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".

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.

Local generalised method of moments: an application to point process-based rainfall models.

Long series of simulated rainfall are required at point locations for a range of applications, including hydrological studies. Clustered point process-based rainfall models have been used for generating such simulations for many decades. These models suffer from a major limitation, however: their stationarity. Although seasonality can be allowed by fitting separate models for each calendar month or season, the models are unsuitable in their basic form for climate impact studies. In this paper, we develop new methodology to address this limitation. We extend the current fitting approach by allowing the discrete covariate, calendar month, to be replaced or supplemented with continuous covariates that are more directly related to the incidence and nature of rainfall. The covariate-dependent model parameters are estimated for each time interval using a kernel-based nonparametric approach within a generalised method-of-moments framework. An empirical study demonstrates the new methodology using a time series of 5-min rainfall data. The study considers both local mean and local linear approaches. While asymptotic results are included, the focus is on developing useable methodology for a complex model that can only be solved numerically. Issues including the choice of weighting matrix, estimation of parameter uncertainty and bandwidth and model selection are considered from this perspective. © 2015 The Authors. Environmetrics Published by John Wiley & Sons Ltd.

Modelling: Understanding pandemics and how to control them.

New disease challenges, societal demands and better or novel types of data, drive innovations in the structure, formulation and analysis of epidemic models. Innovations in modelling can lead to new insights into epidemic processes and better use of available data, yielding improved disease control and stimulating collection of better data and new data types. Here we identify key challenges for the structure, formulation, analysis and use of mathematical models of pathogen transmission relevant to current and future pandemics.

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.

Challenges on the interaction of models and policy for pandemic control.

The COVID-19 pandemic has seen infectious disease modelling at the forefront of government decision-making. Models have been widely used throughout the pandemic to estimate pathogen spread and explore the potential impact of different intervention strategies. Infectious disease modellers and policymakers have worked effectively together, but there are many avenues for progress on this interface. In this paper, we identify and discuss seven broad challenges on the interaction of models and policy for pandemic control. We then conclude with suggestions and recommendations for the future.

Rates of microfilarial production by Onchocerca volvulus are not cumulatively reduced by multiple ivermectin treatments.

Regular distribution of ivermectin reduces onchocerciasis transmission and morbidity by killing, within humans, the microfilarial stage of the parasite (microfilaricidal effect). In addition, ivermectin exerts a so-called embryostatic effect by which microfilarial production by the adult female worm becomes suppressed during a number of weeks after treatment. To assess the overall effect of ivermectin on onchocerciasis transmission and evaluate the likelihood of local elimination of the infection it is important to estimate the magnitude of the anti-fertility effect over the course of a treatment programme. Estimates of the effect of repeated drug treatments on the production of microfilariae by adult Onchocerca volvulus were obtained by developing a model that was fitted to data collected from three hyperendemic communities in Guatemala, where eligible residents received ivermectin twice per year for two and a half years. The data consist of microfilarial load measurements in the skin, collected just before each six-monthly treatment during the programme. The model that is developed describes the dynamics of an individual host's expected microfilarial load over the 30-month study period. We adopt a Bayesian approach and use Markov chain Monte Carlo (McMC) techniques to fit the model to the data. Combining estimates from the three villages, average microfilarial production in the first six months post-treatment was reduced by ~64% of its pre-treatment level, regardless of values chosen for the pre-ivermectin fertility rate within plausible ranges. Increased adult worm death rate after treatment (to mimic removal of macrofilariae via nodulectomy during the programme) resulted in a smaller estimated magnitude of the embryostatic effect (rate of microfilarial production was reduced by ~58% of pre-ivermectin value). After subsequent treatments, the rate of microfilarial production appeared to be similarly decreased. The data and analyses therefore do not support the hypothesis of a cumulative effect of multiple ivermectin treatments on microfilarial production by female worms.

Gaussian process approximations for fast inference from infectious disease data.

We present a flexible framework for deriving and quantifying the accuracy of Gaussian process approximations to non-linear stochastic individual-based models of epidemics. We develop this for the SIR and SEIR models, and we show how it can be used to perform quick maximum likelihood inference for the underlying parameters given population estimates of the number of infecteds or cases at given time points. We also show how the unobserved processes can be inferred at the same time as the underlying parameters.

Response to the Letter to the Editor by Eberhard et al.

In a Letter to the Editor, Eberhard et al. question the validity of our model of skin snip sensitivity and argue against the use of skin snips to evaluate onchocerciasis elimination by mass drug administration. Here we discuss their arguments and compare model predictions with observed data to assess the validity of our model.

Modelling Neglected Tropical Diseases diagnostics: the sensitivity of skin snips for Onchocerca volvulus in near elimination and surveillance settings.

BACKGROUND: The African Programme for Onchocerciasis Control has proposed provisional thresholds for the prevalence of microfilariae in humans and of L3 larvae in blackflies, below which mass drug administration (MDA) with ivermectin can be stopped and surveillance started. Skin snips are currently the gold standard test for detecting patent Onchocerca volvulus infection, and the World Health Organization recommends their use to monitor progress of treatment programmes (but not to verify elimination). However, if they are used (in transition and in parallel to Ov-16 serology), sampling protocols should be designed to demonstrate that programmatic goals have been reached. The sensitivity of skin snips is key to the design of such protocols. METHODS: We develop a mathematical model for the number of microfilariae in a skin snip and parameterise it using data from Guatemala, Venezuela, Ghana and Cameroon collected before the start of ivermectin treatment programmes. We use the model to estimate sensitivity as a function of time since last treatment, number of snips taken, microfilarial aggregation and female worm fertility after exposure to 10 annual rounds of ivermectin treatment. RESULTS: The sensitivity of the skin snip method increases with time after treatment, with most of the increase occurring between 0 and 5 years. One year after the last treatment, the sensitivity of two skin snips taken from an individual infected with a single fertile female worm is 31 % if there is no permanent effect of multiple ivermectin treatments on fertility; 18 % if there is a 7 % reduction per treatment, and 0.6 % if there is a 35 % reduction. At 5 years, the corresponding sensitivities are 76 %, 62 % and 4.7 %. The sensitivity improves significantly if 4 skin snips are taken: in the absence of a permanent effect of ivermectin, the sensitivity of 4 skin snips is 53 % 1 year and 94 % 5 years after the last treatment. CONCLUSIONS: Our model supports the timelines proposed by APOC for post-MDA follow-up and surveillance surveys every 3-5 years. Two skin snips from the iliac region have reasonable sensitivity to detect residual infection, but the sensitivity can be significantly improved by taking 4 snips. The costs and benefits of using four versus two snips should be evaluated.

Five challenges for spatial epidemic models.

Infectious disease incidence data are increasingly available at the level of the individual and include high-resolution spatial components. Therefore, we are now better able to challenge models that explicitly represent space. Here, we consider five topics within spatial disease dynamics: the construction of network models; characterising threshold behaviour; modelling long-distance interactions; the appropriate scale for interventions; and the representation of population heterogeneity.

Eight challenges for network epidemic models.

Networks offer a fertile framework for studying the spread of infection in human and animal populations. However, owing to the inherent high-dimensionality of networks themselves, modelling transmission through networks is mathematically and computationally challenging. Even the simplest network epidemic models present unanswered questions. Attempts to improve the practical usefulness of network models by including realistic features of contact networks and of host-pathogen biology (e.g. waning immunity) have made some progress, but robust analytical results remain scarce. A more general theory is needed to understand the impact of network structure on the dynamics and control of infection. Here we identify a set of challenges that provide scope for active research in the field of network epidemic models.

Seven challenges for metapopulation models of epidemics, including households models.

This paper considers metapopulation models in the general sense, i.e. where the population is partitioned into sub-populations (groups, patches,...), irrespective of the biological interpretation they have, e.g. spatially segregated large sub-populations, small households or hosts themselves modelled as populations of pathogens. This framework has traditionally provided an attractive approach to incorporating more realistic contact structure into epidemic models, since it often preserves analytic tractability (in stochastic as well as deterministic models) but also captures the most salient structural inhomogeneity in contact patterns in many applied contexts. Despite the progress that has been made in both the theory and application of such metapopulation models, we present here several major challenges that remain for future work, focusing on models that, in contrast to agent-based ones, are amenable to mathematical analysis. The challenges range from clarifying the usefulness of systems of weakly-coupled large sub-populations in modelling the spread of specific diseases to developing a theory for endemic models with household structure. They include also developing inferential methods for data on the emerging phase of epidemics, extending metapopulation models to more complex forms of human social structure, developing metapopulation models to reflect spatial population structure, developing computationally efficient methods for calculating key epidemiological model quantities, and integrating within- and between-host dynamics in models.

Seven challenges for modelling indirect transmission: vector-borne diseases, macroparasites and neglected tropical diseases.

Many of the challenges which face modellers of directly transmitted pathogens also arise when modelling the epidemiology of pathogens with indirect transmission--whether through environmental stages, vectors, intermediate hosts or multiple hosts. In particular, understanding the roles of different hosts, how to measure contact and infection patterns, heterogeneities in contact rates, and the dynamics close to elimination are all relevant challenges, regardless of the mode of transmission. However, there remain a number of challenges that are specific and unique to modelling vector-borne diseases and macroparasites. Moreover, many of the neglected tropical diseases which are currently targeted for control and elimination are vector-borne, macroparasitic, or both, and so this article includes challenges which will assist in accelerating the control of these high-burden diseases. Here, we discuss the challenges of indirect measures of infection in humans, whether through vectors or transmission life stages and in estimating the contribution of different host groups to transmission. We also discuss the issues of "evolution-proof" interventions against vector-borne disease.

Modeling infectious disease dynamics in the complex landscape of global health.

Despite some notable successes in the control of infectious diseases, transmissible pathogens still pose an enormous threat to human and animal health. The ecological and evolutionary dynamics of infections play out on a wide range of interconnected temporal, organizational, and spatial scales, which span hours to months, cells to ecosystems, and local to global spread. Moreover, some pathogens are directly transmitted between individuals of a single species, whereas others circulate among multiple hosts, need arthropod vectors, or can survive in environmental reservoirs. Many factors, including increasing antimicrobial resistance, increased human connectivity and changeable human behavior, elevate prevention and control from matters of national policy to international challenge. In the face of this complexity, mathematical models offer valuable tools for synthesizing information to understand epidemiological patterns, and for developing quantitative evidence for decision-making in global health.

Stochastic and spatial dynamics of nematode parasites in farmed ruminants.

Host-parasite systems provide powerful opportunities for the study of spatial and stochastic effects in ecology; this has been particularly so for directly transmitted microparasites. Here, we construct a fully stochastic model of the population dynamics of a macroparasite system: trichostrongylid gastrointestinal nematode parasites of farmed ruminants. The model subsumes two implicit spatial effects: the host population size (the spatial extent of the interaction between hosts) and spatial heterogeneity ('clumping') in the infection process. This enables us to investigate the roles of several different processes in generating aggregated parasite distributions. The necessity for female worms to find a mate in order to reproduce leads to an Allee effect, which interacts nonlinearly with the stochastic population dynamics and leads to the counter-intuitive result that, when rare, epidemics can be more likely and more severe in small host populations. Clumping in the infection process reduces the strength of this Allee effect, but can hamper the spread of an epidemic by making infection events too rare. Heterogeneity in the hosts' response to infection has to be included in the model to generate aggregation at the level observed empirically.

A mathematical model of serotype replacement in pneumococcal carriage following vaccination.

A number of childhood vaccination programmes have recently introduced vaccination against Streptococcus pneumoniae, the pneumococcus, a major cause of pneumonia and meningitis. The pneumococcal conjugate vaccines (PCVs) that are currently in use only protect against some serotypes of the bacterium, and there is now strong evidence that those serotypes not included in the vaccine increase in prevalence among most vaccinated populations. We present a mathematical model for the dynamics of nasopharyngeal carriage of S. pneumoniae that allows for carriage with multiple serotypes. The model is used to predict the prevalence of vaccine type (VT) and non-VT (NVT) serotypes following the introduction of PCV. Parameter estimates for the model are obtained by maximum likelihood using pre-vaccination data from The Gambia. The model predicts that low (1, 6A and 9V) and medium (4, 5, 7F, 14, 18C, 19A and 19F) prevalence serotypes can be eliminated through vaccination, but that the overall prevalence of carriage will be reduced only slightly because of an increase in the prevalence of NVT serotypes. Serotype replacement will be sequential, with high and medium prevalence NVT serotypes dominating initially, followed by an increase of serotypes of low prevalence. We examine the impact of a hypothetical vaccine that provides partial protection against all serotypes, and find that this reduces overall carriage, but is unable to eliminate low or medium prevalence serotypes.

Spread of information and infection on finite random networks.

The modeling of epidemic-like processes on random networks has received considerable attention in recent years. While these processes are inherently stochastic, most previous work has been focused on deterministic models that ignore important fluctuations that may persist even in the infinite network size limit. In a previous paper, for a class of epidemic and rumor processes, we derived approximate models for the full probability distribution of the final size of the epidemic, as opposed to only mean values. In this paper we examine via direct simulations the adequacy of the approximate model to describe stochastic epidemics and rumors on several random network topologies: homogeneous networks, Erdös-Rényi (ER) random graphs, Barabasi-Albert scale-free networks, and random geometric graphs. We find that the approximate model is reasonably accurate in predicting the probability of spread. However, the position of the threshold and the conditional mean of the final size for processes near the threshold are not well described by the approximate model even in the case of homogeneous networks. We attribute this failure to the presence of other structural properties beyond degree-degree correlations, and in particular clustering, which are present in any finite network but are not incorporated in the approximate model. In order to test this "hypothesis"  we perform additional simulations on a set of ER random graphs where degree-degree correlations and clustering are separately and independently introduced using recently proposed algorithms from the literature. Our results show that even strong degree-degree correlations have only weak effects on the position of the threshold and the conditional mean of the final size. On the other hand, the introduction of clustering greatly affects both the position of the threshold and the conditional mean. Similar analysis for the Barabasi-Albert scale-free network confirms the significance of clustering on the dynamics of rumor spread. For this network, though, with its highly skewed degree distribution, the addition of positive correlation had a much stronger effect on the final size distribution than was found for the simple random graph.