P Diogo

Leonor R Godinho

Sophie Rodrigues

Laurane Lefevre

André F Delteil

Inês R Mira

Sara Fragata

Alison B Magalhães

Duncan

Diogo P Godinho
email: diogoprinogodinho@gmail.com

Leonor R Rodrigues

Sophie Lefévre

Laurane Delteil

André F Mira

Inês R Fragata

Sara Magalhães

Alison B Duncan

Limited host availability disrupts the genetic correlation between virulence and transmission

Keywords: Host-parasite interactions, spider-mites, inbred lines, trade-off

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Virulence, the harm inflicted by parasites on their hosts, is a trait with high relevance for human, animal, plant and ecosystem health. Parasites replicate within their host, exploiting their resources and damaging tissues, thus virulence is often positively correlated with replication rate (reviewed in Acevedo et al. 2019). Thus, it is expected that higher replication rates lead to more parasites emerging to infect novel hosts, with a positive impact on parasite fitness (i.e., the reproductive growth rate, R 0 ; [START_REF] Anderson | Coevolution of hosts and parasites[END_REF]. Therefore, selection is expected to also favour higher virulence. This holds as long as virulence is not so high as to induce host mortality, in which case higher virulence shortens the infection period, reducing the chances for transmission, or to cause host morbidity, decreasing the encounter rate with uninfected hosts, which will also hamper transmission. This reduction in transmission opportunities decreases parasite fitness, leading to the so-called virulence-transmission trade-off [START_REF] Anderson | Coevolution of hosts and parasites[END_REF]. This relationship between virulence and parasite fitness is an assumption of most theoretical models of virulence evolution [START_REF] Anderson | Coevolution of hosts and parasites[END_REF][START_REF] Frank | Models of parasite virulence[END_REF][START_REF] Wild | Adaptation and the evolution of parasite virulence in a connected world[END_REF]. Because evolution rests on the existence of genetic variation, it is implicit that the underlying correlation between virulence and transmission is genetic. However, although such correlations have been found in several systems (reviewed in Acevedo et al. 2019), most studies used parasite isolates with different geographic origins and thus that differ not only genetically, but also in their recent ecological and evolutionary history [START_REF] Ebert | Virulence and local adaptation of a horizontally transmitted parasite[END_REF][START_REF] Mackinnon | Genetic relationships between parasite virulence and transmission in the rodent malaria Plasmodium chabaudi[END_REF][START_REF] Roode | Virulence-transmission trade-offs and population divergence in virulence in a naturally occurring butterfly parasite[END_REF][START_REF] Doumayrou | An experimental test of the transmission-virulence trade-off hypothesis in a plant virus[END_REF]. Hence, values for virulence and transmission may hinge upon different environments selecting for particular combinations of traits, and not on hardwired genetic correlations among traits [START_REF] Hochberg | Establishing genetic correlations involving parasite virulence[END_REF]. Disentangling between the environmental and genetic factors affecting the relationship between virulence and transmission is possible using inbred lines derived from the same genetically diverse parasite population [START_REF] Godinho | Creating outbred and inbred populations in haplodiploids to measure adaptive responses in the laboratory[END_REF].

Using this approach is key to identify the conditions under which these parasite traits can evolve independently.

The evolution of virulence is also expected to be contingent upon other factors such as host population structure [START_REF] Thrall | Evolution of virulence in a plant host-pathogen metapopulation[END_REF] and the timing of transmission [START_REF] Day | Virulence evolution and the timing of disease life-history events[END_REF]. Theory predicts that virulence should be reduced in viscous populations (in which individuals tend to remain in the patch where they were born) because in these populations relatedness among parasites increases and the benefits of producing dispersers are reduced [START_REF] Boots | Small worlds' and the evolution of virulence: Infection occurs locally and at a distance[END_REF][START_REF] Wild | Adaptation and the evolution of parasite virulence in a connected world[END_REF][START_REF] Lion | Are parasites '"prudent"' in space?[END_REF]. Experimental evidence for this prediction has been shown in phages [START_REF] Kerr | Local migration promotes competitive restraint in a host-pathogen 'tragedy of the commons[END_REF] and viruses [START_REF] Boots | Local interactions select for lower pathogen infectivity[END_REF]. Selection on virulence is also predicted to be contingent upon the timing of transmission, with early transmission selecting for higher virulence [START_REF] Day | Virulence evolution and the timing of disease life-history events[END_REF], which has been shown empirically in microcosm populations (Berngruber et al 2013). Both these factors, population viscosity and the timing of transmission, may also change within-host parasite dynamics. Indeed, in viscous populations and in populations where individuals disperse at the end of infection only, individuals are expected to remain on their patch for longer periods, which may lead to density dependence. Density dependent parasite growth has been shown in several systems [START_REF] Keymer | Density-dependent mechanisms in the regulation of intestinal helminth populations[END_REF][START_REF] Ebert | Dose effects and density-dependent regulation of two microparasites of Daphnia magna[END_REF][START_REF] Roode | Virulence determinants in a natural butterflyparasite system[END_REF][START_REF] Pollitt | Costs of crowding for the transmission of malaria parasites[END_REF], and it may affect the relationship between virulence and transmission [START_REF] Bonneaud | Experimental evidence for stabilizing selection on virulence in a bacterial pathogen[END_REF], a possibility that remains unexplored. Both population viscosity and the timing of transmission are expected to critically hinge upon host availability during the infection period, as individuals will only effectively disperse when other hosts are attainable. Therefore, host availability during infection is expected to affect virulence, as well as its correlation with transmission.

Here we tease apart the impact of genetic and environmental factors on the relationship between virulence and transmission, using inbred lines of the spider mite Tetranychus urticae, a plant macro-parasite [START_REF] Godinho | Creating outbred and inbred populations in haplodiploids to measure adaptive responses in the laboratory[END_REF]. Spider mites spend their entire life-cycle on their host plants [START_REF] Helle | Spider mites: Their biology, natural enemies and control[END_REF], with high intrinsic growth rates, their dispersal (i.e., transmission between hosts) and damage they inflict resulting in major economic losses in agriculture worldwide [START_REF] Helle | Spider mites: Their biology, natural enemies and control[END_REF]. This damage, presumably negatively correlated with plant fitness [START_REF] Fineblum | Tradeoff between resistance and tolerance to herbivore damage in a morning glory[END_REF] is visible as chlorotic lesions on the leaf surface. Thus, it can be easily quantified [START_REF] Mira | A method to measure the damage caused by cell-sucking herbivores[END_REF]; Fig. S1), being a reliable measure of virulence [START_REF] Noël | The scale of competition impacts parasite virulence evolution[END_REF]. Spider mites can transmit to nearby hosts from early on in an infection, or over-exploit the host plant before transmission occurs [START_REF] Smitley | Photo-oriented aerial-dispersal behavior of Tetranychus urticae (Acari: Tetranychidae) enhances escape from the leaf surface[END_REF][START_REF] Bitume | Density and genetic relatedness increase dispersal distance in a subsocial organism[END_REF]. Such transmission depends on many factors, like within-host parasite density, relatedness and/or the availability of suitable hosts [START_REF] Bitume | Density and genetic relatedness increase dispersal distance in a subsocial organism[END_REF][START_REF] Roissart | Spatial and spatiotemporal variation in metapopulation structure affects population dynamics in a passively dispersing arthropod[END_REF]. Because spider mites are macroparasites, as a proxy for transmission we used the number of transmitting stages (i.e., adult females [START_REF] Anderson | The invasion, persistence and spread of infectious diseases within animal and plant communities[END_REF]), a measure we validate here. We manipulated the opportunity for transmission and the strength of competition of inbred lines of the spider mite T. urticae and quantified how this affected the production of transmitting stages and virulence. We assessed whether a relationship between virulence and transmission was determined by genetic differences among lines, and/or by the build-up of density-dependence within the host.

Additionally, we evaluated whether this relationship was affected by opportunities for transmission during the infection period.

Materials and Methods

Spider mite inbred lines

Tetranychus urticae was collected on different host plants, in Portugal in 2013 (Zélé et al., 2018a), and has since been reared on bean plants (Phaseolus vulgaris, variety Prelude), at the University of Lisbon. In October 2015, 50 individuals from six different field populations (total: 300) were collected and mixed to form an outbred population maintained at high densities (>1000 mites). In October 2016, inbred lines were created from this population by sib mating. This procedure was repeated for 14 generations, ensuring an inbreeding coefficient above 94% [START_REF] Godinho | Creating outbred and inbred populations in haplodiploids to measure adaptive responses in the laboratory[END_REF]. Inbred lines allow simultaneous measurement of many individuals of the same (nearly) homozygous genotype, increasing the accuracy of genetic estimates [START_REF] Godinho | Creating outbred and inbred populations in haplodiploids to measure adaptive responses in the laboratory[END_REF]. Additionally, given that the original population was outbred, genetic variation across lines is expected to be high and because they were derived from the same population, all lines share the same evolutionary and environmental history [START_REF] Godinho | Creating outbred and inbred populations in haplodiploids to measure adaptive responses in the laboratory[END_REF].

Lines were maintained separately on bean leaf patches in Petri dishes. A subset of 15 inbred lines were transferred to the University of Montpellier in January 2018 and maintained on bean leaves (variety Pongo) in small plastic boxes (255 mm length x 183 mm width x 77 mm height) at optimal conditions (25˚C with a 16:8 L: D cycle, at 60% relative humidity). Under these conditions, egg to adult development takes ca. 11 to 13 days. These same conditions were kept throughout all experiments.

Prior to the experiments, cohorts of spider mites from each inbred line were created by isolating 40 to 50 mated females of each line during 48h. After 14 days, the resulting mated females (daughters) were used in the experiments. In this species, mated females are the main dispersers (i.e., transmitting stages) and, additionally, the damage inflicted by males is negligible, thus, does not significantly contribute to virulence [START_REF] Helle | Spider mites: Their biology, natural enemies and control[END_REF]. Not all inbred lines are represented in each experiment due to too few individuals available at the start of the experiment (12 to 14 lines analysed).

Continuous transmission during the infection period

Adult females were randomly assigned to two density treatments (10 or 20 females, respectively), on a four cm 2 bean leaf patch, where they fed and laid eggs for four days. Then, adult females were removed, and a photograph of each patch was taken using a Canon EOS 70D camera. The amount of damage inflicted by spider mites was measured using ImageJ and Ilastik 1.3, using a method developed specifically for spider mites ( [START_REF] Mira | A method to measure the damage caused by cell-sucking herbivores[END_REF]; Fig. S1). Host patches unexposed to spider mites were used as a control. The number of eggs laid was registered, as a measure of parasite replication. On day four, a second leaf patch, uninfected by spider mites, was placed beside the first and connected to it by a 3 x 1 cm Parafilm bridge, thus allowing dispersal of the emerging adult female offspring (Fig. S2). The number of adult daughters on the new host patches was checked on days 11, 12 and 13 in blocks 2 to 4 and on days 12 and 13 in block 1.

When there were more than 15 offspring on the new patch, the latter was replaced by a new one, such that uninfected patches were always available. On day 14, we counted the number of adult daughters on the original host patch and on each of the new patches, i.e., the number of transmitting stages produced (potential transmission). Effective transmission was inferred by the cumulative number of females that infected a new host patch. This set-up mimics the life cycle of a parasite with continuous transmission during the infection period. There were 5 to 16 replicates for each inbred line per density treatment, distributed across four blocks.

Transmission at the end of the infection period Females of each inbred line were randomly assigned to three density treatments, (5, 10 or 20 founding females), on a four cm 2 bean leaf patch, where they laid eggs for four days. They were then removed, a photograph was taken of each leaf to measure the damage inflicted and the number of eggs laid was counted. Fourteen days later, the adult daughters on each patch were counted, i.e. the number of transmitting stages produced (potential transmission). In this set-up, transmission would only be possible after this measurement, i.e., at the end of the infection period (Fig. S2). There were 3-13 replicates for each inbred line per density, distributed across three blocks.

Statistical analysis

We present correlations between different traits (Table 1): virulence (measured as the damage inflicted) and parasite replication (the number of eggs laid), virulence and potential transmission (the number of adult daughters, i.e., transmitting stages). Additionally, to test whether densitydependence rises from the negative effects of intraspecific competition on the survival of juvenile stages, we check the correlation between virulence and juvenile mortality (i.e., proportion of eggs that did not develop into adults). We also validate our measure of transmission by assessing the correlation between the number of transmitting stages (potential transmission) and the number of females colonising novel host patches (effective transmission). To test genetic correlations, we consider total values per host patch, as theoretical predictions are tailored to such population values [START_REF] Anderson | Coevolution of hosts and parasites[END_REF][START_REF] Anderson | The invasion, persistence and spread of infectious diseases within animal and plant communities[END_REF] and most experimental studies on this topic have used them [START_REF] Mackinnon | Genetic relationships between parasite virulence and transmission in the rodent malaria Plasmodium chabaudi[END_REF][START_REF] Roode | Virulence-transmission trade-offs and population divergence in virulence in a naturally occurring butterfly parasite[END_REF][START_REF] Doumayrou | An experimental test of the transmission-virulence trade-off hypothesis in a plant virus[END_REF].

Note however that using per capita data would not modify conclusions on correlations, as density would be present in both numerator and denominator.

Genetic and environmental correlations were measured using a multi-response generalized linear mixed model fitted with an MCMCglmm (package MCMCglmm; [START_REF] Hadfield | MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package[END_REF]). Genetic correlations were determined by including the identity of the line as a random factor in each model and assessing the highest posterior density interval (HPDI) of the genetic (G) structure of the model, which represents the (co)variances between the two traits evaluated across inbred lines [START_REF] Hadfield | MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package[END_REF]. Environmental correlations were obtained by assessing the HPDI of the residual (R) structure in the same model [START_REF] Hadfield | MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package[END_REF]. Effects were considered significant when the HPDI did not include zero [START_REF] Hadfield | MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package[END_REF]. The effect of initial density on correlations was assessed by comparing the deviance information criterion (DIC) of a model not including density with that of a model including density as a covariate (i.e. making it a function-value trait; [START_REF] Gomulkiewicz | Variation and evolution of function-valued traits[END_REF]). Models were considered different when the difference in DIC was higher than 2. Additionally, we report the genetic and environmental correlations for each density level separately (Table 1). We also tested if the relationship between virulence and transmission might be quadratic or saturating, as predicted by theoretical models [START_REF] Anderson | Coevolution of hosts and parasites[END_REF][START_REF] Alizon | Virulence evolution and the trade-off hypothesis: History, current state of affairs and the future[END_REF], by comparing the DIC of models with the number of transmitting stages produced as the response variable with virulence included as a linear term, with models that also included the quadratic and/or square-rooted terms for virulence, respectively.

To measure genetic variance for traits (variance among inbred lines) and the effect of density on this variance we used per capita values, by dividing the value for each host patch by the initial density of adult females (Fig. S3, Table S1). Note that here we did not use population values because heritability measures concern individual, not population traits. We then applied generalized linear mixed models fitted with a Markov Monte Carlo Chain approach [START_REF] Hadfield | MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package[END_REF] with per capita trait values as the response variable and density included in the model as a fixed factor and inbred line as a random factor. Broad-sense heritability, [START_REF] Falconer | Introduction to Quantitative Genetics[END_REF] and the corresponding confidence intervals were extracted from the abovementioned models for each trait. Because heritability is a proportion between 0 and 1, and thus the HPDI is bounded to fluctuate above 0, instead, we determined if there was significant genetic variation by a) assessing the 95% HPDI of the genetic component of each model (i.e.

variance among inbred lines) and b) comparing, for each trait, the DIC of a model including inbred line as a random factor with that of a model without inbred line as a random factor [START_REF] Henriques | The sources of variation for individual prey-to-predator size ratios[END_REF].

All models initially included 300000 iterations, with a burn-in of 10000 iterations, thinning of 100 and a flat prior: For GLMMs (to assess genetic variance), V = 1 and nu = 0.002, assigning a weak informative variance (V) and a low degree of belief (nu) [START_REF] Gelman | Prior distributions for variance parameters in hierarchical models[END_REF]; For Multi-response GLMMs (to assess trait correlations), V = matrix(c(1,0,0,1), ncol = 2, nrow = 2) and nu = 0.002, meaning no covariance between traits, in accordance with the null hypothesis of no correlation.

Additionally, we ran the models using an uninformative prior based on an inverse-Gamma distribution (V= diag(2)*(0.002/1.002), nu=1.002), which gave similar results (not shown) [START_REF] Gelman | Prior distributions for variance parameters in hierarchical models[END_REF]. These priors were used to allow the hyper-parameter values to reflect a reasonable range of values for the traits in question, without any previous information about them or their co-variance. All models were checked for convergence with a stationary test using the heidel.diag function and for autocorrelations in the Markov chain within fixed and random terms using the autocorr.diag function (package coda, [START_REF] Plummer | CODA: convergence diagnosis and output analysis for MCMC[END_REF]) . When models failed one of these tests, the number of iterations was increased to 500000 or 700000 and the burn-in to 20000 or 50000. All figures were produced with the ggplot2 package in R and the regressions fitted with the geom_smooth function [START_REF] Wickham | Ggplot2: Elegant graphics for data analysis[END_REF].

Results

Validation of our measure of transmission

We found a positive correlation, both environmental (i.e., among residuals of the two variables, r e = 0.99) and genetic (i.e., among the factor 'inbred line' for the two variables, r g = 0.99), between the number of transmitting stages produced (i.e., adult daughters) and transmission to uninfected hosts (Table 1a; Fig. 1), indicating that the former is a good proxy for the latter.

Genetic variation for replication rate, virulence, and transmission

To observe genetic correlations among parasite traits, genetic variance for these traits must be present in the population. Thus, we quantified genetic variance for virulence (% of damage inflicted), parasite replication (number of eggs) and transmission (the number of transmitting stages) among the T. urticae inbred lines (Fig. S3), using per capita measurements, which allows the determination of broad sense heritability, H 2 (Table S1, Fig. S4 andS5). We found significant genetic variation for all traits, as the 95% HPDI of the genetic component in the models did not include 0 (Table S1) and the DIC of models including the inbred line as a random factor were lower than those excluding it (Table S2). Additionally, all traits were affected by the initial density of founding females on the host patch (Table S3), but not in a similar way. Indeed, transmission of all lines decreased with density, whereas for parasite replication and virulence the pattern was line-specific (Fig. S3).

Correlation between virulence and parasite replication

We found positive genetic and environmental correlations between virulence (% of damage inflicted) and parasite replication (number of eggs laid) both when transmission occurred continuously during the infection period and when transmission was restricted to the end of the infection period (Table 1b, Fig. 2). These relationships were not affected by the initial parasite density (Table 1b).

Correlation between virulence and transmission

When transmission was continuous, there was no overall environmental or genetic correlation between virulence and transmission across densities. However, at each density (intermediate and high), we found positive environmental and genetic correlations between virulence and transmission (Table 1c; Fig. 3a). In this scenario, selection on virulence will directly affect transmission, therefore parasite fitness is expected to increase with virulence. Accordingly, we did not find any evidence for a quadratic or saturating relationship between virulence and transmission (DIC 2026.7 and 2026.6, respectively, as compared with the linear model 2026.4).

In contrast, when transmission was restricted to the end of the infection, we found a negative environmental correlation between virulence and transmission, but no genetic correlation between these traits (Table 1c, Fig. 3b). Furthermore, although there was no quadratic nor saturating relationship between these traits (DIC 2691.1 and 2691.1, respectively, as compared with the linear model 2690.9), the direction of the correlation between virulence and transmission changes from positive at low density, to absent at intermediate density and negative at high density (Table 1c). This density dependence pertains from the negative effects of intraspecific competition at high densities, which is confirmed by a positive environmental correlation between virulence and juvenile mortality at all densities, which was not the case in the scenario with continuous transmission (Table 1d, Fig. 4). In this scenario, with transmission at the end of the infection period, virulence is not genetically correlated with transmission, but high virulence will lead to strong density dependence within hosts, independently of the genotype considered.

Discussion

In this study we show that opportunities for transmission shape the relationship between virulence and transmission, a key component of parasite fitness. We found a positive genetic correlation between virulence and transmission when the latter was continuous during infection, independently of the initial parasite density. In contrast, when transmission was restricted to the end of the infection period the relationship between these traits no longer had a genetic basis, but instead, was modulated by density dependence: beyond a certain level of virulence, at high initial densities of mites, despite an increase in replication (i.e., number of eggs laid), within-host competition prevents most offspring from becoming adult, such that highly virulent genotypes produce fewer adult daughters. Evidence for density-dependence has been found in this [START_REF] Rotem | Density dependent population growth of the two-spotted spider mite, Tetranychus urticae, on the host plant Leonurus cardiaca[END_REF][START_REF] Roissart | Spatial and spatiotemporal variation in metapopulation structure affects population dynamics in a passively dispersing arthropod[END_REF], and other host-parasite systems, including microparasites [START_REF] Ebert | Dose effects and density-dependent regulation of two microparasites of Daphnia magna[END_REF][START_REF] Roode | Virulence determinants in a natural butterflyparasite system[END_REF][START_REF] Pollitt | Costs of crowding for the transmission of malaria parasites[END_REF].

The positive genetic correlation between virulence and transmission, found when parasites could transmit during the infection period, suggests that there will be selection for high virulence. If this is the case, why then do we still find genetic variance for this trait? We propose three non-mutually exclusive hypotheses. First, T. urticae is a generalist parasite, hence it may frequently shift among hosts, and optimal virulence may vary with the host species [START_REF] Gandon | Evolution of multihost parasites[END_REF][START_REF] Magalhães | Do mites evolving in alternating host plants adapt to host switch?[END_REF]Zélé et al., 2018b). Furthermore, different host plant species, as well as different individuals of the same host plant, may harbour different heterospecific parasites [START_REF] Wimp | Conserving plant genetic diversity for dependent animal communities[END_REF][START_REF] Dinnage | Diversity of plant evolutionary lineages promotes arthropod diversity[END_REF]. Coinfections with other parasites are expected to select for different levels of virulence depending on the direction and mechanism of the interaction [START_REF] Choisy | Mixed infections and the evolution of virulence: Effects of resource competition, parasite plasticity, and impaired host immunity[END_REF][START_REF] Alizon | Multiple infections and the evolution of virulence[END_REF][START_REF] Clay | How parasite interaction strategies alter virulence evolution in multi-parasite communities[END_REF]. Finally, transmission within the infection period relies on the occurrence of hosts to which parasites can transmit. This may not always be possible, as uninfected hosts may be locally absent or become rapidly infected (Hochberg 1991;Crossan et al. 2007). Mites are known to base their decision to leave a host on the perception of cues (e.g., volatiles) from hosts in the environment, including their infection status (Pallini et al., 1997;Kiedrowicz et al., 2017). Therefore, in the absence of uninfected hosts, mites may remain on their host until they overexploit it, which will eventually result in them switching to a parasite life cycle with transmission occurring only at the end of the infection period.

In the latter scenario, when infection was restricted to the end of the infection, we found no genetic correlation between virulence and transmission. Instead, we found a positive environmental correlation at low density, which became negative at high density, leading to a peak of transmission at intermediate levels of virulence, as predicted by theory [START_REF] Anderson | Coevolution of hosts and parasites[END_REF]. However, this relationship between virulence and transmission was not determined by differences among inbred lines, hence it was purely environmental. The fact that the negative correlation between virulence and transmission disappears when transmission occurs continuously during the infection further supports the hypothesis that density dependence modulates the peak of transmission at intermediate levels of virulence only when transmission occurs at the end of the infection period. This effect of the timing of transmission may also explain the mixed empirical evidence for the occurrence of the virulence transmission trade-off, as continuous transmission will only be relevant for some parasite life-cycles and/or experimental designs (Acevedo et al., 2019).

Such modulation of epidemiological trade-offs by the environment has been found in host traits in another system [START_REF] Hall | A power-efficiency trade-off in resource use alters epidemiological relationships[END_REF]. Also, previous work has shown that limited opportunities for transmission can select for lower virulence, ensuring a reduction in kin competition and/or host depletion [START_REF] Wild | Adaptation and the evolution of parasite virulence in a connected world[END_REF][START_REF] Lion | Are parasites '"prudent"' in space?[END_REF]. In line with this, parasite transmission in more structured populations has been shown to select for lower virulence [START_REF] Kerr | Local migration promotes competitive restraint in a host-pathogen 'tragedy of the commons[END_REF][START_REF] Boots | Local interactions select for lower pathogen infectivity[END_REF]Berngruber et al., 2013). We do not find this in our experiment. In fact, at all densities, virulence and parasite replication (i.e., number of eggs) were positively correlated. However, at high densities, we observed high juvenile mortality (Fig. 4). Therefore, genotypes that induce more harm to their host will not necessarily produce more transmitting stages, which explains the absence of a positive genetic correlation between virulence and transmission. The negative environmental correlation found between these traits at high densities may simply be a by-product of density dependence: fewer transmitting stages are produced in plant patches that are more damaged by parasites. Possibly, spider mites are not selected to thrive in highly competitive environments, hence they still produce as many eggs as possible (a trait selected in low-competition environments) but then these individuals die due to absence of resources. Alternatively, spider mites may have been selected in saturated environments, but in this case we would expect them to maximize their competitive ability, rather than resource consumption, which would also not lead to more virulent genotypes producing more transmitting stages. In any case, under this scenario, given the breakdown of the genetic correlation between virulence and transmission, selection on virulence may not affect transmission. In other words, the evolution of virulence may not affect parasite fitness, given that parasite fitness includes both within-host (parasite replication) and between-host (transmission, here the number of transmitting stages) traits [START_REF] Anderson | Coevolution of hosts and parasites[END_REF][START_REF] Alizon | Adaptive virulence evolution: The good old fitness-based approach[END_REF]. Thus, variance for virulence in parasite populations, as observed in this (Fig. S3) and other systems (Dutta et al., 2021;Little et al., 2008;Mackinnon and Read, 1999a) may be maintained, instead, by the heterogeneity pertaining from host availability, variability and connectivity [START_REF] Thrall | Evolution of virulence in a plant host-pathogen metapopulation[END_REF][START_REF] Boots | Local interactions select for lower pathogen infectivity[END_REF][START_REF] Lion | Are parasites '"prudent"' in space?[END_REF]. Moreover, the (co)evolution of host traits, such as resistance, may create temporal heterogeneity for parasites, contributing to the maintenance of variance for virulence in parasite populations [START_REF] Lambrechts | Coevolutionary interactions between host and parasite genotypes[END_REF].

Our results support the prediction that high virulence is advantageous in the beginning of the infection period, when many uninfected hosts are available, but this advantage may be lost later on when fewer hosts are available [START_REF] Day | Virulence evolution and the timing of disease life-history events[END_REF]. Indeed, in our study we find a positive correlation between virulence and transmission, both under continuous transmission and transmission at the end of the infection period, but in the latter case only when parasite density is low, which should typically occur at the beginning of epidemics. At high density, though, this correlation is negative, which suggests that reduced transmission opportunities could result in lower parasite virulence, if parasites respond plastically to the current environment, as has been shown in viruses and Plasmodium [START_REF] Reece | Sex ratio adjustment and kin discrimination in malaria parasites[END_REF][START_REF] Leggett | Experimental evolution of adaptive phenotypic plasticity in a parasite[END_REF]. Besides affecting the relationship between virulence and transmission, host availability may also affect the length of the infection, the other key component of parasite fitness [START_REF] Anderson | Coevolution of hosts and parasites[END_REF], since early transmission may slow host exploitation. However, our design did not allow us to test this hypothesis.

In sum, we show that virulence is genetically correlated with transmission. However, this key assumption of most theoretical models of the evolution of virulence only holds under continuous transmission. Instead, selection for higher virulence in this system is expected to disappear under limited parasite transmission due to within-host density-dependence. This suggests that this fundamental difference in parasite life cycle, i.e., continuous transmission vs transmission at the end if the infection period, has key implications for virulence evolution.

Additionally, it highlights that the effect of within-host density dependence may hinge upon the parasite life cycle. Indeed, whereas for parasites that transmit at the end of the infection period (e.g. obligate killers), within-host density dependence may underlie the predicted optimal transmission at intermediate virulence levels, for parasites that transmit continuously, it may only come into play later in the epidemic, when uninfected hosts become limited. In any case, our results demonstrate that only by including the whole parasite life-cycle in experimental set-ups is it fully possible to understand the ecology and evolution of virulence [START_REF] Alizon | Adaptive virulence evolution: The good old fitness-based approach[END_REF].

Table 1. Genetic and environmental correlations between traits measured. Genetic (rg) and environmental (re) correlations -extracted from the genetic and residual error structure of the models, respectively,were measured between a) the production of transmitting stages (adult daughters) and transmission, and between virulence and b) parasite replication, c) the production of transmitting stages (adult daughters) and d) juvenile mortality (i.e. proportion of eggs that did not develop into adults). All traits were measured per host (as the sum of the contribution of all females initially placed on the leaf). The deviance information criterion (DIC) for models with and without density included are shown. Models with the lowest DIC or the simplest model (when there was no difference in DIC values), are identified using an *. Highest posterior density intervals (HPDI) and correlations coefficients (rg and re) are shown for the models including all data, and separately for each of the different densities. Intervals that do not include zero are shown in bold. Experiment: 1) continuous: continuous transmission; 2) end: transmission at the end of the infection period.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Correlation between transmitting stages and transmission. The number of daughters produced (transmitting stages) and transmission (the number of mites infecting new hosts) for inbred lines of T. urticae infesting hosts at different starting densities (10 females = green circles; 20 females = blue triangles) with the possibility of continuous transmission; dots are the mean for each inbred line ± standard error; the regression (fitted with geom_smooth function) is represented in red.

Figure 4 .

 4 Figure 4. Correlation between virulence and juvenile mortality in inbred lines of T. urticae infecting a host patch. The damage inflicted on the host patch (i.e., virulence) and juvenile mortality (i.e. proportion of eggs that did not develop into adults) were measured in a set-up with a) continuous transmission during the infection period and b) transmission at the end of the infection period. Densities are coded in yellow squares (5 females), green circles (10 females) and blue triangles (20 females); dots are the mean for each inbred line ± standard error; regressions (fitted with geom_smooth function) are shown in red.

Acknowledgments

We deeply thank Élio Sucena, Flore Zélé, Margarida Matos, Oliver Kaltz, Sylvain Gandon and Yannis Michalakis, for their very useful suggestions and comments.

Funding ERC (European Research Council) consolidator grant COMPCON, GA 725419 attributed to SM, PHC-PESSOA grant (38014YC) to ABD and SM, and FCT (Fundação para Ciência e Técnologia) PhD scholarship (PD/BD/114010/2015) to DPG and with the Junior researcher contract (CEECIND/02616/2018) attributed to IF. This is ISEM contribution no ISEM 2022-317.

Data and materials availability

Data is deposited in Dryad: https://doi.org/10.5061/dryad.4xgxd25dv.

Competing interests

Authors declare no competing interests.

Author contributions

Conceptualisation and supervision by SM


Morgane Gerardin

Jean-Philippe Farrugia

Thierry Fournel

Mathieu Hébert

A translucency classification for computer graphics

Translucency is a visual property attributed to objects that light may cross without transmitting a clear image of the scene which is behind. In absence of a more precise definition, this perceptual attribute is often considered as an intermediate between transparency, which is the property of objects that light may cross by transmitting a clear image of the scene behind, and opacity, which is the property of blocking the transmission of light and therefore masking completely the scene behind. If it is rather clear that translucency is closely related to light scattering, it is difficult to classify the translucent appearance according to one scale only, due to the different types of scattering, which can occur as well as the role of absorbance and thickness of the material. Through synthetic images rendered by optical models, we show that surface scattering, volume (or subsurface) scattering, possibly mixed with selective absorption, produce different types of translucency effects and different intermediates between transparency and opacity. We thus propose to represent translucency according to three axes related to these three optical phenomena: surface scattering, volume scattering, and absorption.

Introduction

While it is used to denote the appearance of a wide range of materials such as milk, jade, ceramic, or skin, translucency is a perceptual and physical concept, which is still not well defined. Many models have been developed in computer graphics to render various light effects but translucency has not been fully explored yet.

Translucency is often viewed as the intermediate between transparency and opacity. Among the rare studies dedicated to this concept, the one by Fleming et al. [START_REF] Fleming | Perceiving translucent materials[END_REF] addresses the opposition between translucency and opacity through image synthesis. For example, the authors show that translucency of smooth objects is more noticeable when the object is illuminated from behind than when illuminated on the front side. Moreover, when illuminated by a directional source, an opaque object appears more contrasted than a translucent one. The last point the authors explored is the addition of a blur in the images but this operation is not sufficient to explain the perception of translucency because other phenomena such as depth of field or penumbra effects can be at the origin of similar blurring effect. Another study from Gkioulekas et al. [START_REF] Gkioulekas | Understanding the role of phase function in translucent appearance[END_REF] shows the importance of single scattering within the material to explain differences in translucency. Finally, Ref. [START_REF] Motoyoshi | Highlight-shading relationship as a cue for the perception of translucent and transparent materials[END_REF] stipulates that opacity, translucency and transparency can be described on the same scale to carry out a perceptual experiment.

However, the elements provided in the literature mentioned above are not precise enough to render the different kinds of translucency we can find with objects according to the material in which they are made and their thickness. It is noticeable that the oppositions transparency/translucency and translucency/opacity do not rely on similar perceptual criteria nor similar physical phenomena.

We may agree on the fact that a object is considered as translucent when light can go through it while being scattered. As a result, the scene that is transmitted through the object appears blurred. Two diffusion phenomena may occur: light may be scattered either at the object's surface, which is called surface scattering, or within the object itself, which is known as subsurface scattering, or volume scattering.

Decomposing translucency among absorption, surface and volume scattering would lead to a more complete, physicallybased, intuitive and artist-friendly representation, which may be useful for computer graphics or manufacturing.

In this paper, we define a three-dimensional representation system adapted to a wide range of translucency effects in Section 1. The implementation details are given in Section 2, and some specific points in this representation space are illustrated through generated synthetic images in Section 3. These results are discussed in Section 4, and Section 5 draws the conclusions.

Three-dimensional representation space

In 1225, an English theologian Robert Grosseteste wrote a short text in latin entitled "De Colore" which has been interpreted by Ref. [START_REF] Smithson | A three-dimensional color space from the 13th century[END_REF] as the first intent of a color classification system composed of three axes. The authors have defined these axes based on three oppositions stated in Grosseteste's text: obscura/clara (dark/light), pauca/multa (little/much), impurum/purum (pure/impure); and proposed a representation of Grosseteste's system as a cube, similar to the RGB cube for representing colors in a digital image. This idea of assessing a main visual attribute with independent criteria inspired our approach for assessing translucency.

We have identified three main optical phenomena giving to the objects their transparency, translucency or opacity. To build our representation system, each of these phenomena has been described independently from one another by using a physical law or a model. Some assumptions have been made so that only one parameter is enough to describe the appearance variation over one axis: -absorption, parameterized by the density of absorbers in the material, the absorbers being characterized by an absorption coefficient.

-surface scattering (i.e., the scattering of light at the air-material interface), parameterized by a surface roughness parameter, -volume scattering, i.e., the scattering of light within the material itself, parameterized by a scatterer density.

The three-dimensional representation is featured in Figure 1, and the optical models used to render the three optical phenomena are presented below.

Absorption

The absorbance of an object is described by Beer-Lambert-Bouguer's law [START_REF] Ingle | Spectrochemical analysis[END_REF]. Let's denote A the absorbance of the material and T its transmittance. The Beer-Lambert-Bouguer's law is defined as in Equation ( 1), where σ is the attenuation cross section, n the absorber density and d the thickness that is crossed by the light through the object.

T =10 -A =e -σ nd (1) The transmittance decreases with the thickness of the object. Along this axis, we consider that the absorptivity of the material is fixed, and so we defined the absorber density n as the variable parameter, which controls the absorption of the material.

We assume in our approach that absorption is the only phenomenon responsible for coloration of the material. At the origin of this "absorption" axis, the material is transparent and achromatic (i.e. clear). As the material is more absorbing, its color becomes darker and more saturated. Finally, for an infinite absorbance, the material becomes completely dark and opaque. Along this axis (the other two parameters being zero) are represented the purely absorbing media with a flat surface, which are either transparent or opaque, but not translucent. The appearance along this axis can be seen in Figure 3.

Surface scattering

We use the micro-facet model proposed by Walter [START_REF] Walter | Microfacet models for refraction through rough surfaces[END_REF] to compute surface scattering in both reflection and transmission. The model uses Smith's masking-shadowing function [START_REF] Smith | Geometrical shadowing of a random rough surface[END_REF], and the isotropic GGX micro-facet normal distribution, which provides a close match to the measured data. It allows controlling the surface scattering through a surface roughness parameter, which corresponds to the standard deviation of the slope distribution. At the origin of this axis, the roughness parameter is zero, which corresponds to a smooth and perfectly transparent material. By increasing this parameter, the material becomes rougher and the object less translucent. In Walter's model, inter-reflections between the micro-facets are not taken into account. This results in an energy loss, sensible when the roughness is too high. The correct amount of energy leaving a rough surface can be derived by using the approach that Heitz [START_REF] Heitz | Multiplescattering microfacet BSDFs with the Smith model[END_REF] has developed in order to take into account these inter-reflections as the common micro-facet models does not. Along this "surface scattering" axis are represented the clear materials with rough surface, which are translucent but cannot be opaque (Figure 4).

Sub-surface scattering

For the scattering of light within the material itself, we compute it according to the dipole model proposed by Jensen [START_REF] Jensen | A practical model for subsurface light transport[END_REF], by assuming inclusions in the medium, which scatter the light in an isotropic way. This is the density of this scatterer, which allows us to control the appearance of the material along this axis. At the origin of the "volume scattering" axis, the density of the diffusing inclusions is zero: the material appears as achromatic and transparent. As the inclusion density increases, the lateral diffusion within the material becomes more important. Beyond a certain limit, the density is so high that the free mean path of the light within the material is very short and the material looks totally opaque. This particular case corresponds to the interfaced Lambertian model proposed in Ref. [START_REF] Elias | Bidirectional reflectance of a diffuse background covered by a partly absorbing layer[END_REF]. Along this axis are represented the scattering and non-absorbing materials, whose appearance can vary from transparent to white opaque through various degrees of translucency (Figure 5).

Implementation

In computer graphics, the way materials scatter light is characterized by their BSDF (Bidirectional Scattering Distribution Function). These models are valid if the incident light is only scattered locally at the surface of the object (surface scattering). It is mostly the case for transparent, opaque or thin translucent objects. This function can be decomposed into two light contributions: the light that is reflected by the surface of the object, described by the BRDF (Bidirectional Reflection Distribution Function) and the light that is transmitted through the object described by the BTDF (Bidirectional Transmission Distribution Function). In the case of a completely opaque material, no light is transmitted through the object and only the BRDF is used. As most objects considered in this paper can be crossed by light, we will use the more general term BSDF.

For a thicker translucent material, the scattered light may emerge at a certain distance from where the light has entered into this material. Rendering this effect needs to take into account light scattering within the material with a Bidirectional Surface Scattering Reflection Distribution Function (BSSRDF). The BSSRDF can be modeled by solving the radiative transfer equation [START_REF] Chandrasekhar | Radiative heat transfer[END_REF], generally using the diffusion approximation [START_REF] Ishimaru | Wave propagation and scattering in random media[END_REF]. Taking into account the sub-surface scattering has revolutionized the rendering of translucent materials in animation or video games, particularly thanks to the contribution of Jensen and al. [START_REF] Jensen | A practical model for subsurface light transport[END_REF]. Finally, the BSDF can be viewed as a simplification of the BSSRDF by considering that the incoming point of the light in the object is the same as its exiting point. The BSSRDF model that we use to generate objects with different translucencies in our representation system combines the different models presented above for absorption, surface scattering and volume scattering.

The translucency effect can be visually appreciated through the synthetic images of a spherical object placed in front of a surrounding by an environment map representing a landscape. The images are generated with the open source renderer Mitsuba [START_REF] Ishimaru | Wave propagation and scattering in random media[END_REF]. The first two axes are rendered with existing BSDFs models that are already implemented in Mitsuba: the "absorption" axis can be design by using a model that accounts for the absorption occurring in a dielectric material ("dielectric" plug-in). The "surface scattering axis" is modeled by a micro-facet model that uses the GGX normal distribution ("roughdielectric" plug-in. In order to account for the multiple reflections between the micro-facets, an improvement of this model has been proposed by Heitz [START_REF] Heitz | Multiplescattering microfacet BSDFs with the Smith model[END_REF], which is available as a new Mitsuba plug-in ("roughdielectricGGX" plugin). However, in order to design the sub-surface scattering axis, we need a BSSRDF model handling both perfect transparency and complete opacity. Unfortunately, as far as we know, such a model does not exist. We therefore designed our own model. To describe the volume scattering in the material, we use the "dipole" plug-in that is available in Mitsuba, which applies the BSSRDF according to Jensen's model [START_REF] Jensen | A practical model for subsurface light transport[END_REF]. This plug-in only accounts for the scattering that is happening inside the material. We need to add the effects that are occurring at the air-material interface and which are described by a BSDF.

In the renderer, one BSDF is usable for transparent dielectric materials BSD F 1 ("dielectric" plug-in), and a different one is usable for perfectly opaque materials BSD F 2 ("plastic" plug-in).

To obtain materials from transparent to opaque with a translucent look both in reflection and in transmission, we chose to linearly combine these two BSDFs such as:

BSDF=α 1 BSD F 1 + α 2 BSD F 2 (2)
where α 1 + α 2 =1 in order to guarantee the conservation of energy. The final model is then composed of the BSDF defined by the Equation ( 2), and the BSSRDF obtained with the "dipole" plug-in. The scatterer density which is the parameter describing the "sub-surface scattering axis". On the one hand, in the BSSRDF, it can be modified by a scale parameter. On the other hand, the scatterer density varies with the coefficient α 2 : when α 2 is really small, there is almost no inclusion. The material appears to be transparent as the "dielectric" plug-in does ( ( α 2 =0

) . An increase of α 2 corresponds to an increase of the scatterer density until the material becomes opaque as described by the "plastic" plug-in

( α 2 =1
) .

Illustration through objects in glass

By way of illustration, we propose to consider spherical objects made in glass, whose refractive index n = 1.5 is assumed to be constant over the visible spectrum of light.

All the rendered images are generated by using 1024 samples per pixel. Our representation system has been designed so as the three axes are independent to each another. First, we vary one physical parameter, the other two ones being zero, and observe the influence of this parameter alone on the translucency of the glass bowl.

Figure 3. Absorption axis. The absorber density increases from the left-most object (non-absorbing) to the right-most object (opaque black).

As absorption depends on wavelength, it is responsible for the color of the object. We remind that in our study we assume that there is no other cause to the coloration of the object. In the example shown in Figure 3, we selected a spectral absorption coefficient having a reddish hue. By increasing the density of absorbers, the bowl becomes darker, until opacity (black). In any case, the object remains transparent. The ball behaves like a lens and inverts the image of the background landscape. Apart from the reflection of the sun, the image reflected by the front side of the ball is relatively insignificant. It is only revealed by contrast when the object is opaque. The rendering of balls where only surface scattering occurs is displayed in Figure 4. We accounted for the multiple reflections as proposed by Heitz [START_REF] Heitz | Multiplescattering microfacet BSDFs with the Smith model[END_REF]. Consequently, the ball does not become darker as the roughness increases, in opposition with the results rendered by common surface scattering models. The left-most ball is perfectly smooth, the roughness parameter being set to 0.0. From left to right the four other balls are rendered with a respective roughness of 0.05, 0.1, 0.25, 0.5. We can notice that the rough appearance of the object does not varies linearly with the roughness parameter. Visually, the images in transmission and in reflection (less visible here than for transparent materials) are becoming more and more blurred, up to a certain roughness value from which the appearance of the object becomes almost homogeneous. Although it is possible to get this impression, opacity is never reached: there is always some light transmitted by the From a perceptual point of view, perceived translucency depends on the blurred nature of the transmitted image, and therefore on the distance between the translucent object and the observed scene. For example, a tracing paper sheet allows to replicate an image that is placed underneath and in contact with it, but does not allow to distinguish clearly an image that is placed a few millimeters away.

Regarding to the sub-surface scattering axis, illustrated in Figure 5, the obtained result is coherent with our expectations. The sphere is going from totally transparent to milky or white marbleish aspect, until total opacity.

By comparing Figures 4 and5, the difference between surface scattering and sub-surface scattering is well noticeable. In practice, with real objects, this distinction is not always obvious, especially in case of low scattering. When looking at a white film, it is often difficult to know whether the whiteness is due to volume or surface scattering, and one of the only way to know it is by applying a thin layer of liquid with refractive index close to the one of the object. In case of surface scattering, the oil layer makes a smooth interface and the object becomes transparent. In the case of sub-surface scattering, the oil has no effect on the visual appearance of the translucent object. There is no surface scattering, so the air-material interface is perfectly smooth. In this case, the scatterers are also the absorbers. From left to right the density of these particles is increasing.

Discussion

The three-dimensional space that we have defined for representing translucency enables placing objects with different absorber density, surface roughness and scatterer density in a cube, as shown earlier in Figure 1. Transparency is achieved in absence of scattering and corresponds to the absorption axis (See examples in the picture of Figure 3) except the highly absorbing materials, which make the object opaque. The opaque objects are located near the planes of maximum absorption and maximum volume scattering. All the other states inside the cube correspond to translucent materials. It is even possible to navigate in this threedimensional space in order to choose/design the appearance of an object. This cubic representation is obviously richer than the traditional one-dimensional scale (transparency / translucency / opacity), and has also the advantage of being based on optical properties of the material.

For example, Figure 6 shows different glass bowls whose are located in a plane within the representation system, corresponding to a constant absorber density. Surface scattering increases from left to right, and volume scattering increases from bottom to top. Although the appearances of the different objects evolve in a sensitive way, it is difficult to interpret the materials as transparent or opaque, and diffusing in surface or volume. This confusion is explained by the simultaneous perception of more or less blurred images in transmission and in reflection. If reflection prevails in terms of luminance, the object seems opaque. This impression of opacity increases as surface scattering increases because the image of the landscape in the background is not visible anymore. Hence, some objects look opaque while they still transmit light from the background.

The main argument in favor of this representation system is its simplicity, which lies on the fact that we have selected only three optical phenomena non-correlated to each other. This simple approach implies various assumptions:

-The color of an object is assumed to be only due to its properties of absorption. These assumptions is acceptable for many materials but excludes some other phenomena such as dispersion or chromatic scattering, Rayleigh scattering for example.

-The scattering particles are assumed to be randomly distributed and the effect of light scattering on the surface and/or in volume are assumed to be isotropic. However, textured and anisotropic effects, visible for example on certain openwork textiles, would favor some scattering directions that would significantly modify the perceived translucency.

-Absorption and volume scattering are assumed to be uncorrelated, an assumption, which is rarely satisfied. Moreover, the diffusing particles can also be absorbing. Thus, the representation of the volume diffusion axis (Figure 5) is a very special case where scattering-only particles are in a non-absorbing binder. If the density of scattering particles increases, absorption should also increase in a proportional way. For example, in the case of pigments the absorbers and the scatterers correspond to the same inclusions in the material. As a consequence, when the absorber density increases then the scatterer density increases in the same way. Figure 7 illustrates this effect: tthe object gradually becomes opaque and its color becomes more and more saturated. The color of the opaque object is neither white (no absorption) nor black (no sub-surface scattering).

The proposed representation system can be adapted to the above situations without losing the general idea. On the other hand, taking into account some other optical phenomena such as interference, diffraction, or photo-luminescence would probably require increasing the dimensionality of the representation space. These phenomena can, in some cases, provide information in contradiction with the information related with usual light scattering mechanisms and thus disturb the interpretation of the translucency attribute.

Conclusion

The translucency of an object is often interpreted through the thickness variation of that object, and therefore the variation of the light it transmits. Nevertheless, even an opaque object can be considered as translucent when the penetration of light in the material is perceived. This property is intrinsic of the material and opposed to transparency: On the one hand, the non-scattering and therefore transparent materials possibly colored due to the absorption of light; On the other hand, the wide variety of scattering materials whose translucent appearances depends on whether the light is scattered on the surface or in the volume.

As far as we know, this study is the first intent to classify translucency effects by considering the dimensionality of this visual attribute, based on some physical phenomena that produce it. The representation system for translucency representation suggested is suitable for artists and designers for an intuitive control of the material appearance in computer graphics. Each axis accounts for a different BSDF function that we assumed to be independent from one another, and it may easily be extended to suit any purpose.

Figure 1 .

 1 Figure 1. Three-dimensional space representing translucent materials. Each axis corresponds to one of the physical phenomena among absorption, surface scattering and volume scattering. The absorption axis is described by the absorber density, the surface scattering axis is controlled by the roughness parameter, and the sub-surface scattering axis is described by the scatterer density.

Figure 2 .

 2 Figure 2. Difference between BSDF and BSSRDF. (a) The light is only scattered at the incident point, which can be modeled with a BRDF. (b) The light can emerge at a certain distance from the incident point, which is modeled with a BSSRDF.

Figure 4 .

 4 Figure 4. Surface scattering axis. The roughness increases from the left-most object to the right-most one. The absorbance is zero along this axis.

Figure 5 .

 5 Figure 5. Sub-surface scattering axis. The scatterer density increases from the left-most object to the right-most one. The absorbance is being zero along this axis, a milky appearance is observed.

Figure 6 .

 6 Figure 6. Variation of surface and sub-surface scattering while the absorption is constant. From left to right: increase of the surface scattering. From bottom to top: increase of the subsurface scattering.

Figure 7 .

 7 Figure 7. Evolution in the absorption/sub-surface scattering plane. There is no surface scattering, so the air-material interface is perfectly smooth. In this case, the scatterers are also the absorbers. From left to right the density of these particles is increasing.

Author Biography

Morgane Gerardin completedher Msc in optics from the Institut d'Optique Graduate School, France (2018)