Prioritising older individuals for COVID-19 booster vaccination leads to optimal public health outcomes in a range of socio-economic settings.

The rapid development of vaccines against SARS-CoV-2 altered the course of the COVID-19 pandemic. In most countries, vaccinations were initially targeted at high-risk populations, including older individuals and healthcare workers. Now, despite substantial infection- and vaccine-induced immunity in host populations worldwide, waning immunity and the emergence of novel variants continue to cause significant waves of infection and disease. Policy makers must determine how to deploy booster vaccinations, particularly when constraints in vaccine supply, delivery and cost mean that booster vaccines cannot be administered to everyone. A key question is therefore whether older individuals should again be prioritised for vaccination, or whether alternative strategies (e.g. offering booster vaccines to the individuals who have most contacts with others and therefore drive infection) can instead offer indirect protection to older individuals. Here, we use mathematical modelling to address this question, considering SARS-CoV-2 transmission in a range of countries with different socio-economic backgrounds. We show that the population structures of different countries can have a pronounced effect on the impact of booster vaccination, even when identical booster vaccination targeting strategies are adopted. However, under the assumed transmission model, prioritising older individuals for booster vaccination consistently leads to the most favourable public health outcomes in every setting considered. This remains true for a range of assumptions about booster vaccine supply and timing, and for different assumed policy objectives of booster vaccination.

Modelling the effectiveness of an isolation strategy for managing mpox outbreaks with variable infectiousness profiles.

The global outbreak of mpox in 2022 and subsequent sporadic outbreaks in 2023 highlighted the importance of nonpharmaceutical interventions such as case isolation. Individual variations in viral shedding dynamics may lead to either premature ending of isolation for infectious individuals, or unnecessarily prolonged isolation for those who are no longer infectious. Here, we developed a modeling framework to characterize heterogeneous mpox infectiousness profiles - specifically, when infected individuals cease to be infectious - based on viral load data. We examined the potential effectiveness of three different isolation rules: a symptom-based rule (the current guideline in many countries) and rules permitting individuals to stop isolating after either a fixed duration or following tests that indicate that they are no longer likely to be infectious. Our analysis suggests that the duration of viral shedding ranges from 23 to 50 days between individuals. The risk of infected individuals ending isolation too early was estimated to be 8.8% (95% CI: 6.7-10.5) after symptom clearance and 5.4% (95% CI: 4.1-6.7) after 3 weeks of isolation. While these results suggest that the current standard practice for ending isolation is effective, we found that unnecessary isolation following the infectious period could be reduced by adopting a testing-based rule.

Host behaviour driven by awareness of infection risk amplifies the chance of superspreading events.

We demonstrate that heterogeneity in the perceived risks associated with infection within host populations amplifies chances of superspreading during the crucial early stages of epidemics. Under this behavioural model, individuals less concerned about dangers from infection are more likely to be infected and attend larger sized (riskier) events, where we assume event sizes remain unchanged. For directly transmitted diseases such as COVID-19, this leads to infections being introduced at rates above the population prevalence to those events most conducive to superspreading. We develop an interpretable, computational framework for evaluating within-event risks and derive a small-scale reproduction number measuring how the infections generated at an event depend on transmission heterogeneities and numbers of introductions. This generalizes previous frameworks and quantifies how event-scale patterns and population-level characteristics relate. As event duration and size grow, our reproduction number converges to the basic reproduction number. We illustrate that even moderate levels of heterogeneity in the perceived risks of infection substantially increase the likelihood of disproportionately large clusters of infections occurring at larger events, despite fixed overall disease prevalence. We show why collecting data linking host behaviour and event attendance is essential for accurately assessing the risks posed by invading pathogens in emerging stages of outbreaks.

Optimizing the timing of an end-of-outbreak declaration: Ebola virus disease in the Democratic Republic of the Congo.

Following the apparent final case in an Ebola virus disease (EVD) outbreak, the decision to declare the outbreak over must balance societal benefits of relaxing interventions against the risk of resurgence. Estimates of the end-of-outbreak probability (the probability that no future cases will occur) provide quantitative evidence that can inform the timing of an end-of-outbreak declaration. An existing modeling approach for estimating the end-of-outbreak probability requires comprehensive contact tracing data describing who infected whom to be available, but such data are often unavailable or incomplete during outbreaks. Here, we develop a Markov chain Monte Carlo-based approach that extends the previous method and does not require contact tracing data. Considering data from two EVD outbreaks in the Democratic Republic of the Congo, we find that data describing who infected whom are not required to resolve uncertainty about when to declare an outbreak over.

When should lockdown be implemented? Devising cost-effective strategies for managing epidemics amid vaccine uncertainty.

During an infectious disease outbreak, public health policy makers are tasked with strategically implementing interventions whilst balancing competing objectives. To provide a quantitative framework that can be used to guide these decisions, it is helpful to devise a clear and specific objective function that can be evaluated to determine the optimal outbreak response. In this study, we have developed a mathematical modelling framework representing outbreaks of a novel emerging pathogen for which non-pharmaceutical interventions (NPIs) are imposed or removed based on thresholds for hospital occupancy. These thresholds are set at different levels to define four unique strategies for disease control. We illustrate that the optimal intervention strategy is contingent on the choice of objective function. Specifically, the optimal strategy depends on the extent to which policy makers prioritise reducing health costs due to infection over the costs associated with maintaining interventions. Motivated by the scenario early in the COVID-19 pandemic, we incorporate the development of a vaccine into our modelling framework and demonstrate that a policy maker's belief about when a vaccine will become available in future, and its eventual coverage (and/or effectiveness), affects the optimal strategy to adopt early in the outbreak. Furthermore, we show how uncertainty in these quantities can be accounted for when deciding which interventions to introduce. This research highlights the benefits of policy makers being explicit about the precise objectives of introducing interventions.

Incubation Period and Serial Interval of Mpox in 2022 Global Outbreak Compared with Historical Estimates.

Understanding changes in the transmission dynamics of mpox requires comparing recent estimates of key epidemiologic parameters with historical data. We derived historical estimates for the incubation period and serial interval for mpox and contrasted them with pooled estimates from the 2022 outbreak. Our findings show the pooled mean infection-to-onset incubation period was 8.1 days for the 2022 outbreak and 8.2 days historically, indicating the incubation periods remained relatively consistent over time, despite a shift in the major mode of transmission. However, we estimated the onset-to-onset serial interval at 8.7 days using 2022 data, compared with 14.2 days using historical data. Although the reason for this shortening of the serial interval is unclear, it may be because of increased public health interventions or a shift in the mode of transmission. Recognizing such temporal shifts is essential for informed response strategies, and public health measures remain crucial for controlling mpox and similar future outbreaks.

Mathematical methods for scaling from within-host to population-scale in infectious disease systems.

Mathematical modellers model infectious disease dynamics at different scales. Within-host models represent the spread of pathogens inside an individual, whilst between-host models track transmission between individuals. However, pathogen dynamics at one scale affect those at another. This has led to the development of multiscale models that connect within-host and between-host dynamics. In this article, we systematically review the literature on multiscale infectious disease modelling according to PRISMA guidelines, dividing previously published models into five categories governing their methodological approaches (Garira (2017)), explaining their benefits and limitations. We provide a primer on developing multiscale models of infectious diseases.

Isolation may select for earlier and higher peak viral load but shorter duration in SARS-CoV-2 evolution.

During the COVID-19 pandemic, human behavior change as a result of nonpharmaceutical interventions such as isolation may have induced directional selection for viral evolution. By combining previously published empirical clinical data analysis and multi-level mathematical modeling, we find that the SARS-CoV-2 variants selected for as the virus evolved from the pre-Alpha to the Delta variant had earlier and higher peak in viral load dynamics but a shorter duration of infection. Selection for increased transmissibility shapes the viral load dynamics, and the isolation measure is likely to be a driver of these evolutionary transitions. In addition, we show that a decreased incubation period and an increased proportion of asymptomatic infection are also positively selected for as SARS-CoV-2 mutated to adapt to human behavior (i.e., Omicron variants). The quantitative information and predictions we present here can guide future responses in the potential arms race between pandemic interventions and viral evolution.

Analysis of the risk and pre-emptive control of viral outbreaks accounting for within-host dynamics: SARS-CoV-2 as a case study.

In the era of living with COVID-19, the risk of localised SARS-CoV-2 outbreaks remains. Here, we develop a multiscale modelling framework for estimating the local outbreak risk for a viral disease (the probability that a major outbreak results from a single case introduced into the population), accounting for within-host viral dynamics. Compared to population-level models previously used to estimate outbreak risks, our approach enables more detailed analysis of how the risk can be mitigated through pre-emptive interventions such as antigen testing. Considering SARS-CoV-2 as a case study, we quantify the within-host dynamics using data from individuals with omicron variant infections. We demonstrate that regular antigen testing reduces, but may not eliminate, the outbreak risk, depending on characteristics of local transmission. In our baseline analysis, daily antigen testing reduces the outbreak risk by 45% compared to a scenario without antigen testing. Additionally, we show that accounting for heterogeneity in within-host dynamics between individuals affects outbreak risk estimates and assessments of the impact of antigen testing. Our results therefore highlight important factors to consider when using multiscale models to design pre-emptive interventions against SARS-CoV-2 and other viruses.

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.

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.

A target-cell limited model can reproduce influenza infection dynamics in hosts with differing immune responses.

We consider a hierarchy of ordinary differential equation models that describe the within-host viral kinetics of influenza infections: the IR model explicitly accounts for an immune response to the virus, while the simpler, target-cell limited TEIV and TV models do not. We show that when the IR model is fitted to pooled experimental murine data of the viral load, fraction of dead cells, and immune response levels, its parameters values can be determined. However, if, as is common, only viral load data are available, we can estimate parameters of the TEIV and TV models but not the IR model. These results are substantiated by a structural and practical identifiability analysis. We then use the IR model to generate synthetic data representing infections in hosts whose immune responses differ. We fit the TV model to these synthetic datasets and show that it can reproduce the characteristic exponential increase and decay of viral load generated by the IR model. Furthermore, the values of the fitted parameters of the TV model can be mapped from the immune response parameters in the IR model. We conclude that, if only viral load data are available, a simple target-cell limited model can reproduce influenza infection dynamics and distinguish between hosts with differing immune responses.

Using 'sentinel' plants to improve early detection of invasive plant pathogens.

Infectious diseases of plants present an ongoing and increasing threat to international biosecurity, with wide-ranging implications. An important challenge in plant disease management is achieving early detection of invading pathogens, which requires effective surveillance through the implementation of appropriate monitoring programmes. However, when monitoring relies on visual inspection as a means of detection, surveillance is often hindered by a long incubation period (delay from infection to symptom onset) during which plants may be infectious but not displaying visible symptoms. 'Sentinel' plants-alternative susceptible host species that display visible symptoms of infection more rapidly-could be introduced to at-risk populations and included in monitoring programmes to act as early warning beacons for infection. However, while sentinel hosts exhibit faster disease progression and so allow pathogens to be detected earlier, this often comes at a cost: faster disease progression typically promotes earlier onward transmission. Here, we construct a computational model of pathogen transmission to explore this trade-off and investigate how including sentinel plants in monitoring programmes could facilitate earlier detection of invasive plant pathogens. Using Xylella fastidiosa infection in Olea europaea (European olive) as a current high profile case study, for which Catharanthus roseus (Madagascan periwinkle) is a candidate sentinel host, we apply a Bayesian optimisation algorithm to determine the optimal number of sentinel hosts to introduce for a given sampling effort, as well as the optimal division of limited surveillance resources between crop and sentinel plants. Our results demonstrate that including sentinel plants in monitoring programmes can reduce the expected prevalence of infection upon outbreak detection substantially, increasing the feasibility of local outbreak containment.

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".

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.

Stochastic modelling of African swine fever in wild boar and domestic pigs: Epidemic forecasting and comparison of disease management strategies.

African swine fever (ASF), caused by the African swine fever virus (ASFV), is highly virulent in domestic pigs and wild boar (Sus scrofa), causing up to 100% mortality. The recent epidemic of ASF in Europe has had a serious economic impact and poses a threat to global food security. Unfortunately, there is no effective treatment or vaccine against ASFV, limiting the available disease management strategies. Mathematical models allow us to further our understanding of infectious disease dynamics and evaluate the efficacy of disease management strategies. The ASF Challenge, organised by the French National Research Institute for Agriculture, Food, and the Environment, aimed to expand the development of ASF transmission models to inform policy makers in a timely manner. Here, we present the model and associated projections produced by our team during the challenge. We developed a stochastic model combining transmission between wild boar and domestic pigs, which was calibrated to synthetic data corresponding to different phases describing the epidemic progression. The model was then used to produce forward projections describing the likely temporal evolution of the epidemic under various disease management scenarios. Despite the interventions implemented, long-term projections forecasted persistence of ASFV in wild boar, and hence repeated outbreaks in domestic pigs. A key finding was that it is important to consider the timescale over which different measures are evaluated: interventions that have only limited effectiveness in the short term may yield substantial long-term benefits. Our model has several limitations, partly because it was developed in real-time. Nonetheless, it can inform understanding of the likely development of ASF epidemics and the efficacy of disease management strategies, should the virus continue its spread in Europe.

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.

The African swine fever modelling challenge: Model comparison and lessons learnt.

Robust epidemiological knowledge and predictive modelling tools are needed to address challenging objectives, such as: understanding epidemic drivers; forecasting epidemics; and prioritising control measures. Often, multiple modelling approaches can be used during an epidemic to support effective decision making in a timely manner. Modelling challenges contribute to understanding the pros and cons of different approaches and to fostering technical dialogue between modellers. In this paper, we present the results of the first modelling challenge in animal health - the ASF Challenge - which focused on a synthetic epidemic of African swine fever (ASF) on an island. The modelling approaches proposed by five independent international teams were compared. We assessed their ability to predict temporal and spatial epidemic expansion at the interface between domestic pigs and wild boar, and to prioritise a limited number of alternative interventions. We also compared their qualitative and quantitative spatio-temporal predictions over the first two one-month projection phases of the challenge. Top-performing models in predicting the ASF epidemic differed according to the challenge phase, host species, and in predicting spatial or temporal dynamics. Ensemble models built using all team-predictions outperformed any individual model in at least one phase. The ASF Challenge demonstrated that accounting for the interface between livestock and wildlife is key to increasing our effectiveness in controlling emerging animal diseases, and contributed to improving the readiness of the scientific community to face future ASF epidemics. Finally, we discuss the lessons learnt from model comparison to guide decision making.

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.

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.