection burden is affected by uncertainty in disease severity and vaccine effectiveness estimates. In addition to biological uncertainty, we show that voluntary measures substantially reduce the projected impact of the SARS-CoV-2 Omicron variant but that voluntary measures alone would be unlikely to completely control transmission.

Background: When public health interventions are being loosened after several days of decline in the number of coronavirus disease 2019 (COVID-19) cases, it is of critical importance to identify potential strategies to ease restrictions while mitigating a new wave of more transmissible variants of concern (VOCs). We estimated the necessary enhancements to public health interventions for a partial reopening of the economy while avoiding the worst consequences of a new outbreak, associated with more transmissible VOCs. Methods: We used a transmission dynamics model to quantify conditions that combined public health interventions must meet to reopen the economy without a large outbreak. These conditions are those that maintain the control reproduction number below unity, while accounting for an increase in transmissibility due to VOC. Results: We identified combinations of the proportion of individuals exposed to the virus who are traced and quarantined before becoming infectious, the proportion of symptomatic individuals confirmed and isolated, and individual daily contact rates needed to ensure the control reproduction number remains below unity. Conclusion: Our analysis indicates that the success of restrictive measures including lockdown and stay-at-home orders, as reflected by a reduction in number of cases, provides a narrow window of opportunity to intensify case detection and contact tracing efforts to prevent a new wave associated with circulation of more transmissible VOCs.

Contexte : Lorsque les interventions de santé publique sont assouplies après plusieurs jours de baisse du nombre de cas de maladie à coronavirus 2019 (COVID-19), il est essentiel de déterminer les stratégies potentielles pour assouplir les restrictions tout en atténuant le risque d’une nouvelle vague de variants préoccupants plus transmissibles. Nous avons estimé les améliorations nécessaires aux interventions en santé publique pour une réouverture partielle de l’économie tout en évitant les pires conséquences d’une nouvelle éclosion, associées à des variants préoccupants plus transmissibles. Méthodes : Un modèle dynamique de transmission a été utilisé pour quantifier les conditions auxquelles les interventions de santé publique combinées doivent répondre pour rouvrir l’économie sans risque d’éclosion importante. Ces conditions sont celles qui maintiennent le taux de reproduction de contrôle en dessous de l’unité, tout en tenant compte d’une augmentation de la transmissibilité due aux variants préoccupants. Résultats : Il a été possible de déterminer des combinaisons de la proportion de personnes exposées au virus qui sont retracées et mises en quarantaine avant de devenir infectieuses, la proportion de personnes symptomatiques confirmée et isolée et les taux de contact quotidiens individuels nécessaires pour s’assurer que le taux de reproduction de contrôle demeure inférieur à l’unité. Conclusion : Notre analyse indique que le succès des mesures restrictives, notamment les décrets de confinement et les fermetures, offre une fenêtre d’opportunité étroite pour intensifier les efforts de détection des cas et de recherche des contacts afin d’empêcher une nouvelle vague associée à la circulation de variants préoccupants plus transmissibles, comme en témoigne la réduction du nombre de cas.

Acknowledgements: The authors would like to thank colleagues in SPI-M-O and JUNIPER consortium for various discussions around hospital modelling and forecasting., Funder: National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Emergency Preparedness and Response, Funder: Li Ka Shing Foundation; funder-id: http://dx.doi.org/10.13039/100007421, Funder: National Institute for Health Research Policy Research Programme in Operational Research (OPERA), The first year of the COVID-19 pandemic put considerable strain on healthcare systems worldwide. In order to predict the effect of the local epidemic on hospital capacity in England, we used a variety of data streams to inform the construction and parameterisation of a hospital progression model, EpiBeds, which was coupled to a model of the generalised epidemic. In this model, individuals progress through different pathways (e.g. may recover, die, or progress to intensive care and recover or die) and data from a partially complete patient-pathway line-list was used to provide initial estimates of the mean duration that individuals spend in the different hospital compartments. We then fitted EpiBeds using complete data on hospital occupancy and hospital deaths, enabling estimation of the proportion of individuals that follow the different clinical pathways, the reproduction number of the generalised epidemic, and to make short-term predictions of hospital bed demand. The construction of EpiBeds makes it straightforward to adapt to different patient pathways and settings beyond England. As part of the UK response to the pandemic, EpiBeds provided weekly forecasts to the NHS for hospital bed occupancy and admissions in England, Wales, Scotland, and Northern Ireland at national and regional scales.

We consider a scalar delay differential equation $$\dot{x}(t)=-dx(t)+f((1-\alpha )\rho x(t-\tau )+\alpha \rho x(t-2\tau ))$$ with an instant mortality rate $$d>0$$, the nonlinear Rick reproductive function f, a survival rate during all development stages $$\rho $$, and a proportion constant $$\alpha \in [0, 1]$$ with which population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within $$(3\tau , 6\tau )$$ for a wide range of parameter values. We show this existence of periodic solutions not only for the delay $$\tau $$ near the first critical value $$\tau ^*$$ when a local Hopf bifurcation takes place near the positive equilibrium, but for all $$\tau >\tau ^*$$. We obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li–Muldowney technique to rule out periodic solutions with period $$3\tau $$. We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium.

In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the \begin{document}$ \odot* $\end{document} -space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the \begin{document}$ \odot* $\end{document} -shift operator plays a fundamental role. By applying the pseudospectral discretization technique we derive a system of ordinary differential equations, whose dynamics can be efficiently analyzed by existing bifurcation tools. To better understand to what extent the resulting finite-dimensional system "mimics" the dynamics of the original infinite-dimensional one, we study the pseudospectral approximations of the \begin{document}$ \odot* $\end{document} -shift operator and of the \begin{document}$ \odot* $\end{document} -generator in the supremum norm, which is the natural choice for delay differential equations, when the discretization parameter increases. In this context there are still open questions. We collect the most relevant results from the literature and we present some conjectures, supported by various numerical experiments, to illustrate the behavior w.r.t. the discretization parameter and to indicate the direction of ongoing and future research.

Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of external problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this internal continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.

BackgroundThe Omicron variant of SARS-CoV-2 infection poses substantial challenges to public health. In England, “plan B” mitigation measures were introduced in December 2021 including increased home working and face coverings in shops, but stopped short of restrictions on social contacts. The impact of voluntary risk mitigation behaviours on future SARS-CoV-2 burden is unknown.MethodsWe developed a rapid online survey of risk mitigation behaviours during the winter 2021 festive period and deployed in two longitudinal cohort studies in the UK (Avon Longitudinal Study of Parents and Children (ALSPAC) and TwinsUK/Covid Symptom Study (CSS) Biobank) in December 2021. Using an individual-based, probabilistic model of COVID-19 transmission between social contacts with SARS-CoV-2 Omicron variant parameters and realistic vaccine coverage in England, we describe the potential impact of the SARS-CoV-2 Omicron wave in England in terms of the effective reproduction number and cumulative infections, hospital admissions and deaths. Using survey results, we estimated in real-time the impact of voluntary risk mitigation behaviours on the Omicron wave in England, if implemented for the entire epidemic wave.ResultsOver 95% of survey respondents (NALSPAC=2,686 and NTwins=6,155) reported some risk mitigation behaviours, with vaccination and using home testing kits reported most frequently. Less than half of those respondents reported that their behaviour was due to “plan B”. We estimate that without risk mitigation behaviours, the Omicron variant is consistent with an effective reproduction number between 2.5 and 3.5. Due to the reduced vaccine effectiveness against infection with the Omicron variant, our modelled estimates suggest that between 55% and 60% of the English population could be infected during the current wave, translating into between 15,000 and 46,000 cumulative deaths, depending on assumptions about vaccine effectiveness. We estimate that voluntary risk reduction measures could reduce the effective reproduction number to between 1.8 and 2.2 and reduce the cumulative number of deaths by up to 24%.ConclusionsWe conclude that voluntary measures substantially reduce the projected impact of the SARS-CoV-2 Omicron variant, but that voluntary measures alone would be unlikely to completely control transmission.

The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.

The attack ratio in a subpopulation is defined as the total number of infections over the total number of individuals in this subpopulation. Using a methodology based on an age-stratified transmission dynamics model, we estimated the attack ratio of COVID-19 among children (individuals 0–11 years) when a large proportion of individuals eligible for vaccination (age 12 and above) are vaccinated to contain the epidemic among this subpopulation, or the effective herd immunity (with additional physical distancing measures). We describe the relationship between the attack ratio among children, the time to remove infected individuals from the transmission chain and the children-to-children daily contact rate while considering the increased transmissibility of virus variants (using the Delta variant as an example). We illustrate the generality and applicability of the methodology established by performing an analysis of the attack ratio of COVID-19 among children in the population of Canada and in its province of Ontario. The clinical attack ratio, defined as the number of symptomatic infections over the total population, can be informed from the attack ratio and both can be reduced substantially via a combination of reduced social mixing and rapid testing and isolation of the children.

The attack ratio in a subpopulation is defined as the total number of infections over the total number of individuals in this subpopulation. Using a methodology based on modified age-stratified transmission dynamics model, we estimated the attack ratio of COVID-19 among children (individuals 0-11 years) in Ontario, Canada when a large proportion of individuals eligible for vaccination (age 12 and above) are vaccinated to achieve herd immunity among this subpopulation, or the effective herd immunity with additional physical distancing measures (hence effective herd immunity). We describe the relationship between this attack ratio among children, the time to remove infected individuals from the transmission chain and the children-to-children daily contact rate, while considering the increased transmissibility of virus variants (using the Delta variant as an example). We further illustrate the generality and applicability of the methodology established by performing an analysis of the attack ratio of COVID-19 among children in the Canadian population. The clinical attack ratio, the number of symptomatic infections over the total population can be informed from the attack ratio, and both can be reduced substantially via a combination of higher vaccine coverage in the vaccine eligible population, reduced social mixing among children, and rapid testing and isolation.

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.

The rapid emergence of SARS-CoV-2 mutants with new phenotypic properties is a critical challenge to the control of the ongoing pandemic. B.1.1.7 was monitored in the UK through routine testing and S-gene target failures (SGTF), comprising over 90% of cases by March 2021. Now, the reverse is occurring: SGTF cases are being replaced by an S-gene positive variant, which we associate with B.1.617.2. Evidence from the characteristics of S-gene positive cases demonstrates that, following importation, B.1.617.2 is transmitted locally, growing at a rate higher than B.1.1.7 and a doubling time between 5-14 days. S-gene positive cases should be prioritised for sequencing and aggressive control in any countries in which this variant is newly detected.One-Sentence SummaryThe B.1.617.2 variant of SARS-CoV-2 is replacing B.1.1.7 and emerging as the dominant variant in England, evidenced by sustained local transmission.

We propose a deterministic model capturing essential features of contact tracing as part of public health non-pharmaceutical interventions to mitigate an outbreak of an infectious disease. By incorporating a mechanistic formulation of the processes at the individual level, we obtain an integral equation (delayed in calendar time and advanced in time since infection) for the probability that an infected individual is detected and isolated at any point in time. This is then coupled with a renewal equation for the total incidence to form a closed system describing the transmission dynamics involving contact tracing. We define and calculate basic and effective reproduction numbers in terms of pathogen characteristics and contact tracing implementation constraints. When applied to the case of SARS-CoV-2, our results show that only combinations of diagnosis of symptomatic infections and contact tracing that are almost perfect in terms of speed and coverage can attain control, unless additional measures to reduce overall community transmission are in place. Under constraints on the testing or tracing capacity, a temporary interruption of contact tracing may, depending on the overall growth rate and prevalence of the infection, lead to an irreversible loss of control even when the epidemic was previously contained.

Background. Public Health interventions have succeeded marginally to flatten the COVID-19 epidemic curve of the most recent wave in many regions. Several highly transmissible COVID-19 Variants of Concern (VOCs) have emerged internationally, with variable virulence and as-yet unclear antigenicity. Growing public fatigue highlights the importance of accurately quantifying the speed at which VOCs can replace the resident strain, to inform precise decision-making in terms of both timing and scale of interventions. Methods. Using a simple Susceptible-Infected-Removed (SIR) model, we derive a formula calculating the duration for a VOC to increase its frequency from one level to another. We evaluate the impact of interventions on VOC-related variables. Findings. After introduction of a VOC with a higher transmissibility than the resident strain, its prevalence increases exponentially, leading to eventual replacement by the VOC. Even if the outbreak caused by the resident strain has previously been brought under control, the epidemic is eventually going to increase exponentially, possibly after a transient declining phase. The initial VOC frequency, epidemic mean generation time, and reproduction number of the resident strain all contribute to determining the speed of VOC replacement and the resulting epidemic growth. Interpretation. Following the introduction of a VOC, the risk of a sudden surge of the total cases is hidden behind a transient decline. Maintaining or enhancing Public Health interventions are vital to slowing down the VOC replacement, curbing the epidemic growth, and increasing VOC doubling time.

Social contact mixing plays a critical role in influencing the transmission routes of infectious diseases. Moreover, quantifying social contact mixing patterns and their variations in a rapidly evolving pandemic intervened by changing public health measures is key for retroactive evaluation and proactive assessment of the effectiveness of different age- and setting-specific interventions. Contact mixing patterns have been used to inform COVID-19 pandemic public health decision-making; but a rigorously justified methodology to identify setting-specific contact mixing patterns and their variations in a rapidly developing pandemic, which can be informed by readily available data, is in great demand and has not yet been established. Here we fill in this critical gap by developing and utilizing a novel methodology, integrating social contact patterns derived from empirical data with a disease transmission model, that enables the usage of age-stratified incidence data to infer age-specific susceptibility, daily contact mixing patterns in workplace, household, school and community settings; and transmission acquired in these settings under different physical distancing measures. We demonstrated the utility of this methodology by performing an analysis of the COVID-19 epidemic in Ontario, Canada. We quantified the age- and setting (household, workplace, community, and school)-specific mixing patterns and their evolution during the escalation of public health interventions in Ontario, Canada. We estimated a reduction in the average individual contact rate from 12.27 to 6.58 contacts per day, with an increase in household contacts, following the implementation of control measures. We also estimated increasing trends by age in both the susceptibility to infection by SARS-CoV-2 and the proportion of symptomatic individuals diagnosed. Inferring the age- and setting-specific social contact mixing and key age-stratified epidemiological parameters, in the presence of evolving control measures, is critical to inform decision- and policy-making for the current COVID-19 pandemic.

BackgroundShort-term forecasts of infectious disease can aid situational awareness and planning for outbreak response. Here, we report on multi-model forecasts of Covid-19 in the UK that were generated at regular intervals starting at the end of March 2020, in order to monitor expected healthcare utilisation and population impacts in real time.MethodsWe evaluated the performance of individual model forecasts generated between 24 March and 14 July 2020, using a variety of metrics including the weighted interval score as well as metrics that assess the calibration, sharpness, bias and absolute error of forecasts separately. We further combined the predictions from individual models into ensemble forecasts using a simple mean as well as a quantile regression average that aimed to maximise performance. We compared model performance to a null model of no change.ResultsIn most cases, individual models performed better than the null model, and ensembles models were well calibrated and performed comparatively to the best individual models. The quantile regression average did not noticeably outperform the mean ensemble.ConclusionsEnsembles of multi-model forecasts can inform the policy response to the Covid-19 pandemic by assessing future resource needs and expected population impact of morbidity and mortality.

Since the beginning of the COVID-19 pandemic, most Canadian provinces have gone through four distinct phases of social distancing and enhanced testing. A transmission dynamics model fitted to the cumulative case time series data permits us to estimate the effectiveness of interventions implemented in terms of the contact rate, probability of transmission per contact, proportion of isolated contacts, and detection rate. This allows us to calculate the control reproduction number during different phases (which gradually decreased to less than one). From this, we derive the necessary conditions in terms of enhanced social distancing, personal protection, contact tracing, quarantine/isolation strength at each escalation phase for the disease control to avoid a rebound. From this, we quantify the conditions needed to prevent epidemic rebound during de-escalation by simply reversing the escalation process.

We investigate the effect of school closure and subsequent reopening on the transmission of COVID-19, by considering Denmark, Norway, Sweden and German states as case studies. By comparing the growth rates in daily hospitalizations or confirmed cases under different interventions, we provide evidence that school closures contribute to a reduction in the growth rate approximately 7 days after implementation. Limited school attendance, such as older students sitting exams or the partial return of younger year groups, does not appear to significantly affect community transmission. In countries where community transmission is generally low, such as Denmark or Norway, a large-scale reopening of schools while controlling or suppressing the epidemic appears feasible. However, school reopening can contribute to statistically significant increases in the growth rate in countries like Germany, where community transmission is relatively high. In all regions, a combination of low classroom occupancy and robust test-and-trace measures were in place. Our findings underscore the need for a cautious evaluation of reopening strategies. This article is part of the theme issue ‘Modelling that shaped the early COVID-19 pandemic response in the UK’.

We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations.