Context-specific emergence and growth of the SARS-CoV-2 Delta variant.

The SARS-CoV-2 Delta (Pango lineage B.1.617.2) variant of concern spread globally, causing resurgences of COVID-19 worldwide1,2. The emergence of the Delta variant in the UK occurred on the background of a heterogeneous landscape of immunity and relaxation of non-pharmaceutical interventions. Here we analyse 52,992 SARS-CoV-2 genomes from England together with 93,649 genomes from the rest of the world to reconstruct the emergence of Delta and quantify its introduction to and regional dissemination across England in the context of changing travel and social restrictions. Using analysis of human movement, contact tracing and virus genomic data, we find that the geographic focus of the expansion of Delta shifted from India to a more global pattern in early May 2021. In England, Delta lineages were introduced more than 1,000 times and spread nationally as non-pharmaceutical interventions were relaxed. We find that hotel quarantine for travellers reduced onward transmission from importations; however, the transmission chains that later dominated the Delta wave in England were seeded before travel restrictions were introduced. Increasing inter-regional travel within England drove the nationwide dissemination of Delta, with some cities receiving more than 2,000 observable lineage introductions from elsewhere. Subsequently, increased levels of local population mixing-and not the number of importations-were associated with the faster relative spread of Delta. The invasion dynamics of Delta depended on spatial heterogeneity in contact patterns, and our findings will inform optimal spatial interventions to reduce the transmission of current and future variants of concern, such as Omicron (Pango lineage B.1.1.529).

Investigating the dose-dependency of the midgut escape barrier using a mechanistic model of within-mosquito dengue virus population dynamics.

Arboviruses can emerge rapidly and cause explosive epidemics of severe disease. Some of the most epidemiologically important arboviruses, including dengue virus (DENV), Zika virus (ZIKV), Chikungunya (CHIKV) and yellow fever virus (YFV), are transmitted by Aedes mosquitoes, most notably Aedes aegypti and Aedes albopictus. After a mosquito blood feeds on an infected host, virus enters the midgut and infects the midgut epithelium. The virus must then overcome a series of barriers before reaching the mosquito saliva and being transmitted to a new host. The virus must escape from the midgut (known as the midgut escape barrier; MEB), which is thought to be mediated by transient changes in the permeability of the midgut-surrounding basal lamina layer (BL) following blood feeding. Here, we present a mathematical model of the within-mosquito population dynamics of DENV (as a model system for mosquito-borne viruses more generally) that includes the interaction of the midgut and BL which can account for the MEB. Our results indicate a dose-dependency of midgut establishment of infection as well as rate of escape from the midgut: collectively, these suggest that the extrinsic incubation period (EIP)-the time taken for DENV virus to be transmissible after infection-is shortened when mosquitoes imbibe more virus. Additionally, our experimental data indicate that multiple blood feeding events, which more closely mimic mosquito-feeding behavior in the wild, can hasten the course of infections, and our model predicts that this effect is sensitive to the amount of virus imbibed. Our model indicates that mutations to the virus which impact its replication rate in the midgut could lead to even shorter EIPs when double-feeding occurs. Mechanistic models of within-vector viral infection dynamics provide a quantitative understanding of infection dynamics and could be used to evaluate novel interventions that target the mosquito stages of the infection.

Filter inference: A scalable nonlinear mixed effects inference approach for snapshot time series data.

Variability is an intrinsic property of biological systems and is often at the heart of their complex behaviour. Examples range from cell-to-cell variability in cell signalling pathways to variability in the response to treatment across patients. A popular approach to model and understand this variability is nonlinear mixed effects (NLME) modelling. However, estimating the parameters of NLME models from measurements quickly becomes computationally expensive as the number of measured individuals grows, making NLME inference intractable for datasets with thousands of measured individuals. This shortcoming is particularly limiting for snapshot datasets, common e.g. in cell biology, where high-throughput measurement techniques provide large numbers of single cell measurements. We introduce a novel approach for the estimation of NLME model parameters from snapshot measurements, which we call filter inference. Filter inference uses measurements of simulated individuals to define an approximate likelihood for the model parameters, avoiding the computational limitations of traditional NLME inference approaches and making efficient inferences from snapshot measurements possible. Filter inference also scales well with the number of model parameters, using state-of-the-art gradient-based MCMC algorithms such as the No-U-Turn Sampler (NUTS). We demonstrate the properties of filter inference using examples from early cancer growth modelling and from epidermal growth factor signalling pathway modelling.

Angular reproduction numbers improve estimates of transmissibility when disease generation times are misspecified or time-varying.

We introduce the angular reproduction number Ω, which measures time-varying changes in epidemic transmissibility resulting from variations in both the effective reproduction number R, and generation time distribution w. Predominant approaches for tracking pathogen spread infer either R or the epidemic growth rate r. However, R is biased by mismatches between the assumed and true w, while r is difficult to interpret in terms of the individual-level branching process underpinning transmission. R and r may also disagree on the relative transmissibility of epidemics or variants (i.e. rA > rB does not imply RA > RB for variants A and B). We find that Ω responds meaningfully to mismatches and time-variations in w while mostly maintaining the interpretability of R. We prove that Ω > 1 implies R > 1 and that Ω agrees with r on the relative transmissibility of pathogens. Estimating Ω is no more difficult than inferring R, uses existing software, and requires no generation time measurements. These advantages come at the expense of selecting one free parameter. We propose Ω as complementary statistic to R and r that improves transmissibility estimates when w is misspecified or time-varying and better reflects the impact of interventions, when those interventions concurrently change R and w or alter the relative risk of co-circulating pathogens.

A Bayesian nonparametric method for detecting rapid changes in disease transmission.

Whether an outbreak of infectious disease is likely to grow or dissipate is determined through the time-varying reproduction number, Rt. Real-time or retrospective identification of changes in Rt following the imposition or relaxation of interventions can thus contribute important evidence about disease transmission dynamics which can inform policymaking. Here, we present a method for estimating shifts in Rt within a renewal model framework. Our method, which we call EpiCluster, is a Bayesian nonparametric model based on the Pitman-Yor process. We assume that Rt is piecewise-constant, and the incidence data and priors determine when or whether Rt should change and how many times it should do so throughout the series. We also introduce a prior which induces sparsity over the number of changepoints. Being Bayesian, our approach yields a measure of uncertainty in Rt and its changepoints. EpiCluster is fast, straightforward to use, and we demonstrate that it provides automated detection of rapid changes in transmission, either in real-time or retrospectively, for synthetic data series where the Rt profile is known. We illustrate the practical utility of our method by fitting it to case data of outbreaks of COVID-19 in Australia and Hong Kong, where it finds changepoints coinciding with the imposition of non-pharmaceutical interventions. Bayesian nonparametric methods, such as ours, allow the volume and complexity of the data to dictate the number of parameters required to approximate the process and should find wide application in epidemiology. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Birds invest wingbeats to keep a steady head and reap the ultimate benefits of flying together.

Flapping flight is the most energetically demanding form of sustained forwards locomotion that vertebrates perform. Flock dynamics therefore have significant implications for energy expenditure. Despite this, no studies have quantified the biomechanical consequences of flying in a cluster flock or pair relative to flying solo. Here, we compared the flight characteristics of homing pigeons (Columba livia) flying solo and in pairs released from a site 7 km from home, using high-precision 5 Hz global positioning system (GPS) and 200 Hz tri-axial accelerometer bio-loggers. As expected, paired individuals benefitted from improved homing route accuracy, which reduced flight distance by 7% and time by 9%. However, realising these navigational gains involved substantial changes in flight kinematics and energetics. Both individuals in a pair increased their wingbeat frequency by 18% by decreasing the duration of their upstroke. This sharp increase in wingbeat frequency caused just a 3% increase in airspeed but reduced the oscillatory displacement of the body by 22%, which we hypothesise relates to an increased requirement for visual stability and manoeuvrability when flying in a flock or pair. The combination of the increase in airspeed and a higher wingbeat frequency would result in a minimum 2.2% increase in the total aerodynamic power requirements if the wingbeats were fully optimised. Overall, the enhanced navigational performance will offset any additional energetic costs as long as the metabolic power requirements are not increased above 9%. Our results demonstrate that the increases in wingbeat frequency when flying together have previously been underestimated by an order of magnitude and force reinterpretation of their mechanistic origin. We show that, for pigeons flying in pairs, two heads are better than one but keeping a steady head necessitates energetically costly kinematics.

Estimating the extrinsic incubation period of malaria using a mechanistic model of sporogony.

During sporogony, malaria-causing parasites infect a mosquito, reproduce and migrate to the mosquito salivary glands where they can be transmitted the next time blood feeding occurs. The time required for sporogony, known as the extrinsic incubation period (EIP), is an important determinant of malaria transmission intensity. The EIP is typically estimated as the time for a given percentile, x, of infected mosquitoes to develop salivary gland sporozoites (the infectious parasite life stage), which is denoted by EIPx. Many mechanisms, however, affect the observed sporozoite prevalence including the human-to-mosquito transmission probability and possibly differences in mosquito mortality according to infection status. To account for these various mechanisms, we present a mechanistic mathematical model, which explicitly models key processes at the parasite, mosquito and observational scales. Fitting this model to experimental data, we find greater variation in the EIP than previously thought: we estimated the range between EIP10 and EIP90 (at 27°C) as 4.5 days compared to 0.9 days using existing statistical methods. This pattern holds over the range of study temperatures included in the dataset. Increasing temperature from 21°C to 34°C decreased the EIP50 from 16.1 to 8.8 days. Our work highlights the importance of mechanistic modelling of sporogony to (1) improve estimates of malaria transmission under different environmental conditions or disease control programs and (2) evaluate novel interventions that target the mosquito life stages of the parasite.

Coronary imaging of cardiac allograft vasculopathy predicts current and future deterioration of left ventricular function in patients with orthotopic heart transplantation.

BACKGROUND: Intravascular ultrasound (IVUS) and optical coherence tomography (OCT) improve sensitivity of cardiac allograft vasculopathy (CAV) detection compared to invasive coronary angiography (ICA), but their ability to predict clinical events is unknown. We determined whether severe CAV detected with ICA, IVUS, or OCT correlates with graft function. METHODS: Comparison of specific vessel parameters between IVUS and OCT on 20 patients attending for angiography 12-24 months post-orthotopic heart transplant. Serial left ventricular ejection fraction (EF) was recorded prospectively. RESULTS: Analyzing 55 coronary arteries, OCT and IVUS correlated well for vessel CAV characteristics. A mean intimal thickness (MIT)OCT  > .25 mm had a sensitivity of 86.7% and specificity of 74.3% at detecting Stanford grade 4 CAV. Those with angiographically evident CAV had significant reduction in graft EF over 7.3 years follow-up (median ΔEF -2% vs +1.5%, P = .03). Patients with MITOCT  > .25 mm in at least one vessel had a lower median EF at time of surveillance (57% vs 62%, P = .014). Two MACEs were noted. CONCLUSION: Imaging with OCT correlates well with IVUS for CAV detection. Combined angiography and OCT to screen for CAV within 12-24 months of transplant predicts concurrent and future deterioration in graft function.

Health inequities and clustering of fever, acute respiratory infection, diarrhoea and wasting in children under five in low- and middle-income countries: a Demographic and Health Surveys analysis.

BACKGROUND: Pneumonia, diarrhoea and malaria are responsible for over one third of all deaths in children under the age of 5 years in low and middle sociodemographic index countries; many of these deaths are also associated with malnutrition. We explore the co-occurrence and clustering of fever, acute respiratory infection, diarrhoea and wasting and their relationship with equity-relevant variables. METHODS: Multilevel, multivariate Bayesian logistic regression models were fitted to Demographic and Health Survey data from over 380,000 children in 39 countries. The relationship between outcome indicators (fever, acute respiratory infection, diarrhoea and wasting) and equity-relevant variables (wealth, access to health care and rurality) was examined. We quantified the geographical clustering and co-occurrence of conditions and a child's risk of multiple illnesses. RESULTS: The prevalence of outcomes was very heterogeneous within and between countries. There was marked spatial clustering of conditions and co-occurrence within children. For children in the poorest households and those reporting difficulties accessing healthcare, there were significant increases in the probability of at least one of the conditions in 18 of 21 countries, with estimated increases in the probability of up to 0.23 (95% CrI, 0.06-0.40). CONCLUSIONS: The prevalence of fever, acute respiratory infection, diarrhoea and wasting are associated with equity-relevant variables and cluster together. Via pathways of shared aetiology or risk, those children most disadvantaged disproportionately suffer from these conditions. This highlights the need for horizontal approaches, such as integrated community case management, with a focus on equity and targeted to those most at need.

A Monte Carlo method to estimate cell population heterogeneity from cell snapshot data.

Variation is characteristic of all living systems. Laboratory techniques such as flow cytometry can probe individual cells, and, after decades of experimentation, it is clear that even members of genetically identical cell populations can exhibit differences. To understand whether variation is biologically meaningful, it is essential to discern its source. Mathematical models of biological systems are tools that can be used to investigate causes of cell-to-cell variation. From mathematical analysis and simulation of these models, biological hypotheses can be posed and investigated, then parameter inference can determine which of these is compatible with experimental data. Data from laboratory experiments often consist of "snapshots" representing distributions of cellular properties at different points in time, rather than individual cell trajectories. These data are not straightforward to fit using hierarchical Bayesian methods, which require the number of cell population clusters to be chosen a priori. Nor are they amenable to standard nonlinear mixed effect methods, since a single observation per cell is typically too few to estimate parameter variability. Here, we introduce a computational sampling method named "Contour Monte Carlo" (CMC) for estimating mathematical model parameters from snapshot distributions, which is straightforward to implement and does not require that cells be assigned to predefined categories. The CMC algorithm fits to snapshot probability distributions rather than raw data, which means its computational burden does not, like existing approaches, increase with the number of cells observed. Our method is appropriate for underdetermined systems, where there are fewer distinct types of observations than parameters to be determined, and where observed variation is mostly due to variability in cellular processes rather than experimental measurement error. This may be the case for many systems due to continued improvements in resolution of laboratory techniques. In this paper, we apply our method to quantify cellular variation for three biological systems of interest and provide Julia code enabling others to use this method.

A review of prospective pathways and impacts of COVID-19 on the accessibility, safety, quality, and affordability of essential medicines and vaccines for universal health coverage in Africa.

BACKGROUND: The ongoing pandemic of coronavirus disease 2019 (COVID-19) has the potential to reverse progress towards global targets. This study examines the risks that the COVID-19 pandemic poses to equitable access to essential medicines and vaccines (EMV) for universal health coverage in Africa. METHODS: We searched medical databases and grey literature up to 2 October 2020 for studies reporting data on prospective pathways and innovative strategies relevant for the assessment and management of the emerging risks in accessibility, safety, quality, and affordability of EMV in the context of the COVID-19 pandemic. We used the resulting pool of evidence to support our analysis and to draw policy recommendations to mitigate the emerging risks and improve preparedness for future crises. RESULTS: Of the 310 records screened, 134 were included in the analysis. We found that the disruption of the international system affects more immediately the capability of low- and middle-income countries to acquire the basket of EMV. The COVID-19 pandemic may facilitate dishonesty and fraud, increasing the propensity of patients to take substandard and falsified drugs. Strategic regional cooperation in the form of joint tenders and contract awarding, joint price negotiation and supplier selection, as well as joint market research, monitoring, and evaluation could improve the supply, affordability, quality, and safety of EMV. Sustainable health financing along with international technology transfer and substantial investment in research and development are needed to minimize the vulnerability of African countries arising from their dependence on imported EMV. To ensure equitable access, community-based strategies such as mobile clinics as well as fees exemptions for vulnerable and under-served segments of society might need to be considered. Strategies such as task delegation and telephone triage could help reduce physician workload. This coupled with payments of risk allowance to frontline healthcare workers and health-literate healthcare organization might improve the appropriate use of EMV. CONCLUSIONS: Innovative and sustainable strategies informed by comparative risk assessment are increasingly needed to ensure that local economic, social, demographic, and epidemiological risks and potentials are accounted for in the national COVID-19 responses.

Movement reveals reproductive tactics in male elephants.

Long-term bio-logging has the potential to reveal how movements, and hence life-history trade-offs, vary over a lifetime. Reproductive tactics in particular may vary as individuals' trade-off current investment versus lifetime fitness. Male African savanna elephants (Loxodona africana) provide a telling example of balancing body growth with reproductive fitness due to the combination of indeterminate growth and strongly delineated periods of sexual activity (musth), which results in reproductive tactics that alter with age. Our study aims to quantify the extent to which male elephants alter their movement patterns, and hence energetic allocation, in relation to (a) reproductive state and (b) age, and (c) to determine whether musth periods can be detected directly from GPS tracking data. We used a combination of GPS tracking data and visual observations of 25 male elephants ranging in age from 20 to 52 years to examine the influence of reproductive state and age on movement. We then used a three-state hidden Markov model (HMM) to detect musth behaviour in a subset of sequential tracking data. Our results demonstrate that male elephants increased their daily mean speed and range size with age and in musth. Furthermore, non-musth speed decreased with age, presumably reflecting a shift towards energy acquisition during non-musth. Thus, despite similar speeds and marginally larger ranges between reproductive states at age 20, by age 50, males were travelling 2.0 times faster in a 3.5 times larger area in musth relative to non-musth. The distinctiveness of musth periods over age 35 meant the three-state HMM could automatically detect musth movement with high sensitivity and specificity, but could not for the younger age class. We show that male elephants increased their energetic allocation into reproduction with age as the probability of reproductive success increases. Given that older male elephants tend to be both the target of legal trophy hunting and illegal poaching, man-made interference could drive fundamental changes in elephant reproductive tactics. Bio-logging, as our study reveals, has the potential both to quantify mature elephant reproductive tactics remotely and to be used to institute proactive management strategies around the reproductive behaviour of this charismatic keystone species.

Exploring movement decisions: Can Bayesian movement-state models explain crop consumption behaviour in elephants (Loxodonta africana)?

Animal movements towards goals or targets are based upon either maximization of resource acquisition or risk avoidance, and the way animals move can reveal information about their motivation. We use hidden Markov models (HMMs) fitted in a Bayesian framework and hourly Global Positioning System fixes to distinguish animal movements into distinct states and analyse the influence of environmental variables on being in, and switching to, a particular state. Specifically, we apply our models to understand elephant movement decisions around agricultural fields, and crop consumption. As it is unclear what the role of habitat features are on this complex process, we analyse whether elephants target agricultural crops for consumption, or simply pass through them in search of water. Our HMMs separate elephant movements into two states: exploratory movements that are fast and directional, and encamped movements that are slow and meandering. For each elephant, we ran 16 models with each possible combination of selected habitat features (river, elephant corridor, agricultural field, trees), and repeated these analyses including interaction effects with both season and time of day. We used cross-validation to select the best model. In corridors, exploratory movements are dominant. Elephants mainly showed encamped movements at the river during the dry season, when temporary water sources have dried out and elephants relied on this permanent water source. In fields, males most often exhibited exploratory movements to and from the river, while females showed an increase in the frequency of encamped behaviour during the dry season and at night-the times when most crop consumption and movements through fields occur. Adaptation to risk could explain this behaviour, since foraging in fields is likely less risky under the cover of darkness and during the dry season when farmers are absent. This sex segregation in elephant movement decisions highlights the importance of predation risk in shaping movement patterns, which can result in sex segregation in responses to mitigation methods. The increase in encamped movements in the dry season suggests the importance of agricultural timing, and shows the potential for early ploughing and early-harvest crop types in order to reduce elephant crop consumption. Taking this into account could increase efficiency of elephant crop consumption mitigation.

The use of driving endonuclease genes to suppress mosquito vectors of malaria in temporally variable environments.

BACKGROUND: The use of gene drive systems to manipulate populations of malaria vectors is currently being investigated as a method of malaria control. One potential system uses driving endonuclease genes (DEGs) to spread genes that impose a genetic load. Previously, models have shown that the introduction of DEG-bearing mosquitoes could suppress or even extinguish vector populations in spatially-heterogeneous environments which were constant over time. In this study, a stochastic spatially-explicit model of mosquito ecology is combined with a rainfall model which enables the generation of a variety of daily precipitation patterns. The model is then used to investigate how releases of a DEG that cause a bias in population sex ratios towards males are affected by seasonal or random rainfall patterns. The parameters of the rainfall model are then fitted using data from Bamako, Mali, and Mbita, Kenya, to evaluate release strategies in similar climatic conditions. RESULTS: In landscapes with abundant resources and large mosquito populations the spread of a DEG is reliable, irrespective of variability in rainfall. This study thus focuses mainly on landscapes with low density mosquito populations where the spread of a DEG may be sensitive to variation in rainfall. It is found that an introduced DEG will spread into its target population more reliably in wet conditions, yet an established DEG will have more impact in dry conditions. In strongly seasonal environments, it is thus preferable to release DEGs at the onset of a wet season to maximize their spread before the following dry season. If the variability in rainfall has a substantial random component, there is a net increase in the probability that a DEG release will lead to population extinction, due to the increased impact of a DEG which manages to establish in these conditions. For Bamako, where annual rainfall patterns are characterized by a long dry season, it is optimal to release a DEG at the start of the wet season, where the population is growing fastest. By contrast release timing is of lower importance for the less seasonal Mbita. CONCLUSION: This analysis suggests that DEG based methods of malaria vector control can be effective in a wide range of climates. In environments with substantial temporal variation in rainfall, careful timing of releases which accounts for the temporal variation in population density can substantially improve the probability of mosquito suppression or extinction.

Bayesian inference of agent-based models: a tool for studying kidney branching morphogenesis.

The adult mammalian kidney has a complex, highly-branched collecting duct epithelium that arises as a ureteric bud sidebranch from an epithelial tube known as the nephric duct. Subsequent branching of the ureteric bud to form the collecting duct tree is regulated by subcellular interactions between the epithelium and a population of mesenchymal cells that surround the tips of outgrowing branches. The mesenchymal cells produce glial cell-line derived neurotrophic factor (GDNF), that binds with RET receptors on the surface of the epithelial cells to stimulate several subcellular pathways in the epithelium. Such interactions are known to be a prerequisite for normal branching development, although competing theories exist for their role in morphogenesis. Here we introduce the first agent-based model of ex vivo kidney uretic branching. Through comparison with experimental data, we show that growth factor-regulated growth mechanisms can explain early epithelial cell branching, but only if epithelial cell division depends in a switch-like way on the local growth factor concentration; cell division occurring only if the driving growth factor level exceeds a threshold. We also show how a recently-developed method, "Approximate Approximate Bayesian Computation", can be used to infer key model parameters, and reveal the dependency between the parameters controlling a growth factor-dependent growth switch. These results are consistent with a requirement for signals controlling proliferation and chemotaxis, both of which are previously identified roles for GDNF.

Power Analysis for Human Melatonin Suppression Experiments.

In humans, the nocturnal secretion of melatonin by the pineal gland is suppressed by ocular exposure to light. In the laboratory, melatonin suppression is a biomarker for this neuroendocrine pathway. Recent work has found that individuals differ substantially in their melatonin-suppressive response to light, with the most sensitive individuals being up to 60 times more sensitive than the least sensitive individuals. Planning experiments with melatonin suppression as an outcome needs to incorporate these individual differences, particularly in common resource-limited scenarios where running within-subjects studies at multiple light levels is costly and resource-intensive and may not be feasible with respect to participant compliance. Here, we present a novel framework for virtual laboratory melatonin suppression experiments, incorporating a Bayesian statistical model. We provide a Shiny web app for power analyses that allows users to modify various experimental parameters (sample size, individual-level heterogeneity, statistical significance threshold, light levels), and simulate a systematic shift in sensitivity (e.g., due to a pharmacological or other intervention). Our framework helps experimenters to design compelling and robust studies, offering novel insights into the underlying biological variability in melatonin suppression relevant for practical applications.

Estimating the effects of temperature on transmission of the human malaria parasite, Plasmodium falciparum.

Despite concern that climate change could increase the human risk to malaria in certain areas, the temperature dependency of malaria transmission is poorly characterized. Here, we use a mechanistic model fitted to experimental data to describe how Plasmodium falciparum infection of the African malaria vector, Anopheles gambiae, is modulated by temperature, including its influences on parasite establishment, conversion efficiency through parasite developmental stages, parasite development rate, and overall vector competence. We use these data, together with estimates of the survival of infected blood-fed mosquitoes, to explore the theoretical influence of temperature on transmission in four locations in Kenya, considering recent conditions and future climate change. Results provide insights into factors limiting transmission in cooler environments and indicate that increases in malaria transmission due to climate warming in areas like the Kenyan Highlands, might be less than previously predicted.

Understanding the impact of numerical solvers on inference for differential equation models.

Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers that seem sufficiently accurate for the forward problem, i.e. for obtaining an accurate simulation, might not be sufficiently accurate for the inverse problem, i.e. for inferring the model parameters from data. We show that for both fixed step and adaptive step ODE solvers, solving the forward problem with insufficient accuracy can distort likelihood surfaces, which might become jagged, causing inference algorithms to get stuck in local 'phantom' optima. We demonstrate that biases in inference arising from numerical approximation of ODEs are potentially most severe in systems involving low noise and rapid nonlinear dynamics. We reanalyse an ODE change point model previously fit to the COVID-19 outbreak in Germany and show the effect of the step size on simulation and inference results. We then fit a more complicated rainfall run-off model to hydrological data and illustrate the importance of tuning solver tolerances to avoid distorted likelihood surfaces. Our results indicate that, when performing inference for ODE model parameters, adaptive step size solver tolerances must be set cautiously and likelihood surfaces should be inspected for characteristic signs of numerical issues.

COVID-19 testing and reporting behaviours in England across different sociodemographic groups: a population-based study using testing data and data from community prevalence surveillance surveys.

BACKGROUND: Understanding underlying mechanisms of heterogeneity in test-seeking and reporting behaviour during an infectious disease outbreak can help to protect vulnerable populations and guide equity-driven interventions. The COVID-19 pandemic probably exerted different stresses on individuals in different sociodemographic groups and ensuring fair access to and usage of COVID-19 tests was a crucial element of England's testing programme. We aimed to investigate the relationship between sociodemographic factors and COVID-19 testing behaviours in England during the COVID-19 pandemic. METHODS: We did a population-based study of COVID-19 testing behaviours with mass COVID-19 testing data for England and data from community prevalence surveillance surveys (REACT-1 and ONS-CIS) from Oct 1, 2020, to March 30, 2022. We used mass testing data for lateral flow device (LFD; data for approximately 290 million tests performed and reported) and PCR (data for approximately 107 million tests performed and returned from the laboratory) tests made available for the general public and provided by date and self-reported age and ethnicity at the lower tier local authority (LTLA) level. We also used publicly available data on mean population size estimates for individual LTLAs, and data on ethnic groups, age groups, and deprivation indices for LTLAs. We did not have access to REACT-1 or ONS-CIS prevalence data disaggregated by sex or gender. Using a mechanistic causal model to debias the PCR testing data, we obtained estimates of weekly SARS-CoV-2 prevalence by both self-reported ethnic groups and age groups for LTLAs in England. This approach to debiasing the PCR (or LFD) testing data also estimated a testing bias parameter defined as the odds of testing in infected versus not infected individuals, which would be close to zero if the likelihood of test seeking (or seeking and reporting) was the same regardless of infection status. With confirmatory PCR data, we estimated false positivity rates, sensitivity, specificity, and the rate of decline in detection probability subsequent to reporting a positive LFD for PCR tests by sociodemographic groups. We also estimated the daily incidence, allowing us to calculate the fraction of cases captured by the testing programme. FINDINGS: From March, 2021 onwards, individuals in the most deprived regions reported approximately half as many LFD tests per capita as individuals in the least deprived areas (median ratio 0·50 [IQR 0·44-0·54]). During the period October, 2020, to June, 2021, PCR testing patterns showed the opposite trend, with individuals in the most deprived areas performing almost double the number of PCR tests per capita than those in the least deprived areas (1·8 [1·7-1·9]). Infection prevalences in Asian or Asian British individuals were considerably higher than those of other ethnic groups during the alpha (B.1.1.7) and omicron (B.1.1.529) BA.1 waves. Our estimates indicate that the England Pillar 2 COVID-19 testing programme detected 26-40% of all cases (including asymptomatic cases) over the study period with no consistent differences by deprivation levels or ethnic groups. Testing biases for PCR were generally higher than those for LFDs, in line with the general policy of symptomatic and asymptomatic use of these tests. Deprivation and age were associated with testing biases on average; however, the uncertainty intervals overlapped across deprivation levels, although the age-specific patterns were more distinct. We also found that ethnic minorities and older individuals were less likely to use confirmatory PCR tests through most of the pandemic and that delays in reporting a positive LFD test were possibly longer in populations self-reporting as "Black; African; Black British or Caribbean". INTERPRETATION: Differences in testing behaviours across sociodemographic groups might be reflective of the higher costs of self-isolation to vulnerable populations, differences in test accessibility, differences in digital literacy, and differing perceptions about the utility of tests and risks posed by infection. This study shows how mass testing data can be used in conjunction with surveillance surveys to identify gaps in the uptake of public health interventions both at fine-scale levels and across sociodemographic groups. It provides a framework for monitoring local interventions and yields valuable lessons for policy makers in ensuring an equitable response to future pandemics. FUNDING: UK Health Security Agency.