Combining models to generate consensus medium-term projections of hospital admissions, occupancy and deaths relating to COVID-19 in England (preprint)

Mathematical modelling has played an important role in offering informed advice during the COVID-19 pandemic. In England, a cross government and academia collaboration generated Medium-Term Projections (MTPs) of possible epidemic trajectories over the future 4-6 weeks from a collection of epidemiological models.In this paper we outline this collaborative modelling approach and evaluate the accuracy of the combined and individual model projections against the data over the period November 2021-December 2022 when various Omicron subvariants were spreading across England. Using a number of statistical methods, we quantify the predictive performance of the model projections for both the combined and individual MTPs, by evaluating the point and probabilistic accuracy. Our results illustrate that the combined MTPs, produced from an ensemble of heterogeneous epidemiological models, were a closer fit to the data than the individual models during the periods of epidemic growth or decline, with the 90% confidence intervals widest around the epidemic peaks. We also show that the combined MTPs increase the robustness and reduce the biases associated with a single model projection. Learning from our experience of ensemble modelling during the COVID-19 epidemic, our findings highlight the importance of developing cross-institutional multi-model infectious disease hubs for future outbreak control.

Spatio-temporal surveillance and early detection of SARS-CoV-2 variants of concern (preprint)

The SARS-CoV-2 pandemic has been characterized by the repeated emergence of genetically distinct virus variants of increased transmissibility and immune evasion compared to pre-existing lineages. In many countries, their containment required the intervention of public health authorities and the imposition of control measures. While the primary role of testing is to identify infection, target treatment, and limit spread (through isolation and contact tracing), a secondary benefit is in terms of surveillance and the early detection of new variants. Here we study the spatial invasion and early spread of the Alpha, Delta, and Omicron (BA.1 and BA.2) variants in England from September 2020 to February 2022 using the random neighbourhood covering (RaNCover) method, a statistical technique for the detection of aberrations in spatial point processes, applied to community PCR (polymerase-chain-reaction) test data where the TaqPath kit provides a proxy measure of the switch between variants. The application of RaNCover method could rapidly detect outbreaks of future SARS-CoV-2 variants of concern and hence inform optimal spatial interventions.

The Impact of SARS-CoV-2 Vaccine Dose Separation and Dose Targeting on Hospital Admissions and Deaths from COVID-19 in England (preprint)

In late 2020, the JCVI (the Joint Committee on Vaccination and Immunisation, which provides advice to the Department of Health and Social Care, England) made two important recommendations for the initial roll-out of the COVID-19 vaccine. The first was that vaccines should be targeted to the elderly and vulnerable, with the aim of maximally preventing disease rather than infection - a prioritisation that has become the adopted practice in many countries. The second was to increase the interval between first and second doses for 3 to 12-weeks, which both accounted for the observation that the ChAdOx vaccine appeared to have a higher efficacy with this longer dose interval, and that by delaying second doses the capacity to deliver first doses was increased. Here, using the latest data on vaccine efficacy we re-examine these recommendations through a mathe- matical model, to understand their short and medium-term impacts in England. In particular, we show that targeting the most vulnerable had the biggest immediate impact (compared to targeting younger individuals who may be more responsible for transmission). The 12-week delay was also highly beneficial, estimated to have averted between 32-72 thousand hospital admissions and 4-9 thousand deaths over the first ten months of the campaign (December 2020 - September 2021) - depending on the assumed interaction between dose interval and efficacy.

The impacts of increased global vaccine sharing on the COVID-19 pandemic; a retrospective modelling study (preprint)

Background: The SARS-CoV-2 pandemic has generated considerable morbidity and mortality worldwide. While the protection offered by vaccines (and booster doses) offers a method of mitigating the worst effects, by the end of 2021 the distribution of vaccine was highly heterogeneous with some countries achieving over 90% coverage in adults by the end of 2021, while others have less than 2%. In part, this is due to the availability of sufficient vaccine, although vaccine hesitancy also plays a role. Methods: We use an age-structured model of SARS-CoV-2 dynamics, matched to national data from 152 countries, to investigate the global impact of different vaccine sharing protocols during 2021. We assume a direct relationship between the emergence of variants with increased transmissibility and the cumulative amount of global infection, such that lower global prevalence leads to a lower reproductive number within each country. We compare five vaccine sharing scenarios, from the current situation, through sharing once a particular within-country threshold is reached (e.g. all over 40s have received 2 doses), to full sharing where all countries achieve equal age-dependent vaccine deployment. Findings: Compared to the observed distribution of vaccine uptake, we estimate full vaccine sharing would have generated a 1.5% (PI -0.1 - 4.5%) reduction in infections and a 11.3% (PI 0.6 - 23.2%) reduction in mortality globally by January 2022. The greatest benefit of vaccine sharing would have been experienced by low and middle income countries, who see an average 5.2% (PI 2.5% - 10.4%) infection reduction and 26.8% (PI 24.1% - 31.3%) mortality reduction. Many high income countries, that have had high vaccine uptake (most notably Canada, Chile, UK and USA), suffer increased infections and mortality under most of the sharing protocols investigated, assuming no other counter measures had been taken. However, if reductions in vaccine supply in these countries had been offset by prolonged use of non-pharmaceutical intervention measures, we predict far greater reductions in global infection and mortality of 64.5% (PI 62.6% - 65.4%) and 62.8% (PI 44.0% - 76.3%), respectively. Interpretation: By itself, our results suggest that although more equitable vaccine distribution would have had limited impact on overall infection numbers, vaccine sharing would have substantially reduced global mortality by providing earlier protection of the most vulnerable. If increased vaccine sharing from high income nations had been combined with slower easing of non pharmaceutical interventions to compensate for this, a large reduction in both infection and mortality globally would be expected, confounded by a lower risk of new variants arising.

A computational framework for modelling infectious disease policy based on age and household structure with applications to the COVID-19 pandemic (preprint)

The widespread, and in many countries unprecedented, use of non-pharmaceutical interventions (NPIs) during the COVID-19 pandemic has highlighted the need for mathematical models which can estimate the impact of these measures while accounting for the highly heterogeneous risk profile of COVID-19. Models accounting either for age structure or the household structure necessary to explicitly model many NPIs are commonly used in infectious disease modelling, but models incorporating both levels of structure present substantial computational and mathematical challenges due to their high dimensionality. Here we present a modelling framework for the spread of an epidemic that includes explicit representation of age structure and household structure. Our model is formulated in terms of tractable systems of ordinary differential equations for which we provide an open-source Python implementation. Such tractability leads to significant benefits for model calibration, exhaustive evaluation of possible parameter values, and interpretability of results. We demonstrate the flexibility of our model through four policy case studies, where we quantify the likely benefits of the following measures which were either considered or implemented in the UK during the current COVID-19 pandemic: control of within- and between-household mixing through NPIs; formation of support bubbles during lockdown periods; out-of-household isolation (OOHI); and temporary relaxation of NPIs during holiday periods. Our ordinary differential equation formulation and associated analysis demonstrate that multiple dimensions of risk stratification and social structure can be incorporated into infectious disease models without sacrificing mathematical tractability. This model and its software implementation expand the range of tools available to infectious disease policy analysts.

Bayesian Estimation of real-time Epidemic Growth Rates using Gaussian Processes: local dynamics of SARS-CoV-2 in England (preprint)

Quantitative assessments of the recent state of an epidemic and short-term projections into the near future are key public health tools that have substantial policy impacts, helping to determine if existing control measures are sufficient or need to be strengthened. Key to these quantitative assessments is the ability to rapidly and robustly measure the speed with which the epidemic is growing or decaying. Frequently, epidemiological trends are addressed in terms of the (time-varying) reproductive number R. Here, we take a more parsimonious approach and calculate the exponential growth rate, r, using a Bayesian hierarchical model to fit a Gaussian process to the epidemiological data. We show how the method can be employed when only case data from positive tests are available, and the improvement gained by including the total number of tests as a measure of heterogeneous testing effort. Although the methods are generic, we apply them to SARS-CoV-2 cases and testing in England, making use of the available high-resolution spatio-temporal data to determine long-term patterns of national growth, highlight regional growth and spatial heterogeneity.

The effect of notification window length on the epidemiological impact of COVID-19 contact tracing mobile applications (preprint)

The reduction in SARS-CoV-2 transmission from contact tracing applications (apps) depends both on the number of contacts notified and on the probability that those contacts quarantine after notification.Referring to the number of days preceding a positive test that contacts are notified as an app's notification window, we use an epidemiological model of SARS-CoV-2 transmission that captures the profile of infection to consider the trade-off between notification window length and active app-usage. We focus on 5-day and 2-day windows, the lengths used by the NHS COVID-19 app in England and Wales before and after 2nd August 2021, respectively. Short windows can be more effective at reducing transmission if they are associated with higher levels of active app usage and adherence to isolation upon notification, demonstrating the importance of understanding adherence to control measures when setting notification windows for COVID-19 apps.

Possible future waves of SARS-CoV-2 infection generated by variants of concern with a range of characteristics (preprint)

Ongoing infection with, and associated viral reproduction of, SARS-CoV-2 provides opportunities for the virus to acquire advantageous mutations, which may alter viral transmissibility and disease severity, and allow escape from natural or vaccine-derived immunity. The number of countries reporting Variants of Concern (VOCs) with such mutations continues to rise. Here, we investigate two scenarios for third waves of the COVID pandemic: one driven by increased transmissibility, and another driven by immune escape. We do this using three mathematical models: a parsimonious susceptible-latent-infectious-recovered (SEIR) deterministic model with homogeneous mixing, an age-structured SARS-CoV-2 transmission model and a stochastic importation model. We calibrated our models to the situation in England in May 2021, although the insights will generalise to other contexts. We therefore accurately captured infection dynamics and vaccination rates, and also used these to explore the potential impact of a putative new VOC-targeted vaccine. Epidemiological trajectories for putative VOCs are wide-ranging and heavily dependent on their transmissibility, immune escape capability, and the time at which a postulated VOC-targeted vaccine may be introduced. We demonstrate that a VOC with either a substantial transmission advantage over resident variants, or the ability to evade vaccine-derived and prior immunity, is expected to generate a wave of infections and hospitalisations comparable to those seen in the winter 2020-21 wave. Moreover, a variant that is less transmissible, but shows partial immune-escape could provoke a wave of infection that would not be revealed until control measures are further relaxed.

Optimal health and economic impact of non-pharmaceutical intervention measures prior and post vaccination in England: a mathematical modelling study (preprint)

Background Even with good progress on vaccination, SARS-CoV-2 infections in the UK may continue to impose a high burden of disease and therefore pose substantial challenges for health policy decision makers. Stringent government-mandated physical distancing measures (lockdown) have been demonstrated to be epidemiologically effective, but can have both positive and negative economic consequences. The duration and frequency of any intervention policy could, in theory, could be optimised to maximise economic benefits while achieving substantial reductions in disease. Methods Here we use a pre-existing SARS-CoV-2 transmission model to assess the health and economic implications of different strengths of control through time in order to identify optimal approaches to non-pharmaceutical intervention stringency in the UK, considering the role of vaccination in reducing the need for future physical distancing measures. The model is calibrated to the COVID-19 epidemic in England and we carry out retrospective analysis of the optimal timing of precautionary breaks in 2020 and the optimal relaxation policy from the January 2021 lockdown, considering the willingness to pay for health improvement. Results We find that the precise timing and intensity of interventions is highly dependent upon the objective of control. As intervention measures are relaxed, we predict a resurgence in cases, but the optimal intervention policy can be established dependent upon the willingness to pay (WTP) per QALY loss avoided. Our results show that establishing an optimal level of control can result in a reduction in net monetary loss of billions of pounds, dependent upon the precise WTP value. Conclusions It is vital, as the UK emerges from lockdown, but continues to face an on-going pandemic, to accurately establish the overall health and economic costs when making policy decisions. We demonstrate how some of these can be quantified, employing mechanistic infectious disease transmission models to establish optimal levels of control for the ongoing COVID-19 pandemic.

An analysis of school absences in England during the Covid-19 pandemic (preprint)

The introduction of SARS-CoV-2, the virus that causes COVID-19 infection, in the UK in early 2020, resulted in the UK government introducing several control policies in order to reduce the spread of disease. As part of these restrictions, schools were closed to all pupils in March (except for vulnerable and key worker children), before re-opening to certain year groups in June. Finally all school children returned to the classroom in September. In this paper, we analyse the data on school absences from September 2020 to December 2020 as a result of COVID-19 infection and how that varied through time as other measures in the community were introduced. We utilise data from the Educational Settings database compiled by the Department for Education and examine how pupil and teacher absences change in both primary and secondary schools. Our results show that absences as a result of COVID-19 infection rose steadily following the re-opening of schools in September. Cases in teachers were seen to decline during the November lockdown, particularly in those regions that had previously been in tier 3, the highest level of control at the time. Cases in secondary school pupils increased for the first two weeks of the November lockdown, before decreasing. Since the introduction of the tier system, the number of absences owing to confirmed infection in primary schools was observed to be significantly lower than in secondary schools across all regions and tiers. In December, we observed a large rise in the number of absences per school in secondary school settings in the South East and Greater London, but such rises were not observed in other regions or in primary school settings. We conjecture that the increased transmissibility of the new variant in these regions may have contributed to this rise in cases in secondary schools. Finally, we observe a positive correlation between cases in the community and cases in schools in most regions, with weak evidence suggesting that cases in schools lag behind cases in the surrounding community. We conclude that there is not significant evidence to suggest that schools are playing a significant role in driving spread in the community and that careful monitoring may be required as schools re-open to determine the effect associated with open schools upon community incidence.

Contrasting factors associated with COVID-19-related ICU and death outcomes: interpretable multivariable analyses of the UK CHESS dataset. (preprint)

BackgroundIdentifying factors associated with severe COVID-19 is a priority to guide clinical care and resource use in this pandemic. MethodsThis cohort comprised 13954 in-patients with confirmed COVID-19. Study outcomes were death and intensive care unit admission (ICUA). Multivariable logistic regression estimated odd ratios adjusted for 37 covariates (comorbidities, demographic, and others). Gradient boosted decision tree (GBDT) classification generated Shapley values evaluating the impact of covariates for each patient. FindingsDeaths due to COVID-19 were associated with immunosuppression due to disease (Odds Ratio 1.39, 95%CI [1.10-1.76]), type-2 diabetes (1.31, [1.17-1.46]), chronic respiratory disease (1.19, [1.05-1.35]), obesity (1.16, [1.01-1.33], age (1.56/10-year increment, [1.52-1.61]), and male sex (1.54, [1.42-1.68]). Associations with ICUA differed in direction (e.g., age, chronic respiratory disease) and in scale, e.g., obesity (3.37, [2.90-3.92]) for some factors. Ethnicity was strongly but variably associated with both outcomes, for example Irish ethnicity is negatively with death but not ICUA. GBDTs had similar performance (ROC-AUC, ICUA 0.83, death 0.68 for GBDT; 0.80 and 0.68 for logistic regression). Shapley explanations overall were consistent with odds ratios. Chronic heart disease, hypertension, other comorbidities, and some ethnicities had Shapley impacts on death ranging from positive to negative among different patients, although consistently associated with ICUA for all. Immunosuppressive disease, type-2 diabetes, and chronic liver and respiratory diseases had positive impacts on death with either positive or negative on ICUA. InterpretationVery different association of some factors, e.g., obesity, with death and ICUA may guide review of practice. Shapley explanation identified varying effects among patients emphasising the importance of individual patient assessment.

Spatially resolved simulations of the spread of COVID-19 in European countries (preprint)

We explore the spatial and temporal spread of the novel SARS-CoV-2 virus under containment measures in three European countries based on fits to data of the early outbreak. Using data from Spain and Italy, we estimate an age dependent infection fatality ratio for SARS-CoV-2, as well as risks of hospitalization and intensive care admission. We use them in a model that simulates the dynamics of the virus using an age structured, spatially detailed agent based approach, that explicitly incorporates governamental interventions, changes in mobility and contact patterns occurred during the COVID-19 outbreak in each country.Our simulations reproduce several of the features of its spatio-temporal spread in the three countries studied. They show that containment measures combined with high density are responsible for the containment of cases within densely populated areas, and that spread to less densely populated areas occurred during the late stages of the first wave. The capability to reproduce observed features of the spatio-temporal dynamics of SARS-CoV-2 makes this model a potential candidate for forecasting the dynamics of SARS-CoV-2 in other settings, and we recommend its application in low and lower-middle countries which remain understudied.

A network modelling approach to assess non-pharmaceutical disease controls in a worker population: An application to SARS-CoV-2 (preprint)

BackgroundAs part of a concerted pandemic response to protect public health, businesses can enact non-pharmaceutical controls to minimise exposure to pathogens in workplaces and premises open to the public. Amendments to working practices can lead to the amount, duration and/or proximity of interactions being changed, ultimately altering the dynamics of disease spread. These modifications could be specific to the type of business being operated. MethodsWe use a data-driven approach to parameterise an individual-based network model for transmission of SARS-CoV-2 amongst the working population, stratified into work sectors. The network is comprised of layered contacts to consider the risk of spread in multiple encounter settings (workplaces, households, social and other). We analyse several interventions targeted towards working practices: mandating a fraction of the population to work from home; using temporally asynchronous work patterns; and introducing measures to create COVID-secure workplaces. We also assess the general role of adherence to (or effectiveness of) isolation and test and trace measures and demonstrate the impact of all these interventions across a variety of relevant metrics. ResultsThe progress of the epidemic can be significantly hindered by instructing a significant proportion of the workforce to work from home. Furthermore, if required to be present at the workplace, asynchronous work patterns can help to reduce infections when compared with scenarios where all workers work on the same days, particularly for longer working weeks. When assessing COVID-secure workplace measures, we found that smaller work teams and a greater reduction in transmission risk reduced the probability of large, prolonged outbreaks. Finally, following isolation guidance and engaging with contact tracing without other measures is an effective tool to curb transmission, but is highly sensitive to adherence levels. ConclusionsIn the absence of sufficient adherence to non-pharmaceutical interventions, our results indicate a high likelihood of SARS-CoV-2 spreading widely throughout a worker population. Given the heterogeneity of demographic attributes across worker roles, in addition to the individual nature of controls such as contact tracing, we demonstrate the utility of a network model approach to investigate workplace-targeted intervention strategies and the role of test, trace and isolation in tackling disease spread.

Short-term forecasts to inform the response to the COVID-19 epidemic in the UK (preprint)

Background: Short-term forecasts of infectious disease can create situational awareness and inform 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. Methods: We 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 to ensemble forecasts using a simple mean as well as a quantile regression average that aimed to maximise performance. We further compared model performance to a null model of no change. Results: In 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. Conclusions: Ensembles 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.

Predicting the impact of COVID-19 interruptions on transmission of gambiense human African trypanosomiasis in two health zones of the Democratic Republic of Congo (preprint)

Many control programmes against neglected tropical diseases have been interrupted due to COVID-19 pandemic, including those that rely on active case finding. In this study we focus on gambiense human African trypanosomiasis (gHAT), where active screening was suspended in the Democratic Republic of Congo (DRC) due to the pandemic. We use two independent mathematical models to predict the impact of COVID-19 interruptions on transmission and reporting, and the achievement of 2030 elimination of transmission (EOT) goal for gHAT in two moderate-risk regions of DRC. We consider different interruption scenarios, including reduced passive surveillance in fixed health facilities, and whether this suspension lasts until the end of 2020 or 2021. Our models predict an increase in the number of new infections in the interruption period only if both active screening and passive surveillance were suspended, and with slowed reduction - but no increase - if passive surveillance remains fully functional. In all scenarios, the EOT may be slightly pushed back if no mitigation such as increased screening coverage is put in place. However, we emphasise that the biggest challenge will remain in the higher prevalence regions where EOT is already predicted to be behind schedule without interruptions unless interventions are bolstered.

Modelling SARS-CoV-2 transmission in a UK university setting (preprint)

Background: The higher education system in the United Kingdom comprises a large student population. Around 40% of school leavers attend university and individual universities generally host thousands of students each academic year. In the setting of the COVID-19 pandemic, bringing together these student communities presents questions regarding the strength of interventions required to control transmission. Prior modelling analysis of SARS-CoV-2 transmission within universities has usually adopted a compartmental modelling approach, whose projections suggest an almost inevitable likelihood of outbreaks. Methods: We constructed a network-based model to capture the interactions of a student population in different settings (housing, social and study). For a representative campus-based university, we ran a susceptible-latent-infectious-recovered type epidemic process, parameterised according to available estimates for SARS-CoV-2. Over the course of a single academic term, we investigated the impact on infection control of adherence to (or effectiveness of) isolation, test and trace measures, the additional use of room isolation as an intervention and supplementary mass testing. Results: Incorporating uncertainty in the fraction of cases that are asymptomatic and their associated infectivity, in the absence of interventions our model estimated that 16% (2% - 38%) of the student population could be infected during the autumn term. In contrast, with full adherence to isolation measures and engagement with test-and-trace, predictions of the cumulative infection count were lower, 1.4% (0.4% - 5%). Irrespective of the adherence to isolation measures, on average a higher proportion of students resident on-campus became infected compared with students resident off-campus. Widespread adherence of interventions led to reductions in the average fraction of time those individuals adhering to measures were expected to be isolated, with room isolation as an additional intervention generating minimal benefits. The model found that a one-off instance of mass testing would not drastically reduce the term-long case load or end-of-term prevalence, but regular weekly or fortnightly testing could reduce both measures by more than 50% (compared to having no mass testing). Conclusions: Our findings suggest SARS-CoV-2 may readily transmit amongst a student population within a university setting if there is limited adherence to nonpharmaceutical interventions and there are delays present in receiving test results. Following isolation guidance and effective contact tracing both curbed transmission and reduced the expected time an adhering student would spend in isolation. Additionally, widespread adherence throughout the term suppresses the amount of unwitting asymptomatic transmission to family and community members in the students' domicile regions at the end of term.

Precautionary breaks: planned, limited duration circuit breaks to control the prevalence of COVID-19 (preprint)

The COVID-19 pandemic in the UK has been characterised by periods of exponential growth and decline, as different non-pharmaceutical interventions (NPIs) are brought into play. During the early uncontrolled phase of the outbreak (early March 2020) there was a period of prolonged exponential growth with epidemiological observations such as hospitalisation doubling every 3-4 days (growth rate r{approx}0.2). The enforcement of strict lockdown measures led to a noticeable decline in all epidemic quantities (r{approx}-0.06) that slowed during the summer as control measures were relaxed (r{approx}-0.02). Since August, infections, hospitalisations and deaths have been rising (precise estimation of the cur-rent growth rate is difficult due to extreme regional heterogeneity and temporal lags between the different epidemiological observations) and various NPIs have been applied locally throughout the UK in response. Controlling any rise in infection is a compromise between public health and societal costs, with more stringent NPIs reducing cases but damaging the economy and restricting freedoms. Currently, NPI imposition is made in response to the epidemiological state, are of indefinite length and are often imposed at short notice, greatly increasing the negative impact. An alternative approach is to consider planned, limited duration periods of strict NPIs aiming to purposefully reduce prevalence before such emergency NPIs are required. These 'precautionary breaks' may offer a means of keeping control of the epidemic, while their fixed duration and the forewarning may limit their society impact. Here, using simple analysis and age-structured models matched to the unfolding UK epidemic, we investigate the action of precautionary breaks. In particular we consider their impact on the prevalence of infection, as well as the total number of predicted hospitalisations and deaths. We find that precautionary breaks provide the biggest gains when the growth rate is low, but offer a much needed brake on increasing infection when the growth rate is higher, potentially allowing other measures (such as contact tracing)to regain control.

Modelling optimal vaccination strategy for SARS-CoV-2. (preprint)

The COVID-19 outbreak has highlighted our vulnerability to novel infections. Faced with this threat and no effective treatment, most countries adopted some form of enforced social distancing (lockdown) to reduce transmission - in most cases successfully reducing the reproductive number,R, below one. However, given the large pool of susceptible individuals that remain, complete relaxation of controls is likely to generate a substantial second wave. Vaccination remains the only foreseeable means of both containing the infection and returning to normal interactions and behaviour. Here, we consider the optimal targeting of vaccination with the aim of minimising future deaths or quality adjusted life year (QALY) losses. We show that, for a range of assumptions on the action and efficacy of the vaccine, targeting older age groups first is optimal and can avoid a second wave if the vaccine prevents transmission as well as disease.