Getting the most out of maths: How to coordinate mathematical modelling research to support a pandemic, lessons learnt from three initiatives that were part of the COVID-19 response in the UK.

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

The importance of vaccinated individuals to population-level evolution of pathogens.

Virus evolution shapes the epidemiological patterns of infectious disease, particularly via evasion of population immunity. At the individual level, host immunity itself may drive viral evolution towards antigenic escape. Using compartmental SIR-style models with imperfect vaccination, we allow the probability of immune escape to differ in vaccinated and unvaccinated hosts. As the relative contribution to selection in these different hosts varies, the overall effect of vaccination on the antigenic escape pressure at the population level changes. We find that this relative contribution to escape is important for understanding the effects of vaccination on the escape pressure and we draw out some fairly general patterns. If vaccinated hosts do not contribute much more than unvaccinated hosts to the escape pressure, then increasing vaccination always reduces the overall escape pressure. In contrast, if vaccinated hosts contribute significantly more than unvaccinated hosts to the population level escape pressure, then the escape pressure is maximised for intermediate vaccination levels. Past studies find only that the escape pressure is maximal for intermediate levels with fixed extreme assumptions about this relative contribution. Here we show that this result does not hold across the range of plausible assumptions for the relative contribution to escape from vaccinated and unvaccinated hosts. We also find that these results depend on the vaccine efficacy against transmission, particularly through the partial protection against infection. This work highlights the potential value of understanding better how the contribution to antigenic escape pressure depends on individual host immunity.

Coughs, colds and "freshers' flu" survey in the University of Cambridge, 2007-2008.

Universities provide many opportunities for the spread of infectious respiratory illnesses. Students are brought together into close proximity from all across the world and interact with one another in their accommodation, through lectures and small group teaching and in social settings. The COVID-19 global pandemic has highlighted the need for sufficient data to help determine which of these factors are important for infectious disease transmission in universities and hence control university morbidity as well as community spillover. We describe the data from a previously unpublished self-reported university survey of coughs, colds and influenza-like symptoms collected in Cambridge, UK, during winter 2007-2008. The online survey collected information on symptoms and socio-demographic, academic and lifestyle factors. There were 1076 responses, 97% from University of Cambridge students (5.7% of the total university student population), 3% from staff and <1% from other participants, reporting onset of symptoms between September 2007 and March 2008. Undergraduates are seen to report symptoms earlier in the term than postgraduates; differences in reported date of symptoms are also seen between subjects and accommodation types, although these descriptive results could be confounded by survey biases. Despite the historical and exploratory nature of the study, this is one of few recent detailed datasets of influenza-like infection in a university context and is especially valuable to share now to improve understanding of potential transmission dynamics in universities during the current COVID-19 pandemic.

Commentary on the use of the reproduction number R during the COVID-19 pandemic.

Since the beginning of the COVID-19 pandemic, the reproduction number [Formula: see text] has become a popular epidemiological metric used to communicate the state of the epidemic. At its most basic, [Formula: see text] is defined as the average number of secondary infections caused by one primary infected individual. [Formula: see text] seems convenient, because the epidemic is expanding if [Formula: see text] and contracting if [Formula: see text]. The magnitude of [Formula: see text] indicates by how much transmission needs to be reduced to control the epidemic. Using [Formula: see text] in a naïve way can cause new problems. The reasons for this are threefold: (1) There is not just one definition of [Formula: see text] but many, and the precise definition of [Formula: see text] affects both its estimated value and how it should be interpreted. (2) Even with a particular clearly defined [Formula: see text], there may be different statistical methods used to estimate its value, and the choice of method will affect the estimate. (3) The availability and type of data used to estimate [Formula: see text] vary, and it is not always clear what data should be included in the estimation. In this review, we discuss when [Formula: see text] is useful, when it may be of use but needs to be interpreted with care, and when it may be an inappropriate indicator of the progress of the epidemic. We also argue that careful definition of [Formula: see text], and the data and methods used to estimate it, can make [Formula: see text] a more useful metric for future management of the epidemic.

SARS-CoV-2 infection in UK university students: lessons from September-December 2020 and modelling insights for future student return.

In this paper, we present work on SARS-CoV-2 transmission in UK higher education settings using multiple approaches to assess the extent of university outbreaks, how much those outbreaks may have led to spillover in the community, and the expected effects of control measures. Firstly, we found that the distribution of outbreaks in universities in late 2020 was consistent with the expected importation of infection from arriving students. Considering outbreaks at one university, larger halls of residence posed higher risks for transmission. The dynamics of transmission from university outbreaks to wider communities is complex, and while sometimes spillover does occur, occasionally even large outbreaks do not give any detectable signal of spillover to the local population. Secondly, we explored proposed control measures for reopening and keeping open universities. We found the proposal of staggering the return of students to university residence is of limited value in terms of reducing transmission. We show that student adherence to testing and self-isolation is likely to be much more important for reducing transmission during term time. Finally, we explored strategies for testing students in the context of a more transmissible variant and found that frequent testing would be necessary to prevent a major outbreak.

Vaccine escape in a heterogeneous population: insights for SARS-CoV-2 from a simple model.

As a countermeasure to the SARS-CoV-2 pandemic, there has been swift development and clinical trial assessment of candidate vaccines, with subsequent deployment as part of mass vaccination campaigns. However, the SARS-CoV-2 virus has demonstrated the ability to mutate and develop variants, which can modify epidemiological properties and potentially also the effectiveness of vaccines. The widespread deployment of highly effective vaccines may rapidly exert selection pressure on the SARS-CoV-2 virus directed towards mutations that escape the vaccine-induced immune response. This is particularly concerning while infection is widespread. By developing and analysing a mathematical model of two population groupings with differing vulnerability and contact rates, we explore the impact of the deployment of vaccines among the population on the reproduction ratio, cases, disease abundance and vaccine escape pressure. The results from this model illustrate two insights: (i) vaccination aimed at reducing prevalence could be more effective at reducing disease than directly vaccinating the vulnerable; (ii) the highest risk for vaccine escape can occur at intermediate levels of vaccination. This work demonstrates a key principle: the careful targeting of vaccines towards particular population groups could reduce disease as much as possible while limiting the risk of vaccine escape.

Epidemic interventions: insights from classic results.

Analytical expressions and approximations from simple models have performed a pivotal role in our understanding of infectious disease epidemiology. During the current COVID-19 pandemic, while there has been proliferation of increasingly complex models, still the most basic models have provided the core framework for our thinking and interpreting policy decisions. Here, classic results are presented that give insights into both the role of transmission-reducing interventions (such as social distancing) in controlling an emerging epidemic, and also what would happen if insufficient control is applied. Though these are simple results from the most basic of epidemic models, they give valuable benchmarks for comparison with the outputs of more complex modelling approaches. This article is part of the theme issue 'Modelling that shaped the early COVID-19 pandemic response in the UK'.

Key questions for modelling COVID-19 exit strategies.

Combinations of intense non-pharmaceutical interventions (lockdowns) were introduced worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement exit strategies that relax restrictions while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute 'Models for an exit strategy' workshop (11-15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, would allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. This roadmap requires a global collaborative effort from the scientific community and policymakers, and has three parts: (i) improve estimation of key epidemiological parameters; (ii) understand sources of heterogeneity in populations; and (iii) focus on requirements for data collection, particularly in low-to-middle-income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.

How you can help with COVID-19 modelling.

Many physicists want to use their mathematical modelling skills to study the COVID-19 pandemic. Julia Gog, a mathematical epidemiologist, explains some ways to contribute.

Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: a mathematical modelling study.

BACKGROUND: The isolation of symptomatic cases and tracing of contacts has been used as an early COVID-19 containment measure in many countries, with additional physical distancing measures also introduced as outbreaks have grown. To maintain control of infection while also reducing disruption to populations, there is a need to understand what combination of measures-including novel digital tracing approaches and less intensive physical distancing-might be required to reduce transmission. We aimed to estimate the reduction in transmission under different control measures across settings and how many contacts would be quarantined per day in different strategies for a given level of symptomatic case incidence. METHODS: For this mathematical modelling study, we used a model of individual-level transmission stratified by setting (household, work, school, or other) based on BBC Pandemic data from 40 162 UK participants. We simulated the effect of a range of different testing, isolation, tracing, and physical distancing scenarios. Under optimistic but plausible assumptions, we estimated reduction in the effective reproduction number and the number of contacts that would be newly quarantined each day under different strategies. RESULTS: We estimated that combined isolation and tracing strategies would reduce transmission more than mass testing or self-isolation alone: mean transmission reduction of 2% for mass random testing of 5% of the population each week, 29% for self-isolation alone of symptomatic cases within the household, 35% for self-isolation alone outside the household, 37% for self-isolation plus household quarantine, 64% for self-isolation and household quarantine with the addition of manual contact tracing of all contacts, 57% with the addition of manual tracing of acquaintances only, and 47% with the addition of app-based tracing only. If limits were placed on gatherings outside of home, school, or work, then manual contact tracing of acquaintances alone could have an effect on transmission reduction similar to that of detailed contact tracing. In a scenario where 1000 new symptomatic cases that met the definition to trigger contact tracing occurred per day, we estimated that, in most contact tracing strategies, 15 000-41 000 contacts would be newly quarantined each day. INTERPRETATION: Consistent with previous modelling studies and country-specific COVID-19 responses to date, our analysis estimated that a high proportion of cases would need to self-isolate and a high proportion of their contacts to be successfully traced to ensure an effective reproduction number lower than 1 in the absence of other measures. If combined with moderate physical distancing measures, self-isolation and contact tracing would be more likely to achieve control of severe acute respiratory syndrome coronavirus 2 transmission. FUNDING: Wellcome Trust, UK Engineering and Physical Sciences Research Council, European Commission, Royal Society, Medical Research Council.