Evolutionary competition between types is often studied through changes in their relative frequencies, and the slope of the logit-transformed frequency of the invading type serves as an estimate of selective advantage. However, observed frequencies usually do not directly reflect the underlying process: events are recorded after sample collection and processing, which introduces delays. If delays are the same for all types, they merely shift the trajectory in time, but different delays lead to comparing types at different phases of the dynamics. This work sits alongside studies of evolutionary dynamics, horizontal gene transfer, and phylogenetic networks, where temporal distortions in inference are also important. The authors use the replacement of SARS-CoV-2 variants in the UK as a concrete example where genomic surveillance data have delays from infection to sample collection.
The authors analyze two types of models. In the simplest exponential replacement model, they mathematically show that fixed or random observation delays change only the intercept of the logit trajectory, not its slope. Then they use two-variant SIR models with full cross-immunity and its extensions (waning immunity, no cross-immunity, combined model), where the relative growth rate depends on time due to depletion of susceptibles. For numerical experiments, they set parameters mimicking epidemic waves and introduce observation delays by shifting the daily incidence of each variant by a fixed number of days or convolving with a delay distribution. They then estimate the slope of the logit curve from the delayed data and compare it with the undelayed value, computing the relative error.
In the exponential model, delays do not change the slope of the logit trajectory if delayed counts are aggregated before forming frequencies and boundary effects are absent. However, in SIR models, delays can substantially bias the slope estimate. For example, in the basic model with full cross-immunity, the relative error in slope ranged from -47.3% to +60.7% depending on the difference in delays and the timing of variant introduction. A key result: even when the delayed trajectory remains almost perfectly linear on the logit scale (R² = 0.9999), the bias can be about 31%. This means that a good logistic fit does not guarantee an unbiased estimate. In extended models with waning immunity and combined immunity, similar effects were observed, while in the model without cross-immunity, in some regimes the logit linearity was disrupted so much that the slope estimate lost meaning.
The work shows that the standard method of estimating selective advantage from the slope of the logit curve is sensitive to type-dependent observation delays. Unlike neighboring studies that focus on reconstructing full incidence (nowcasting) or correcting right truncation, this work examines the direct impact of delays on the estimate of relative fitness. This is the first systematic check of how type-specific delays distort the logit slope in nonlinear competition models. The results caution against interpreting empirical slopes as true growth advantages without accounting for delays, especially in epidemiology and microbial evolution.
The topic could develop toward statistical methods that correct for delay-induced bias. It may be possible to create models that simultaneously estimate delays and relative fitness from observational data. Also promising is studying the effect of delays in more complex models with multiple variants, spatial structure, and time-varying selection intensity. Another path is applying Bayesian approaches to account for uncertainty in delay distributions.
The results are important for epidemiology, especially for analyzing pathogen variant replacement using genomic surveillance data. They also apply to experimental evolution of microorganisms, where delays between sampling and frequency measurement can distort fitness estimates. Furthermore, the findings are relevant to population genetics and ecology when analyzing time series of allele or species frequencies.
The next step could be developing a method to correct the bias caused by delays, based on estimates of delay distributions for each type. It is also necessary to test how strongly delays affect estimates in real data with known delays, for example, in virus competition experiments.
In evolutionary biology, the question of how to accurately estimate selective coefficients from time series of frequencies under imperfect observation remains open. This work connects the problem of observation delays to the classical task of inferring fitness from type replacement data, showing that ignoring delays can lead to systematic errors. It underscores the need for methods robust to such distortions, which is part of the broader problem of identifying parameters in evolutionary processes.
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What the work does
The authors investigate how observation delays that depend on the variant type bias the estimate of selective advantage derived from the slope of the logit frequency curve of an invading type. Using the replacement of SARS-CoV-2 variants in the UK as an example and SIR models, they show that delays cause variants to be compared at different epidemic phases, and even when the logit trajectory is almost perfectly linear, the bias can go unnoticed.
Where its strength is
The strength of the work lies in clearly demonstrating that a good logistic fit does not guarantee an unbiased estimate, and in providing concrete numerical examples based on realistic epidemiological models.
Where it could go next
Worth trying
Why this may matter
Accounting for observation delays is critical for correctly estimating selective advantages in real time, which affects the prediction of pathogen evolution and public health decision-making.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.50, against a usual 0.64. It looks like few have drilled nearby.
In GISAID data for the UK, the logit slope for the Alpha→Delta replacement was 0.107 per day, and for Delta→Omicron BA.1 it was 0.237 per day, corresponding to a doubling of the new variant's proportion roughly every 6.5 and 2.9 days, respectively.