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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

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 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.

Early assessments of the growth rate of COVID-19 were subject to significant uncertainty, as expected with limited data and difficulties in case ascertainment, but as cases were recorded in multiple countries, more robust inferences could be made. Using multiple countries, data streams and methods, we estimated that, when unconstrained, European COVID-19 confirmed cases doubled on average every 3 days (range 2.2–4.3 days) and Italian hospital and intensive care unit admissions every 2–3 days; values that are significantly lower than the 5–7 days dominating the early published literature. Furthermore, we showed that the impact of physical distancing interventions was typically not seen until at least 9 days after implementation, during which time confirmed cases could grow eightfold. We argue that such temporal patterns are more critical than precise estimates of the time-insensitive basic reproduction number R 0 for initiating interventions, and that the combination of fast growth and long detection delays explains the struggle in countries' outbreak response better than large values of R 0 alone. One year on from first reporting these results, reproduction numbers continue to dominate the media and public discourse, but robust estimates of unconstrained growth remain essential for planning worst-case scenarios, and detection delays are still key in informing the relaxation and re-implementation of interventions. This article is part of the theme issue ‘Modelling that shaped the early COVID-19 pandemic response in the UK’.

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.

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.

Early assessments of the spreading rate of COVID-19 were subject to significant uncertainty, as expected with limited data and difficulties in case ascertainment, but more reliable inferences can now be made. Here, we estimate from European data that COVID-19 cases are expected to double initially every three days, until social distancing interventions slow this growth, and that the impact of such measures is typically only seen nine days - i.e. three doubling times - after their implementation. We argue that such temporal patterns are more critical than precise estimates of the basic reproduction number for initiating interventions. This observation has particular implications for the low- and middle-income countries currently in the early stages of their local epidemics., Main text: 13 pages (1-13), 3 figures, 1 table; Supplementary Information: 9 pages (14-22), 4 figures, 1 table

Background. After the declaration of COVID-19 pandemic on March 11th 2020, local transmission chains starting in different countries including Canada are forcing governments to take decisions on public health interventions to mitigate the spread of the epidemic. Methods. We conduct data-driven and model-free estimations for the growth rates of the COVID-19 epidemics in Italy and Canada, by fitting an exponential curve to the daily reported cases. We use these estimates to predict epidemic trends in Canada under different scenarios of public health interventions. Results. In Italy, the initial growth rate (0.22) has reduced to 0.1 two weeks after the lockdown of the country on March 8th 2020. This corresponds to a reduction of the doubling time from about 3.15 to almost 7 days. In comparison, the growth rate in Canada has increased from 0.13 between March 1st and 13th, to 0.25 between March 13th to 22nd. This current growth rate corresponds to a doubling time of 2.7 days, and therefore, unless further public health interventions are escalated in Canada, we project 15,000 cases by March 31st. However, the case number can be reduced to 4,000 if escalated public health interventions can be implemented instantly to reduce the growth rate to 0.1, the same level achieved in Italy. Interpretation. Intervention measures implemented so far in different countries worldwide have been effective in reducing the growth rate and increasing the doubling time, but their effects come with a substantial delay as long as 2 weeks. Prompt and farsighted interventions are critical to counteract the very rapid initial growth of the COVID-19 epidemic in Canada. Mitigation plans must take into account the delayed effect of interventions by up to 2-weeks and the short doubling time of 3-4 days.

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’.