Samuel Alizon
email: samuel.alizon@cnrs.fryannis.michalakis@ird.fr

Yannis Michalakis

Adaptive virulence evolution: the good old fitness-based approach

Keywords: virulence, evolution, trade-off, epidemiology, fitness, pathogen Highlights:, Parasite virulence can be adaptive

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Why do infectious diseases harm their hosts?

The maintenance of parasite virulence, which is most generally defined as a decrease in host fitness due to the infection, is a longstanding puzzle [START_REF] Smith | Some problems in the life-history of pathogenic microorganism[END_REF][START_REF] Ball | Parasitism and evolution[END_REF][START_REF] Anderson | Coevolution of hosts and parasites[END_REF][START_REF] Ewald | Host-parasite relations, vectors, and the evolution of disease severity[END_REF][START_REF] Read | The evolution of virulence[END_REF][START_REF] Frank | Models of parasite virulence[END_REF][START_REF] Ebert | The evolution of parasitic diseases[END_REF][START_REF] Schmid-Hempel | Evolutionary Parasitology: The Integrated Study of Infections, Immunology[END_REF][START_REF] Bull | Theory and empiricism in virulence evolution[END_REF]. Even if there are exceptions known as obligate killing parasites, host death appears as an a priori costly event for parasites as it also results in their own death and thus the end of the infectious period and transmission.

One theoretical possibility to explain that a parasite population does not eventually evolve towards complete avirulence is that the trait cannot evolve due to the lack of relevant genetic variation. Virulence can also persist if it is neutral (i.e. neither selected for or against). This could occur for example if the host is a 'dead-end' to the parasite, in which case any harm to the host does not affect parasite spread. The final possibility, on which we focus here, is that virulence is maintained through natural selection because this trait directly or indirectly confers an advantage to the parasite strain expressing it.

The most common hypothesis to explain the adaptive scenario is the transmission-virulence trade-off hypothesis, which states that strains with increased transmission ability cause shorter infections. This hypothesis has been so successful that most empirical tests of whether virulence can be adaptive for the parasite have focused on testing the existence of a positive relationship between proxies for virulence and transmission [START_REF] Dwyer | A simulation model of the population dynamics and evolution of myxomatosis[END_REF][START_REF] Mackinnon | Virulence in malaria: an evolutionary viewpoint[END_REF][START_REF] Jensen | Empirical support for optimal virulence in a castrating parasite[END_REF][START_REF] Fraser | Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis[END_REF][START_REF] Råberg | Infection intensity and infectivity of the tick-borne pathogen Borrelia afzelii[END_REF][START_REF] Doumayrou | An experimental test of the transmission-virulence trade-off hypothesis in a plant virus[END_REF][START_REF] Williams | Evidence of trade-offs shaping virulence evolution in an emerging wildlife pathogen[END_REF].

We argue here that the way the transmission-virulence trade-off hypothesis is currently formulated entails three major problems when attempting to prove that virulence is adaptive. First, the hypothesis relies very strongly on the way transmission and virulence are defined. Second, it neglects 'hidden' trade-offs and thus paves the way to pan-adaptationist interpretations. Third, testing the existence of the relationship between virulence and transmission is complicated. Here, we propose to recast the problem in its original formulation: looking at the relationship between virulence and parasite fitness, as a few experimental studies have already done and as we propose here, curtails the limitations of the transmission-virulence trade-off hypothesis and answers directly the question of whether virulence is adaptive or not. Adequately defining parasite fitness requires addressing all the relevant life-history traits of a parasite, and mathematical modeling can greatly help in that stage.

Adaptive bases of virulence The transmission-virulence trade-off

In 1943, Gordon Ball [START_REF] Ball | Parasitism and evolution[END_REF] challenged the avirulence theory, which states that given enough time parasites should coevolve with their hosts to become avirulent. His opinion was based on evidence that some parasites have maintained their virulence over long periods of time, but he lacked an alternative theory. The latter emerged in the early 80s when Ewald [START_REF] Ewald | Host-parasite relations, vectors, and the evolution of disease severity[END_REF] and Anderson and May [START_REF] Anderson | Coevolution of hosts and parasites[END_REF] argued that virulence can be adaptive if its value correlates with other infection traits that are beneficial to parasite transmission. implicit idea is that in order to be transmitted, parasites need to exploit (and hence harm) their host. This is now known as the transmission-virulence trade-off hypothesis.

The fact that the term 'trade-off' is used to describe a positive correlation between virulence and transmission is often confusing to evolutionary biologists, a trade-off between two traits intuitively implying a negative relationship. The use of the term probably comes from the fact that virulence is inversely proportional to the duration of the infection, and that an intermediate optimum arises only if at some point the benefits (increased transmission rate) increase less rapidly than the costs (increase host mortality; [START_REF] Van Baalen | The dynamics of multiple infection and the evolution of virulence[END_REF]).

Several studies have provided empirical support to the transmission-virulence relationship (reviewed in [START_REF] Alizon | Virulence evolution and the trade-off hypothesis: history, current state of affairs and the future[END_REF]). Among these, three host-parasite associations stand out: myxoma virus in rabbits [START_REF] Anderson | Coevolution of hosts and parasites[END_REF][START_REF] Dwyer | A simulation model of the population dynamics and evolution of myxomatosis[END_REF], HIV in humans [START_REF] Fraser | Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis[END_REF] and Cauliflower mosaic virus in Brassica rapa [START_REF] Doumayrou | An experimental test of the transmission-virulence trade-off hypothesis in a plant virus[END_REF]. Not only do these show a positive correlation between virulence and transmission but they also find that there exists an intermediate optimal level of virulence. Indeed, they find that the transmission-virulence trade-off curve saturates, which implies that at some point the benefits, that is increased transmission, increase more slowly than the costs (Figure 1A).

Other explanations

Another adaptive explanation that accounts for the evolution of parasite virulence involves multiple infections (reviewed in [START_REF] Alizon | Multiple infections and the evolution of virulence[END_REF]). It states that if more virulent strains are more competitive within a host, then allowing for parasites to infect already-infected hosts can maintain virulence, even in absence of a transmission-virulence trade-off [START_REF] Levin | Selection of intermediate rates of increase in parasite-host systems[END_REF].

Increased virulence could also occur if parasites can evolve over the course of an infection and if more virulent strains are better adapted to the within-host level. This process has been termed 'short-sighted' evolution [START_REF] Levin | Short-sighted evolution and the virulence of pathogenic microorganisms[END_REF] because virulence can evolve even if it does not provide any benefit for transmission (or, more generally, any benefit at the between-host level).

Finally, if one allows for more complicated parasite life cycles, several other sources for the maintenance of parasite virulence emerge. For instance, if parasites have a free-living state, there can be coincidental selection and virulence in the focal host can be maintained if it provides an advantage to the parasite in the environment [START_REF] Levin | Selection and evolution of virulence in bacteria: an ecumenical excursion and modest suggestion[END_REF][START_REF] Brown | Evolution of virulence in opportunistic pathogens: generalism, plasticity, and control[END_REF].

Three limitations of the current trade-off theory Virulence and transmission definitions are very stringent

The initial formalization that led to the transmission-virulence trade-off hypothesis, and the great majority of subsequent formalizations, considered horizontally and directly transmitted parasites [START_REF] Anderson | Coevolution of hosts and parasites[END_REF][START_REF] Anderson | The population dynamics of microparasites and their invertebrate hosts[END_REF]. As a consequence, by 'transmission' the theory means transmission rate (that is the product between the contact rate between hosts and the transmission probability per contact) and by 'virulence' it means the increase in host mortality rate due to the infection. Indeed, in horizontally transmitted parasites, infection effects on host reproduction do not affect parasite fitness, and thus should not influence virulence evolution unless the contacts in the host populations are structured [START_REF] O'keefe | Playing by different rules: the evolution of virulence in sterilizing pathogens[END_REF][START_REF] Lion | Evolution of spatially structured host-parasite interactions[END_REF].

The fact that virulence is expressed as the host mortality rate in the theoretical models on which the transmission-virulence hypothesis was built on is not a secondary issue as measuring virulence experimentally through host case mortality, lethal dose or host life expectancy (which are all related to host mortality) can yield different results. For instance, if the host background mortality rate increases, parasite-induced mortality rate, i.e. the 'virulence' of the mathematical models, is expected to increase though, perhaps counter-intuitively, case mortality might decrease [START_REF] Day | On the evolution of virulence and the relationship between various measures of mortality[END_REF]. This result makes it easy to envision how sub-lethal measures of virulence (e.g. anemia or weight loss), which are often used for practical reasons, could yield very different predictions than those produced by the measure of virulence used in the theoretical models underlying the trade-off hypothesis.

There can be multiple trade-offs

Even for parasites that follow the simple life cycle assumed by the models underlying the transmission-virulence trade-off hypothesis (with horizontal and direct transmission), there might be 'hidden' trade-offs. For instance, one might conclude from the absence of a relationship between the probability of transmission and the parasite-induced mortality rate that there is no trade-off. However, it could be that the cost of an increased probability of transmission is paid through a decrease in contact rate (a hypothesis put forward originally by Ewald [START_REF] Ewald | Host-parasite relations, vectors, and the evolution of disease severity[END_REF]).

The problem becomes even more acute for parasites with life cycles that depart from the classical simple one. Let us consider spore-transmitted parasites for instance. Their life cycle can be described as follows: infected hosts produce spores, these spores need to survive long enough to encounter a susceptible host and infect this host upon contact. We could of course consider the trade-off between the probability of infection and virulence but by doing so we could be ignoring a great deal of the biology. What if the trade-off involves virulence and spore production rate or spore survival rate instead?

This multiplicity of trade-offs raises two problems. First, even if we do find a trade-off between two variables of the life cycle, how can we be sure that this is not going to be canceled by another trade-off that involves other variables? For instance, if we find that more virulent spores cause infections that produce less spores, we might conclude that virulence is not adaptive. But what if the quality of the spores (how long they persist in the environment) increases with virulence, as supported by data on human respiratory pathogens [START_REF] Walther | Pathogen survival in the external environment and the evolution of virulence[END_REF]? Then, perhaps a more virulent strain that produces less spores but of extremely high quality would be fitter than a less virulent strain. The inadequacy of studying trade-offs between specific fitness components rather than 'total fitness' has already been highlighted in other fields of evolutionary biology [START_REF] Hunt | What is genetic quality?[END_REF].

The converse problem occurs if one does not find a trade-off between two variables because proponents of the trade-off hypothesis could be tempted to argue that this is because not all the possible trade-offs have been investigated. Such a pan-adaptationist argument would make the trade-off hypothesis virtually untestable for many parasites with a complicated life-cycle.

Measuring the trade-off is complicated

A final issue with the current theory occurs when trying to assess the shape of the transmissionvirulence trade-off curve (if any) for a given host-parasite interaction. Importantly, the trade-off curve arises from within-host interactions. Although variations in epidemiological factors such as the number of susceptible hosts have no effect on the trade-off shape, host differences do. In order to minimise these variations, it is highly desirable to study parasites while minimizing environmental variation and variation due to differences due to the hosts, be they genetic or environmental [START_REF] Alizon | Emergence of a convex trade-off between transmission and virulence[END_REF].

Then, one needs parasite strains that differ in their virulence. These should be sufficiently related because the implicit assumption is that they compete for the same niche and can replace one another. A potential problem is that different isolates might have different evolutionary histories and might thus be more or less (mal)adapted to exploiting the host in the environmental conditions set by the experiment. For example, assume a researcher is using a set of parasitic isolates collected in different parts of the distribution of the species under study. Assuming that the transmission-virulence trade-off hypothesis holds for this species, some of these isolates might be well adapted to the experimental conditions and the variation they will exhibit will lie around the transmission-virulence curve that the experiment aims at inferring. However, other isolates might be particularly maladapted, and appear as outliers of the curve that we try to infer.

A fitness-based approach

We argue that adopting an approach based on the fitness of the parasite can address the limitations listed above. By doing so, we go back to the roots of the transmission-virulence trade-off hypothesis, which stems from an epidemiological model (p. 499 in [START_REF] Anderson | The population dynamics of microparasites and their invertebrate hosts[END_REF]).

Obviously this approach requires a definition of parasite fitness (see Box 1 for a formal definition) and ways to measure it (see Box 2). One of the key features of this approach is that the entire life cycle of the parasite is closed and explicitly considered. Therefore, by definition, we are sure that a trait value (usually the virulence) that maximises fitness is evolutionarily stable.

The fitness-based approach has already been applied to empirical data in a very small number of cases. HIV provides a striking example where intermediate virulence values maximise parasite fitness [START_REF] Fraser | Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis[END_REF]. Arguably, HIV offers a simple setting since parasite fitness has two components, the duration of the infection and the transmission rate. However, several examples from the literature lead us to think that this approach can be widely applicable to biological systems.

For instance, Chapuis et al. [START_REF] Chapuis | Trade-offs shape the evolution of the vector-borne insect pathogen Xenorhabdus nematophila[END_REF] showed that intermediate virulence maximises parasite fitness in the case of a bacterium that can alternate between two hosts (an insect and a nematode vector). Another example is provided by Mackinnon and Read [START_REF] Mackinnon | Virulence in malaria: an evolutionary viewpoint[END_REF], who expressed the fitness of Plasmodium falciparum infections by analysing several components separately. In the latter case, they found that the composition of the host population (the proportion of children) strongly affects parasite fitness.

But the best example of the wide applicability of the fitness-based framework is probably the study by de Roode et al. [START_REF] De Roode | Virulence-transmission trade-offs and population divergence in virulence in a naturally occurring butterfly parasite[END_REF]. In their system, the parasite (the protozoan Ophryocystis elektroscirrha) infects monarch butterflies (Danaus plexippus). Typically, adults do not die from the disease but the infection is still virulent because it can decrease the probability of successful emergence from pupal cases. The transmission of the parasite mainly occurs when infected females lay eggs. Parasite spores are then found on the egg and the leaf and larvae become infected by ingesting these spores. In this system, directly measuring the number of secondary infections is complicated so the authors define fitness as the product between five 'components': the probability of adult emergence of an infected larva, the probability of mating, the proportion of eggs that acquire parasite spores and the proportion of offspring that will become infected.

By doing so, they close the life-cycle of the parasite and thus measure the number of secondary infections. Overall, they show that intermediate spore loads (their proxy for virulence) can maximise parasite fitness (Figure 1B).

The fitness-based framework

The idea

For any parasite, it is possible to investigate the relationship between the harm it does to its host and the efficiency with which it spreads. Under this approach, the first quantity, i.e. the parasite's virulence, can be measured in any way that is relevant to the biology of the hostparasite interaction. The second quantity is the parasite's fitness (Box 1). This approach is sufficient to show whether a given level of virulence is maintained by natural selection, which would be the conclusion if fitness is maximised for non-minimal levels of virulence. It would open the way to investigations on the various mechanisms throughout the host and the parasite's life cycles responsible for the benefits virulence confers to the parasite.

We argue that looking at the correlation between parasite fitness and virulence instead of the correlation between transmission rate and virulence addresses all the three major limitations currently faced by the virulence evolution theory.

Wide applicability

The current transmission-virulence trade-off theory can (and should) only be used for parasites that are transmitted directly. This is frustrating because if virulence can be adaptive for such parasites, it seems logical to assume that it could also be adaptive for parasites transmitted by spores or by vectors. However, a direct translation of the theory to parasites with different life cycles can lead to erroneous results. It is necessary to 'patch' the 'standard' model such that the predictions tested are generated by models that adequately take into account the biology of the systems under study. This may sound like a trivial remark, but unfortunately it often happens that the predictions of the model assuming horizontally transmitted parasites with a direct life cycle are applied to systems with very different biology. Models addressing various deviations from the 'default' horizontally and directly transmitted parasites exist (e.g. [START_REF] Anderson | The population dynamics of microparasites and their invertebrate hosts[END_REF]) but have rarely been used to make predictions on the evolution of virulence. The advantage of the fitness-based approach is that it relies on a quantity, fitness (Box 1), that can be measured for any parasite with a known life cycle (Box 2), and will yield reliable results provided the entire life cycle is studied, i.e. fitness, and not simply a component of, is measured. In fact, even if part of the life cycle of the parasite is unknown, as is the case for many opportunistic pathogens that have a free-living stage [START_REF] Brown | Evolution of virulence in opportunistic pathogens: generalism, plasticity, and control[END_REF], a fitness expression can still be written, even if it might lack in accuracy. Interestingly, the fitness-based approach encompasses almost all the adaptive interpretations of virulence mentioned earlier because as long as parasite fitness can be measured, it is possible to study its relationship to virulence.

Another situation that fits into the framework in a way that could be seen as less intuitive is multiple infections. As shown in previous studies, parasite fitness can be written for superinfection models and for most coinfection models [START_REF] Van Baalen | The dynamics of multiple infection and the evolution of virulence[END_REF][START_REF] Levin | Selection of intermediate rates of increase in parasite-host systems[END_REF][START_REF] Choisy | Mixed infections and the evolution of virulence: effects of resource competition, parasite plasticity and impaired host immunity[END_REF]]. An intuitive interpretation of the underlying trade-offs is that virulence correlates with within-host competitiveness. Importantly, if multiple infections are allowed, the fitness of a strain de facto becomes a relative measure as it depends on the nature of the resident strain already present in the system. One possibility to circumvent this problem is to use experimental evolution and compare scenarios with or without multiple infections [START_REF] Alizon | Multiple infections and the evolution of virulence[END_REF].

Get the main picture before the details Currently, many studies only investigate the existence of a potential trade-off between virulence and transmission. This approach can lead to false negative results, not only because some level of virulence may be adaptive even if there is no correlation between transmission and virulence but also because the relationship between virulence and transmission can be blurred by correlations with other traits. It can also lead to false positive results, because the effects of a transmissionvirulence trade-off might be cancel out by correlations with other elements of the parasite's fitness.

These effects are difficult to predict. For instance, we mentioned that data on human respiratory pathogens find a negative correlation between virulence and survival time in the environment [START_REF] Walther | Pathogen survival in the external environment and the evolution of virulence[END_REF]. However, data on phages suggests a negative correlation between virus persistence in the environment and growth rate in host cells [START_REF] De Paepe | Viruses' life history: towards a mechanistic basis of a trade-off between survival and reproduction among phages[END_REF][START_REF] Heineman | Experimental evolution of a bacteriophage virus reveals the trajectory of adaptation across a fecundity/longevity trade-off[END_REF]. In fact, Goldhill and Turner [START_REF] Goldhill | The evolution of life history trade-offs in viruses[END_REF] argue that that this difference between viruses of Eucaryotes and phage might be more general.

The fitness-based approach addresses directly the question of whether virulence is adaptive. Investigating the detailed correlations between various elements of the parasite life cycle is only the second step. This is particularly helpful, at least conceptually, for parasites with complex life cycles, since the multitude of potential trade-offs between all the elements of the parasite's life cycle are merged into a single expression (parasite's fitness).

Solving the virulence definition problem

Embracing the entire parasite life cycle also addresses the recurrent problem of choosing an appropriate measure of virulence. In evolutionary biology, virulence is a quantitative trait most generally defined as the decrease in host fitness caused by the infection [START_REF] Read | The evolution of virulence[END_REF]. Since host fitness combines reproduction and survival, there are many ways to measure virulence. The problem is that the formulation of the transmission-virulence trade-off hypothesis is very specific as to how virulence should be measured, and depends on the biology of the system: considering components which are not relevant to parasite fitness, such as e.g. any other measure than the host increase in mortality rate due to the infection for the "canonical" horizontally and directly transmitted parasites, can lead to erroneous interpretations. Furthermore, there is an ongoing debate as to whether sub-lethal measures of virulence such as decline in red-blood cells [START_REF] Mackinnon | Genetic relationships between parasite virulence and transmission in the rodent malaria Plasmodium chabaudi[END_REF] or castration [START_REF] Jensen | Empirical support for optimal virulence in a castrating parasite[END_REF] are adequate proxies of mortality rate to study parasite evolution.

The fitness-based approach solves this problem because we can study the relationship between parasite fitness and any virulence measure (or any relevant trait for that matter).

Graphical representation

Arguably, part of the misunderstandings in the field of virulence evolution come from the difficulties to interpret the transmission-virulence trade-off curve. More specifically, showing a positive correlation between transmission and virulence is not sufficient to conclude that selection favours an intermediate level of virulence. Indeed, if the relationship is linear, theory predicts that ever increasing levels of virulence will always be favoured. It is only if the relationship saturates, that is, if at some point the benefits (increased transmission rate) increase less rapidly than the costs (increase host mortality), that there can exist an intermediate optimal level of virulence [START_REF] Van Baalen | The dynamics of multiple infection and the evolution of virulence[END_REF]. Note that transmission rate does not need to decrease with virulence for there to be a finite optimal level of virulence.

The fitness-based approach solves these problems by providing a more intuitive representation of the situation in that it directly shows whether parasite fitness is maximised for an intermediate value of virulence or not. In the case of the study by de Roode et al. [START_REF] De Roode | Virulence-transmission trade-offs and population divergence in virulence in a naturally occurring butterfly parasite[END_REF], a classical trade-off approach would for instance have considered the correlation between the probability of adult emergence (a good proxy for virulence) and the proportion of infected offspring, which seems the closest to a transmission function (note that by doing so one would clearly violate the assumptions of the transmission-virulence trade-off model regarding the underlying model). This is shown in Figure 1A and it is difficult to interpret. Conversely, analysing the parasite fitness as a function of parasite load provides a much more precise idea of the adaptiveness of virulence (Figure 1B).

Conclusion

Rooting virulence evolution in the ecology and epidemiology of the parasite allows addressing directly and conclusively the adaptive nature of virulence. In passing, it also allows solving many of the ongoing controversies regarding the validity of the trade-off hypothesis. More generally, once the adaptive, or not, nature of virulence is clarified, studying trade-offs is certainly useful for understanding the mechanisms linking virulence to fitness, and potentially for controlling virulence.

[48] C. Fraser, K. Lythgoe, G. E. Leventhal, G. Shirreff, T. D. Hollingsworth, S. Alizon, S. Bonhoeffer, Virulence and pathogenesis of HIV-1 infection: an evolutionary perspective, Science 343 (6177) (2014) 1243727.

Box 1: What is parasite fitness?

In the case of parasites, several definitions can be found in the literature. For instance, some will equate parasite fitness with parasite load, others with replication rate, others with virulence. The measures we listed above are all restricted to the individual host level but here we want to identify the parasite strain that is most capable of invading a host population. Therefore, fitness must be defined at the host population level.

One of the easiest ways to estimate fitness in this situation is to measure the ability of the parasite to spread through the host population. In epidemiology, this is also known as the basic reproduction ratio (R 0 ) and corresponds to the expected number of secondary infections caused over the whole course of an infection (akin to the number of offspring for free living organisms). Note that this definition implicitly accounts for the whole life-cycle of the parasite.

As all fitness measures, this one has some limitations. For instance, there can be differences between the invasion fitness in a naive host population and in a host population already infected by a resident strain of the parasite [START_REF] Dieckmann | Adaptive dynamics of pathogen-host interactions[END_REF]. Furthermore, the definition does not account for out of equilibrium dynamics [START_REF] Sabelis | Taking stock: relating theory to experiment[END_REF]. However, this invasion fitness has the advantage to be applicable to most host-parasite systems and to allow confrontation to experimental data (see Box 2).

Box 2: Measuring parasite fitness

Empirical approach

The first way to measure parasite fitness is to 'close' the life-cycle in the laboratory. Practically, one would start from a host infected by a parasite strain and count how many hosts have been infected in the population from this initial host. The fittest strain is the strain that generates the highest number of secondary infections.

This experimental might not be feasible for practical reasons in some host-parasite systems. A second way to estimate parasite fitness then consist in dividing the fitness into 'components' that are easier to measure. Note that writing down the mathematical expression of parasite fitness can be instrumental in defining the components and identifying potential correlations between them. For instance, in the case of HIV it is difficult to count the number of secondary infections for a given host. However, the problem can be overcome by noticing that the fitness is equal to the product between two components, which are the duration of the infection and the transmission rate [START_REF] Fraser | Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis[END_REF].

The main limitation of dividing fitness into components is that one has to be careful as to when the value of a component is measured. Again in the case of HIV the transmission probability per contact varies widely over the course of an infection [START_REF] Hollingsworth | HIV-1 transmission, by stage of infection[END_REF] so ideally this should be reflected on the 'transmission' component. One possibility for instance is to measure values at several time point during the infection.

Mathematical approach

Practically, a parasite's life cycle can be captured by a system of differential equations [START_REF] Otto | A biologist's guide to mathematical modeling in ecology and evolution[END_REF]. One can then calculate parasite fitness by differentiating this system to obtain what is known as the Jacobian matrix [START_REF] Diekmann | Mathematical epidemiology of infectious diseases: model building, analysis, and interpretation[END_REF]. The dominant eigenvalue of this matrix indicates if the parasite can spread or not in the host population (see also [START_REF] Hurford | Next-generation tools for evolutionary invasion analyses[END_REF]). In the classical case assumed implicitly by the trade-off hypothesis, parasites are only transmitted upon contacts between hosts and the fitness R is:

R = β µ + α + γ ( 1
)
where β is the transmission rate (which is the product of a contact rate between hosts and a probability of transmission per contact), µ is the host baseline mortality rate, α is the diseaseinduced mortality (or virulence) and γ is the recovery rate. Note that, as explained in Supporting Text S1, R is often equivalent to the basic reproduction ratio R 0 but not always [START_REF] Dieckmann | Adaptive dynamics of pathogen-host interactions[END_REF].

In addition to providing an intuition regarding the fitness components to measure, the mathematical expression of parasite fitness helps to make predictions regarding virulence evolution. From equation 1, strains with low virulence (α) should be favored because they have a higher fitness, R (the avirulence theory). The transmission-virulence trade-off hypothesis challenges this by stating that virulence (α) and transmission rate (β) are positively linked, such that decreasing one decreases the other. As a result parasites will evolve either to an intermediate level of virulence, if the relationship between the two traits saturates, or to the maximal possible value otherwise.

In the Supporting Text S1, we give further details about how to derive mathematical expressions for parasites with a variety of transmission routes. Ideally these should be derived for each system in order to capture the specificities of this system and guide empirical measurements.

Box 3: Open problems Practical limitations

It is not always easy or even sometimes possible to study the entire life cycle empirically. For instance for opportunistic parasites, we have little (or no) idea of what their ecology outside the host is [START_REF] Brown | Evolution of virulence in opportunistic pathogens: generalism, plasticity, and control[END_REF]. Some parasites have complex life cycles; and some of their hosts cannot be raised in the lab, or experiments on them are not ethically acceptable.

One can go even further by making the point that lab conditions will never re-create real ecological conditions. Think for instance of a spore-producing parasite that would face a tradeoff between virulence and spore longevity (as shown for human respiratory viruses [START_REF] Walther | Pathogen survival in the external environment and the evolution of virulence[END_REF]). Even if we manage to close the life cycle in the lab, how can this trade-off be tested unless the experiment runs for a very long time or unless the input of susceptible hosts is varied? Arguably, this is where mathematical modeling can help. We first need to identify which parameters of the life cycle depend on the epidemiological state of the population in order to infer how optimal levels of virulence can vary in response to environmental changes. Of course, this is assuming that the parasite's life cycle recreated in the lab can be compared to that in nature.

Timing problem

In vector-borne diseases for instance, the probability that a vector becomes infected after biting an infected host can vary with the age of the infection. One possibility is to study the components of parasite fitness over time and average them to perform the analysis. There even exist some more elaborate methods inspired from quantitative genetics that allow incorporating within-host time series into an epidemiological model [START_REF] Day | Bridging scales in the evolution of infectious disease life histories: theory[END_REF]. By applying such a framework to Plasmodium chabaudi infections in mice, Mideo et al. [START_REF] Mideo | Bridging scales in the evolution of infectious disease life histories: application[END_REF] showed that transmission early in the infection can be traded-off against transmission later in the infection.

Within-host evolution

Although multiple infections can readily be included in this framework, it is more challenging to include within-host evolution. The main problem with this process is that the parasite strains that come out of the host can strongly differ from those that infected it [START_REF] Luciani | The evolutionary dynamics of a rapidly mutating virus within and between hosts: the case of hepatitis C virus[END_REF]. However, recent models that include biological features of the parasite seem to offer promising perspectives by assuming an explicit within-host fitness landscape and deriving a next-generation epidemiological model [START_REF] Lythgoe | Is HIV short-sighted? Insights from a multistrain nested model[END_REF]. Also, recent findings suggest that even in rapidly evolving parasites causing persistent infections, virulence could be highly 'heritable' from one infection to the next [48], thus suggesting that within-host evolution could be less of a problem than previously hypothesized [START_REF] Levin | Short-sighted evolution and the virulence of pathogenic microorganisms[END_REF].

Figure 1 :

 1 Figure1: Graphical representation of the trade-off curve (A) and of the parasite fitness (B). Data originates from the work by de Roode et al.[START_REF] De Roode | Virulence-transmission trade-offs and population divergence in virulence in a naturally occurring butterfly parasite[END_REF] on a protozoan parasite infecting monarch butterflies. The lines show the best model that fit the data best based on the Akaike Information Criterion. A) A proxy of transmission as a function of a proxy of virulence. The evolutionarily stable (ES) virulence cannot be inferred directly from this curve because the host baseline mortality is unknown. B) Parasite fitness as estimated by de Roode et al. through the measurement of four fitness components as a function of the same proxy of virulence. The dashed line shows the ES virulence, which by definition maximises parasite fitness. Note that fitness could not be computed for all the isolates. Further details about the figure and this system are available in Supplementary Text S1.

Ackowledgments

We are very grateful to Jaap de Roode for sharing his data with us.


Tom Jeltes

John M Mcnamara

Wim Hogervorst

Wim Null Vassen

Valentina Krachmalnicoff

Martijn Schellekens

Aurélien Perrin

Hong Chang

Denis Boiron

W Vassen

A Aspect

C I Westbrook

Comparison of the Hanbury Brown-Twiss effect for bosons and fermions

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effects linked to the quantum statistics in a many body system. It may thus find applications to study more exotic situations [START_REF] Altman | Probing many-body states of ultracold atoms via noise correlations[END_REF] . Two-particle correlation analysis is an increasingly important method for studying complex quantum phases of ultracold atoms 7,[START_REF] Spielman | The Mott insulator transition in two dimensions[END_REF][START_REF] Greiner | Probing Pair-Correlated Fermionic Atoms through Correlations in Atom Shot Noise[END_REF][START_REF] Altman | Probing many-body states of ultracold atoms via noise correlations[END_REF][START_REF] Grondalski | Spatial correlation diagnostics for atoms in optical lattices[END_REF][START_REF] Hellweg | Measurement of the spatial correlation function of phase fluctuating Bose-Einstein condensates[END_REF][START_REF] Estève | Observations of density fluctuations in an elongated Bose gas: ideal gas and quasicondensate regimes[END_REF] . It goes back to the discovery by Hanbury Brown and Twiss 1 , that photons emitted by a chaotic (incoherent) light source tend to be bunched: the joint detection probability is enhanced, compared to that of statistically independent particles, when the two detectors are close together.

Although the effect is easily understood in the context of classical wave optics [START_REF] Loudon | The Quantum Theory of Light, 3rd Ed[END_REF] , it took some time to find a clear quantum interpretation 3,[START_REF] Fano | Quantum theory of interference effects in the mixing of light from phase independent sources[END_REF] . The explanation relies upon interference between the quantum amplitude for two particles, emitted from two source points S 1 and S 2 , to be detected at two detection points D 1 and D 2 (see fig. 1). For bosons, the two amplitudes D S D S must be added, which yields a factor of 2 excess in the joint detection probability, if the two amplitudes have the same phase. The sum over all pairs (S 1 ,S 2 ) of source points washes out the interference, unless the distance between the detectors is small enough that the phase difference between the amplitudes is less than one radian, or equivalently if the two detectors are separated by a distance less than the coherence length. Study of the joint detection rates vs. detector separation along the i-direction then reveals a bump whose width l i is the coherence length along that axis 1,5, [START_REF] Hanbury Brown | A Test of a New Stellar Interferometer on Sirius[END_REF][START_REF] Baym | The physics of Hanbury Brown-Twiss intensity interferometry: From stars to nuclear collisions[END_REF][START_REF] Boal | Intensity interferometry in subatomic physics[END_REF][START_REF] Viana Gomes | Theory for a Hanbury Brown Twiss experiment with a ballistically expanding cloud of cold atoms[END_REF] . For a source size s i along i (standard half width at e -1/2 of a Gaussian density profile), one has a half width at 1/e of l i = ht / 2πms i , where m is the mass of the particle, t the time of flight from the source to the detector, and h Planck's constant. This formula is the analogue of the formula l i = Lλ / 2πs i for photons if one identifies λ = h / mv with the de Broglie wavelength for particles travelling at velocity v = L / t from the source to the detector.

For indistinguishable fermions, the two-body wave function is antisymmetric, and the two amplitudes must be subtracted, yielding a null probability for joint detection in the same coherence volume. In the language of particles, it means that two fermions cannot have momenta and positions belonging to the same elementary cell of phase space. As a result, for fermions the joint detection rate vs. detector separation is expected to exhibit a dip around the null separation. Such a dip for a fermion ensemble must not be confused with the antibunching dip that one can observe with a single particle (boson or fermion) quantum state, e.g., resonance fluorescence photons emitted by an individual quantum emitter [START_REF] Kimble | Photon antibunching in resonance fluorescence[END_REF] . In contrast to the HBT effect for bosons, the fermion analogue cannot be interpreted by any classical model, either wave or particle, and extensive efforts have been directed toward an experimental demonstration.

Experiments have been performed with electrons in solids [START_REF] Henny | The fermionic Hanbury Brown and Twiss experiment[END_REF][START_REF] Oliver | Hanbury Brown and Twiss-type experiment with electrons[END_REF] and in a free beam [START_REF] Kiesel | Observation of Hanbury Brown-Twiss anticorrelations for free electrons[END_REF] , and with a beam of neutrons [START_REF] Iannuzzi | Direct Experimental Evidence of Free-Fermion Antibunching[END_REF] , but none has allowed a detailed study and a comparison of the pure fermionic and bosonic HBT effects for an ideal gas. A recent experiment using fermions in an optical lattice [START_REF] Rom | Free fermion antibunching in a degenerate atomic fermi gas released form an optical lattice[END_REF] however, does permit such a study and is closely related to our work.

Here, we present an experiment in which we study the fermionic HBT effect for a sample of polarised, metastable 3 He atoms ( 3 He*), and we compare it to the bosonic HBT effect for a sample of polarised, but not Bose condensed, metastable 4 He atoms ( 4 He*) produced in the same apparatus at the same temperature. We have combined the position and time resolved detector previously used 5,26 for 4 He*, with an apparatus with which ultracold samples of 3 He* or 4 He* have recently been produced [START_REF] Mcnamara | Degenerate Bose-Fermi Mixture of Metastable Atoms[END_REF] . Fermions or bosons at thermal equilibrium in a magnetic trap are released onto the detector which counts individual atoms (see Fig. 1) with an efficiency of approximately 10%. The detector allows us to construct the normalised correlation function g (2) (Δr), i.e. the probability of joint detection at two points separated by Δr, divided by the product of the single detection probabilities at each point. Statistically independent detection events result in a value of 1 for g (2) (Δr). A value larger than 1 indicates bunching, while a value less than 1 is evidence of antibunching.

We produce gases of pure 3 He* or pure 4 He* by a combination of evaporative and sympathetic cooling in an anisotropic magnetic trap (see Methods). Both isotopes are in pure magnetic substates, with nearly identical magnetic moments and therefore nearly identical trapping potentials, so that trapped non degenerate and non interacting samples Figure 2 shows clearly the contrasting behaviours of bosons and fermions. In both cases one observes a clear departure from statistical independence at small separation.

Around zero separation, the fermion signal is lower than unity (antibunching) while the boson signal is higher (bunching). Since the sizes of the 3 He* and 4 He* clouds at the same temperature are the same, as are the times of flight (pure free fall), the ratio of the correlation lengths is expected to be equal to the inverse of the mass ratio, 4/3. The observed ratio of the correlation lengths along the z axis in the data shown is 1.3 ±0.2.

The individual correlation lengths are also in good agreement with the formula l z =ht/2πms z . Due to the finite resolution, the contrast in the signal, which should ideally go to 0 or 2 is reduced by a factor of order ten. The amount of contrast reduction is slightly different for bosons and fermions and the ratio should be about 1.5. The measured ratio is 2.4 ± 0.2. This discrepancy has several possible explanations. First, the magnetic field switch-off is not sudden (time scale ~1 ms) and this could affect bosons and fermions differently. Second, systematic errors may be present in our estimate of the resolution function. The resolution however, does not affect the widths of the observed correlation functions along z, and thus we place the strongest emphasis on this ratio as a test of our understanding of boson and fermion correlations in an ideal gas. More information on uncertainties, systematic errors as well as a more complete summary of the data are given in the supplementary material.

Improved detector resolution would allow a more detailed study of the correlation function, and is thus highly desirable. One can circumvent the effect of the resolution using a diverging atom lens to demagnify the source 4 . According to the formula l = ht/2πms, a smaller effective source size gives a larger correlation length. We have tried such a scheme by creating an atomic lens with a blue detuned, vertically propagating, laser beam, forcing the atoms away from its axis (see Methods). The laser waist was not large compared to the cloud size and therefore our "lens" suffered from strong aberrations, but a crude estimate of the demagnification, neglecting aberrations, gives about 2 in the x-y plane. Figure 3 shows a comparison of g (2) (Δz) for fermions with and without the defocusing lens. We clearly see a greater antibunching depth, consistent with larger correlation lengths in the x-y plane (we have checked that l y is indeed increased) and therefore yielding a smaller reduction of the contrast when convolved with the detector resolution function. As expected, the correlation length in the z-direction is unaffected by the lens in the x-y plane. Although our atomic lens was far from ideal, the experiment shows that it is possible to modify the HBT signal by optical means.

To conclude, we emphasise that we have used samples of neutral atoms at a moderate density in which interactions do not play any significant role. Care was taken to manipulate bosons and fermions in conditions as similar as possible. Thus the observed differences can be understood as a purely quantum effect associated with the exchange symmetries of wave functions of indistinguishable particles.

The possibility of having access to the sign of phase factors in a many body wave function opens fascinating perspectives for the investigation of intriguing analogues of condensed matter systems, which can now be realised with cold atoms. For instance, one could compare the many body state of cold fermions and that of "fermionised" bosons in a 1D sample [START_REF] Girardeau | Relationship between systems of impenetrable bosons and fermions in one dimension[END_REF][START_REF] Olshanii | Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons[END_REF] . Our successful manipulation of the HBT signal by interaction with a laser suggests that other lens configurations could allow measurements in position space (by forming an image of the cloud at the detector) or in any combination of momentum and spatial coordinates.

Methods

Experimental sequence. Clouds of cold 4 He* are produced by evaporative cooling of a pure 4 He* sample, loaded into a Ioffe-Pritchard magnetic trap [START_REF] Tychkov | Metastable helium Bose-Einstein condensate with a large number of atoms[END_REF] . The trapped state is 2 3 S 1 , m J = 1 and the trap frequency values are 47Hz and 440Hz, for axial and radial confinement respectively. The bias field is 0.75 G corresponding to a frequency of 2.1

MHz for a transition between the m J = 1 and m J = 0 states at the bottom of the trap.

After evaporative cooling, we keep an RF knife on at constant frequency for 500 ms, then wait for 100 ms before switching off the trap. In contrast to the experiments of Ref.

5, atoms are released in a magnetic field sensitive state.

To prepare 3 He* clouds, we simultaneously load 3 He* and 4 He* atoms in the magnetic trap [START_REF] Mcnamara | Degenerate Bose-Fermi Mixture of Metastable Atoms[END_REF] . The trapping state for 3 He* is 2 3 S 1 , F=3/2, m F =3/2, and axial and radial trap frequencies are 54 Hz and 506 Hz -the difference compared to 4 He* is only due to the mass. The two gases are in thermal equilibrium in the trap, so that 3 He* is sympathetically cooled with 4 He* during the evaporative cooling stage. Once the desired temperature is reached, we selectively eliminate 4 He* atoms from the trap using the RF knife. The gyromagnetic ratios for 4 He* and 3 He* are 2 and 4/3 respectively, so that the resonant frequency of the m=1 to m=0 transition for 4 He* is 3/2 times larger than the m=3/2 to m=1/2 transition for 3 He*. An RF ramp from 3 MHz to 1.9 MHz expels all the 4 He* atoms from the trap without affecting 3 He*. We then use the same trap switch-off procedure to release the 3 He* atoms onto the detector, also in a magnetic field sensitive state. We can apply magnetic field gradients to check the degree of spin polarisation of either species.

Correlation Function. The detailed procedure leading to this correlation is given in Ref. 5. Briefly, we convert arrival times to z positions and then use the 3-dimensional positions of each atom to construct a histogram of pair separations Δr in a particular cloud. We then sum the pair distribution histograms for 1000 successive runs at the same temperature. For separations much larger than the correlation length, this histogram reflects the Gaussian spatial distribution of the cloud. To remove this large scale shape and obtain the normalised correlation function we divide the histogram by the autoconvolution of the sum of the 1000 single particle distributions.

Atom lens experiment.

A 300 mW beam with an elliptical waist of approximately 100×150 µm 2 propagates vertically through the trap. The laser frequency is detuned by 300 GHz from the 2 3 S 1 to 2 3 P 2 transition. After turning off the magnetic trap, and waiting 500 μs for magnetic transients to die away, the defocusing laser is turned on for 500 μs.

Fit results

If one neglects finite resolution effects, the normalised correlation function should be well described by a Gaussian function:

g (2) (Δx,Δy,Δz) = 1± exp - Δx l x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 + Δy l y ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 2 + Δz l z ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ , (1)
where the + sign refers to bosons and the -sign to fermions. We denote the correlation lengths in the 3 different spatial directions i, by l i . In practice this function must be convolved with the resolution function of the detector. The resolution function is determined by the method discussed in Ref. 26. The resolution along the z direction is approximately 3 nm and is neglected. The convolution in the x-y plane is described in Ref. 19 for the case of a Gaussian resolution function. Careful measurements have revealed that the wings of the resolution function are broader than those of a Gaussian and we thus use an empirically determined analytical function to approximate the pair resolution function. Its 1/e halfwidth is about 500 µm. Since the correlation length in the x direction is more than an order of magnitude smaller than the resolution, we set l x = 0.

The convolution also affects the height of the signal so that g (2) (0,0,0) = 1 ± η. The parameter η is referred to as the contrast. The fit parameters are thus l y , l z and η.

Data were taken for fermions ( 3 He*) at 0.5 µK, 1.0 µK and 1.4 µK. The corresponding fit results for l z and η are plotted in Supplementary Figure 2. In addition,

we have data for two other situations, one using 4 He* at 0.5 µK, and another using 3 He* at 0.5 µK and a diverging lens. All 5 runs are summarised in table 1. In the graphs, we have plotted the formula l z =ht/2πms z , extracting the size s z from the measured temperature, trap oscillation frequency and assuming the cloud is an ideal gas. For the contrast η, we plot the expected variation based on the measured resolution function.

main text. This is because, as in Ref. 5, we computed the correlation function along the z axis over an area slightly larger than the width of the resolution function. This procedure improves the signal to noise ratio and preserves the form and the width of the correlation function but slightly modifies its height.

We observe three small anomalies: first, the contrast η for both bosons and fermions at 0.5 µK is below the prediction and the ratio, after correction for the resolution, is 2.4 ± 0.2 instead of the expected value 1.5. Second, the correlation length l y for fermions at 0.5 µK seems quite low resulting in a ratio of fermions to bosons of 1.0 ± 0.2 instead of 1.3. Third, the width of the antibunching dip in the normalised pair separation histogram along y for fermions at 1.4 µK is smaller than the width of the measured resolution function, meaning that the fitted value of l y is consistent with zero.

A systematic error may be present in the estimation of the detector resolution.

After moving the detector from Orsay to Amsterdam, we noticed that the detector resolution differed by up to 30% from day to day. A systematic error in the resolution has approximately the same relative effect on the value of η. It would also have an effect on the value of l y . Uncontrolled variations in the resolution may thus account for the above anomalies. The correlation length in the vertical direction l z however, should not be affected by an imprecise knowledge of the resolution in the x-y plane. The good agreement we find with our expectations along this axis is the strongest argument that the correlations we observe are consistent with the ideal gas model.

A second possible source of systematic error is related to the switch-off of the magnetic trap. Eddy currents cause a typical time scale of 1 ms in this turn-off [START_REF] Tychkov | Metastable helium Bose-Einstein condensate with a large number of atoms[END_REF] . Since, unlike in Ref. 5, the released atoms are in a magnetic field sensitive state, partially adiabatic effects or focussing by residual curvatures could affect our measurement of the temperature or of the effective source size viewed from the detector. We have no independent estimate of the magnitude of these effects and can simply conclude that the reasonable agreement with our model means that these effects are not very large.

  have the same size at the same temperature. The temperatures of the samples yielding the results of Fig.2, as measured by the spectrum of flight times to the detector, are 0.53 ± 0.03 µK and 0.52±0.05 µK for 3 He* and 4 He* respectively. The uncertainties correspond to the standard deviation of each ensemble. In a single realisation, we typically produce 7 × 10 4 atoms of both 3 He* and 4 He*. The atom number permits an estimate of the Fermi and BEC temperatures of approximately 0.9 µK and 0.4 µK respectively. Consequently, Fermi pressure in the trapped 3 He* sample has a negligible (3%) effect on the trap size and repulsive interactions in the 4 He* sample have a similarly small effect. The trapped samples are therefore approximately Gaussian ellipsoids elongated along the x-axis with an rms size of about 110×12×12 µm 3 . To release the atoms, we turn off the current in the trapping coils and atoms fall under the influence of gravity. The detector, placed 63 cm below the trap centre (see fig.1), then records the x-y position and arrival time of each detected atom.The normalised correlation functions g (2) (0,0,Δz) along the z (vertical) axis, for 3 He* and 4 He* gases at the same temperature, are shown in Fig 2. Each correlation function is obtained by analysing the data from about 1000 separate clouds for each isotope (see Methods). Results analogous to those of Fig. 2 are obtained for correlation functions along the y-axis, but the resolution of the detector in the x-y plane (about 500 µm halfwidth at 1/e for pair separation) broadens the signals. Along the x-axis (the long axis of the trapped clouds), the expected widths of the HBT structures are one order of magnitude smaller than the resolution of the detector and are therefore not resolved.

Caption for figure 1 :

 1 The experimental setup. A cold cloud of metastable helium atoms is released at the switch-off of a magnetic trap. The cloud expands and falls under the effect of gravity onto a time resolved and position sensitive detector (micro-channel plate and delay-line anode), that detects single atoms.The inset shows conceptually the two 2-particle amplitudes (in black or grey) that interfere to give bunching or antibunching: S 1 and S 2 refer to the initial positions of two identical atoms jointly detected at D 1 and D 2 . Caption for figure 2: Normalised correlation functions for 4 He* (bosons) in the upper graph, and 3 He* (fermions) in the lower graph. Both functions are measured at the same cloud temperature (0.5 µK), and with identical trap parameters. Error bars correspond to the root of the number of pairs in each bin. The line is a fit to a Gaussian function. The bosons show a bunching effect;the fermions anti-bunching. The correlation length for 3 He* is expected to be 33% larger than that for 4 He* due to the smaller mass. We find 1/e values for the correlation lengths of 0.75±0.07 mm and 0.56±0.08 mm for fermions and bosons respectively. Caption for figure 3: Effect of demagnifying the source size. We show normalised correlation functions along the z (vertical) axis for 3 He*, with (dark blue squares) and without (light blue circles) a diverging atomic lens in the x-y plane. The dip is deeper with the lens because of the increase of the correlation lengths in the x-y plane leading to less reduction of contrast when convolved by the resolution function in that plane.

Fundamental Research on Matter (FOM) and by the Space Research Organization Netherlands (SRON).

The Atom Optics group of LCFIO is member of the IFRAF instute and of the Fédération LUMAT of the CNRS, and is supported by the french ANR and by the SCALA programme of the European Union.

Supplementary Information accompanies the paper on www.nature.com/nature. Acknowledgements: This work was supported by the access programme of LASERLAB EUROPE. The LCVU group in Amsterdam is supported by the "Cold Atoms" program of the Dutch Foundation for Competing Interests statement: The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to christoph.westbrook@institutoptique.fr or w.vassen@few.vu.nl Supplementary material for "Hanbury Brown Twiss effect for bosons versus fermions"

Unnormalised pair histogram

In order to give the reader an idea of the "raw" data, we show in Supplementary figure 1 some unnormalised pair histograms. The data correspond to the normalised plots shown in Fig. 2

Supplementary table 1. Summary of fit results for all data sets

Generally the data are in good agreement with the predictions of the ideal gas model. In the run with the lens, we have made no quantitative comparison with a calculation because it would involve taking into account the severe aberrations of the lens. Qualitatively however, we see that, as expected, the correlation length along z is unchanged while that along y, as well as the contrast, are increased. The fitted values of η do not correspond to those one would deduce from the data in Figs. 2 and 3 in the