Contact-number-driven virus evolution: A multi-level modeling framework for the evolution of acute or persistent RNA virus infection.

Viruses evolve in infected host populations, and host population dynamics affect viral evolution. RNA viruses with a short duration of infection and a high peak viral load, such as SARS-CoV-2, are maintained in human populations. By contrast, RNA viruses characterized by a long infection duration and a low peak viral load (e.g., borna disease virus) can be maintained in nonhuman populations, and the process of the evolution of persistent viruses has rarely been explored. Here, using a multi-level modeling approach including both individual-level virus infection dynamics and population-scale transmission, we consider virus evolution based on the host environment, specifically, the effect of the contact history of infected hosts. We found that, with a highly dense contact history, viruses with a high virus production rate but low accuracy are likely to be optimal, resulting in a short infectious period with a high peak viral load. In contrast, with a low-density contact history, viral evolution is toward low virus production but high accuracy, resulting in long infection durations with low peak viral load. Our study sheds light on the origin of persistent viruses and why acute viral infections but not persistent virus infection tends to prevail in human society.

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".

Non-pharmaceutical interventions and the emergence of pathogen variants.

Non-pharmaceutical interventions (NPIs), such as social distancing and contact tracing, are important public health measures that can reduce pathogen transmission. In addition to playing a crucial role in suppressing transmission, NPIs influence pathogen evolution by mediating mutation supply, restricting the availability of susceptible hosts, and altering the strength of selection for novel variants. Yet it is unclear how NPIs might affect the emergence of novel variants that are able to escape pre-existing immunity (partially or fully), are more transmissible or cause greater mortality. We analyse a stochastic two-strain epidemiological model to determine how the strength and timing of NPIs affect the emergence of variants with similar or contrasting life-history characteristics to the wild type. We show that, while stronger and timelier NPIs generally reduce the likelihood of variant emergence, it is possible for more transmissible variants with high cross-immunity to have a greater probability of emerging at intermediate levels of NPIs. This is because intermediate levels of NPIs allow an epidemic of the wild type that is neither too small (facilitating high mutation supply), nor too large (leaving a large pool of susceptible hosts), to prevent a novel variant from becoming established in the host population. However, since one cannot predict the characteristics of a variant, the best strategy to prevent emergence is likely to be an implementation of strong, timely NPIs.

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.

The impact of cross-reactive immunity on the emergence of SARS-CoV-2 variants.

Introduction: A key feature of the COVID-19 pandemic has been the emergence of SARS-CoV-2 variants with different transmission characteristics. However, when a novel variant arrives in a host population, it will not necessarily lead to many cases. Instead, it may fade out, due to stochastic effects and the level of immunity in the population. Immunity against novel SARS-CoV-2 variants may be influenced by prior exposures to related viruses, such as other SARS-CoV-2 variants and seasonal coronaviruses, and the level of cross-reactive immunity conferred by those exposures. Methods: Here, we investigate the impact of cross-reactive immunity on the emergence of SARS-CoV-2 variants in a simplified scenario in which a novel SARS-CoV-2 variant is introduced after an antigenically related virus has spread in the population. We use mathematical modelling to explore the risk that the novel variant invades the population and causes a large number of cases, as opposed to fading out with few cases. Results: We find that, if cross-reactive immunity is complete (i.e. someone infected by the previously circulating virus is not susceptible to the novel variant), the novel variant must be more transmissible than the previous virus to invade the population. However, in a more realistic scenario in which cross-reactive immunity is partial, we show that it is possible for novel variants to invade, even if they are less transmissible than previously circulating viruses. This is because partial cross-reactive immunity effectively increases the pool of susceptible hosts that are available to the novel variant compared to complete cross-reactive immunity. Furthermore, if previous infection with the antigenically related virus assists the establishment of infection with the novel variant, as has been proposed following some experimental studies, then even variants with very limited transmissibility are able to invade the host population. Discussion: Our results highlight that fast assessment of the level of cross-reactive immunity conferred by related viruses against novel SARS-CoV-2 variants is an essential component of novel variant risk assessments.

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

Background: The reduction in SARS-CoV-2 transmission facilitated by mobile contact tracing applications (apps) depends both on the proportion of relevant contacts notified and on the probability that those contacts quarantine after notification. The proportion of relevant contacts notified depends upon the number of days preceding an infector's positive test that their contacts are notified, which we refer to as an app's notification window. Methods: 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 use. We focus on 5-day and 2-day windows, the notification windows of the NHS COVID-19 app in England and Wales before and after 2nd August 2021, respectively. Results: Our analyses show that at the same level of active app use, 5-day windows result in larger reductions in transmission than 2-day windows. However, short notification windows can be more effective at reducing transmission if they are associated with higher levels of active app use and adherence to isolation upon notification. Conclusions: Our results demonstrate the importance of understanding adherence to interventions when setting notification windows for COVID-19 contact tracing apps.

After submitting a positive SARS-CoV-2 test result, mobile contact-tracing apps identify 'recent' high-risk encounters with other app users, who are then notified of potential exposure. An app's success at limiting further transmission depends on the proportion of infected contacts notified. This depends on what counts as 'recent', e.g. notifying contacts from 5 days prior to the positive test can capture more infections than notifying contacts from 2 days prior. We call this number of days an app's notification window. However, an app's effectiveness also depends on whether or not exposed contacts use the app and adhere to isolation if notified. If shorter windows are associated with higher levels of active app use, they can be more effective at reducing transmission than longer windows, demonstrating the importance of considering the potential impact on active app use when setting an app's notification window length.

Estimation of heterogeneous instantaneous reproduction numbers with application to characterize SARS-CoV-2 transmission in Massachusetts counties.

The reproductive number is an important metric that has been widely used to quantify the infectiousness of communicable diseases. The time-varying instantaneous reproductive number is useful for monitoring the real-time dynamics of a disease to inform policy making for disease control. Local estimation of this metric, for instance at a county or city level, allows for more targeted interventions to curb transmission. However, simultaneous estimation of local reproductive numbers must account for potential sources of heterogeneity in these time-varying quantities-a key element of which is human mobility. We develop a statistical method that incorporates human mobility between multiple regions for estimating region-specific instantaneous reproductive numbers. The model also can account for exogenous cases imported from outside of the regions of interest. We propose two approaches to estimate the reproductive numbers, with mobility data used to adjust incidence in the first approach and to inform a formal priori distribution in the second (Bayesian) approach. Through a simulation study, we show that region-specific reproductive numbers can be well estimated if human mobility is reasonably well approximated by available data. We use this approach to estimate the instantaneous reproductive numbers of COVID-19 for 14 counties in Massachusetts using CDC case report data and the human mobility data collected by SafeGraph. We found that, accounting for mobility, our method produces estimates of reproductive numbers that are distinct across counties. In contrast, independent estimation of county-level reproductive numbers tends to produce similar values, as trends in county case-counts for the state are fairly concordant. These approaches can also be used to estimate any heterogeneity in transmission, for instance, age-dependent instantaneous reproductive number estimates. As people are more mobile and interact frequently in ways that permit transmission, it is important to account for this in the estimation of the reproductive number.

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

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, be optimized to maximize 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 (WTP) 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 WTP per quality adjusted life year 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. Conclusion. 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.

Emergence potential of monkeypox in the Western Pacific Region, July 2022.

Although new cases of monkeypox have been expected in the Western Pacific Region (WPR) since the virus emerged in Europe earlier this year, there have been only a few reported cases across the WPR (New Zealand 2, Singapore 6, South Korea 1, Taiwan 2), other than a limited number of cases (compared to numbers of cases seen elsewhere in the world) in Australia (33), as of July 15, 2022. In our short communication, we highlight two key reasons for this: i) international travel has still not fully resumed in the WPR following the COVID-19 pandemic, and ii) local public health measures to counter the spread of COVID-19 have not been completely relaxed. We provide supporting evidence for both of these reasons.

Are epidemic growth rates more informative than reproduction numbers?

statistics, often derived from simplified models of epidemic spread, inform public health policy in real time. The instantaneous reproduction number, R t , is predominant among these statistics, measuring the average ability of an infection to multiply. However, R t encodes no temporal information and is sensitive to modelling assumptions. Consequently, some have proposed the epidemic growth rate, r t , that is, the rate of change of the log-transformed case incidence, as a more temporally meaningful and model-agnostic policy guide. We examine this assertion, identifying if and when estimates of r t are more informative than those of R t . We assess their relative strengths both for learning about pathogen transmission mechanisms and for guiding public health interventions in real time.

Assessing the impact of lateral flow testing strategies on within-school SARS-CoV-2 transmission and absences: A modelling study.

Rapid testing strategies that replace the isolation of close contacts through the use of lateral flow device tests (LFTs) have been suggested as a way of controlling SARS-CoV-2 transmission within schools that maintain low levels of pupil absences. We developed an individual-based model of a secondary school formed of exclusive year group bubbles (five year groups, with 200 pupils per year) to assess the likely impact of strategies using LFTs in secondary schools over the course of a seven-week half-term on transmission, absences, and testing volume, compared to a policy of isolating year group bubbles upon a pupil returning a positive polymerase chain reaction (PCR) test. We also considered the sensitivity of results to levels of participation in rapid testing and underlying model assumptions. While repeated testing of year group bubbles following case detection is less effective at reducing infections than a policy of isolating year group bubbles, strategies involving twice weekly mass testing can reduce infections to lower levels than would occur under year group isolation. By combining regular testing with serial contact testing or isolation, infection levels can be reduced further still. At high levels of pupil participation in lateral flow testing, strategies replacing the isolation of year group bubbles with testing substantially reduce absences, but require a high volume of testing. Our results highlight the conflict between the goals of minimising within-school transmission, minimising absences and minimising testing burden. While rapid testing strategies can reduce school transmission and absences, they may lead to a large number of daily tests.

The risk of SARS-CoV-2 outbreaks in low prevalence settings following the removal of travel restrictions.

Background: Countries around the world have introduced travel restrictions to reduce SARS-CoV-2 transmission. As vaccines are gradually rolled out, attention has turned to when travel restrictions and other non-pharmaceutical interventions (NPIs) can be relaxed. Methods: Using SARS-CoV-2 as a case study, we develop a mathematical branching process model to assess the risk that, following the removal of NPIs, cases arriving in low prevalence settings initiate a local outbreak. Our model accounts for changes in background population immunity due to vaccination. We consider two locations with low prevalence in which the vaccine rollout has progressed quickly - specifically, the Isle of Man (a British crown dependency in the Irish Sea) and the country of Israel. Results: We show that the outbreak risk is unlikely to be eliminated completely when travel restrictions and other NPIs are removed. This general result is the most important finding of this study, rather than exact quantitative outbreak risk estimates in different locations. It holds even once vaccine programmes are completed. Key factors underlying this result are the potential for transmission even following vaccination, incomplete vaccine uptake, and the recent emergence of SARS-CoV-2 variants with increased transmissibility. Conclusions: Combined, the factors described above suggest that, when travel restrictions are relaxed, it may still be necessary to implement surveillance of incoming passengers to identify infected individuals quickly. This measure, as well as tracing and testing (and/or isolating) contacts of detected infected passengers, remains useful to suppress potential outbreaks while global case numbers are high.

Inference of the SARS-CoV-2 generation time using UK household data.

The distribution of the generation time (the interval between individuals becoming infected and transmitting the virus) characterises changes in the transmission risk during SARS-CoV-2 infections. Inferring the generation time distribution is essential to plan and assess public health measures. We previously developed a mechanistic approach for estimating the generation time, which provided an improved fit to data from the early months of the COVID-19 pandemic (December 2019-March 2020) compared to existing models (Hart et al., 2021). However, few estimates of the generation time exist based on data from later in the pandemic. Here, using data from a household study conducted from March to November 2020 in the UK, we provide updated estimates of the generation time. We considered both a commonly used approach in which the transmission risk is assumed to be independent of when symptoms develop, and our mechanistic model in which transmission and symptoms are linked explicitly. Assuming independent transmission and symptoms, we estimated a mean generation time (4.2 days, 95% credible interval 3.3-5.3 days) similar to previous estimates from other countries, but with a higher standard deviation (4.9 days, 3.0-8.3 days). Using our mechanistic approach, we estimated a longer mean generation time (5.9 days, 5.2-7.0 days) and a similar standard deviation (4.8 days, 4.0-6.3 days). As well as estimating the generation time using data from the entire study period, we also considered whether the generation time varied temporally. Both models suggest a shorter mean generation time in September-November 2020 compared to earlier months. Since the SARS-CoV-2 generation time appears to be changing, further data collection and analysis is necessary to continue to monitor ongoing transmission and inform future public health policy decisions.

Quantifying pupil-to-pupil SARS-CoV-2 transmission and the impact of lateral flow testing in English secondary schools.

A range of measures have been implemented to control within-school SARS-CoV-2 transmission in England, including the self-isolation of close contacts and twice weekly mass testing of secondary school pupils using lateral flow device tests (LFTs). Despite reducing transmission, isolating close contacts can lead to high levels of absences, negatively impacting pupils. To quantify pupil-to-pupil SARS-CoV-2 transmission and the impact of implemented control measures, we fit a stochastic individual-based model of secondary school infection to both swab testing data and secondary school absences data from England, and then simulate outbreaks from 31st August 2020 until 23rd May 2021. We find that the pupil-to-pupil reproduction number, Rschool, has remained below 1 on average across the study period, and that twice weekly mass testing using LFTs has helped to control pupil-to-pupil transmission. We also explore the potential benefits of alternative containment strategies, finding that a strategy of repeat testing of close contacts rather than isolation, alongside mass testing, substantially reduces absences with only a marginal increase in pupil-to-pupil transmission.

Generation time of the alpha and delta SARS-CoV-2 variants: an epidemiological analysis.

BACKGROUND: In May, 2021, the delta (B.1.617.2) SARS-CoV-2 variant became dominant in the UK, superseded by the omicron (B.1.1.529) variant in December, 2021. The delta variant is associated with increased transmissibility compared with the alpha variant, which was the dominant variant in the UK between December, 2020, and May, 2021. To understand transmission and the effectiveness of interventions, we aimed to investigate whether the delta variant generation time (the interval between infections in infector-infectee pairs) is shorter-ie, transmissions are happening more quickly-than that of the alpha variant. METHODS: In this epidemiological analysis, we analysed transmission data from an ongoing UK Health Security Agency (UKHSA) prospective household study. Households were recruited to the study after an index case had a positive PCR test and genomic sequencing was used to determine the variant responsible. By fitting a mathematical transmission model to the data, we estimated the intrinsic generation time (which assumes a constant supply of susceptible individuals throughout infection) and the household generation time (which reflects realised transmission in the study households, accounting for susceptible depletion) for the alpha and delta variants. FINDINGS: Between February and August, 2021, 227 households consisting of 559 participants were recruited to the UKHSA study. The alpha variant was detected or assumed to be responsible for infections in 131 households (243 infections in 334 participants) recruited in February-May, and the delta variant in 96 households (174 infections in 225 participants) in May-August. The mean intrinsic generation time was shorter for the delta variant (4·7 days, 95% credible interval [CI] 4·1-5·6) than the alpha variant (5·5 days, 4·7-6·5), with 92% posterior probability. The mean household generation time was 28% (95% CI 0-48%) shorter for the delta variant (3·2 days, 95% CI 2·5-4·2) than the alpha variant (4·5 days, 3·7-5·4), with 97·5% posterior probability. INTERPRETATION: The delta variant transmits more quickly in households than the alpha variant, which can be attributed to faster depletion of susceptible individuals in households and a possible decrease in the intrinsic generation time. Interventions such as contact tracing, testing, and isolation might be less effective if transmission of the virus occurs quickly. FUNDING: National Institute for Health Research, UK Health Security Agency, Engineering and Physical Sciences Research Council, and UK Research and Innovation.

Challenges for modelling interventions for future pandemics.

Mathematical modelling and statistical inference provide a framework to evaluate different non-pharmaceutical and pharmaceutical interventions for the control of epidemics that has been widely used during the COVID-19 pandemic. In this paper, lessons learned from this and previous epidemics are used to highlight the challenges for future pandemic control. We consider the availability and use of data, as well as the need for correct parameterisation and calibration for different model frameworks. We discuss challenges that arise in describing and distinguishing between different interventions, within different modelling structures, and allowing both within and between host dynamics. We also highlight challenges in modelling the health economic and political aspects of interventions. Given the diversity of these challenges, a broad variety of interdisciplinary expertise is needed to address them, combining mathematical knowledge with biological and social insights, and including health economics and communication skills. Addressing these challenges for the future requires strong cross-disciplinary collaboration together with close communication between scientists and policy makers.

Estimating local outbreak risks and the effects of non-pharmaceutical interventions in age-structured populations: SARS-CoV-2 as a case study.

During the COVID-19 pandemic, non-pharmaceutical interventions (NPIs) including school closures, workplace closures and social distancing policies have been employed worldwide to reduce transmission and prevent local outbreaks. However, transmission and the effectiveness of NPIs depend strongly on age-related factors including heterogeneities in contact patterns and pathophysiology. Here, using SARS-CoV-2 as a case study, we develop a branching process model for assessing the risk that an infectious case arriving in a new location will initiate a local outbreak, accounting for the age distribution of the host population. We show that the risk of a local outbreak depends on the age of the index case, and we explore the effects of NPIs targeting individuals of different ages. Social distancing policies that reduce contacts outside of schools and workplaces and target individuals of all ages are predicted to reduce local outbreak risks substantially, whereas school closures have a more limited impact. In the scenarios considered here, when different NPIs are used in combination the risk of local outbreaks can be eliminated. We also show that heightened surveillance of infectious individuals reduces the level of NPIs required to prevent local outbreaks, particularly if enhanced surveillance of symptomatic cases is combined with efforts to find and isolate nonsymptomatic infected individuals. Our results reflect real-world experience of the COVID-19 pandemic, during which combinations of intense NPIs have reduced transmission and the risk of local outbreaks. The general modelling framework that we present can be used to estimate local outbreak risks during future epidemics of a range of pathogens, accounting fully for age-related factors.

Statistical Estimation of the Reproductive Number From Case Notification Data.

The reproductive number, or reproduction number, is a valuable metric in understanding infectious disease dynamics. There is a large body of literature related to its use and estimation. In the last 15 years, there has been tremendous progress in statistically estimating this number using case notification data. These approaches are appealing because they are relevant in an ongoing outbreak (e.g., for assessing the effectiveness of interventions) and do not require substantial modeling expertise to be implemented. In this article, we describe these methods and the extensions that have been developed. We provide insight into the distinct interpretations of the estimators proposed and provide real data examples to illustrate how they are implemented. Finally, we conclude with a discussion of available software and opportunities for future development.

A hospital-related outbreak of SARS-CoV-2 associated with variant Epsilon (B.1.429) in Taiwan: transmission potential and outbreak containment under intensified contact tracing, January-February 2021.

OBJECTIVES: A hospital-related cluster of 22 cases of coronavirus disease 2019 (COVID-19) occurred in Taiwan in January-February 2021. Rigorous control measures were introduced and could only be relaxed once the outbreak was declared over. Each day after the apparent outbreak end, we estimated the risk of future cases occurring in order to inform decision-making. METHODS: Probabilistic transmission networks were reconstructed, and transmission parameters (the reproduction number R and overdispersion parameter k) were estimated. The reporting delay during the outbreak was estimated (Scenario 1). In addition, a counterfactual scenario with less effective interventions characterized by a longer reporting delay was considered (Scenario 2). Each day, the risk of future cases was estimated under both scenarios. RESULTS: The values of R and k were estimated to be 1.30 ((95% credible interval (CI) 0.57-3.80) and 0.38 (95% CI 0.12-1.20), respectively. The mean reporting delays considered were 2.5 days (Scenario 1) and 7.8 days (Scenario 2). Following the final case, ttthe inferred probability of future cases occurring declined more quickly in Scenario 1 than Scenario 2. CONCLUSIONS: Rigorous control measures allowed the outbreak to be declared over quickly following outbreak containment. This highlights the need for effective interventions, not only to reduce cases during outbreaks but also to allow outbreaks to be declared over with confidence.