Correction: Practical considerations for measuring the effective reproductive number, Rt.

[This corrects the article DOI: 10.1371/journal.pcbi.1008409.].

Practical considerations for measuring the effective reproductive number, Rt.

Estimation of the effective reproductive number Rt is important for detecting changes in disease transmission over time. During the Coronavirus Disease 2019 (COVID-19) pandemic, policy makers and public health officials are using Rt to assess the effectiveness of interventions and to inform policy. However, estimation of Rt from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of Rt, we recommend the approach of Cori and colleagues, which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis, are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to the spread. We advise caution when using methods derived from the approach of Bettencourt and Ribeiro, as the resulting Rt estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in Rt estimation.

Practical considerations for measuring the effective reproductive number, R t.

Estimation of the effective reproductive number, R t , is important for detecting changes in disease transmission over time. During the COVID-19 pandemic, policymakers and public health officials are using R t to assess the effectiveness of interventions and to inform policy. However, estimation of R t from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of R t , we recommend the approach of Cori et al. (2013), which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis (2004), are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to spread. We advise against using methods derived from Bettencourt and Ribeiro (2008), as the resulting R t estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in R t estimation.

Designing Robust N-of-1 Studies for Precision Medicine: Simulation Study and Design Recommendations.

BACKGROUND: Recent advances in molecular biology, sensors, and digital medicine have led to an explosion of products and services for high-resolution monitoring of individual health. The N-of-1 study has emerged as an important methodological tool for harnessing these new data sources, enabling researchers to compare the effectiveness of health interventions at the level of a single individual. OBJECTIVE: N-of-1 studies are susceptible to several design flaws. We developed a model that generates realistic data for N-of-1 studies to enable researchers to optimize study designs in advance. METHODS: Our stochastic time-series model simulates an N-of-1 study, incorporating all study-relevant effects, such as carryover and wash-in effects, as well as various sources of noise. The model can be used to produce realistic simulated data for a near-infinite number of N-of-1 study designs, treatment profiles, and patient characteristics. RESULTS: Using simulation, we demonstrate how the number of treatment blocks, ordering of treatments within blocks, duration of each treatment, and sampling frequency affect our ability to detect true differences in treatment efficacy. We provide a set of recommendations for study designs on the basis of treatment, outcomes, and instrument parameters, and make our simulation software publicly available for use by the precision medicine community. CONCLUSIONS: Simulation can facilitate rapid optimization of N-of-1 study designs and increase the likelihood of study success while minimizing participant burden.

Identifying functional groups among the diverse, recombining antigenic var genes of the malaria parasite Plasmodium falciparum from a local community in Ghana.

A challenge in studying diverse multi-copy gene families is deciphering distinct functional types within immense sequence variation. Functional changes can in some cases be tracked through the evolutionary history of a gene family; however phylogenetic approaches are not possible in cases where gene families diversify primarily by recombination. We take a network theoretical approach to functionally classify the highly recombining var antigenic gene family of the malaria parasite Plasmodium falciparum. We sample var DBLα sequence types from a local population in Ghana, and classify 9,276 of these variants into just 48 functional types. Our approach is to first decompose each sequence type into its constituent, recombining parts; we then use a stochastic block model to identify functional groups among the parts; finally, we classify the sequence types based on which functional groups they contain. This method for functional classification does not rely on an inferred phylogenetic history, nor does it rely on inferring function based on conserved sequence features. Instead, it infers functional similarity among recombining parts based on the sharing of similar co-occurrence interactions with other parts. This method can therefore group sequences that have undetectable sequence homology or even distinct origination. Describing these 48 var functional types allows us to simplify the antigenic diversity within our dataset by over two orders of magnitude. We consider how the var functional types are distributed in isolates, and find a nonrandom pattern reflecting that common var functional types are non-randomly distinct from one another in terms of their functional composition. The coarse-graining of var gene diversity into biologically meaningful functional groups has important implications for understanding the disease ecology and evolution of this system, as well as for designing effective epidemiological monitoring and intervention.

Networks of genetic similarity reveal non-neutral processes shape strain structure in Plasmodium falciparum.

Pathogens compete for hosts through patterns of cross-protection conferred by immune responses to antigens. In Plasmodium falciparum malaria, the var multigene family encoding for the major blood-stage antigen PfEMP1 has evolved enormous genetic diversity through ectopic recombination and mutation. With 50-60 var genes per genome, it is unclear whether immune selection can act as a dominant force in structuring var repertoires of local populations. The combinatorial complexity of the var system remains beyond the reach of existing strain theory and previous evidence for non-random structure cannot demonstrate immune selection without comparison with neutral models. We develop two neutral models that encompass malaria epidemiology but exclude competitive interactions between parasites. These models, combined with networks of genetic similarity, reveal non-neutral strain structure in both simulated systems and an extensively sampled population in Ghana. The unique population structure we identify underlies the large transmission reservoir characteristic of highly endemic regions in Africa.

Recurring infection with ecologically distinct HPV types can explain high prevalence and diversity.

The high prevalence of human papillomavirus (HPV), the most common sexually transmitted infection, arises from the coexistence of over 200 genetically distinct types. Accurately predicting the impact of vaccines that target multiple types requires understanding the factors that determine HPV diversity. The diversity of many pathogens is driven by type-specific or "homologous" immunity, which promotes the spread of variants to which hosts have little immunity. To test for homologous immunity and to identify mechanisms determining HPV transmission, we fitted nonlinear mechanistic models to longitudinal data on genital infections in unvaccinated men. Our results provide no evidence for homologous immunity, instead showing that infection with one HPV type strongly increases the risk of infection with that type for years afterward. For HPV16, the type responsible for most HPV-related cancers, an initial infection increases the 1-year probability of reinfection by 20-fold, and the probability of reinfection remains 14-fold higher 2 years later. This increased risk occurs in both sexually active and celibate men, suggesting that it arises from autoinoculation, episodic reactivation of latent virus, or both. Overall, our results suggest that high HPV prevalence and diversity can be explained by a combination of a lack of homologous immunity, frequent reinfections, weak competition between types, and variation in type fitness between host subpopulations. Because of the high risk of reinfection, vaccinating boys who have not yet been exposed may be crucial to reduce prevalence, but our results suggest that there may also be large benefits to vaccinating previously infected individuals.

Host population structure and treatment frequency maintain balancing selection on drug resistance.

It is a truism that antimicrobial drugs select for resistance, but explaining pathogen- and population-specific variation in patterns of resistance remains an open problem. Like other common commensals, Streptococcus pneumoniae has demonstrated persistent coexistence of drug-sensitive and drug-resistant strains. Theoretically, this outcome is unlikely. We modelled the dynamics of competing strains of S. pneumoniae to investigate the impact of transmission dynamics and treatment-induced selective pressures on the probability of stable coexistence. We find that the outcome of competition is extremely sensitive to structure in the host population, although coexistence can arise from age-assortative transmission models with age-varying rates of antibiotic use. Moreover, we find that the selective pressure from antibiotics arises not so much from the rate of antibiotic use per se but from the frequency of treatment: frequent antibiotic therapy disproportionately impacts the fitness of sensitive strains. This same phenomenon explains why serotypes with longer durations of carriage tend to be more resistant. These dynamics may apply to other potentially pathogenic, microbial commensals and highlight how population structure, which is often omitted from models, can have a large impact.

Limits to Causal Inference with State-Space Reconstruction for Infectious Disease.

Infectious diseases are notorious for their complex dynamics, which make it difficult to fit models to test hypotheses. Methods based on state-space reconstruction have been proposed to infer causal interactions in noisy, nonlinear dynamical systems. These "model-free" methods are collectively known as convergent cross-mapping (CCM). Although CCM has theoretical support, natural systems routinely violate its assumptions. To identify the practical limits of causal inference under CCM, we simulated the dynamics of two pathogen strains with varying interaction strengths. The original method of CCM is extremely sensitive to periodic fluctuations, inferring interactions between independent strains that oscillate with similar frequencies. This sensitivity vanishes with alternative criteria for inferring causality. However, CCM remains sensitive to high levels of process noise and changes to the deterministic attractor. This sensitivity is problematic because it remains challenging to gauge noise and dynamical changes in natural systems, including the quality of reconstructed attractors that underlie cross-mapping. We illustrate these challenges by analyzing time series of reportable childhood infections in New York City and Chicago during the pre-vaccine era. We comment on the statistical and conceptual challenges that currently limit the use of state-space reconstruction in causal inference.

Trade-offs in antibody repertoires to complex antigens.

Pathogens vary in their antigenic complexity. While some pathogens such as measles present a few relatively invariant targets to the immune system, others such as malaria display considerable antigenic diversity. How the immune response copes in the presence of multiple antigens, and whether a trade-off exists between the breadth and efficacy of antibody (Ab)-mediated immune responses, are unsolved problems. We present a theoretical model of affinity maturation of B-cell receptors (BCRs) during a primary infection and examine how variation in the number of accessible antigenic sites alters the Ab repertoire. Naive B cells with randomly generated receptor sequences initiate the germinal centre (GC) reaction. The binding affinity of a BCR to an antigen is quantified via a genotype-phenotype map, based on a random energy landscape, that combines local and distant interactions between residues. In the presence of numerous antigens or epitopes, B-cell clones with different specificities compete for stimulation during rounds of mutation within GCs. We find that the availability of many epitopes reduces the affinity and relative breadth of the Ab repertoire. Despite the stochasticity of somatic hypermutation, patterns of immunodominance are strongly shaped by chance selection of naive B cells with specificities for particular epitopes. Our model provides a mechanistic basis for the diversity of Ab repertoires and the evolutionary advantage of antigenically complex pathogens.

Seasonality in the migration and establishment of H3N2 Influenza lineages with epidemic growth and decline.

BACKGROUND: Influenza A/H3N2 has been circulating in humans since 1968, causing considerable morbidity and mortality. Although H3N2 incidence is highly seasonal, how such seasonality contributes to global phylogeographic migration dynamics has not yet been established. In this study, we incorporate time-varying migration rates in a Bayesian MCMC framework. We focus on migration within China, and to and from North-America as case studies, then expand the analysis to global communities. RESULTS: Incorporating seasonally varying migration rates improves the modeling of migration in our regional case studies, and also in a global context. In our global model, windows of increased immigration map to the seasonal timing of epidemic spread, while windows of increased emigration map to epidemic decline. Seasonal patterns also correlate with the probability that local lineages go extinct and fail to contribute to long term viral evolution, as measured through the trunk of the phylogeny. However, the fraction of the trunk in each community was found to be better determined by its overall human population size. CONCLUSIONS: Seasonal migration and rapid turnover within regions is sustained by the invasion of 'fertile epidemic grounds' at the end of older epidemics. Thus, the current emphasis on connectivity, including air-travel, should be complemented with a better understanding of the conditions and timing required for successful establishment. Models which account for migration seasonality will improve our understanding of the seasonal drivers of influenza, enhance epidemiological predictions, and ameliorate vaccine updating by identifying strains that not only escape immunity but also have the seasonal opportunity to establish and spread. Further work is also needed on additional conditions that contribute to the persistence and long term evolution of influenza within the human population, such as spatial heterogeneity with respect to climate and seasonality.

Homology blocks of Plasmodium falciparum var genes and clinically distinct forms of severe malaria in a local population.

BACKGROUND: The primary target of the human immune response to the malaria parasite Plasmodium falciparum, P. falciparum erythrocyte membrane protein 1 (PfEMP1), is encoded by the members of the hyper-diverse var gene family. The parasite exhibits antigenic variation via mutually exclusive expression (switching) of the ~60 var genes within its genome. It is thought that different variants exhibit different host endothelial binding preferences that in turn result in different manifestations of disease. RESULTS: Var sequences comprise ancient sequence fragments, termed homology blocks (HBs), that recombine at exceedingly high rates. We use HBs to define distinct var types within a local population. We then reanalyze a dataset that contains clinical and var expression data to investigate whether the HBs allow for a description of sequence diversity corresponding to biological function, such that it improves our ability to predict disease phenotype from parasite genetics. We find that even a generic set of HBs, which are defined for a small number of non-local parasites: capture the majority of local sequence diversity; improve our ability to predict disease severity from parasite genetics; and reveal a previously hypothesized yet previously unobserved parasite genetic basis for two forms of severe disease. We find that the expression rates of some HBs correlate more strongly with severe disease phenotypes than the expression rates of classic var DBLα tag types, and principal components of HB expression rate profiles further improve genotype-phenotype models. More specifically, within the large Kenyan dataset that is the focus of this study, we observe that HB expression differs significantly for severe versus mild disease, and for rosetting versus impaired consciousness associated severe disease. The analysis of a second much smaller dataset from Mali suggests that these HB-phenotype associations are consistent across geographically distant populations, since we find evidence suggesting that the same HB-phenotype associations characterize this population as well. CONCLUSIONS: The distinction between rosetting versus impaired consciousness associated var genes has not been described previously, and it could have important implications for monitoring, intervention and diagnosis. Moreover, our results have the potential to illuminate the molecular mechanisms underlying the complex spectrum of severe disease phenotypes associated with malaria--an important objective given that only about 1% of P. falciparum infections result in severe disease.

Long-lasting transition toward sustainable elimination of desert malaria under irrigation development.

In arid areas, people living in the proximity of irrigation infrastructure are potentially exposed to a higher risk of malaria due to changes in ecohydrological conditions that lead to increased vector abundance. However, irrigation provides a pathway to economic prosperity that over longer time scales is expected to counteract these negative effects. A better understanding of this transition between increased malaria risk and regional elimination, in particular whether it is slow or abrupt, is relevant to sustainable development and disease management. By relying on space as a surrogate for stages of time, we investigate this transition in a semidesert region of India where a megairrigation project is underway and expected to cover more than 1,900 million hectares and benefit around 1 million farmers. Based on spatio-temporal epidemiological cases of Plasmodium vivax malaria and land-use irrigation from remote sensing sources, we show that this transition is characterized by an enhanced risk in areas adjacent to the trunk of the irrigation network, despite a forceful and costly insecticide-based control. Moreover, this transition between climate-driven epidemics and sustained low risk has already lasted a decade. Given the magnitude of these projects, these results suggest that increased health costs have to be planned for over a long time horizon. They further highlight the need to integrate assessments of both health and environmental impacts to guide adaptive mitigation strategies. Our results should help to define and track these transitions in other arid parts of the world subjected to similar tradeoffs.

Spatial guilds in the Serengeti food web revealed by a Bayesian group model.

Food webs, networks of feeding relationships in an ecosystem, provide fundamental insights into mechanisms that determine ecosystem stability and persistence. A standard approach in food-web analysis, and network analysis in general, has been to identify compartments, or modules, defined by many links within compartments and few links between them. This approach can identify large habitat boundaries in the network but may fail to identify other important structures. Empirical analyses of food webs have been further limited by low-resolution data for primary producers. In this paper, we present a Bayesian computational method for identifying group structure using a flexible definition that can describe both functional trophic roles and standard compartments. We apply this method to a newly compiled plant-mammal food web from the Serengeti ecosystem that includes high taxonomic resolution at the plant level, allowing a simultaneous examination of the signature of both habitat and trophic roles in network structure. We find that groups at the plant level reflect habitat structure, coupled at higher trophic levels by groups of herbivores, which are in turn coupled by carnivore groups. Thus the group structure of the Serengeti web represents a mixture of trophic guild structure and spatial pattern, in contrast to the standard compartments typically identified. The network topology supports recent ideas on spatial coupling and energy channels in ecosystems that have been proposed as important for persistence. Furthermore, our Bayesian approach provides a powerful, flexible framework for the study of network structure, and we believe it will prove instrumental in a variety of biological contexts.