Christelle Baunez
email: christelle.baunez@univ-amu.fr

Mickael Degoulet
email: mickael.degoulet@univ-amu.fr

Stéphane Luchini
email: stephane.luchini@univ-amu.fr

Matteo L Pintus
email: matteo.pintus@agroparistech.fr

Patrick A Pintus
email: patrick.pintus@univ-amu.fr

Miriam Teschl
email: miriam.teschl@ehess.fr

The Acceleration Index as a Test-Controlled Reproduction Number: Application to COVID-19 in France *

Keywords: JEL Classification Numbers: I18, H12 COVID-19, Reproduction Number, Lock-down, Acceleration Index, Real-time Analysis, France

We show that the acceleration index, a novel yet simple indicator that measures acceleration of viral spread (Baunez et al., 2021), corrects the reproduction number for the time-varying amount of tests.

Not doing so results in the reproduction number being a biased estimate of viral acceleration and we provide a formal decomposition of the resulting bias. When applied to French data for the COVID-19 pandemic (from May 13 to November 19, 2020), our decomposition shows that the reproduction number consistently underestimates the resurgence of the pandemic since the summer of 2020, compared to the acceleration index which accounts for the time-varying volume of tests. Because the acceleration index aggregates all the relevant information and captures in real time the sizable time variation featured by viral circulation, it is a sufficient statistic to track the pandemic's propagation. As such it is a more accurate indicator to track the dynamics of an infectious disease outbreak in real time.

Introduction

The reproduction number is a widely used measure of how fast a pathogen propagates both at the outset and during an infectious disease outbreak (see for example May and Anderson [START_REF] May | Infectious Diseases of Humans: Dynamics and Control[END_REF]). One of its major shortcomings, however, is that it does not control for the quantity of tests (or any diagnostics) performed in real time. Doing so is of crucial importance for two reasons. One is the fact that accurate empirical estimates of reproduction numbers are time-varying in nature (see e.g. Fraser [11] among many others). A considerable source of time variation comes from the fact that the amount of tests changes substantially across time and hence affects the number of known cases, due to demand and supply effects. Second, testing acts as a magnifying lens on viral activity at least on the part of the population that has effectively been tested. The reproduction number however does not rely on that information but rather makes assumptions on the infectivity of positive cases based on the observation of onset of symptoms and transmission in closed systems such as households (see e.g. Cori et al. [START_REF] Ferguson | A New Framework and Software to Estimate Time-Varying Reproduction Numbers during Epidemics[END_REF]). But this information, again, depends on tests (or any diagnostics more generally). Hence inferring infectivity from (assumed) transmissions is only secondary information based on the availability of tests. The logical consequence of these observations is that any measure of virus reproduction should take tests (or any available diagnostics) into account and this is exactly the issue we address in this paper.

In Baunez et al. [START_REF] Baunez | An Early Assessment of Curfew and Second COVID-19 Lock-down on Virus Propagationin France[END_REF][START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF][START_REF] Baunez | COVID-19 Acceleration and Vaccine Status in France -Summer[END_REF], we have introduced an alternative and novel measure of viral spread in the context of COVID-19 -the acceleration index. This measure considers the variation of cases relative to the variation of tests and thus it avoids the shortcomings mentioned above. The purpose of this article is to discuss the reproduction number in the light of our acceleration index, and to show that the former is actually a special case of the latter and in fact a less accurate metric of the pandemic's time-varying speed.

We examine this important issue in two steps. In Section 2, we start from the very definition of the reproduction number as a gross rate of growth of infected people, traditionally denoted R, and derive a general formula that connects it to our acceleration index that we denote ε. The acceleration index is an elasticity that measures the proportional responsiveness of cases to tests, and it can also be thought of as the ratio between the current and average viral speeds. More specifically, we present an explicit measure of the ratio between R and ε, the interpretation of which is further discussed in terms of the infectivity and test contributions. Our theoretical inquiry stresses that while the acceleration index is a ratio of growth rates -that of cases divided by that of tests -the reproduction number tracks only the growth rate of cases. In other words, the acceleration index corrects the reproduction number for the time-varying amount of tests. Not doing so results in the reproduction number being a biased estimate of viral acceleration and we provide a formal decomposition of the resulting bias.

In Section 3, we apply such an analysis to French data and we show that there is a sizeable difference between both measures. Indeed that the reproduction number R largely under-estimates the spread of the virus, compared to our test-controlled measure of acceleration. In Appendix B, we provide a similar analysis for five other countries to show that this result is not limited to the French case. It is in this sense that we say that the reproduction number is biased. This has obviously important consequences if the reproduction number is used as the basis for which public health decisions such as entering or exiting a lock-down. We also look at the effects of the second lock-down period in France, which started October 30, 2020, through the lens of both indicators, as a further example that illustrates the bias unavoidably implied by not adjusting for the volume of tests over time when measuring the pandemic's acceleration.

Materials and Methods

About a century ago, a series of seminal articles by Kermack and McKendrick [12,[START_REF] Kermack | A Contribution to the Mathematical Theory of Epidemics II. The Problem of Endemicity[END_REF][START_REF] Kermack | A Contribution to the Mathematical Theory of Epidemics. III. Further Studies of the Problem of Endemicity[END_REF] have laid the foundations for a mathematical theory of epidemics. More specifically, their compartmental (that is, Susceptible, Infected and Removed or SIR-type) and time-since-infection models have been extensively used and refined in the academic literature about infectious and emerging diseases. A core concept in this paradigm is the reproduction number, usually noted R, which roughly captures how many secondary cases originate, on average, from a pool of primary cases who is still currently infectious (see May and Anderson [17]).

As evident from publications by health agencies around the world since the onset of COVID-19, much of the guidance for designing policy measures to curb the pandemic relies prominently on estimates of R, among other things. The reproduction number is initially a theoretical concept, conceived to understand the transmissibility of an epidemic. Many efforts have been put into defining ways to empirically estimate it. Broadly speaking, estimation strategies fall into two broad categories. The first one rests on the basic SIR model (see e.g. Weiss [22] for a clear exposition), which predicts that the reproduction number R is the product of four parameters: the duration of infection, the number of contacts per case and the fraction of contacts who are in turn infected, on average, and finally the fraction of total population susceptible to infection. Although each of these parameters could be estimated in real-time, this turns out to be a gigantic task, in particular when a novel pathogen like SARS-Cov-2 emerges. A short-cut to avoid such a demanding procedure is to fit a SIR model using the number of cases, so as to estimate R directly, given the infection duration (see, among many others, Althaus [START_REF] Althaus | Estimating the Reproduction Number of Ebola Virus (EVOB) during the 2014 Outbreak in West Africa[END_REF] for a recent example related to Ebola using maximum likelihood estimation). This is feasible, even in real-time, provided that enough data is available to ensure precision and structural assumptions about the time-dependency of R are made. A caveat, though, is that such fitting procedures have limitations (see e.g. Cori et al. [START_REF] Donnelly | Key Data for Outbreak Evaluation: building on the Ebola Experience[END_REF]). An additional issue arising from estimation based on compartmental models is the sizeable range of estimates. See Chris et al. [START_REF] Bauch | Dynamically Modeling SARS and Other Newly Emerging Respiratory Illnesses: Past, Present, and Future[END_REF] for SARS, and Viceconte and Petrosillo [START_REF] Viceconte | COVID-19 R0: Magic Number or Conundrum?[END_REF] for the early stages of COVID-19. The second estimation strategy addresses more directly the time-varying dimension of R, which is more in line with epidemiological and clinical data. Many health agencies rely on such estimates of time-since-infection transmission models rather than SIR-type models. Here the basic idea is that R is essentially (1+) the growth rate of infected, which is the ratio between the number of new (that is, secondary) cases arising, say, within 24 hours, and the number of primary cases (see Fraser [11]). For example, the French agency in charge of health statistics uses the Cori method, after Cori et al. [START_REF] Ferguson | A New Framework and Software to Estimate Time-Varying Reproduction Numbers during Epidemics[END_REF] (see https://www.santepubliquefrance.fr/content/download/266456/2671953). Other European health agencies are also using this method, e.g. Austria (see https://www.ages.at/download/0/0/ e03842347d92e5922e76993df9ac8e9b28635caa/fileadmin/AGES2015/Wissen-Aktuell/COVID19/ Methoden_zur_Schätzung_der_epi_Parameter.pdf) and Germany (see https://www.rki.de/DE/ Content/InfAZ/N/Neuartiges_Coronavirus/Projekte_RKI/R-Wert-Erlaeuterung.pdf?__blob= publicationFile).

Our task here is to relate the acceleration index defined in Baunez et al. [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF][START_REF] Baunez | An Early Assessment of Curfew and Second COVID-19 Lock-down on Virus Propagationin France[END_REF] and the reproduction number that is estimated using the time-since-infection approach just described. The main purpose of this section is to derive a theoretical relationship between both concepts, which helps both to explain why they are different, to give a sense of the magnitude of their difference, and to state the conditions under which they are equivalent. We then turn, in the next section, to data to gauge whether the difference between the two matters to track the COVID-19 pandemic. As stressed in Baunez et al. [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF], accurate information about the dynamics of a pandemic rests on both the number of cases and the number of tests, and the former cannot be properly understood without the latter. In that paper, we introduce an acceleration index, denoted ε T at date T , which is an elasticity that measures the proportional responsiveness of cases with respect to tests. Given that the number of cases and tests are not necessarily varying at the same rate across time, groups and also space, the acceleration index measures the percentage change of cases with respect to a percentage change of tests and is thus unit-free. The acceleration index is defined as follows:

ε T = P T -P T -1 P T ÷ D T -D T -1 D T (1)
Rearranging the terms of the latter equation, we see that the acceleration index relates to the daily and average positivity rates, in the following way:

P T -P T -1 D T -D T -1
daily positivity rate

= P T D T average positivity rate × ε T acceleration index (2)
Equation [START_REF] Bauch | Dynamically Modeling SARS and Other Newly Emerging Respiratory Illnesses: Past, Present, and Future[END_REF] shows that the acceleration index is an elasticity, which is a concept widely used by economists since Marshall [START_REF] Marshall | Principles of Economics[END_REF], to study the responsiveness of demand with respect to a change in price of a good. The reason we call such an elasticity an index of viral acceleration can be clarified using an analogy with linear body motions. Given that our analysis relies on data about cases and tests only, with the latter as units of measurement, one can think of the acceleration index as the ratio between current and average viral speed. With tests as the unit-measurement, the daily positivity rate p T /d T becomes a measure of current viral speed at date T , that is, the fraction of tested people that turn out to be positive on that day. The average positivity rate P T /D T at date T can be thought of as average viral speed, taken over the entire data sample. In Appendix A we illustrate through an example why we do not average over daily positivity rates in the usual way, but rather take the ratio of cumulated cases to cumulated tests.

where the weights w's capture the generation time distribution, with n j=0 w j = 1. This means that the time-independent function f that follows from the denominator in equation ( 3) is, in that case, assumed to be linear in the number of cases p (which does not imply that p t is linear in time of course). Note that such an assumption implies that, given the reproduction number R, the dynamics of new cases follow an auto-regressive process AR(n).

Even though it might go unnoticed a first sight, we should stress that a major difference between the rather general definition of R in equation ( 3) and the usual definition in equation ( 4) is that the function f t implicitly depends on calendar time t. That is, what is taken account of are simply the number of cases detected on a particular day, but not the fact that the number of cases detected will depend on the diagnostic effort or number of tests that has been realized. The fact that the diagnostics dimension is largely ignored in the literature about SIR-type models surfaces, for instance, in Wallinga and Lipsitch [START_REF] Wallinga | How Generation Intervals Shape the Relationship between Growth Rates and Reproductive Numbers[END_REF], who relate the epidemic growth rate to incidence and generation time interval only. It seems reasonable to assume that infectious and emerging diseases involve a diversity of pathogens, which require a variety of technologies to diagnose. In the context of COVID-19, PCR and antigen testing is of course key. This difference in accounting for cases turns out to be important to understand the connection between the acceleration index and the reproduction number, as we now show.

Using equations ( 2) and (3), we can relate our acceleration index and the reproduction number in the following way:

ε T = R T × A T B T with A T = f T [p T , p T -1 , . . . , p T -n ] P T and B T = d T D T (5)
Equation [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF] shows that the ratio between the acceleration index and the basic reproduction number can be thought of itself as a ratio. The numerator of this latter ratio, A, can be thought of as the current infectivity contribution to cumulated cases, that is, the ratio of the average number of primary cases up to period T who can originate infections in T as a fraction of the total number of persons who have been infected since the outset of the pandemic. The denominator, B, on the other hand, represents the fraction of tests in period T out of cumulated tests up to T , that is, the current test contribution to cumulated tests. To sum up, the ratio of the acceleration index to the reproduction number is, in any period, the ratio of the infectivity contribution to the test contribution.

From Equation [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF] we see that both indicators are equivalent, that is, ε = R, if and only if at all dates t:

f t [p t , p t-1 , . . . , p t-n ] = d t × P t D t (6)
Equation ( 6) is very important to conceptualize the core idea of this paper: in order to properly control for the (time-varying) volume of tests/diagnostics, one needs to use the appropriate function f t , that is, one which depends on calendar time because tests do. Said differently, the function f t should be specified in such a way that it takes account of the fact that cases are produced by tests or any other diagnostics. The linear form with no time dependence which appears in the denominator of equation ( 4) is therefore problematic, as it assumes away tests which are however key to measure the pandemic's dynamics. In this sense, the acceleration index ε nests the basic reproduction number R: if the function f t is specified as in equation ( 6), R is equivalent to ε as it takes account of testing; in any other case, ε is more general than R as in equation ( 4) that takes account of cases only.

To better understand the relation between R and ε as indicated in equation ( 5), a more theoretical analysis and its implications may be helpful. First of all, as it also becomes clear from equation [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF], when A = B -that is when equation ( 6) holds -then ε = R. This basically means that if the test contribution B follows the dynamics of the infectivity contribution A, there is enough testing to capture viral activity. In fact, seen from this perspective, we have a clear testing strategy:

the daily tests d t need to offset the assumed infectiousness captured by function f t , and more specifically equation [START_REF] Baunez | COVID-19 Acceleration and Vaccine Status in France -Summer[END_REF]. The smaller the total number of cases, the easier it will be to match that testing requirement in particular through contact tracing. As total cases go up, contact tracing and sufficient testing may come to its structural and systemic limits. This in itself is a sign that additional health policies will need to be promoted.

If A > B, then ε > R, whilst when A < B, ε < R. In the former case, the infectivity rate A of the pandemic cannot be captured by the test contribution B in place. That is, R does not give the appropriate picture of the infectiousness of the pandemic, in fact it underestimates it. To alleviate this bias, either testing would need to be increased, or viral spread would need to be cut by establishing policies that reduce contacts or a mixture of both. In any case, it shows that equation

(3) that composes R does depend on more than past and current cases, because they themselves depend on tests and other factors that may favour or not transmissibility. Equivalently in the latter case, R will overestimate the speed of the pandemic if the test contribution B is greater than the infectivity contribution A. In such a situation, greater testing than underlying infectivity will necessarily find more cases, actually too many to reflect the correct transmissibility. To capture the correct picture, either testing would need to be reduced, which however seems counterproductive at least to the extent that testing is a way to look at the underlying viral dynamics, or the infectivity function f of equation (3) needs to be adapted to reflect reduced transmissibility.

A further analysis of equation ( 5) requires structural assumptions, in particular to generate predictions about how both ε and R, rather than their ratio, move over time. Just for the sake of illustration, we now resort to the handy example of deterministic exponential growth. Time is assumed to be continuous, to ease derivation of results, and the number of cases grows exponentially over time, as usually assumed in epidemiological models, of SIR type and related for example. In such a case, R as defined in equation ( 4) is constant and any difference between R and ε is due to differences between A and B. For ε, we also have to introduce tests and we assume that they also grow exponentially.

Under those assumptions, we show in Appendix A that while the reproduction number is constant over time, the acceleration index is not, as it features different regimes depending on how the growth rate of daily cases compares with the growth rate of daily tests. For example, when the former is larger than the latter, the acceleration index first rises and then approaches a plateau, where it equals the ratio of growth rates, which is larger than 1 in that case. In contrast, the reproduction number stays constant over time. We can visualize this more easily in the simple setting of exponential growth (see Appendix A), but it also holds more generally that the difference Test rate (Bt) Infectivity rate (At) between both indicators is essentially due to the fact that while the acceleration index is the ratio of two growth rates, that of cases divided by that of tests, the reproduction number tracks only the former, thus ignoring the latter. It is also for this reason that we say the the acceleration index ε nests the basis reproduction number R, which is simply a special case of ε. Different configurations may in principle occur, therefore, over time, depending on how fast cases grow compared to the growth of tests.

To further illustrate what happens in the case of exponential growth outlined above and studied in more details in Appendix A, we now provide an illustration such that the growth rate of daily cases is twice as large as the growth rate of daily tests. Figure 1 illustrates how the acceleration index and the reproduction number, as well as the infectivity and test rates, evolve over time in this particular example.

In Figure 1, panel (a), we report the evolution over time of the time-varying acceleration index ε(t)

and the constant reproduction number R that follow from the numerical example. In Appendix A, we show that while R is constant, ε tends to the ratio of growth rates, which is equal to 2 in the example. As a consequence, a first regime with the reproduction number exceeding the acceleration number happens, followed by a second regime that features the reverse configuration.

Not surprisingly, panel (b) in Figure 1 shows that the first regime materializes when

B(t) > A(t)
-that is the test contribution exceeds the infectivity contribution -while the second regime is associated with A(t) > B(t). Panel (b) reveals in particular that the plateau for the acceleration index that is featured in panel (a) comes from the fact that both A(t) and B(t) themselves plateau, with the infectivity contribution exceeding the test contribution. Overall, therefore, the acceleration tracks the ratio of grow rates -which equals 2 in our example -while the reproduction number underestimates that ratio because it roughly reflects only its numerator.

In the next section, we apply the theoretical decomposition outlined above to capture how the

Results and Discussion

In Figure 2, panel (a), we report both the acceleration index (blue curve) and the reproduction number (green curve) over time, using data for France (May 13 to November 19, 2020). The acceleration index ε is computed using equation (1) while the reproduction number R is computed from equation ( 4) with n = 7 and equal weights w. Note that the infection kernel could be adapted to account for sub-exponential growth as in Chowell et al. [START_REF] Chowell | Characterizing the Reproduction Number of Epidemics with Early Subexponential Growth Dynamics[END_REF]. The lower spikes of the reproduction number R are due to lower amount of testing during week-ends. This can clearly be seen in the panel (d) of Figure 2 that presents the number of daily tests (in pink). We represent here the raw data rather any smoothed estimates in order to avoid any additional layer of interpretation.

In panel (c) of Figure 2, we present local polynomial regressions for R and ε of panel (a), that use the Savitzky-Golay filter also known as a locally estimated scatter-plot smoothing method in modern statistics (see Cleveland and Devlin [START_REF] Cleveland | Locally-Weighted Regression: An Approach to Regression Analysis by Local Fitting[END_REF]). The blue line is again our acceleration indicator. In red, we

show the reproduction number as used by Santé Publique France, whereas the green line represents our own estimation. As can be noted, the reproduction number estimated by Santé Publique France falls within the confidence bands of our own estimate which are the dotted green lines. Even though Santé Publique France refers to the Cori method, we have not found public information about the weights attached to past values for the number of cases in computing infectivity. In addition, as can be seen from the confidence bands, the acceleration index is estimated more precisely than the reproduction number, because we take account of variations of tests and thus cases due to the weekend effect. In effect, being a ratio of growth rates (that of cases over that of tests), the acceleration index turns out to be smoother than the reproduction number (which is closer to the growth rate of cases).

If we concentrate on panel (c) of Figure 2, we see that right after the end of the first lock-down, both indicators are roughly "plateauing" just under 1, with R being slightly above ε. We concentrate here on the green line, i.e. our estimation of R. This estimated R then rises quickly to a higher plateau in the first half of July to indicate greater transmissibility, and stays at a level of about 1.2 until mid-August. At that same time ε first remains put at a level smaller than 1 and becomes greater than 1 a few days later, effectively crossing R at the beginning of August and accelerating all along until about mid-August.

The difference in dynamics of both indicators for France can easily be explained by looking at panel (b) in Figure 2. Here, we report the two terms that appear in equation [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF], that is, A, the infectivity contribution (orange curve), and B, the test contribution (red curve). The latter graph exhibits spikes, due to the fact that much less tests, if any, are performed during week-ends. The test contribution follows a downward trend that simply reveals the fact that tests being done in a given period constitute, over time, a smaller and smaller fraction of the cumulated amount of diagnostics. What we see in panel (b) in particular is that before August 5, the test contribution B is greater than the infectivity contribution A that has, at first, also a downward trend. A greater testing rate implies that more cases will be found. This corresponds to the period when R is greater As a consequence, ε captures more accurately the considerable time variation of virus propagation than R.

To make clear that the discrepancy between the reproduction number and the acceleration index shown in Figure 2 is not specific to France, we report in Appendix B a similar decomposition for five other countries. Such a comparison reveals that the reproduction number might either underestimate or underestimate viral acceleration, depending on the country and the time period, due to not correcting for the time-varying amount of testing. This readily suggests that this issue might be highly relevant in many other countries as well, where public health authorities are also in dire need of accurate indicators to track epidemics. As such, this observation should also preoccupy international bodies that design cooperation strategies to fight pandemics, including of course the World Health Organization and other regional agencies.

To futher illustrate the differences between R and ε and their consequences for public health decisions, we consider Figure 3. We use a background map that is in the public domain and can be down- loaded via https://commons.wikimedia.org/wiki/File:D%C3%A9partements_de_France-simple.

svg. Figure 3 is then constructed by incorporating our own data into this background map and gives an overview of how the second lock-down, which started on October 30, 2020, has contributed to reduce virus circulation across French départements. Looking at the two bottom maps shows that the acceleration index has been reduced everywhere during the period from October 30 to November 19. While a similar improvement is indicated by the reproduction number, as one concludes from the top maps, that measure of virus spread tells an altogether different story. Three weeks after the beginning of the second lock-down in France, the acceleration index suggests that the acceleration regime still prevails, except for 6 départements which happen to be the happy few, but with values still close to unity. However, one concludes rather wrongly from the reproduction number that, at the same date, deceleration is underway in most départements in green. In view of the discussion about the ε/R ratio in Section 3, this comes as no surprise since the reproduction number under-estimates virus circulation, due to the fact that it does not take tests into account.

To sum up, two main differences between the reproduction number and the index appear in panel (c). The first being that R crosses unity earlier than the acceleration index, which starts to increase around July 6. This is most likely due to the infectivity rate reaching its lowest point around that date (as seen in panel (b)). Passing that date, the infectivity rate begins to increase, while the test rate continues downwards. This explains why the acceleration index could not start growing before July 6. Overall, therefore, R is larger than ε before August 5. The second difference is a sudden decrease of the reproduction number in the second half of September, while the acceleration index stays at a plateau. This can be explained by panel (d), in which we see a sharp plummet in the number of daily tests around that period. Less tests equals less detected cases, which R relies heavily on for its calculation. Seeing R rising sharply from about 1 in early October is all the more surprising when seen in isolation. In contrast, the acceleration index, which accounts for variations of both cases and tests, consistently shows a succession of periods of steep rise followed by plateaus over the summer and until the second lock-down.

In practice, many public health agencies report (daily or weekly) positivity rates, to complement the information contained in the reproduction rate R. In light of the connection with ε that we have highlighted in this note, both formally and empirically, we argue that the acceleration index is closer to a sufficient statistic that helps tracking the rapidly changing dynamics of any pandemic, because it explicitly takes into account the dynamics of diagnostics. In the context of COVID-19, diagnostics equal tests, but our claim is valid more generally when this is not the case. This means that the index can potentially be applied to any effort designed at detecting infected people, no matter what the pathogen agent triggering the infectious disease turns out to be. In real time, this is quite valuable, we believe, to guide health policies and to assess containment measures, especially in the context of a new pathogen appearing (such as SARS-Cov-2), with unforeseeable pandemic dynamics.

Conclusion

We show in this paper that the reproduction number R is a special case of the the acceleration index proposed in Baunez et al. [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF]. While the former only considers the growth rate of cases, the latter measures variations of cases in relation to that of tests, and it does so in a unit-free manner since it is an elasticity. Most importantly, the acceleration index is a sufficient statistic of viral spread in the sense that it aggregates all the relevant information in a synthetic manner. In contrast, looking at pieces of information, like positivity or prevalence rates, separately might lead to the misleading conclusion that there is conflicting evidence about whether the epidemic worsens or improves. As such, a test-controlled reproduction number like the acceleration index should be part of any data dashboard to track an epidemic, and especially to guide public policy in the design of the most efficient methods to curb it. For example, we have shown in Baunez et al [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF] that an accurate measure of virus circulation is a key input to feed algorithms that are designed to efficiently allocate the diagnostic effort across space.

The result that the reproduction number is, as a measure of viral spread, subject to a considerable bias is not specific either to France or to the period considered, as illustrated using data from five other countries. Such a comparison reveals that the reproduction number might either underestimate or underestimate viral acceleration, depending on the country and the time period, due to not correcting for the time-varying amount of testing. This readily suggests that this issue might be highly relevant in many other countries as well, where public health authorities are also in dire need of accurate indicators to track epidemics. As such, this observation should also preoccupy international bodies that design cooperation strategies to fight pandemics, including of course the World Health Organization and other regional agencies.

Such an observation makes the acceleration index a much more useful indicator to track a pandemic in real-time, as it is context-dependent: the acceleration index takes into account the effort to diagnose people who have been infected by the pathogen. In the case of COVID-19, diagnostics equal PCR (and other types of biological) tests, but this might not be the case for other diseases where diagnostics require even greater effort. However, our analysis makes a strong case for incorporating in any measure of pathogen circulation the observed effort to diagnose the agent that makes people sick.

Even though there is a variety of infectious (and emerging) diseases, with different pathogens and various ways to diagnose them, we claim that our conceptual approach is general enough to shed light, not only on the current pandemic, but also on any future ones which may come.

Finally, we would like to mention some limitations of our analysis. Some important issues, beyond the scope of our paper, have however been addressed by the literature. First, the fact that unaccounted cases arise when testing is not compulsory (see Pullano, Di Domenico, Sabbatini et al. [START_REF] Pullano | Underdetection of Cases of COVID-19 in France threatens Epidemic Control[END_REF] for France). Second, the coexistence of symptomatic and asymptotic cases during COVID-19 has led to additional statistical methods (see Khailaie et al. [15]).

A Exponential Growth: an Illustrative Example

We now illustrate the relationship between the acceleration index and the reproduction number when time is assumed, to ease derivation of results, to be continuous and when the number of cases grows exponentially over time, as usually assumed in epidemiological models, of SIR type and related for example. Although typically absent in the latter strand of literature, we have to introduce tests and we assume that they also grow exponentially. More formally, using the notation in the previous section, suppose that the number of cases per unit of time is denoted by p(t) = αe βt while the number of tests per unit of time is d(t) = γe νt , where the growth rates β and ν are assumed to be positive for the sake of illustration. The reproduction number is then constant over time, as we now show when the infection kernel is uniform, that is, if the weights w are constant and equal to 1/δ to ensure that w δ 0 ds = 1. Note that it is not difficult to show that the constancy of R holds for other infection distributions as well. The analog of equation ( 4) is as follows:

R(t) = p(t) w δ 0 p(t -s)ds (7)
which implies that, given p(t) = αe βt , one has:

R(t) = R = βδ 1 -e -βδ (8)
It is easily seen that the reproduction is the growth rate of daily cases (adjusted for the delay δ) only, since it obviously does not take into account tests. In fact, R is the growth rate of the denominator in equation [START_REF] Cleveland | Locally-Weighted Regression: An Approach to Regression Analysis by Local Fitting[END_REF], that is, what we note f in the main text.

Cumulated cases and tests are then noted P (t) = t 0 p(τ ) dτ and D(t) = t 0 d(τ ) dτ , respectively. It is easy to derive, by straight integration, the expressions:

P (t) = α β e βt -1 , D(t) = γ ν e νt -1 (9)
It follows that our acceleration index is given, as function of time, by:

ε(t) = p(t)/P (t) d(t)/D(t) = β ν 1 -e -νt 1 -e -βt (10)
From equation [START_REF] Donnelly | Key Data for Outbreak Evaluation: building on the Ebola Experience[END_REF], one concludes that the acceleration index, which is an elasticity that measures the responsiveness of cases to tests, is essentially the ratio of the growth rate of cumulated cases divided by that of cumulated tests, since p(t) = dP (t)/dt and d(t) = dD(t)/dt. In addition, ε tends to the ratio of the growth rate of daily cases to that of daily cases β/ν. This means that the acceleration index tracks that ratio over time and converges to it eventually.

It follows that three cases occur. When β = ν, that is, when both daily cases and daily tests grow at the exact same rate, then our acceleration index equals 1 at all dates. When the two growth rates differ, however, ε(t) converges, when t goes to infinity, to the ratio of growth rates β/ν, independently of the scale parameters α and γ. As an illustrative example, suppose that β > ν, so that positives grow faster than tests. Then the pattern of our acceleration index ε(t) over time will have two regimes: it first grows almost linearly and eventually reaches the upper bound β/ν > 1.

Obviously, in that case both the daily positivity rate p(t)/d(t) and the average positivity P (t)/D(t) grow over time, and the latter quantity exceeds the former all the time so that acceleration prevails.

The symmetric case when β < ν is easily adapted.

The property that the reproduction number is constant under our assumptions essentially means that the dynamics of the acceleration index is driven by that of A(t)/B(t), which we now decompose to help understand how R and ε compare over time. It follows from the definition of A and B in equation ( 5) that:

A(t) = e βt -e β(t-δ)
δ(e βt -1) and

B(t) = ν e νt e νt -1 (11)
with both functions A(t) and B(t) decreasing with time t.

To further illustrate what happens in the theoretical case outlined above with β > ν, let us take a numerical example. Suppose that the growth rate of cases β equals 20% while the growth rate of tests ν = 10%. In addition suppose that the delay parameter δ = 4 so that the weight w = 1/4. Figure 1 illustrates how the acceleration index and the reproduction number, as well as the infectivity and test rates, evolve over time in this particular example.

The exponential example is also useful to illustrate the formal relationship between viral speed and the acceleration index. To do that, let us now make an analogy with linear body motions to relate our indicators to speed and acceleration. If one think of the positivity rate as viral speed and define it as s(t) = p(t) d(t) then straightforward computations give s(t) = α γ e (β-ν)t . By analogy, viral acceleration could be defined as the derivative of viral speed, that is, a(t) = s (t) = α γ (β -ν)e (β-ν)t . Not surprisingly, then, it turns out that the sign of a(t) indicates acceleration in the sense that a positive sign reveals acceleration (this happens when β > ν since speed goes up over time in that case) while a negative sign reveals deceleration (when β < ν).

However, a drawback of a(t) as a measure of acceleration is that it is scale-dependent since it depends on scale parameters α and ν, though their ratio. In other words, a(t) depends on the absolute level of speed, that is, of the positivity rate. On the contrary, the acceleration index given by Equation [START_REF] Donnelly | Key Data for Outbreak Evaluation: building on the Ebola Experience[END_REF] does not and is unit-free, since it is an elasticity. Note that a possible, also unit-free, alternative would be to define acceleration as the semi-elasticity a(t)/s(t) = β -ν. However, since it relates the percentage change of positives case to the absolute change of tests, it is arguably less amenable to interpretation than the elasticity. It is easily shown that the semi-elasticity would be obtained if the average positivity rate would alternatively be defined as the average of the daily positivity rates up to date T , as opposed to the ratio of number of cumulated cases to the number of cumulated tests as in Equation (2) which leads to the acceleration index as an elasticity. Finally, note that the acceleration index equals one when viral speed stays constant over time.

B Test-Controlling the Reproduction Number in Practice:

Five Countries

In this appendix, we provide estimates for both the acceleration index and the reproduction number, for Argentina, Austria, South Africa, South Korea and the United Kingdom, using the same method that is applied in Section 3 for France and over the same period (May 13 to November 19, 2020).

The public data source for the five countries comes from Our World in Data (Ritchie et al. [START_REF] Ritchie | Coronavirus Pandemic (COVID-19)[END_REF]).

The resulting estimates, depicted in Figures 4 to 8 (that are hence directly comparable to Figure 2 for France), unambiguously reveal that the substantial bias due to time-varying tests, which is corrected by the acceleration index but not by the reproduction number, is not specific to France or to a particular period. In fact, the corresponding bias forces the reproduction number to either underestimate (Argentina over the whole period is a case in point) or overestimate viral acceleration, depending on the country and the period considered. This observation suggests that a similar bias should come as no surprise for other countries as well.

  Suppose that data is available about the number of tested and positive persons, up to end date T . Denote {p 1 , . . . , p T } the historical times series of the new (per period) number of positive persons from date t = 1 to end date t = T . Similarly, {d 1 , . . . , d T } is the historical times series of new (per period) diagnosed/tested persons. Denote P t = t τ =1 p τ and D t = t τ =1 d τ the cumulative numbers of positive and diagnosed persons up to date t.

Figure 1 :

 1 Figure 1: Numerical example for exponential growth of both daily cases and daily tests. Panel (a) Acceleration index (blue curve) and Reproduction number (green curve). Panel (b) Infectivity contribution (orange curve) and Test contribution (red curve) (a) Acc. Index and Rep. Number

Figure 2 :

 2 Figure 2: France -Panel (a) Acceleration index (blue curve) vs reproduction number (green curve). Panel (b) Infectivity contribution (orange curve) vs test contribution (red curve). Panel (c) Kernel estimates with confidence bands (dashed lines). Panel (d) Daily tests (purple line) and infectivity function (black curve). Beige area depicts the lock-down period. Source: Agence Santé Publique France and authors' computations.

  than ε. But R basically overestimates viral activity because it does not consider tests and focuses on cases only, while ε takes account of this because it looks at the ratio of both infectivity and test contributions. The opposite is true for the period after August 5, when the infectivity contribution A becomes greater than the test contribution B, and ε is greater than R. Despite growing daily tests until the end of August, as we can see in panel (d) of Figure2, R remains first at a plateau but then steadily declines until the second half of September.This clearly shows that R has been unable to represent viral acceleration basically during both summer months because, as we can see from panel (b), the infectivity contribution is rising more quickly than the test contribution. Therefore, R overlooks the testing dimension and only captures the number of cases, but those are undervalued given a lower test rate B. Even worse, testing then declines at the end of August while the infectivity function f , indicated in black in panel (d) starts going up. This affects R that declines to reach a level of about 1 by the end of September. This is in very stark contrast to ε that accelerates from early July onwards and then hovers at a plateau of about 2 up to end of September. It takes appropriately into account the relationship between the changing growth rates of testing and infectivity.Both indicators go up again from the end of September onwards as testing rises again. But while R reaches a plateau again as the first curfew measures where put into place to cut transmission, ε further indicates acceleration. Both indicators then start declining when, at the end of October, the second lock-down was put into place. However, R < 1 since the beginning of November, whilst at the same time, our indicator ε still shows an ongoing acceleration, although a reduced one with respect to the time before the lockdown. What we see very clearly from panel (d) is that lock-down coincides with a great reduction of testing. Obviously, lock-down is aimed at reducing contacts and thus viral spread. This will necessarily reduce cases and hence R declines. But if at the same time testing is reduced as well, which is the only way to get a clearer picture of the viral activity this necessarily influences R more dramatically and explains the under-evaluation of the viral spread than ε that continues to indicate acceleration. More specifically, ε indicates what happens to cases when we reduce testing by some percentage change. The fact that the percentage change of cases goes in the same direction as the percentage change of testing, i.e. that both decrease, is a good sign and indicates that lock-down measures have their effect. But looking at ε does not yet allow to give an all-clear such as R does.

Figure 3 :

 3 Figure 3: Acceleration index and reproduction number for French départements at different dates. October 30 dates the start of the second lock-down in France. Data source: Agence Santé Publique France and authors' computations.

Figure 4 :

 4 Figure 4: Argentina -Panel (a) Acceleration index (blue curve) vs reproduction number (green curve). Panel (b) Infectivity contribution (orange curve) vs test contribution (red curve). Panel (c) Kernel estimates with confidence bands (dashed lines). Panel (d) Daily tests (purple line) and infectivity function (black curve). Source: Our World in Data and authors' computations.

Figure 5 :

 5 Figure 5: Austria -Panel (a) Acceleration index (blue curve) vs reproduction number (green curve). Panel (b) Infectivity contribution (orange curve) vs test contribution (red curve). Panel (c) Kernel estimates with confidence bands (dashed lines). Panel (d) Daily tests (purple line) and infectivity function (black curve). Source: Our World in Data and authors' computations.

Figure 6 :

 6 Figure 6: South Africa -Panel (a) Acceleration index (blue curve) vs reproduction number (green curve). Panel (b) Infectivity contribution (orange curve) vs test contribution (red curve). Panel (c) Kernel estimates with confidence bands (dashed lines). Panel (d) Daily tests (purple line) and infectivity function (black curve). Source: Our World in Data and authors' computations.

Figure 7 :

 7 Figure 7: South Korea -Panel (a) Acceleration index (blue curve) vs reproduction number (green curve). Panel (b) Infectivity contribution (orange curve) vs test contribution (red curve). Panel (c) Kernel estimates with confidence bands (dashed lines). Panel (d) Daily tests (purple line) and infectivity function (black curve). Source: Our World in Data and authors' computations. Testing data for South Korea were incomplete: five dates were missing. Missing values were completed by a linear interpolation.

Figure 8 :

 8 Figure 8: United Kingdom -Panel (a) Acceleration index (blue curve) vs reproduction number (green curve). Panel (b) Infectivity contribution (orange curve) vs test contribution (red curve). Panel (c) Kernel estimates with confidence bands (dashed lines). Panel (d) Daily tests (purple line) and infectivity function (black curve). Source: Our World in Data and authors' computations.

  If then current viral speed is larger than average viral speed, we are in a situation of viral acceleration and the pandemic is on the loose.

	In that case, our acceleration index ε T is larger than one, which means that increasing tests by 1%
	leads to more than 1% of new cases. An arguably legitimate goal of public health policy would
	therefore be to make sure that the acceleration index gets smaller than one, i.e. that current viral
	speed becomes smaller than average viral speed: this would indicate that the pandemic decelerates
	and becomes under control. Ideally, one would like to find ever fewer cases the more one tests. This
	reasoning also shows why it is not sufficient to look at positivity rates alone -they only indicate
	viral speed. What matters for public health is to understand whether speed becomes greater or
	smaller compared to its historical average as tests increase, which is what our acceleration index
	measures. In Appendix A, we give an example using exponential growth, for which closed-form
	solutions are derived and can be used to further illustrate the interpretation of the acceleration
	index as a unit-free elasticity that relates to how the current viral speed compares to its historical
	average.
	Regarding the reproduction number, our rather general assumption is, in accordance with the
	mathematical literature on epidemics, that the reproduction number is essentially a gross rate
	of growth and, as such, can be written at date t as:
	R t =	p t f t [p t , p t-1 , . . . , p t-n ]	(3)
	where f t is a function of new cases from date t to date t-n, which can be thought of as the infectious
	potential, that is, the average number of people who have been infected at t and before, and who
	can infect people at t. The parameter n is related to infection duration. Specifications for f t have
	been used in the literature. For example, Fraser's [11] equation (9), on page 3 of his paper, defines
	the time-varying effective reproduction number as follows:
	R t =	p t j=0 w j p t-j n

The authors thank the Editor and three referees for suggestions that prompted us to improve the paper. This work was supported by the French National Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University -A*MIDEX.


D Schanne

A Degiron

Making Quantum Emitters Emit Vector Vortex Beams

How far can we go in shaping the luminescence of quantum emitters? It is well known that by using cavities or nano-antennas, one can beam their luminescence into wellcontrolled directions of space. Here, we push this control to a level of complexity usually achieved with laser radiation: we make an assembly of colloidal quantum dots spontaneously emit vector vortex beams, which are swirls of light combining a non-zero angular momentum and an inhomogeneous field structure. The key ingredient to obtain this result is to increase the spatial coherence of the emitted photons. To fulfill this condition, we design and use plasmonic holograms that also impart a non-zero orbital angular momentum and a radial polarization state to the beam.

I. Introduction

Scalar vortex beams are characterized by a chiral helical wavefront associated with a nonzero orbital momentum [START_REF] Dennis | Chapter 5 Singular optics: Optical vortices and polarization singularities[END_REF]. The phase varies spatially with the azimuthal angle e imθ , where m is the topological charge of the vortex, resulting in a phase singularity at the center of the beam. This phase singularity implies a null point of intensity, giving to the vortex an annular intensity profile. Vector vortex beams are even more complex because they also possess a polarization singularity in addition to the phase singularity. Both scalar and vector versions of optical vortices are associated with a trove of discoveries in fluorescence imaging, optical trapping, high-speed telecommunications and quantum technologies [START_REF] Carpentier | Making optical vortices with computer-generated holograms[END_REF]- [START_REF] Willner | Optical communications using orbital angular momentum beams[END_REF].

A necessary condition to construct optical vortices is to use light that possesses spatial coherence, so as to create the specific set of interferences that defines them. It is therefore perhaps not surprising, then, that the realizations found in the literature are focused on lasing light. For example, phase plates, diffractive holograms, SLM, or optical metasurfaces can produce a phase singularity, while birefringent crystals or polarizing prisms can impart a polarization singularity to the laser beam.

Contrarily to lasers, the spontaneous emission of quantum emitters has little spatial coherence. However, there is a well-known remedy to this problem that consists in coupling them with propagating surface plasmons. Surface plasmons have coherence lengths that are typically limited by their propagation distance, which can be tens of microns in the near infrared. These long coherence lengths explain why one can make surface plasmons emit light into very narrow beams using gratings carved at the metal surface [START_REF] Lupton | Bragg scattering from periodically microstructured light emitting diodes[END_REF]. Here, we revisit this approach to shape the spontaneous infrared emission of PbS colloidal quantum dots (cQDs) into vector vortex beams.

II. Results and Discussion

As illustrated on the inset of Fig. 1(a), our sample consists of a gold hologram on a gold substrate, with a thin but compact layer of cQDs coated on top. The samples are fabricated in a cleanroom with electron beam lithography followed by metal evaporation, the thickness of the gold hologram being equal to 25nm. First, it is crucial to notice that we do not observe a quenching of the photoluminescence, usually observed when a single quantum emitter is put in direct contact with a metallic surface. This unintuitive behavior has been recently explained by the realization that an ensemble of cQDs is very different from the single emitter case and that they respond to a new regime of light-matter interactions described by a local Kirchhoff law [START_REF] Wojszvzyk | Light emission by a thermalized ensemble of emitters coupled to a resonant structure[END_REF]. Then, as explained in the introduction, in order to reshape the spontaneous of cQDs, we need to maximize their spatial coherence. Here the quantum emitters are coupled to spatially coherent surface plasmon modes. More especially, after pumping the cQDs with a HeNe laser excitation at the center of the spiral, they populate surface plasmon modes that propagate radially at the surface of the sample and interact with the hologram. To make the cQDs outcouple as a vector vortex beam, the hologram must possess a specific design which is obtained by computing the interference pattern between the radial plasmonic wave and the optical vortex:

H(r,θ)=|exp[i Re(2πrneff /λ)] +exp(imθ)|², ( 1
)
where neff is the effective refractive index seen by the plasmons. The resulting pattern is a spiral with a number of arms equal to the topological charge of the vortex. A SEM picture of a symmetric 1-arm spiral is displayed on Fig. 1(a). To characterize experimentally the sample, we use a 4f optical setup to measure the angular distribution of the photoluminescence beam in the Fourier plane. An objective collects the photoluminescence signal, a dichroic mirror isolates it from the laser pump and then a pair of lenses images the Fourier plane of the objective on an IR InGaAs camera. A narrow spectral filter is inserted in front of the IR camera to select one single wavelength. As illustrated in the Fig. 1(b), the photoluminescent beam appears with an annular distribution. Due to the dispersive nature of our structure, the diameter of the ring changes with the wavelength filter, as discussed in Ref. [START_REF] Schanne | Spontaneous emission of vector vortex beams[END_REF] for vector vortex beams with higher topological charges (NB: the results presented in this paper are fully original). We have also established an analytical model that calculates the intensity and phase distribution of the far-field diffraction pattern in the scalar Fraunhofer approximation. As depicted in Fig. 1(c-d), we notice a very good agreement between the theoretical and experimental intensity distributions, while the theoretical phase varies from 0 to 2π and exhibits the expected characteristics of a 1st order vortex. Finally, we have performed polarization measurements to probe the field structure of the photoluminescence beam. As depicted on Fig. 3, when a polarizer is added in front of the camera to select one specific polarization state, the intensity pattern is reduced to two lobes, and these two lobes rotate with the angle of the polarizer. Hence, the photoluminescent vortex exhibits a radial polarization distribution associated with a polarization singularity in its center. In other words, the beam displays all the features of a vector vortex beam. The polarization properties are explained by the field structure of the radial surface plasmons that are involved in the process [START_REF] Ghaemi | Surface plasmons enhance optical transmission through subwavelength holes[END_REF]. This result can be generalized to arbitrary propagation directions, as shown for example on Fig. 3(b) for a sample that produces a vector vortex beam at an oblique incidence of 10 degrees with respect to the normal.

III. Conclusion

In conclusion, we have experimentally demonstrated the generation of vector vortex beams from the incoherent photoluminescence of cQDs-a feat usually achieved with lasers.

Following the same approach, other complex forms of light could be generated including vortex knots or links [START_REF] Wang | Ultrasmall optical vortex knots generated by spin-selective metasurface holograms[END_REF]. Moreover, the optical excitation of the cQDs could be replaced by electrical pumping, opening up the way to LED sources generating optical vortices. Such vortex emitting diodes would offer tremendous advantages for integrated systems.

Fig. 1 .

 1 Fig. 1. a) Inset: Schematic cross-section of the sample. SEM image of a 1-branch spiral hologram, scale bar: 1µm. The radial pitch of the spiral is equal to λ/neff = 1095nm. b) Experimental image (in false colors) of the photoluminescent beam intensity distribution in the Fourier plane. The spectral filter in front of the IR camera is centered on λ=1050nm. The dashed circle corresponds to the numerical aperture of the objective NA=0.65. c-d) Theoretical intensity and phase distributions of the photoluminescent beam in the Fourier plane.

Fig. 2 .

 2 Fig. 2. a) SEM image of the interferometric spiral hologram, scale bar: 1µm. The geometry consists of two 4-arm spirals with the same radial pitch 1.095 µm but with opposite winding. b) Interference pattern of the photoluminescence beam visualized experimentally in the Fourier plane and c) comparison with the theoretical predictions.

Fig. 3 .

 3 Fig. 3. a) Experimental Fourier plane images measured for the same structure and with the same spectral filter as the one used in Fig. 1 but with a linear polarizer inserted in front of the IR camera. The orientation of the polarizer is indicated with double arrows. b) Same observations for an asymmetric spiral generating a vortex with a 10° angle with respect to the normal.

Acknowledgements

We acknowledge E. Lhuillier for synthetizing and providing the PbS colloidal quantum dots, as well as S. Suffit and P. Filloux for their precious help concerning the cleanroom nanofabrication. We acknowledge support from the European Research Council grant FORWARD (reference: 771688).


Michael Lattorff

Silvio Erler

Patrick Lhomme

Alessandro Fisogni

Gherardo Bogo

François Massol

Cuckoo bumblebee males might reduce plant fitness

published or not. The documents may come

Cuckoo bumblebee males might reduce plant fitness

three groups of bees visited the same number of plants, males of both types visited more flowers within a whorl, but cuckoo males spent more time on flowers within a whorl and the whole plant than the free-living bumblebees.

The flower visits of bumblebee workers are optimised, aiming at collecting as much pollen as possible within a short time frame. This, in turn, has consequences for the pollination process by enhancing cross-pollination between different plants. By contrast, males and especially cuckoo bumblebee males, are not selected for an optimised foraging pattern. Instead, they spend more time on flowers, eventually resulting in higher levels of pollen transfer within a plant (geitonogamy), which might lead to reduced plant fitness. This is the first study to relate the foraging behaviour of cuckoo bumblebees to pollination and plant fitness.

-Table 1 and main text: Please name the cuckoo bumble bee "Psithyrus" instead of "Bombus" to make clear that these are the cuckoo species.

Reviewed by Patrick Lhomme, 05 Sep 2021

The authors took into account the comments from all reviewers. I'm satisfied with this new version.

Author's Reply

Download tracked changes file Dear Dr. Lattorff, thank you for handling our manuscript. We have addressed all the experts' comments and modified the text accordingly. We have provided precise replies to each comment in the attached file. We have uploaded the revised version of the manuscript, supporting information and R scripts on Zenodo.

Decision by Michael Lattorff

Dear authors, your subission was evaluated by four experts in the field. All of them were very convinced about your study and enjoyed reading your ´well written manuscript. However, they also raised some concerns about methodology and data anaylsis. I think these issues need to be addressed to improve your mansucript. Please, read all reviewer reports carefully and respond to the points raised in a point-by-point response document incl. how the points raised affected your manuscript.

Many thanks for submitting your work to PCI Zoology, we are looking forward to receiving you revision.

Reviewed by Patrick Lhomme, 14 Mar 2021

Dear authors, I have enjoyed reading your ms entitled "Cuckoo bumblebees perform slower and longer flower visits than true bumblebees", it is a well written MS that is pleasant to read.

Males are generally considered to visit fewer flowers than workers, to be slower paced and to have a lower flower fidelity however most of the pollen they carry is freely available for pollination so I believe that their contribution to pollination services is often overlooked.

One strength of the study is to highlight this contribution and show that what we generally think about bumblebee male foraging is maybe not always true and probably more context-dependent.

Among the weakness, I feel that the authors are stretching their data a bit too much in trying to compare foraging behavior of cuckoo bumble bees and free-living bumble bees using only cuckoo male versus host worker/male foraging data. Including data from Bombus queens and Psithyrus females would have been a plus.

Unlike Bombus queens, Psithyrus females do not forage to feed their colony but only for themselves, their incentive to forage is thus very different although both have also offspring to produce, the comparison would thus have been very interesting. I noticed that females of certain species of cuckoo bumble bees like B. vestalis are regularly observed foraging while others like B. rupestris females are rarely encountered foraging on flowers although both species population density seem pretty similar (interestingly it's opposite for males, as you show in your dataset). I believe females of B. rupestris feed directly on the resources gathered by their hosts and don't bother much foraging, so the parasitic strategy can I'm sure explain differences in foraging behaviours between cuckoo and free-living bumble bees but also between the different cuckoo species.

The comparison of both cuckoo and free-living males makes sense too, although they might have very similar physiological requirements, the parasitic strategy of the females could probably explain some the minor differences between cuckoo and free-living male foraging behaviors, as discussed by the authors. I also noticed that mating durations between cuckoo and host bumblebees were drastically different (see Lhomme et al 2013) for reasons that I can't really explain, but this could somehow impact male foraging behavior. As mating duration is almost 10 times shorter in cuckoos their energetic needs might also be much lower I don't know.

I'm also a bit frustrated not to have more qualitative data to interpret the results. You measured visiting durations but it would have been useful to measure foraging durations, it's not clear to me if the males are feeding or just resting on the flowers, quantifying resting / nectar foraging / pollen foraging would have also been a plus.

Here are some minor comments: Title : This title is misleading, you can't really state this based on your study. You are comparing bumble bee workers with cuckoo males. It would have made sense if you were comparing Bombus queens and Psithyrus female foraging behaviors, but it's not the case here. Please provide a more accurate title or at least specify that cuckoo bumblebees are males and that true bumblebees are workers L 26-28 : Many studies have investigated differences between cuckoo bumble bees and their hosts in terms of social behaviors, pre-mating behaviors etc... You should be more specific on the behavior (flower visiting behavior) you talk about.

Keywords : In general, avoid using words that are already in the title.. I would for example replace "cuckoo bumble bees" by Psithyrus, "behavior" is also too vague, use flower visiting behavior L 52 : Williams 1998 actually cites 29 species, it was before Lecocq et al. (2011) show conspecificity between B. barbutellus and B. maxillosus. I don't like doing that (self citation) but we also confirmed in a recent paper (in press) that B. flavidus and B. fernaldae are conspecific, so the number of species is now 27. You can just simply stick to 28, but if you wish to specify 27 and if you're allowed to cite papers in press with DOI then feel free to cite "Lhomme P, Williams SD, Ghisbain G, Martinet B, Gérard M, Hines HH (2021) Diversification pattern of the widespread Holarctic cuckoo bumble bee Bombus flavidus (Hymenoptera : Apidae) : The East Side Story. Insect Systematics and Diversity, in press, doi: 10.1093/isd/ixab007" L. 56 : To be honest I'm not a big fan of the adjective "true" to refer to non-parasitic bumblebees, Psithyrus species are also true bumble bees.. I would rather use the terms "cuckoo vs free-living" or "cuckoo vs host", but it's just a personal preference feel free to keep "true" if you wish.

L. 211-214 : I agree but it doesn't really make sense here. If you were comparing bumblebee queens and cuckoo females I would agree but here you compare workers versus males, the parasitic behavior of females has nothing to do with male flower visiting behavior.

L. 271 : were cuckoo bumblebee males.

Best wishes, Patrick Lhomme

Reviewed by anonymous reviewer, 06 Apr 2021

Fisogni and co-authors present an interesting study comparing behavior of cuckoo and free-living bumblebees. They studied floral visit of different individuals on patches of Gentiana lutea and they show that behaviors are different. We need more empirical studies like this one to better understand the ecology of the pollinators, and their conservation. I recommend the publication of this article but I present here different propositions to make the presentation of this study more accurate.

Change everywhere in the manuscript the term "true bumblebees", including in the title. As you know, Psithyrus are now included in the genus Bombus. I recommend to use the term "free-living bumblebees" or "non-parasitic" if you want to distinguish the cuckoo bumblebees with the other bumblebees.

You have to include in the abstract and in the introduction the interest of your study. Why is it important to compare the two groups? Why is it important to compare their behavior? What's your hypothesis? You should indicate, in the abstract and in the introduction, that the evaluation of the behavior during the floral visit is important to estimate the efficiency of a pollinator and its fitness. We might think that the cuckoo bees are not good pollinators because they don't forage on pollen. Line 128: why you don't consider species level? Difference among free-living bumblebees species might be more important than between cuckoo versus free-living forms. I would test first differences at species level, and if you don't have differences, you can pool the data. Aggregation of data has to be justified. Line 156. Line 161. There is no justification to lump the data from different species. In other words, are the differences related to species or to "trait" (cuckoo versus free-living)? Line 167. The accuracy to study these metrics was not introduced. Lines 204-205; lines 216-217. If it's not significant, don't mention it. It does not make sense.

Line 247: Please clarify. How? overdispersed). Second, some of thse counts seem to have quite a bit of missing data (NAs) in the data file. More explanation of thse missing data should be provided.

How well do these correspond with individual identifications. I find it a bit hard to connect the two. When recording visits on a plant, one has to make spot judgements. The sampling afterwards would help confirm your confidence of identification in general but wont actually validate your decision at the time. For the species here, it is pretty straightforward to distinguish. But, for identifying sex it can be trickier. For eg. B. terrestris males look very similar to females. People with experience can vey reliably identify which are males and females (and I'm sure you are) but it relies on a clear view of the bee, and in particular, their head (which can be obscured if it's deep in a a flower). # Discussion.

L235 is an interesting suggestion, that phenology determines foraging activity. This is intuitive. A correlate of this is that the bees differ in age. If sexuals are produced earlier for cuckoo bumblebees then we would expect them to be older on average than the other bumblebees. This could be estimated by wing wear from the collected samples. L241 this would be supported by phenology data.

  DOI or URL of the preprint: https://doi.org/10.5281/zenodo.4489067 Version of the preprint: 1.0

Line 34 :

 34 Significant differences? Line 37; Lines 239-241: I did not understand this mechanistic hypothesis to explain the different behavior by the difference in colony development. Where are the data on the timing of colony development? What about other traits like the size? The physiological needs? The flight metabolism? Lines 51-53: evolution of inquilinism in 3 subgenera (see Lhomme & Hines 2019) Lines 65-67: to be developed. What makes a pollinator efficient or not? Cleaning behavior for exemple should be introduced. Pollen on corbiculae is not available for pollination. Line 90: size of the flower? Line 94: Rossi et al. 2014 did not explore the present question? Line 108: How did you determine the species in the field?

Table 1 :

 1 The subgenus system presented here is outdated. Please followWilliams et al. 2008. All cuckoo species listed here are included in the subgenus Psithyrus.

	Williams P.H., Cameron S.A., Hines H.M., Cederberg B. & Rasmont P. (2008) A simplified subgeneric
	classification of the bumblebees (genus Bombus). Apidologie, 39, 46-74

Cite this recommendation as:

Michael Lattorff (2021) Cuckoo bumblebee males might reduce plant fitness. Peer Community in Zoology, . https://zool.peercommunityin.org/articles/rec?id=44

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Reviewed by Silvio Erler, 16 Sep 2021

The authors nicely revised their manuscript by including most reviewer suggestions and addressing their concern. The point-by-point authors response sufficiently answered the reviewer concerns. I have only some minor comments that might improve the manuscript.

-I suggest to change the writing from "bumblebee" to "bumble bee".

Reviewed by anonymous reviewer, 28 Feb 2021

This is an nice example of rare studies asking the relevance and importance of bumble bee males for pollination and plant reproductive success. In particular socially parasitic cuckoo bumble bees are completely neglected in this context. Here, the authors found that cuckoo males stay much longer on flowers and behave slower than host bumble bee workers and males, most likely related to the different foraging strategies and colony development.

The study design, data analysis and presentation is clear and easy to follow. The authors use all relevant literature to discuss and explain their data. However, a major downside of the study is the very limited observation time with only 3 days per year (6 in total). It is well known that climate (temperature, hours of sunshine, rain, etc.) significantly impact flower nectar production and bumble bee foraging behaviour. It is needed to include climate data during observation to discuss the pros and cons of the data. Slower behaviour and resulting longer flower visits might also be explained by the climate data, this discussion is missing so far.

It might also be worth separating data for the host bumble bee species, having a huge difference between sexes for B. terrestris, B. lapidarius and B. pratorum. For each species, colony life cycles differ strongly among months and might be included in the discussion of the data.

The efficiency of pollination success can be easily supported by having pollen data of the foraging bees. Did the authors count pollen grains per cuckoo and host bumble bee males and workers while foraging? Table 2 and 3 might move to the suppl. material.

Line 208-214: Is there anything known on nectar and pollen quality, nutritional values, of G. lutea. If yes, this would be relevant for the discussion on energy intake and foraging behaviour strategies.

Line 216-223: Seems to be redundant compared to previous sections and should be shortened. Generally, the authors should double-check for redundancy between the introduction and discussion.

The authors highlighted several times that only males of cuckoo bumble bees have been found, but no queens. Do they have any data that at the same time queens were foraging on other plants in the same vicinity and that young queens already emerged? Otherwise I would reduce the discussion on this fact.

Reviewed by anonymous reviewer, 06 Apr 2021

In general, it's a well written and interesting manuscript. I do see a few areas where the ms could be improved and clarified.

# Methods

Line 110: the authors verify visual identification by capturing and keying out species after visual surveying the visitations. My concern here is that this approach will validate the experimenter's eye for identifying species/sex but not whether they were successful in the instance of observation. IDing the bees while visiting plants is absolutely achievable by experienced observers but it's not always trivial and especially when they're moving around and have their head buried in a flower etc. For example, B. terrestris males look largely similar to females but have a different 'face'. Being confident in sexing can require a close look.

Stats: I think that some mention of species variation in foraging is warranted. I have gone through their well annotated code and included bumblebee species. In none of the models I tried did bumblebee species significantly explain the response variables. This then makes it sensible to ignore, but it should be mentioned explicitly in text that they did not vary by species (unless I missed something).

I do find, however, some issue with two things. First, I suspect that a generalized linear model would be more appropriate for the number of flowers/whorls visited with a Poisson error distribution (or quasi if