Author response: Selection for infectivity profiles in slow and fast epidemics, and the rise of SARS-CoV-2 variants
Year: 2022
Type: peer-review
Abstract: Article Figures and data Abstract Editor's evaluation eLife digest Introduction Results Discussion Materials and methods Data availability References Decision letter Author response Article and author information Metrics Abstract Evaluating the characteristics of emerging SARS-CoV-2 variants of concern is essential to inform pandemic risk assessment. A variant may grow faster if it produces a larger number of secondary infections (“R advantage”) or if the timing of secondary infections (generation time) is better. So far, assessments have largely focused on deriving the R advantage assuming the generation time was unchanged. Yet, knowledge of both is needed to anticipate the impact. Here, we develop an analytical framework to investigate the contribution of both the R advantage and generation time to the growth advantage of a variant. It is known that selection on a variant with larger R increases with levels of transmission in the community. We additionally show that variants conferring earlier transmission are more strongly favored when the historical strains have fast epidemic growth, while variants conferring later transmission are more strongly favored when historical strains have slow or negative growth. We develop these conceptual insights into a new statistical framework to infer both the R advantage and generation time of a variant. On simulated data, our framework correctly estimates both parameters when it covers time periods characterized by different epidemiological contexts. Applied to data for the Alpha and Delta variants in England and in Europe, we find that Alpha confers a+54% [95% CI, 45–63%] R advantage compared to previous strains, and Delta +140% [98–182%] compared to Alpha, and mean generation times are similar to historical strains for both variants. This work helps interpret variant frequency dynamics and will strengthen risk assessment for future variants of concern. Editor's evaluation This manuscript will be of broad interest to readers interested in understanding the characteristics of variants in ongoing epidemics that lead to faster (or slower) growth rates and will be of particular interest to those wishing to understand the factors leading to the selection of SARS–CoV–2 variants. The selective advantage of a novel strain of a pathogen depends not only on its relative transmissibility but also on its generation time relative to other strains; the relation between transmissibility, transmission advantage and generation time changes across different phases of the epidemic. Key innovations in this paper are a robust framework for using this relationship to make statistical inferences about both the transmissibility advantage and generation time of an emerging variant and conceptual novelty in the general investigation of selection on infectivity profiles. The approach is supported by simulation studies and applied to the Alpha and Delta SARS–CoV–2 variants to show that selection was likely driven by changes in transmissibility rather than changes in the generation time. https://doi.org/10.7554/eLife.75791.sa0 Decision letter Reviews on Sciety eLife's review process eLife digest Mutations in genes of the SARS-CoV-2 virus have generated new variants of concern, like Alpha, Delta, and more recently Omicron. These strains contain genetic modifications that help the virus spread more easily as well as altering the severity of the illness it causes. This has led to rising numbers of infections, known as epidemic waves, in many parts of the world. Tracking new variants of concern is crucial to protecting the public. To do this, scientists monitor how many people one person with the virus can infect, also known as the number of secondary infections. They may also measure when in the course of the illness an individual may pass along the virus to others. Together, these metrics help determine how fast and large an outbreak caused by a new variant will grow. The more people the new variant infects and the quicker it spreads, the more likely it is to replace existing strains of the virus. So far, most studies have assumed that the growth rate of a new variant solely depends on the number of secondary infections, and the timing of secondary infections is often not considered. To address this, Blanquart et al. built a mathematical model that combines both these parameters to determine the growth rate of new viral strains. The model showed that variants which rapidly cause secondary infections have a larger growth advantage over existing strains when the virus is more easily transmitted between individuals and the epidemic spreads rapidly. But when there is less transmission and the epidemic is declining, variants that generate secondary infections after a longer time have an advantage. For example, when control measures like mask wearing or social distancing are in place, delayed secondary infections may be more advantageous. Blanquart et al. then applied their model to data from the Alpha and Delta variant outbreaks in the United Kingdom. They found that Alpha and Delta did not change the timing of secondary infections compared to previously circulating strains. But the Alpha variant had a 54% transmission advantage over previous strains and the Delta variant had a 140% transmission advantage over Alpha. Taken together, these findings suggest that the timing of secondary infections and transmission rates both play an important role in how quickly a virus spreads. The new mathematical model created by Blanquart et al. may help epidemiologists better predict the trajectory of new SARS-CoV-2 variants and determine how to best control their spread. Introduction Human pathogens rapidly adapt to their hosts. During the short evolutionary history of SARS-CoV-2 with humans, since the host shift in late 2019, several selected mutations and combinations of mutations (variants) have emerged: for example, the D614G mutation in Spring 2020 (Volz et al., 2021a), the variant Alpha in Fall 2020 (B.1.1.7), and Delta in Spring 2020 (B.1.617.2). Variant Alpha, first detected in England in September 2020, rapidly rose in frequency in October 2020 in England and spread to multiple countries, causing important rebounds of the epidemic (Volz et al., 2021b; Davies et al., 2021; Borges et al., 2021; Washington et al., 2021; Gaymard et al., 2021). Later on, the Delta variant became dominant in many parts of the world, leading to new surges in infections and hospitalizations in 2021. Each time a variant of concern emerges, it is essential to determine whether it has acquired a competitive advantage compared to circulating SARS-CoV-2 strains, in order to anticipate the potential effects on the epidemic trajectory and hospital admissions. The advantage can be caused by an increased capacity to transmit or to escape the immune response acquired by previous infection or vaccination. The transmission advantage, hereafter R advantage, of an emerging variant has been defined as the relative increase in the effective reproduction number R (average number of secondary cases). It has often been estimated by analysing the rise in the variant’s frequency, under the assumption that all strains shared the same generation time distribution (i.e. delay between infection in the primary case and the people they infect). However, this has sometimes led to unstable estimates of the R advantage. For example, in the United Kingdom, the R advantage of the Alpha variant was estimated to decline from +89 to +54% from December to mid-January. Estimates of R advantage of Alpha and Delta across countries also exhibit substantial variability (Campbell et al., 2021). The reasons why estimates of the R advantage would vary with the epidemiological context remain obscure. The instability this generates is problematic for planning since it affects medium and long-term evaluations of the epidemic trajectory. It also suggests that methods currently used to ascertain the risks posed by emerging variants may miss important drivers of emergence. The evolutionary fate of an emerging variant is ultimately determined by its exponential growth rate relative to that of the circulating strains. When evaluating the growth advantage of a variant, the focus has so far largely been on estimating differences in R. However, the growth rate of an emerging variant depends not only on the effective reproduction number but also on the generation time distribution. Natural selection is therefore expected to act on the full infectivity profile, which combines both R and the generation time. Previous theoretical work has explored how selection acts on various pathogen life-history traits (Lenski and May, 1994; Day and Proulx, 2004; Day and Gandon, 2007; Day et al., 2020), with most often a focus on selection for transmissibility and virulence (disease-induced mortality). Recent work on SARS-CoV-2 mentioned the possibility of selection on the time interval between infection and infectiousness, but did not characterize the sign and magnitude of this component of selection in detail (Day et al., 2020). Here, we develop a mathematical model to investigate how the growth advantage may vary with the epidemiological context when variants have different generation times. We use the relationship between infectivity profile and growth rate to explore conceptually how infectivity profiles are selected for. We find that the growth advantage of a variant generally depends on the epidemiological context, more precisely on whether the transmission is low (declining or slow epidemic) or high (fast epidemic). We call “transmission” the rate of infectious contacts in the community, which depends on the epidemiological context, for example, social distancing, mask-wearing, population immunity, seasonal effects, etc. We use the time-varying reproduction number of the historical strain (taken as a reference), denoted RHt , as a proxy for transmission. Our conceptual exploration suggests that it is possible to infer changes in infectivity profile associated with an emerging variant, and not only its growth rate, in settings where transmission changes over time. We thus develop an inference framework to estimate changes in infectivity profile associated with a variant and more precisely characterize the R advantage of the variant and its dependency to the epidemiological context. We apply our approach to data on Alpha and Delta variants frequency over time in England in Fall 2020 and Spring 2021. On the methodological front, the originality of our work lies in the use of the Euler-Lotka equation to explore generally how infectivity profiles are selected for (in contrast to previous work using ordinary differential equation models), and in the development of an inference framework. Results General Principle We consider a situation where a variant, or a set of strains with similar properties, has been circulating for some time (historical strains H) and is challenged by an emerging variant E. When investigating the rise of variant E relative to historical strains H, we typically only observe the exponential growth rates of H and E strains, which translates into a growth advantage (or disadvantage). More precisely, if the historical strains grow with an exponential rate rH and the emerging variant grows with rate rE , the logit of the frequency of the emerging variant will grow linearly with slope rE-rH . This slope is called the growth advantage or selection coefficient. The selection coefficient of the emerging variant relative to historical strains is obtained by a linear regression of the logit frequency of the variant over time. It is then typically assumed that both variant and historical strains share the same distribution of generation time. The effective reproduction number of the emerging variant relative to historical strains, called the R advantage (RE/RH-1), can thus be deduced. Yet natural selection acts on the full infectivity profile of a strain, not only on the effective reproduction number. The infectivity profile is a function βτ=R w(τ) of the time since infection τ, where R is the effective reproduction number, while w(τ) characterizes the distribution of generation time and satisfies ∫τ=0∞wτdτ=1. Selection for Infectivity Profiles in Slow and Fast Epidemics We first examine how the selection coefficient rE-rH varies with transmission for various infectivity profiles of the variant compared to historical strains (Figure 1). To do so, we use the Euler-Lotka equation, relating the growth rate to the effective reproduction number and the distribution of the generation time (Methods). This equation assumes that each variant grows or declines exponentially and that the distribution of the time since infection is at any time at its equilibrium value: an exponential distribution with rate equal to the exponential growth rate (Wallinga and Lipsitch, 2007). This equilibrium distribution naturally follows from the assumption of exponential growth, which implies that individuals infected t+Δt days ago are er Δt less numerous than individuals infected t days ago, with r the exponential growth rate. Figure 1 Download asset Open asset Variation of the selection coefficient as a function of transmission, for several infectivity profiles of emerging variants. The Panels A and B show several variant infectivity profiles with the same effective reproduction number as historical strains (A) or an effective reproduction number increased by +10% (B). For variants with shorter or longer mean generation time, the relative mean generation time is –15 or +15% compared to historical strains. For variants with shorter or longer standard deviation (sd) in generation time, the relative sd in generation time is –40 or +40% compared to historical strains. The Panels C and D show the selection coefficient of these variants as a function of the proxy for transmission (effective reproduction number of the historical strains). The gray lines show the selection coefficient if the variant had the same infectivity profile as historical strains, with the infectivity profile increased uniformly by a constant as commonly assumed. We first note that even for variants with an R advantage but not affecting the distribution of generation time, the selection coefficient increases with transmission (Day and Proulx, 2004; Day and Gandon, 2007; Day et al., 2020). This relationship is shown in Figure 1 as gray lines for a series of variants conferring a R advantage of +10 −+50% but not affecting the gamma-distributed generation time. Several more general infectivity profiles of the emerging variant are now considered. We first consider variants with the same effective reproduction number as historical strains, which can be selected for if secondary infections occur at different timings (Figure 1A and C). A variant with a shorter mean generation time is selected for in a growing epidemic because it produces the same number of secondary infections in a shorter time, while the same variant is counter-selected in a declining epidemic (Figure 1C). The opposite holds for variants with a longer mean generation time. A variant with a larger standard deviation (sd) in generation time is always selected for because it enjoys both an excess of early secondary infections and late secondary infections compared to historical strains. The opposite holds for a variant with smaller sd in generation time. Generally, in a growing epidemic, more individuals have been infected recently and there is a greater advantage to early transmission, while in a declining epidemic, more individuals have late time since infection and therefore strains that transmit at later times have a greater advantage. Again, this is a direct demographic consequence of growth or decline of the epidemic. Selected variants could thus enjoy a more favorable timing of secondary infections, even in the absence of R advantage (Park et al., 2021). Additionally, an advantage in the timing of secondary infections can of course be combined with a R advantage. We, therefore, consider hypothetical emerging variants with a range of generation time distributions and a+10% R advantage (Figure 1B and D). These variants generally have a stronger growth advantage, but the variation in selection coefficient with the transmission is similar to that of the analogous variants without the R advantage (Figure 1C and D). Notably, in both cases, there is a level of transmission (expressed in terms of the effective reproduction number of the historical strain RH) at which all variants perform similarly regardless of their distribution of generation time. This level of transmission is precisely where the size of the variant epidemic is constant in time (variant growth rate is 0). For example, this level of transmission is RH=1 for a variant not affecting R (Figure 1C). For a 10% R advantage, this level is RH=1/1.1 (Figure 1D). Generally, for a variant conferring a R advantage δ1 , the level of transmission is RH=1/(1+δ1). At this level of transmission, the distribution of time since infection is uniform—all ages of infection are equally represented—and therefore the infectivity profile does not matter for the selection of the variant. While these results assume that the time since infection is fixed at its equilibrium distribution, it is not necessarily the case in a realistic scenario where transmission changes over time. We applied these analytical results to simulations where the time since infection structure emerges from the model dynamics. The epidemiological dynamics are simulated with discrete-time renewal equations including time-varying transmission, the build-up of population immunity (assumed to be identical for historical strains and emerging variants), and detection and isolation of cases (Methods). In the context of progressive decline of levels of transmission, variants with the same reproduction number but distinct generation time distributions can exhibit a variety of epidemiological and frequency dynamics (Figure 2A and C). The epidemiological dynamics, that is the daily number of variant cases, are also strongly affected by the generation time distribution of the variant (Figure 2A). Frequency trajectories can be non-monotonous: a variant with shorter mean generation time could be selected, increase in frequency, but then decline in frequency as transmission declines (Figure 2C). Several variants all sharing the same 10% R advantage exhibit a range of frequency trajectories in the simulations: the trajectory of one of them initially resembles that of a variant with a+30% advantage and same generation time distribution as historical strains (Figure 2D, light blue line compared to +30% curve), while others stagnate at low frequency (Figure 2D, blue and green lines). Figure 2 with 1 supplement see all Download asset Open asset Epidemiological and evolutionary trajectories of several types of emerging variants competing with historical strains. The Panels A and B show epidemiological dynamics (daily cases number of historical strains in black, of variants in colours). Historical strains display several slightly different curves when competing with each of the variants because of the weak competition brought about by the build-up of population immunity. The Panels C and D show evolutionary trajectories, with daily variant frequency as colored lines. The light gray curves show the frequency dynamics of variants with a +10 to +50%R advantage, with the same generation time as historical strains. The inset shows the transmission through time in these simulations, given by the basic reproduction number of historical strains (see details of simulation model in Methods). Inference of the Infectivity Profiles of Variants The variety of variant trajectories shown in Figure 2 suggests that it may be possible to infer the changes in infectivity profiles of variants by analyzing their growth advantage as a function of transmission when we have accumulated sufficient data documenting growth rates in different epidemiological situations. We first tested this idea on simulated data, in scenarios of emergence and progressive decline in transmission. Our inference framework is based on the relationship between growth rates of historical strains (rH) and of the emerging variant (rE)). Our aim is to infer three parameters δ1 , δ2 , δ3 , characterizing respectively the R advantage, the mean generation time and the sd of generation time of the emerging strain relative to that of historical strains. The relationship can be used to infer properties of the variant infectivity profile, but not with a classical linear regression because both rH and rE are measured with errors with a complex covariance structure. Furthermore, large errors can blur the subtle distinction between different profiles (Figure 2—figure supplement 1). The error variance is inversely related to the number of cases and the size of the sample used to assay variant frequency (“sample size”). When the total number of cases and the sample size are both large, the joint distribution of the time series of estimated r^H and r^E is approximately multivariate normal (Methods). The mean vector of the multivariate normal depends on the parameters RH(t) (the time series of effective reproduction number), on the mean and sd of the historical strains’ generation time (fixed to 6.5 and 4 days respectively), and on the parameters of interest δ1 , δ2 , δ3 . The variance-covariance structure of the multivariate normal distribution is fixed and depends on the daily total number of cases and the sample size. It is also possible to derive the multivariate normal distribution of r^H and r^E across different independent regions instead of over time (Methods). With this statistical framework at hand, we conducted a simulation study to infer jointly the R advantage of the variants and their relative mean generation time (Figure 3). In these simulations, we systematically varied the infectivity profile of the emerging strain: R advantage (δ1) from +0 to +50% and relative mean generation time (δ2) from –40 to +40%. We assumed the emerging strain had the same sd of generation time as historical strains (δ3=0). We jointly inferred the two parameters δ1 and δ2 for different sample sizes (sample size, N=1000, 3000, 10,000 per day) and for different scenarios of variability in epidemiological context, from a strong to a weak decline. In the explicit simulations, the parameter that we tune is the basic reproductive number R0,Ht assumed to decline from 1.5 to 0.5, 1.3 to and to over days in the three and the inferred parameters the effective reproductive number after for population immunity and case detection and isolation (Figure supplement 1). Figure with see all Download asset Open asset Inference of the R advantage and relative mean generation time in the simulation The show the between inferred and The three show different sample sizes used to infer variant 10,000 and daily by The show the R advantage, the the mean generation time and show inference for large, and variability in The simulations are with and For the inference of the R advantage, we used a variant relative mean generation time of inferred by the while for the inference of relative mean generation time we used the We used a greater R advantage when the mean generation time is or longer to the emerging variant a frequency over the days time considered. For each parameter simulations with the same the same epidemiological dynamics, but with different error on data error on cases number and error on The R advantage was always precisely inferred even with sample size and variability in effective reproduction number (Figure In the relative mean generation time was well inferred only for a large sample size and large variability in effective reproduction number (Figure This was because the large error on rE and rH to the sample size, in with the variability in inference of the relationship between rE and rH For sample data by of by for the same inference (Figure green We tested the of the inference framework in several we whether between infections and cases the scenario where the mean time from to case detection was days instead of days (Figure supplement The does not the of inference with Figure as expected since the time series of number of detected cases and variant frequency share the same with infections. we applied our framework to simulations where the generation time is not gamma-distributed as in our we assumed that transmission is not possible the first two this is by a gamma-distributed timing of secondary infections. This between assumption and the generation time distribution led to a of the relative mean generation time (Figure supplement we inferred parameters when there is a decline in RHt from 1.5 to at a fixed instead of a progressive decline. In this the distribution of time since infection may not after the the assumption of our analytical correctly inferred when the time series after the decline was With only days of data after the the inferred relative mean generation time was inferred if cases to measure the variant frequency (Figure supplement 3). was better for variants with a shorter generation time than the historical strains relative mean generation This is because their distribution of time since infection faster after the decline. to the Variants Alpha and Delta in England We used this framework to infer the R advantage and mean generation time of Alpha in We used data from England on cases numbers and the frequency of the Alpha variant 1). In the frequency of the Alpha variant could be quickly inferred because some of the for SARS-CoV-2 detection a caused by the of Alpha The variant Alpha through from to frequency over data of for the (Figure our we found that Alpha had a R advantage equal to and the same mean generation time as previous strains, (Figure Figure 4 Download asset Open asset Data used for A and of the Alpha variant frequency in England (A) and of the Delta variant frequency in England estimated through These are shown with the dynamics of cases over September for Alpha, and for the dynamics of the Delta variant frequency in For each we used the growth rate of historical strains and emerging Delta variant when the frequency with larger and line on the Figure with see all Download asset Open asset Inference of the infectivity profiles of the Alpha and Delta variants. (A) for how the relationship depends on the mean generation time for hypothetical variants with R advantage and relative mean generation time of and in Figures 1 and The parameter the coefficient of variation of the historical strains distribution of generation time, equal to the standard deviation (sd) by the mean of the distribution. The between estimated growth rates of historical strains and variants in For Alpha this is a in England over September 2020 to 2021. For Delta is a across The curves with are the model for each variant and show that the selection coefficient (the to the is constant across epidemiological that the mean generation time is not in these variants. The inferred infectivity profile of each variant with with the infectivity profile of historical strains circulating the rise of the Alpha variant This inference assumes a gamma-distributed generation time. The
Source: eLife
Institutions: Centre National de la Recherche Scientifique, Inserm, Collège de France, Université Paris Sciences et Lettres, Centre Interdisciplinaire de Recherche en Biologie, Institut Pasteur, Université Paris Cité, Génomique évolutive, modélisation et santé, Laboratory of Data Discovery for Health, University of Hong Kong, Université Paris-Est Créteil, Sorbonne Université, Institut National de Recherche pour l'Agriculture, l'Alimentation et l'Environnement, Institut d'écologie et des sciences de l'environnement de Paris, Institut de Recherche pour le Développement
Language: English
Subfield: Infectious Diseases
Field: Medicine
Domain: Health Sciences
Open Access status: gold
Estimated APC paid: $2,500
OpenAlex ID: W4283029485