Miguel Nicolau
email: miguel.nicolau@inria.fr

Marc Schoenauer
email: marc.schoenauer@inria.fr

Evolving Specific Network Statistical Properties using a Gene Regulatory Network Model

The generation of network topologies with specific, userspecified statistical properties is the aim of this paper. This is achieved through the use of an artifical Gene Regulatory Network Model, shown previously to be able to correctly seed the population of an evolutionary algorithm, with the aim of steering the evolution towards the desired topologies. This method had previously been shown to be able to evolve scale-free topologies; the results obtained in this paper reinforce the applicability of the method, showing that the evolution of small-world topologies is also possible.

Introduction

Network generative procedures targeted toward specific topology properties are generally either iterative processes that sequentially add nodes using problemdependent rules [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] Barabási | Emergence of scaling in random networks[END_REF], or ad hoc stochastic procedures modifying existing networks [START_REF] Pastor-Satorras | Evolving protein interaction networks through gene duplication[END_REF]. However, such procedures need to be designed anew whenever a new target property is wanted. Goal-directed procedure, on the other hand, only require a measure of the desired properties, and some (generally stochastic) optimization method able to search the space of network topologies. Unfortunately, such approaches generally suffer from two main drawbacks: the curse of dimensionality, making it untractable to optimize large networks; the bootstrap problem, in that random topologies barely indicate any meaningful path toward good solutions of the problem at hand.

An alternative is to use specific initialisation procedures that are able to produce diverse enough, though not random, topologies as a starting point for further optimization based topology design. Artificial Genetic Regulatory Networks offer such a procedure: previous work has demonstrated that simple models of GRNs were able to boost the evolutionary optimization of topologies in order to create scale-free topologies.

Building on this work, the present paper uses similar ideas to extend the generality of the approach, and introduces an original generative procedure to design Small World topologies [START_REF] Watts | Small Worlds: The Dynamics of Networks Between Order and Randomness[END_REF]. This is a specific kind of topology, where most nodes are not directly connected, yet the average connection distance between any two nodes is very low. This kind of topology has been shown to exist in many real-life networks, such as biological transcriptional networks [START_REF] Vera Van Noort | The yeast coexpression network has a small-world, scale-free architecture and can be explained by a simple model[END_REF], computer networks [START_REF] Serguei | Evolution of Networks: From Biological Nets to the Internet and WWW[END_REF], or social networks [START_REF] Guare | Six Degrees of Separation[END_REF].

The results obtained in the current study show that the presented method is applicable to the evolution of small-world networks, therefore reinforcing the usability of such approaches to the design of network topologies with user-specified statistical properties.

This paper is structured as follows: Section 2 presents the gene regulatory model used; Section 3 introduces the definition of small-world topologies, and the method used to extract topologies for the regulatory model. Then Section 4 presents and analyses the results obtained, and finally section 5 draws conclusions and future work directions.

The Gene Regulatory Network Model

The expression of genes in a genome is regulated by Transcription Factors. These are special proteins, produced by other genes, which can enhance or inhibit the production of their target genes; the networks of interactions between genes and the proteins they produce are termed Gene Regulatory Networks (GRN).

The model used in this work was first proposed by Wolfgang Banzhaf [START_REF] Banzhaf | Artificial regulatory networks and genetic programming[END_REF]. It represents a genome as a bit string, and uses specific bit sequences as promoter sites, identifying the location of a gene. If a gene is found, the 5 × 32 bits following it represent the protein it produces, and the the 2 × 32 bits upstream from the promoter site represent the enhancer and inhibitor sites for this gene, respectively. Fig. 1 illustrates the model. The promoter site can be any sequence of bits; in this case, it is the sequence XYZ01010101, where X, Y and Z are any 8 bit sequence. The protein produced by the gene is a 32 bit binary sequence, extracted by a majority rule between all 5 sequences of 32 bits that compose it.

In this model, all proteins produced are transcription factors, and therefore they all regulate the expression of all genes, including their own producing gene. Regulation works by matching a protein and the regulating sites of a gene with the XOR operation: the result is the regulating strength. The enhancing and inhibiting signals regulating the production of protein p i are then calculated as:

e i , h i = 1 N N j=1 c j exp(β(u i,j -u i,max )) ( 1
)
where N is the number of proteins, c j is the concentration of protein j, u i,j is the XOR result between the regulating site of gene i and protein j, u i,max is the maximum match achieved for gene i, and β is a positive scaling factor. Given these signals, the production of protein i is calculated via the following differential equation:

dc i dt = δ(e i -h i )c i -Φ (2)
where δ is a positive scaling factor (representing a time unit), and Φ is a term that proportionally scales protein production, ensuring that i c i = 1 at all times, which results in competition between binding sites for proteins.

Initialisation Method

Although the binary genomes used within the model can be randomly created, an initialisation method has been proposed [START_REF] Banzhaf | Artificial regulatory networks and genetic programming[END_REF], based on a Duplication and Mutation (DM) process. It involves creating a random 32 bit sequence, followed by a series of length duplications associated with a low mutation rate. It has been shown [START_REF] Wolfe | Molecular evidence for an ancient duplication of the entire yeast genome[END_REF] that this process of growing genomes can also occur in nature.

Network Topologies

Even though the model used is overly simplified compared to what is known of biological GRNs, an interesting issue is to find out whether or not the resulting interaction network exhibits particular properties resembling those found in natural networks. Previous work [START_REF] Nicolau | Evolving scale-free topologies using a gene regulatory network model[END_REF] has shown that evolution of linear genomes to achieve Scale-Free topologies [11-13, 5, 14] is possible; the current work concentrates on Small-World topologies [START_REF] Watts | Small Worlds: The Dynamics of Networks Between Order and Randomness[END_REF][START_REF] Vera Van Noort | The yeast coexpression network has a small-world, scale-free architecture and can be explained by a simple model[END_REF].

Small-World Networks

Small-World networks are characterised by a short average distance between any two nodes. Watts [START_REF] Watts | Small Worlds: The Dynamics of Networks Between Order and Randomness[END_REF] describes them in terms of their characteristic path length L, and clustering coefficient C, with C i being the percentage of all nodes connected to node i that are also connected to each other1 . Given these definitions, a graph with n nodes and average vertex degree k is a small-world network, if it satisfies

L ≥ L rand ∼ ln(n) ln(k) and C ≫ C rand ∼ k n , with n ≫ k ≫ ln(n) ≫ 1.

Artificial Regulatory Networks

Once a genome has been constructed with the model described (see Section 2), its regulatory network topology can be analysed: genes will be represented as nodes, and the proteins they produce will be directed edges towards the genes they regulate; the weight of those edges will be the number of complementary bits between the regulating protein and the target gene (as seen in Eq. 1).

As all produced proteins regulate all genes, the resulting graph is complete. However, a threshold can be set on the minimum weight a connection must have, before being considered as an edge on the resulting regulating network.

Using different thresholds will therefore result in different networks. For example, Fig. 2 and3 show regulating networks of the same random genome for two different thresholds (23 and 24). While almost all nodes are connected on Fig. 2, increasing the threshold by one removes many connections, and the graph on Fig. 3 is only a small sub-graph of the previous one (nodes which become isolated are not shown, which explains the smaller number of genes).

G36 G3 G41 G42 G25 G5 G49 G6 G18 G39 G48 G12 G7 G24 G26 G34 G44 G17 G8 G28 G45 G19 G9 G38 G10 G31 G53 G46 G37 G11 G50 G33 G13 G29 G32 G22 G15 G52 G40 G1 G43 G20 G21 G51 G30 G14 G2 G35
Fig. 2. Regulatory network for a random genome of length LG = 32768, at a threshold of 23 bits. Solid edges are enhancing interactions, dotted edges are inhibiting ones.

Networks extracted from genomes initialised with the DM technique have remarkably different topologies, as seen in Fig. 4. The use of DM steps with low mutation result in a much shallower hierarchy of genes, with a few master genes, and most other genes poorly connected. Increasing the threshold removes connections, but the same master genes are still present, as can be seen in Fig. 5.

G45 G8 G28 G50 G12 G26 G33 G13 G29 G32 G41 G38 G15 G18 G42 G1 G43 G21 G25 G49 G34 G30 G48
Fig. 3. Gene regulatory network for the genome from Fig. 2, at a threshold of 24 bits.

G27 G1 G2 G4 G5 G7 G10 G13 G21 G22 G23 G17 G18 G19 G26 G11 G20 G6 G8 G15 G24
Fig. 4. Gene regulatory network for a genome of length LG = 32768, created using 10 duplication events and a mutation rate of 1%, at a threshold of 16 bits.

G27 G2 G4 G5 G13 G21 G23 G18 G19 G1 G11 G7 G22 G6 G8 G15 G17 G20 G24
Fig. 5. Gene regulatory network for the genome from Fig. 4, at a threshold of 17 bits.

Connectivity variance

In order to analyse the effect of the threshold value on the network connectivity, an approach similar to that of Kuo et al. [START_REF] Kuo | Network topology and the evolution of dynamics in an artificial regulatory network model created by whole genome duplication and divergence[END_REF] has been used: 100 genomes have been generated, using 14 duplication events, and the network connectivity (fraction of edges) has been computed for each threshold. The network connectivity is defined as:

N C = #edges 2n 2 (3)
where #edges is the number of edges in the network, and n is the number of nodes, or genes (2n 2 is hence the maximum number of possible edges, as each node can be connected twice to any other node, including itself). Fig. 6 shows the connectivity as a function of the threshold, for mutation rates of 1%, 5%, 10%, and 50%. It is a clear illustration of the very different behaviors with respect to connectivity depending on the mutation rate used during the DM process:

-A high mutation rate (or, equivalently, the completely random generation of the genome) creates a network which stays fully connected with a wide range of threshold values; then, there is a sharp transition to no connectivity. Moreover, there is a very small variance between different networks. -A low mutation rate creates a network which quickly loses full connectivity; however, its transition to no connectivity is much smoother than that of random networks. Moreover, there is very large variance between different networks generated with the same mutation rate. Fig. 6. Fraction of edges in a graph as compared to a fully connected network (and standard deviations), versus threshold parameter, based on samples of 100 genomes, created using 14 duplication events, and mutation rates of 1%,5%, 10%, and 50%.

Evolution of Statistical Properties

The objective of this section is to test whether genomes created with the model can be evolved, such that their regulating networks exhibit precise statistical properties, consistent with the definition of small-world topologies.

The Evolutionary Algorithm

A population of bit-string genomes is evolved, using the simple bit-flip mutation as the only variation operator. The evolution is a straightforward (25 + 25)-ES: 25 parents give birth to 25 offspring, and the best 25 of the 50 parents+offspring become the parents of next generation. The mutation rate is adapted following the 1/5 rule of Evolution Strategies [START_REF] Rechengerg | Evolutionsstrategie '94. Frommann-Holzboog[END_REF]: its rate is initially set to 1% (per bit), and when the rate of successful mutations is higher than 1/5 (i.e. when more than 20% mutation events result in an increase of fitness), the mutation rate is doubled; it is halved in the opposite case 2 . In order to compare the evolvability of DM initialised genomes and completely random populations, 50 independent runs of 50 generations were performed with each type of genome, for each of the following fitness functions.

Fitness Function

The objective is to make "as true as possible" the relationship n ≫ k ≫ ln(n) ≫ 1, as seen in Section 3.1. Candidate objective functions thus should try to maximise the three (normalized) following terms:

n -k n ; k -ln(n) k ; ln(n) -1 ln(n)
Furthermore, in order to be considered small-world, a network must satisfy L ≥ L rand , and C ≫ C rand , so networks with C < C rand should be discarded, and relevant objective functions should maximise the following terms:

(L rand -L) + ; C -C rand C rand
A possible fitness function for an Evolutionary Algorithm is thus given by a weighted sum of those terms, whose weights should be tailored to the problem at hand:

F (x) = α 1 n -k n + α 2 k -ln(n) k + α 3 ln(n) -1 ln(n) + α 4 (L rand -L) + + α 5 C -C rand C rand (4
) Note that, in the current work, all these weights were set to 1.

Results

Fig. 7 shows the best fitness in the population averaged over the 50 runs, for both initialisations procedures, using both fitness cases.

The results obtained show that DM initialised genomes get off their starting blocks with topologies which are much closer to being small-world, as defined in equation 4. Not only that, but their structure actually allows them to continuosly improve their fitness score through evolution.

Random genomes, on the other hand, have starting topologies which score badly. Furthermore, their regular topologies have very similar scores across different runs, and show little signs of evolution over time. Table 1 shows the relevant statistical measures of random and DM initialised networks, both without or with evolution. Of relevant interest is the fact that, after evolution, DM initialised networks exhibit a high clustering coefficient, while keeping a characteristic path length similar to that of randomly initialised genomes.

Analysis

The shallow hierarchies observed in initialised genomes exhibit characteristics similar to those of Small-World topologies, leading to the results observed. To analyse the reasons leading to such a different in the extracted network topologies, a sequence of DM steps was analysed, as it took place. The original random 32 bit sequence was as follows:

This sequence was then subjected to a series of DM events, with a probability of mutation of 1% per bit. After 6 DM events, the first gene appeared, and after 7 events, there are already four genes. The resulting networks were drawn (Fig. 8 and9) using a threshold of 133 .

G1

Fig. 8. Gene network after 6 duplication events.

G1 G2 G3 G4
Fig. 9. Gene network after 7 duplication events.

The shape of the genes determines their starting location, when mapped to the original 32 bit sequence. For example, for the genome in Fig. 9, the starting locations for its genes were 905, 1929, 2377 and 2761, respectively. If we divide these by 32 and take the remainder, we see that they all start at the 9 th bit of a duplication of the original sequence, so they are all drawn using the same shape.

This also explains why there are no connections between genes in Fig. 9. As all genes originate from the same initial sequence of bits, the few mutations that occurred during the 7 DM steps did not create enough differences between regulating sites and produced proteins, to trigger a connection at threshold 13.

After the 8 th DM event, the network takes on a different topology (Fig. 10). Most genes are still duplications of the 9 th bit of the original sequence; however, G7 starts at a different location, and is thus drawn using a rectangular shape.

As the connectivity between genes is established by the difference between regulation sites and proteins, genes originating from different locations are more likely to be connected, even using lower threshold values. This can be seen in Fig. 10: genes labelled with different shapes do not connect to each other.

In this DM step one can also see pure duplications of genes, that is, genes that are created as duplications of other genes appearing upstream in the genome sequence: in those cases, the genes are labelled with their originating gene between brackets (e.g. G6(1)). But even pure duplications can generate slightly different genes, because mutation events can occur during the duplication process. G6(1) is an example: it only has an outwards inhibiting connection to G7, whereas G1 also has an inwards inhibiting connection originating from the same gene.

G1 G7 G2 G3 G4 G5 G6(1) G8(2) G10(4) G11 G9 (3)
Fig. 10. Gene network after 8 duplication events.

With 9 DM steps, the network becomes a lot more complex (Fig. 11). There are still only two relative gene origins (triangles and rectangles), but either through pure duplications or discovery of new genes, there are now 25 genes.

One can see that triangles still do not mix with rectangles (due to the threshold value chosen). Therefore, since there are a lot more triangles than rectangles, the latter become highly connected, and can be seen acting as connection hubs.

Finally, a last DM step is performed (Fig. 12), creating a network with 50 genes. Although hard to analyse at the naked eye, one can clearly see its shallow hierarchy, with a few highly connected nodes, to which most other nodes connect to. One can also see the appearance of a third type of gene, labelled with a pentagonal shape, which becomes the most connected gene. Table 2 shows a list of the gene families, along with their count, initial location, corresponding initial bit, and average number of inwards, outwards, and total connections.

G1 G7 G17 G20(7) G24 G2 G3 G4 G5 G6(1) G8(2) G10(4) G11 G12 G13(1) G14(2) G16(4) G18(5) G21(8) G23(10) G25 G9(3) G15(3) G19(6) G22 (9)
Fig. 11. Gene network after 9 duplication events.

Although this network is just an example, many networks were found to follow the same mechanics while being extracted from genomes grown with DM steps. It shows that the tendency of initialised genomes to generate shallow hierarchies comes from the fact that genes starting at the same bit from the duplicated initial sequence tend not to connect, due to the use of the XOR operator (see Section 2). As duplications of the first gene(s) represent the majority of the genes present in the genome, they will not be connected (when choosing an appropriate threshold value), and genes discovered in later DM steps (in smaller numbers) will be highly connected.

Conclusions and Future Work

The experimental results obtained with this work show that the use of a gene regulatory network model allows for the construction of network topologies with specific statistical properties, in this case small-world topologies. Once again the initialisation procedure plays a vital role in the seeding of potential models, not only giving them a head-start when compared to random models, but also leading to better final statistical measures, with specific emphasis on clustering coefficient values (as seen in Table 1).

This way of evolving topologies also allows one to fine-tune the objective function, such that specific relationships between n, k and ln(n) (see Section 3) are possible. A few problems remain with the current approach. The set of weights used in the fitness function (see 4) gives equal importance to the differences between the terms n, k, ln(n) and 1, regardless of them having a few orders of magnitude of difference. Furthermore, the relationship C ≫ C rand , which fundamentally defines a small-world topology, is given the same fitness weight as the differences between the previous four terms. Although there is evolution with this setup, a better fitness function could be designed in the future.

Another interesting future work direction could be a multi-objective approach to this problem. Even more promising could be a multi-objective approach to join the current work and the previous work on scale-free networks [START_REF] Nicolau | Evolving scale-free topologies using a gene regulatory network model[END_REF], evolving network topologies which are both scale-free and small-world.

Fig. 1 .

 1 Fig. 1. Close-up view of the representation of a gene within the model.

Fig. 7 .

 7 Fig. 7. Average best fitness per generation across 50 independent runs, for random and 1% DM initialised genomes. Error bars plot the standard deviation across runs.

Fig. 12 .

 12 Fig. 12. Gene network after 10 duplication events.

Table 1 .

 1 Initial and evolved results, for random and DM initialised genomes.

		Generation Threshold	N	k	ln(N) L	C Fitness
	Random	0	22.199 1072.400 129.300 6.978 2.016 0.121 2.812
	Genomes	50	23.000 1093.099 42.899 6.996 2.140 0.040 2.662
	Initialised	0	12.900 1910.799 90.700 7.554 1.962 0.404 2.274
	Genomes	50	11.699 1782.400 44.399 7.479 1.976 0.810 1.936

Table 2 .

 2 List of all genes after 10 duplication events.Family # genes 1 st loc. 1 st seq. bit Avg. in Avg. out Avg. total

	Triangle	39	905	9	9	8.8	17.8
	Quadrangle	10	5713	17	33	31.6	64.6
	Pentagon	1	27872	0	37	59	96

The phenomenon Ci = 1.0 is also known as a clique.

Because of the possibility of neutral mutations (especially with low mutation rates), if there were more than 50% neutral mutations, the rate was doubled in any case.

This value was chosen deliberately, based on the resulting network after all DM events, to illustrate the propose discussed.

Acknowledgment

This work was supported by the Sixth European Research Framework (proposal number 034952, GENNETEC project).


Lorenza N C Dezanet

Raisha Kassime

Patrick Miailhes

Caroline Lascoux-Combe

Julie Chas

Sarah Maylin

Audrey Gabassi

Hayette Rougier

Constance Delaugerre

Karine Lacombe
email: karine.lacombe2@aphp.fr

Dr Anders Boyd
email: anders.boyd@iplesp.upmc.fr

Pr Karine Lacombe

Effect of viral replication and liver fibrosis on all-cause mortality in HIV/HBV coinfected individuals: a retrospective analysis of a 15-year longitudinal cohort

Keywords: hepatitis B virus, HIV, mortality, tenofovir, joint models

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Roughly 8% of individuals living with human immunodeficiency virus (HIV) are chronically co-infected with hepatitis B virus (HBV) [START_REF] Leumi | Global Burden of Hepatitis B Infection in People Living With Human Immunodeficiency Virus: A Systematic Review and Metaanalysis[END_REF]. Without effective treatment, HIV-HBV co-infected individuals are at increased risk of liver-related and all-cause mortality when compared to HIV-positive individuals without HBV infection [START_REF] Falade-Nwulia | Comparative Risk of Liver-Related Mortality From Chronic Hepatitis B Versus Chronic Hepatitis C Virus Infection[END_REF].

Treating co-infected individuals primarily involves controlling both HIV and HBV replication. Higher levels of serum HIV-RNA have been associated with increased risk of AIDS-related morbidity and mortality in HIV-positive individuals [START_REF] Mellors | Plasma viral load and CD4+ lymphocytes as prognostic markers of HIV-1 infection[END_REF]. With effective antiretroviral therapy (ART), HIV replication is suppressed and the risk of HIV-related morbidity and mortality greatly decreases [START_REF]The antiretroviral therapy cohort collaboration (ART-CC). Prognosis of patients treated with cART from 36 months after initiation, according to current and previous CD4 cell count and plasma HIV-1 RNA measurements[END_REF][START_REF] Smith | Trends in underlying causes of death in people with HIV from 1999 to 2011 (D:A:D): a multicohort collaboration[END_REF]. Likewise, higher levels of circulating HBV-DNA have been linked to fibrosis, hepatocellular carcinoma (HCC), and liver-related and overall death in HBV mono-infected individuals [START_REF] Iloeje | Risk and Predictors of Mortality Associated With Chronic Hepatitis B Infection[END_REF][START_REF] Hawkins | Assessment of Liver Fibrosis by Transient Elastography in Patients With HIV and Hepatitis B Virus Coinfection in Nigeria[END_REF][START_REF] Kim | Risk of Hepatocellular Carcinoma with Hepatitis B Viremia among HIV/Hepatitis B Virus-Coinfected Persons in North America[END_REF]. Currently available nucleoside/nucleotide analogues (NA) can suppress HBV-DNA replication, which coincides with liver fibrosis regression [START_REF] Van Welzen | Decreased All-Cause and Liver-Related Mortality Risk in HIV/Hepatitis B Virus Coinfection Coinciding With the Introduction of Tenofovir-Containing Combination Antiretroviral Therapy[END_REF] and reduced risk of HCC [START_REF] Papatheodoridis | Risk of hepatocellular carcinoma in chronic hepatitis B: assessment and modification with current antiviral therapy[END_REF]. Conveniently, some antiretroviral agents, such as tenofovir (TDF) and tenofovir alafenamide (TAF), possess potent anti-HIV and anti-HBV activity [START_REF] Boyd | Persistent viremia in human immunodeficiency virus/hepatitis B coinfected patients undergoing long-term tenofovir: Virological and clinical implications[END_REF][START_REF] Huang | Efficacy and safety of elvitegravir/cobicistat/emtricitabine/tenofovir alafenamide as maintenance treatment in HIV/HBVcoinfected patients[END_REF], and are thus ideal therapeutic options for HIV-HBV co-infected individuals [13]. It could be hypothesized that reductions in HIV and HBV replication would give way to lower incidence of both HIV-and HBV-related causes of morbidity and mortality.

Nevertheless, epidemiological studies in co-infected populations are not entirely clear on the relationship between active replication of these viruses and mortality. Recent research has demonstrated clear reductions in AIDS-related, liver-related and overall mortality in the years coinciding with widespread TDF-use [START_REF] Van Welzen | Decreased All-Cause and Liver-Related Mortality Risk in HIV/Hepatitis B Virus Coinfection Coinciding With the Introduction of Tenofovir-Containing Combination Antiretroviral Therapy[END_REF][START_REF] Tsai | Impact of antiretroviral therapy containing tenofovir disoproxil fumarate on the survival of patients with HBV and HIV coinfection[END_REF]. However, incidence of HCC in TDF-treated co-infected individuals above the age of 45 remains high enough to warrant increased HCC screening [START_REF] Wandeler | Incidence of hepatocellular carcinoma in HIV/HBVcoinfected patients on tenofovir therapy: Relevance for screening strategies[END_REF]. Furthermore, large studies from Tanzania and Côte d'Ivoire have shown that HIV-HBV co-infected individuals, particularly when their HBV-DNA levels are high, are still at increased risk of overall mortality despite TDF-containing ART [START_REF] Christian | Hepatitis B virus coinfection is associated with high early mortality in HIV-infected Tanzanians on antiretroviral therapy[END_REF][START_REF] Kouamé | Higher Mortality Despite Early Antiretroviral Therapy in Human Immunodeficiency Virus and Hepatitis B Virus (HBV)-Coinfected Patients With High HBV Replication[END_REF]. The principal causes of death in these studies seem related to HIV-related illness or invasive bacterial infections. The common limitation shared across the studies conducted thus far is the lack of consistently collected data on both HIV-and HBVreplication.

The aim of the present study was then to describe detailed causes of mortality in a cohort of HIV-HBV co-infected individuals followed for up to 15 years with highly effective anti-HIV and anti-HBV treatment. We further develop our analysis by exploring the effect of HBV-DNA and HIV-RNA replication over time and liver fibrosis on all-cause mortality in this study population.

Methods

Study population

We analyzed HIV-HBV co-infected participants of the French HIV-HBV Cohort Study. Briefly, this was a closed, longitudinal cohort study including 308 HIV-positive patients with chronic HBV infection from four centers located in Paris and Lyon, France. Individuals were included if they had an HIVpositive serological result confirmed by western blot and HBsAg-positive serological results for >6 months. Participants were recruited in 2002-2003 and followed every 6-12 months until 2017-2018.

The cohort design and procedures are described elsewhere [START_REF] Boyd | Subclinical and Clinical Outcomes in Patients Coinfected With HIV and Chronic Hepatitis B Virus From Clinical Outpatient Centers in France: Protocol for an Ambispective, Longitudinal Cohort Study[END_REF]. For this analysis, we included participants who had at least two consecutive visits and available information on vital status.

All individuals provided written informed consent to participate in the study and the protocol was approved by a Hospital Ethics Committee (Paris, France) in accordance with the Helsinki Declaration.

Data collection

Demographic information was collected at study inclusion. Medical history on antiretroviral and anti-HBV treatments, alcohol consumption and the presence of comorbidities, including diabetes, cardiovascular disease (CVD), renal and other liver diseases, were collected at study entry and at each follow-up visit.

Laboratory data were collected at study entry and at each follow-up visit. Commercial polymerase chain reaction (PCR)-based assays were used to quantify HBV-DNA viral load (VL; COBAS AmpliPrep/COBAS TaqMan; detection limit=12 or 38 IU/mL; COBAS Amplicor; detection limit=60 IU/mL; Roche Diagnostic Systems, Meylan, France). We defined undetectable HBV-DNA at the highest threshold (<60 IU/mL). HIV-RNA VL was measured using a commercial PCR-based assay and CD4 + cell count using standard methods. Antibodies to hepatitis C virus (HCV) and hepatitis D virus (HDV) were measured with an ELISA-based assay and if positive, serum HCV-RNA and/or HDV-RNA was quantified by either commercial PCR-based assay (for HCV-RNA) or in-house assay (for HDV-RNA).

Liver fibrosis was assessed at study entry and each yearly interval using the FibroTest® [START_REF] Poynard | Noninvasive Markers of Hepatic Fibrosis in Chronic Hepatitis B[END_REF].

METAVIR equivalents for HIV-HBV coinfected individuals were used to grade liver fibrosis (F2=0.48-0.58, F3=0.59-0.73, F4>0.74) [START_REF] Bottero | Performance of 11 biomarkers for liver fibrosis assessment in HIV/HBV co-infected patients[END_REF].

Mortality outcome assessment

Deaths observed during follow-up, along with the underlying cause of death and date of death, were reported by the treating physician. To obtain vital status for individuals lost to follow-up (LTFU), a trusted third party (Inserm U1018) was requested to link data from the French HIV-HBV cohort to a national identification registry (Répertoire national d'identification des personnes physiques). For individuals reported as deceased, the cause of death was then obtained by a separate trusted third party (Centre d'épidémiologie sur les causes médicales de décès, CépiDc), linking data from the French HIV-HBV cohort to a national registry of death certificates and death notifications. Both registries are managed by the Institut National de la Statistique et des Etudes Economiques. Causes of death were classified by ICD-10 codes. We recategorized causes of death as liver-related, AIDSrelated, non-AIDS and non-liver-related malignancies, CVD-related, other, or unknown [START_REF] Smith | Trends in underlying causes of death in people with HIV from 1999 to 2011 (D:A:D): a multicohort collaboration[END_REF].

Statistical analysis

Baseline was defined as the date of study entry. Follow-up began at baseline and continued until the date of death or the last study visit.

We assessed treatment efficacy with undetectable HBV-DNA (<60 IU/mL) and HIV-RNA (<50 copies/mL) VLs. The extent of replication was assessed with HBV-DNA (log10 IU/mL) and HIV-RNA (log10 copies/mL) levels. Finally, the historical extent of replication was assessed with time-averaged, cumulative copy-years over follow-up time (log10 copy-yearsTAVG), as detailed elsewhere [START_REF] Boyd | Rates and determinants of hepatitis B 'e' antigen and hepatitis B surface antigen seroclearance during long-term follow-up of patients coinfected with HIV and hepatitis B virus[END_REF].

To analyze the contribution of HIV and HBV replication on mortality during follow-up, we simultaneously modeled (i) HBV replication, (ii) HIV replication, and (iii) all-cause mortality. We carried out a generalized, multivariate, joint longitudinal-survival model approach by which the link between these outcomes could be taken into account. First, we ran two submodels (on HBV-DNA and HIV-RNA) for three separate sets of replication outcomes (detectable VL, log-transformed VL, and log10 copy-yearsTAVG). The probability of having a detectable VL was assumed to be Bernoullidistributed and modeled using logistic regression, while mean log-transformed VL and mean log10 copy-yearsTAVG were assumed to be continuous Poisson-distributed and modeled using Poisson regression. We adjusted, in the model regressing HBV-DNA, for HBeAg serostatus at baseline and cumulative tenofovir use (as a cubic-spline function using 4 knots) and, in the model regressing HIV-RNA, for squared CD4 + cell count and HIV treatment era (2002-2007, ≥2008). Second, the hazards of death were assumed to have an exponential survival function and were estimated using a parametric survival model. We included the two submodels of HBV-DNA and HIV-RNA to estimate the hazards ratio (HR) and 95% confidence intervals (CI) of increasing expected probability of detectable VL, expected mean log-transformed VL, or expected mean log10 copy-yearsTAVG on allcause mortality. The survival model also included age, previous history of an AIDS-defining illness and level of fibrosis at study entry (F0-F1-F2 and F3-F4) as covariates, selected from the analysis described in the Supplementary Materials. We included a random-intercept across all models to account for between-patient variance at baseline, while the random-intercepts were constrained at 1 for the two submodels. Parameters from the three models were jointly estimated via maximum likelihood using the 'merlin' program in STATA [START_REF] Crowther | merlin-A unified modeling framework for data analysis and methods development in Stata[END_REF].

All statistical analyses were performed using STATA (v15.1; College Station, Texas, USA). Significance was defined as a p value <0.05.

Results

Description of the study population

Of the 308 cohort participants, 8 were excluded (only one visit, n=7; no information on vital status, n=1). The 300 included participants were mostly male (84.0%), with a median age of 40 years (IQR=35-45) at study entry. Participants had a median CD4+ count of 400/mm 3 (IQR=268-557) and 160 (53.5%) had undetectable HIV-RNA. Participants were mostly HBeAg-positive (52.0%) and 63.3% (n=190/300) had detectable HBV-DNA. Of the 281 individuals (94.9%) with previous lamivudine (LAM) exposure, median LAM duration was 3.5 years (IQR=1.4-5.5) at study entry and 90 (32.0%) harbored LAM-resistant mutations.

Participants were followed for a median 10.5 years (IQR=4.0-14.6), totaling 2934.5 person-years. 111 (37.0%) were LTFU, including 41 with known vital status and 70 with unknown vital status. Individuals LTFU had a significantly higher median HBV-DNA (for those with detectable levels), ALT and AST levels at inclusion (supplementary table 2). The proportion of participants undergoing ART was high at study inclusion (90.0%) and increased to 100% from the first year until the end of follow-up. Consequently, improvements in CD4 + cell counts (p for trend <0.001) were observed over time (Figure 1A). In addition, the proportion of individuals undergoing TDF-based ART increased from 18.7% at baseline to 40.1% at the first year and 79.1% at the end of follow-up (Figure 1B, p for trend <0.001).

Description of all-cause mortality

42 deaths (cumulative incidence=14.0%; 95%CI=10.3%-18.4%) occurred after a median 6.2 years (IQR=3.4-7.9) of follow-up (incidence=1.43/100 person-years) 7 of these deaths were obtained through linkage.

The most common causes of death were non-AIDS/non-liver related malignancies (n=12 [28.6%]; 0.41/100 person-years), liver-related (n=7 [16.7%]; 0.24/100 person-years), AIDS-related (n=7 [16.7%]; 0.24/100 person-years) and CVD-related (n=6 [14.3%]; 0.20/100 person-years) (Table 1).

HCC and hepatic failure accounted for most liver-related deaths (n=4 and 1, respectively). 7 individuals (16.7%; 0.24/100 person-years) died from others causes of death, while for three (7.0%), the cause of death was unknown.

At study entry, individuals who died, compared to those alive, were older (p<0.001), more likely to come from zones of low/moderate HBV-prevalence (p<0.004), have acquired HIV infection by injecting drug use (IDU) (p<0.02), have other liver diseases or hepatic decompensation (p=0.001), have an AIDS-defining event (p<0.001), have longer duration since first positive HIV test (p=0.01), lower nadir CD4+ cell counts (p=0.03), longer duration of ART (p=0.05), higher levels of AST (p=0.01), HBeAg-positive status (p=0.02), and F3-F4 fibrosis (p<0.001) (Table 2).

During follow-up, individuals who died, when compared to those alive, were more often diagnosed with HDV coinfection (p=0.01), had shorter duration of cumulative tenofovir use (p<0.001), lower CD4+ cell counts (p=0.001), and detectable HBV-DNA at last follow-up visit (p=0.03) (Table 2).

HBV and HIV viral replication and all-cause mortality

The average individual proportion of detectable HBV-DNA VL during follow-up was higher in deceased versus alive individuals (52.6%, 95%CI=25.0-77.8 versus 25.0%, 95%CI=6.7-58.8; respectively, p<0.003). When HBV-DNA was detectable, median levels were at 5646 IU/mL (IQR=446-3,802,281). Deceased individuals were exposed to a higher level of time-averaged copyyears of HBV-DNA (log10copy-yearsTAVG) than those remaining alive (Figure 2) (overall p-value <0.001). Conversely, the average individual proportion of detectable HIV-RNA VL during follow-up was no different between alive and deceased individuals (15.4%, 95%CI=4.0-48.1 versus 10.0%, 95%CI=0.0-60.0, respectively, p=0.6). When HIV-RNA was detectable, median levels were at 232 copies/mL (IQR=3,159-22,972). Time-averaged copy-years of HIV-RNA was similarly low in both groups during follow-up (p=0.6, supplementary Figure S1).

When jointly modeling HBV and HIV replication on all-cause mortality, we found no significant associations with undetectable HBV-DNA on mortality rates (p=0.08, Table 3). However, we did observe a higher rate of all-cause mortality with higher expected mean log-transformed HBV-DNA VL (p=0.03) and cumulative time-averaged copy-years of HBV-DNA (p=0.03) after adjustment for age, AIDS-defining illness and F3-F4 fibrosis level at study entry (Table 3). F3-F4 fibrosis level at study entry was the only covariate consistently and significantly associated with overall mortality. No significant association was found between any of the HIV-RNA outcomes and all-cause mortality (Table 3).

Discussion

In this long-term, prospective study of treated HIV-HBV co-infected individuals, the most common cause of death observed in our study was non-AIDS/non-liver-related malignancies, representing approximately one-third of deaths. Coupled with the high proportion of deaths due to CVD, the spectrum of mortality causes in HIV-HBV co-infection would appear to mirror that of aging HIVpositive individuals in general [START_REF]HIV/AIDS. HIV and aging[END_REF]. Nevertheless, liver-related and AIDS-related causes together represented one-third of deaths, which occurred even during periods when use of TDF-containing ART and suppression of HIV-RNA and HBV-DNA were common at the population level.

Interestingly, the concurrent and historical extent of HBV viremia over time, but not HIV, seemed to play a major role in all-cause mortality. This effect was not observed when modeling undetectable HBV-DNA over time. Other studies in HIV-HBV co-infected individuals have shown an association between higher HBV-DNA at treatment initiation and all-cause mortality [START_REF] Kouamé | Higher Mortality Despite Early Antiretroviral Therapy in Human Immunodeficiency Virus and Hepatitis B Virus (HBV)-Coinfected Patients With High HBV Replication[END_REF][START_REF] Velen | Chronic hepatitis B increases mortality and complexity among HIV-coinfected patients in South Africa: a cohort study[END_REF]. The curious part of this previous finding was that it was observed in the presence of almost exclusively TDF-treated individuals, with assumedly extensive HBV DNA suppression. Given the findings in our study, perhaps the historical exposure of high HBV DNA explained the increased risk in mortality in these studies despite effective anti-HIV and anti-HBV treatment.

Conversely, HIV-RNA had no effect on all-cause mortality in our study. This result was somewhat surprising considering the well-described effects of HIV replication on AIDS and non-AIDS-related death [START_REF] Phillips | The role of HIV in serious diseases other than AIDS[END_REF]. Although a high proportion of individuals had detectable HIV-RNA levels at inclusion, most were able to achieve HIV-RNA suppression early in the cohort study. Under these conditions, AIDS-related mortality is generally associated with the time spent at lower CD4+ cell counts [START_REF] Battegay | Immunological recovery and antiretroviral therapy in HIV-1 infection[END_REF][START_REF] Eurocoord | CD4 cell count and the risk of AIDS or death in HIV-Infected adults on combination antiretroviral therapy with a suppressed viral load: a longitudinal cohort study from COHERE[END_REF], while most individuals in our cohort attained levels of CD4+ associated with reduced risk of both AIDS-related and overall mortality. The low overall proportion of AIDS-related deaths is also in accordance with findings from European cohorts of HIV-positive individuals [START_REF] Vandenhende | Cancer-Related Causes of Death among HIV-Infected Patients in France in 2010: Evolution since 2000[END_REF]. When comparing HIV and HBV replication, there were comparable degrees of HIV-RNA suppression for individuals who remained alive and died during follow-up, while HBV-DNA was more frequently detectable in deceased than alive individuals. These differences were similarly observed in historical exposure and perhaps drove HBV-DNA, rather than HIV-RNA, to contribute more towards overall mortality in our study.

The question is then what aspects of HBV-DNA replication are driving overall mortality. Naturally, immunological responses against prolonged HBV infection are responsible for liver-related disease [START_REF] Singh | HIV-hepatitis B virus coinfection: epidemiology, pathogenesis, and treatment[END_REF], which partly explains the association between advanced liver fibrosis and all-cause mortality.

Although liver fibrosis is mostly known to affect liver-related mortality (i.e. death due to HCC or endstage liver disease), we observed few liver-related deaths. These results are rather surprising. Most individuals were undergoing TDF-containing ART and hence this study population represents those with well-controlled HBV-DNA replication. The risk of HCC would be reduced at these HBV-DNA levels [START_REF] Papatheodoridis | Risk of hepatocellular carcinoma in chronic hepatitis B: assessment and modification with current antiviral therapy[END_REF]. Nevertheless, individuals with liver cirrhosis do have a high risk of death due to invasive bacterial pathogens [START_REF] Nahon | Bacterial infection in compensated viral cirrhosis impairs 5-year survival (ANRS CO12 CirVir prospective cohort)[END_REF] and many of the other causes of death in our cohort were related to bacteremia. Although effective anti-HBV treatment is expected to decrease fibrosis in HBV monoinfected individuals [START_REF] Marcellin | Regression of cirrhosis during treatment with tenofovir disoproxil fumarate for chronic hepatitis B: a 5-year open-label follow-up study[END_REF], fibrosis levels are mostly unchanged or can progress during TDF-based ART in co-infected individuals [START_REF] Boyd | Liver fibrosis regression and progression during controlled hepatitis B virus infection among HIV-HBV patients treated with tenofovir disoproxil fumarate in France: a prospective cohort study[END_REF]. Continued risk of death due to this comorbidity should be elucidated in larger cohorts.

One intriguing finding was that certain extra-hepatic malignant tumors caused several deaths in this study population. In fact, 40% of the 20 individuals who died from cancer developed an extrahepatic tumor (i.e. anal cancer, cholangiocarcinoma, pancreatic adenocarcinoma, and non-Hodgkin lymphoma [NHL]). Previous research has found that HBsAg-positive individuals with high-risk human papillomavirus had an increased risk of high-grade anal squamous intraepithelial lesions compared to those who were HBsAg-negative [START_REF] Mccloskey | Syndemic synergy of HPV and other sexually transmitted pathogens in the development of high-grade anal squamous intraepithelial lesions[END_REF]. Furthermore, HBsAg-positive individuals exhibiting high levels of HBV activity (i.e. HBeAg-positivity and detectable HBV-DNA) have a significantly higher risk of pancreatic carcinoma compared to HBsAg-negative individuals [START_REF] Iloeje | Risk of pancreatic cancer in chronic hepatitis B virus infection: data from the REVEAL-HBV cohort study[END_REF]. Finally, although NHL is normally related to immunocompromised individuals and is classified as an AIDS-defining illness, growing evidence suggests that ART-treated individuals with chronic HBV infection are at increased risk for NHL [START_REF] Wang | Chronic Hepatitis B and C Virus Infection and Risk for Non-Hodgkin Lymphoma in HIV-Infected Patients: A Cohort Study[END_REF]. How HBV activity participates in the tumorigenesis of extra-hepatic tumors is not clearly understood. Similar to others [START_REF] Béguelin | Hepatitis delta-associated mortality in HIV/HBVcoinfected patients[END_REF], HDV-co-infected individuals had a higher all-cause mortality rate.

Unfortunately, due to its close association with fibrosis, we decided not to include HDV-co-infection in multivariable analysis. Of the 22 individuals in our cohort with HIV-HBV-HDV infection, 7 died.

Despite the fact that 6 of these deceased individuals had advanced liver fibrosis and at least one liver-related complication, only one died from liver-related diseases. In contrast to others [START_REF] Béguelin | Hepatitis delta-associated mortality in HIV/HBVcoinfected patients[END_REF], only one HCV-positive individual (without HDV) died in our cohort, which could be a reflection of the low overall proportion of IDU (7.7%) and the increasingly effective direct acting antivirals available by the end of follow-up [START_REF] Smit | HCV micro-elimination in individuals with HIV in the Netherlands 4 years after universal access to direct-acting antivirals: a retrospective cohort study[END_REF]. Given the very few individuals with tri-/quad-infection who remained in follow-up, generalizability of our data to this population would be limited.

Our study has certain limitations. First, the study population involves HIV-positive individuals with extensive ART-experience and larger degrees of immunosuppression compared to contemporary patient populations, but still actively seen in outpatient settings. Second, HBsAg-seroclearance has been shown to reduce all-cause mortality in HBV mono-infected individuals [START_REF] Arase | Long-term outcome after hepatitis B surface antigen seroclearance in patients with chronic hepatitis B[END_REF], yet we had an insufficient number of events to validate this in our cohort. Third, there was a rather high rate of LTFU. Individuals who were LTFU had higher HBV DNA, ALT/AST, but not fibrosis levels at cohort inclusion, suggesting a higher risk of more HBV activity. It is difficult to assess the direction of this differential LTFU bias without knowing whether they were virally suppressed after being LTFU.

Fourth, we had limited to no data on HBV-DNA and HIV-RNA replication prior to inclusion, alcohol use, smoking, treatment adherence, and metabolic diseases, all of which could not be considered in analysis. We also used a non-invasive measure to assess liver fibrosis, which involves some error [START_REF] Bottero | Performance of 11 biomarkers for liver fibrosis assessment in HIV/HBV co-infected patients[END_REF]; however, the FibroTest® does accurately predict liver fibrosis evolution [START_REF] Poynard | Validation of liver fibrosis biomarker (FibroTest) for assessing liver fibrosis progression: proof of concept and first application in a large population[END_REF] and overall survival [START_REF] De Lédinghen | Non-invasive tests for fibrosis and liver stiffness predict 5-year survival of patients chronically infected with hepatitis B virus[END_REF] in individuals with chronic HBV infection. Finally, we did not have enough power to determine which causes of death were more associated with HBV-DNA replication.

In conclusion, our findings provide strong evidence that HIV-HBV co-infected individuals who have been exposed to higher levels of HBV-DNA over time are at elevated risk for all-cause mortality. The lack of association with HIV-RNA replication could be due to the more extensive viral suppression overall compared to HBV-DNA. Accompanied by the strong association between advanced liver fibrosis and overall mortality, monitoring liver fibrosis and HBV-DNA VL should be an essential component to help assess the prognosis of co-infected individuals. The noticeably common deaths due to extra-hepatic malignancies should be further studied and perhaps increased screening is called for in the HIV-HBV co-infected patient population.

Crohn's disease 1 1 0 Quadriplegia 1 0 1 Subdural hematoma 1 0 1 Suicide 1 1 0 Unknown 3 (7.0%) 1 2
*4 patients died from multiple causes: hepatocellular carcinoma and sepsis (N=1, considered as a liver-related death), non-Hodgkin's lymphoma and sepsis (N=1, considered as an AIDS-related death), prostate cancer and sepsis (N=1, considered as a non-AIDS and non-liver malignancy death), and empyema and sepsis (N=1, considered as a death related to other diseases). b HBV replication was measured as longitudinal HBV-DNA VL detectability (<60 IU/mL versus ≥60 IU/mL); whereas HIV replication as longitudinal HIV-RNA detectability (<50 copies/mL versus ≥50 copies/mL).

c HBV replication was measured as longitudinal HBV-DNA (per log10 IU/mL); whereas HIV replication as longitudinal HIV-RNA (per log10 copies/mL).

d HBV replication was measured as time-averaged cumulative HBV-DNA (per log10copy-yearsTAVG); whereas HIV replication as cumulative HIV-RNA (per log10copy-yearsTAVG).

e Risk estimates are additionally adjusted for the cumulative duration of tenofovir according to a spline function restricted by 4 knots (0.021, 3, 8 and 11.978 years). § Time-updated covariate.

ɸ Variable included percentage of visits during two HIV treatment eras, 2002-2007 and ≥2008.

# Estimated using the Fibrotest®.

¥ Expected value as estimated from the respective submodels (HBV replication and HIV replication) as outcomes.

Abbreviations: AIDS, acquired immunodeficiency syndrome; HBV, hepatitis B virus; HIV, human immunodeficiency virus;

Figure 1 .

 1 Figure 1. Evolution of HIV, hepatitis B virus (HBV), and antiviral treatment against hepatitis B virus over time.

Figure 2 .

 2 Figure 2. Evolution of the cumulative extent of viral replication over time

Table 1 . Causes of death observed in the French HIV-HBV cohort, 2002-2018.

 1

		All periods	2002-2007	2008-2018
	Cause of death
		(N=42)	(N=10)	(N=32)
	Liver-related disease	7 (16.7%)	2	5
	Hepatocellular carcinoma	4	1	3
	Cholangiocarcinoma	1	0	1
	Liver failure	2	1	1
	AIDS-related disease	7 (16.7%)	2	5
	Kaposi sarcoma	1	1	0
	Burkitt lymphoma	1	0	1
	Other non-Hodgkin's lymphoma	1	0	1
	Progressive multifocal leukoencephalopathy	1	1	0
	HIV-associated neurocognitive disorder	1	0	1
	Opportunistic infections	2	0	2
	Non-AIDS and non-liver-related malignancy	12 (28.6%)	0	12
	Anal cancer	5	0	5
	Colorectal cancer	1	0	1

Table 2 . Characteristics of study population (at cohort inclusion or during follow-up)

 2 Number (%). § Significance determined using Kruskal-Wallis test for continuous variables and Pearson χ 2 test or Fisher exact test for categorical variables. In a subgroup of 138 individuals, liver biopsies were performed within 12 months before or at study entry, based on concomitant guidelines from the European Association for the Study of the Liver. Histological fibrosis and activity were scored with the METAVIR classification. Established by a positive ELISA-based assay for HCV or HDV, and confirmed by a positive PCR-based assay for HCV-RNA or HDV-RNA, respectively. ‡ In 156 HBeAg-positive individuals.

		Vital status
	Characteristics
		Alive (n=258)	Died (n=42)	p §
	Demographics
	Gender, male/female (% male)	213/45 (83)	39/3 (93)	0.09
	Age at baseline, years*	39.5 (34.5-43.7)	43.5 (38.0-53.2)	<0.001
	From zone of high HBV-prevalence †	81 (31.4)	4 (9.5)	0.004
	Mode of HIV transmission †
	Heterosexual	95 (36.8)	10 (23.8)	0.10
	MSM	143 (55.4)	25 (59.5)	0.6
	IDU	16 (6.2)	7 (16.7)	0.02
	Other/Unknown	4 (1.6)	0 (0)	0.9
	Clinical characteristics at study entry
	BMI, Kg/m 2 [N = 284]*	22.6 (21.0-24.3)	21.9 (20.6-23.5)	0.11
	Alcohol consumption, glasses/day [N = 288]*	1 (0 -2.7)	1 (0 -2)	0.4
	Comorbidities †
	Cardiovascular disease	28 (10.9)	6 (14.3)	0.5

† # ɸ

Table 3 . Association of HBV-DNA and HIV-RNA replication with all-cause mortality (joint models) a

 3 Analysis included 298 individuals, 41 of whom were deceased. Two individuals were not considered in this analysis due to missing data for level of fibrosis at baseline; one of them died during study follow-up.

						Model: historical extent of
		Model: treatment efficacy b	Model: extent of replication c
						replication d
	Parameter
		Risk estimate e		Risk estimate e		Risk estimate e
		(95% CI)	p	(95% CI)	p	(95% CI)	p
	HBV replication	OR		RR		RR
	HBeAg at baseline	0.04 (0.03-0.05)	<0.001	1.88 (1.79-1.97)	<0.001	2.02 (1.93-2.11)	<0.001
	HIV replication	OR		RR		RR
	CD4+ cell count (√/mm 3 ) §	1.10 (1.08-1.12)	<0.001	0.987 (0.984-0.991) <0.001	0.988 (0.985-0.992)	<0.001
	HIV treatment era from 2002 to 2007 ɸ	0.14 (0.11-0.17)	<0.001	1.22 (1.17-1.27)	<0.001	1.16 (1.11-1.20)	<0.001
	Time to all-cause mortality	HR		HR		HR
	Age §	1.06 (1.02-1.10)	0.001	1.06 (1.02-1.10)	0.004	1.05 (1.01-1.09)	0.007
	AIDS-defining illness at baseline	1.97 (1.00-3.87)	0.05	1.83 (0.94-3.54)	0.07	1.92 (1.00-3.71)	0.05
	F3-F4 fibrosis at baseline #	2.33 (1.16-4.70)	0.02	2.35 (1.16-4.76)	0.02	2.37 (1.17-4.81)	0.02
	HBV replication ¥	0.30 (0.08-1.09)	0.08	1.41 (1.04-1.93)	0.03	1.37 (1.03-1.83)	0.03

a

  TAVG, time-averaged.

	HIV replication ¥	0.42 (0.02-10.53)	0.6	2.43 (0.45-13.05)	0.30	2.80 (0.49-15.85)	0.25

Supplementary Table S2. Characteristics of the study population at cohort inclusion, stratified by lost to follow-up.

	Characteristics	Patients	p §
		Completed follow-up	Lost to follow-up
		(n=147)	(n=118) ¶
	Demographics
	Gender, male/female (% male)	119/28 (81.0)	99/19 (84.7)	0.53
	Age at baseline, years*	40.0 (34.6-44.1)	38.1 (34.1-42.0)	0.10
	From zone of high HBV-prevalence †	47 (32.0)	35 (29.7)	0.69
	Mode of HIV transmission †			0.42
	Heterosexual	57 (38.8)	42 (35.6)
	MSM	80 (54.4)	64 (54.2)
	IDU	7 (4.8)	11 (9.3)
	Other/Unknown	3 (2.0)	1 (0.9)
	Clinical characteristics at study entry
	BMI, Kg/m 2 [N = 251]*	22.4 (21.0-24.3)	22.7 (21.0-24.6)	0.58
	Alcohol consumption, glasses/day [N = 254]*	1 (0-2)	1 (0-2.9)	0.67
	Comorbidities †
	Cardiovascular disease	17 (11.6)	11 (9.3)	0.55
	Diabetes	3 (3.0)	2 (1.7)	1.00
	Renal disease	3 (2.0)	3 (2.5)	1.00
	Other liver diseases ¥	4 (2.7)	2 (1.7)	0.58
	HIV infection variables at study entry
	Time since first HIV-positive test, years [N = 264]*	9.4 (3.8-13.1)	9.5 (2.8-14.5)	0.93
	AIDS-defining illness †	30 (20.4)	29 (24.6)	0.42
	HIV-RNA > 50 copies/mL †	64 (43.5)	63 (53.4)	0.14
	HIV-RNA, log10 copies/mL [N = 127]*	3.9 (4.5-4.4)	3.9 (2.8-4.5)	0.74
	CD4 + cell count, cells/µL*	403 (283-557)	406 (249-586)	0.87
	Nadir CD4 + cell count, cells/µL [N = 235]*	212 (107-309)	236 (110-372)	0.20
	Initiated ART †	135 (91.8)	102 (86.4)	0.17
	Duration of ART, years [N = 230]*	5.9 (2.4-7.4)	4.8 (2.7-6.8)	0.33
	Viral hepatitis at study entry
	HBV-genotype [N = 142] †
	A	51 (68.0)	36 (53.7)	0.09
	D	7 (9.3)	5 (7.5)	0.77
	E	8 (10.7)	11 (16.4)	0.32
	G	9 (12.0)	13 (19.4)	0.22
	HBeAg-positive †	73 (49.7)	60 (50.9)	0.85
	HBV-DNA > 60 IU/mL †	95 (64.6)	74 (62.7)	0.75
	HBV-DNA, log10 IU/mL [N = 169]*	4.3 (2.9-6.6)	5.3 (3.6-6.9)	0.04
	ALT level, IU/mL [N = 260]*	38 (22-64)	43 (28-88)	0.02
	AST level, IU/mL [N = 260]*	33 (25-52)	38 (29-59)	0.03
	Metavir F3-F4 fibrosis †
	Estimated using Fibrotest® [N = 264]	38 (26.0)	27 (22.9)	0.56
	Determined by liver biopsy [N = 119] #	22 (31.0)	13 (27.1)	0.65
	Metavir F4 fibrosis †
	Estimated using Fibrotest® [N = 264]	21 (14.4)	13 (11.0)	0.42
	Determined by liver biopsy [N = 119] #	6 (8.5)	5 (10.4)	0.72
	Ever HCV coinfected † ɸ	13 (8.8)	8 (6.8)	0.54
	Ever HDV coinfected † ɸ	7 (4.8)	9 (7.6)	0.33
	Cumulative lamivudine use at study entry, years*	2.8 (0.9-5.5)	3.1 (0.4-5.3)	0.74
	*Median (IQR).

† Number (%). ¶ Does not include the 35 deceased individuals whose deaths were observed during follow-up. § Significance determined using Kruskal-Wallis' test for continuous variables and Pearson's χ 2 test or Fisher's exact test for categorical variables. ¥ Other liver diseases or hepatic decompensation: acute, subacute or unspecified hepatic failure; haemorrhagic necrosis of liver; fatty liver disease; portal hypertension; and hepatocellular carcinoma.

Acknowledgements

The authors are grateful to the participants and the clinical teams for their commitment to the French HIV-HBV Cohort. This study was sponsored by the Institut de Médecine et d'Epidémiologie Appliquée (IMEA). L.N.C.D. was awarded a post-doctoral fellowship from the ANRS. We thank Amir Moheb Mohareb for editing parts of the manuscript. Role of each author. L.N.C.D. was responsible for the statistical analysis, interpretation of the data, and drafting the manuscript. R.K. obtained and verified vital status on participants, assisted in the statistical analysis, and gave critical revisions of the manuscript. H.R., P.M., C. L-C., and J.C. acquired data for the cohort, assisted in interpreting data, and gave critical revisions of the manuscript. S.M., A.G. and C.D. were responsible for interpretation of the data and drafting the manuscript. K.L. helped design, conceptualize, and obtain funding for the French HIV-HBV cohort study, coordinated data collection, and drafted the manuscript. A.B. coordinated data analysis, gave important comments on data interpretation, drafted parts of the manuscript, and provided critical revisions of the manuscript. All authors have approved the final version of the article.

Funding

This work was supported by SIDACTION (AO 19) and the France REcherche Nord&sud Sida-hiv Hépatites (ANRS). Gilead Sciences, Inc. provided an unrestricted grant for the French HIV-HBV cohort and was not involved in any part of the data collection, analysis and manuscript writing.

Potential conflicts of interest

None to declare.

In multivariable modeling, BMI had too many missing data and was not considered further; AST levels, HDV coinfection, F3-F4 fibrosis (estimated using FibroTest) and time-updated F3-F4 fibrosis levels were collinear and we preferred F3-F4 fibrosis (estimated using FibroTest) at inclusion; CD4 + cell count at the last follow-up visit, initiation of ART and nadir CD4 + cell count were collinear and we preferred nadir CD4 + cell count; HBeAg status and time-updated CD4 + cell count were included in the submodels of the joint models analysis and were not considered further. The following variables were removed as their association was no longer significant in the multivariable model: male gender (p=0.63), from zone of high prevalence (p=0.17), and nadir CD4 + cell count (p=0.42). The final multivariable model was adjusted by all covariates listed in the column. # In a subgroup of 138 patients, liver biopsies were performed within 12 months before or at study entry, based on concomitant guidelines from the European Association for the Study of the Liver. Histological fibrosis and activity were scored with the METAVIR classification. Abbreviations: ALT, alanine aminotransferase; ART, antiretroviral therapy; AST, aspartate aminotransferase; BMI, body mass index; HBeAg, hepatitis B "e" antigen; HBsAg, hepatitis B surface antigen; HBV, hepatitis B virus; HIV, human immunodeficiency virus; IDU, injection drug use; MSM, men who have sex with men; ULN, upper limit of normal.

Supplementary Figure S1. Evolution of cumulative HIV-RNA (log10 copy-yearsTAVG) over time according to mortality outcome

Evolution of HIV-RNA log10 copy-yearsTAVG is given for alive patients in the left panel and for deceased patients in the right panel. Means are expressed as bold lines from a LOWESS curve and individual levels are expressed as gray lines. Abbreviation: TAVG, time-averaged.


Laurent Théry
email: thery@sophia.inria.fr

Sudoku in Coq

Keywords: , permutation l2 l1 -> permutation, l1) | permutation_swap, forall a b l, permutation -> permutation l2 l3 -> permutation l1 l3

This note presents a formalisation done in Coq of Sudoku. We formalise what is a Sudoku, a Sudoku checker and a Sudoku solver. The Sudoku solver uses a naive Davis-Putnam procedure.

The Grid

The grid is parametrised by two integers h and w. It is composed of hw subrectangles, each rectangle is composed of h lines and w columns. If we take h = 3 and w = 3, we get the usual grid composed of 9 subsquares.

We are going to formalize the grid as a list of integers. In order to express what rows, columns, subrectangles are, we first need to have some basic list operations. The function take takes the n first elements of a list l. Fixpoint take (n: nat) (l: list A) : list A := match l with nil => nil | a::l1 => match n with O => nil | S n1 => a:: take n1 l1 end end.

The function jump skips the first n elements of a list l. Fixpoint progression (n m: nat) : list nat := match n with O => nil | S n1 => m :: progression n1 (S m) end. Definition ref_list := progression size 1. 2 2 The Sudoku A way to define what is a Sudoku is to use the notion of permutation. Inductive permutation : list A -> list A -> Prop := | permutation_nil: permutation nil nil | permutation_skip: forall a l1 l2

Fixpoint jump (n: nat) (l: list A) : list A := match l with nil => nil | a::l1 => match n with O => l | S n1 => jump n1 l1 end end.

The function take_and_jump takes the t first elements then skips j elements and repeats this n times.

Fixpoint take_and_jump (t j n: nat) (l: list A) : list A := match n with 0 => nil | S n1 => take t l ++ take_and_jump t j n1 (jump j l) end.

Now we are ready to consider an arbitrary grid. We take h as the number of rows of a subrectangle and w as the number of columns of a subrectangle. So a subrectangle contains exactly size cells.

Variable h w: nat. Definition size := h * w.

In a grid, there are size rows. The function row returns the row number i.

Definition row i (l: list nat) := take size (jump (i * size) l).

The function column returns the column number i.

Definition column i (l: list nat) := take_and_jump 1 size size (jump i l).

The function rect returns the subrectangle number i.

Definition rect i (l: list nat) := take_and_jump w size h (jump (w * (mod i h) + h * (div i h) * size) l).

In the list that represents the grid, we take the convention that the integer 0 indicates an empty cell. We define the list ref_list that contains all the possible values of a cell: 1, 2, . . . , size.

Definition sudoku l := length l = size * size /\ (forall i, i < size -> permutation (row i l) ref_list) /\ (forall i, i < size -> permutation (column i l) ref_list) /\ (forall i, i < size -> permutation (rect i l) ref_list).

The Sudoku checker

We can easily decide if a list is a a permutation of another one with the function permutation_dec1.

Definition permutation_dec1 :

(forall a b : A, {a = b} + {a <> b}) -> forall l1 l2, {permutation l1 l2} + {~permutation l1 l2}.

It follows that we can define a Sudoku checker.

Definition check: forall l, {sudoku l} + {~sudoku l}.

The Sudoku solver

In order to solve Sudoku, we are going to translate the problem into a constraint problem and use a Davis-Putnam procedure [START_REF] Davis | A computing procedure for quantification theory[END_REF] to solve it.

Positions, literals, clause and clauses

Positions

A literal represents the fact that a given cell contains a given value. We first define the type pos for positions.

Inductive pos: Set := Pos (x: nat) (y: nat).

Positions starts at (0,0) till (size-1, size-1). A position is valid if it is inside the grid.

Definition valid_pos (p: pos) := let (x,y) := p in x < size /\ y < size.

To enumerate all the positions in a left-right top-down manner, we use the function next. We can turn a position into its index in the list that represents the grid with the function pos2n

Definition pos2n (p: pos) := let (x,y) := p in x * size + y.

The function get retrieves from the list s that represents the grid the value inside the cell at position p Definition get p l := nth 0 (jump (pos2n p) l) 0.

The function update puts inside the cell at position p the value v.

Literals

A literal is composed of a position and a value.

Inductive lit: Set := L (p: pos) (v: nat).

Clause

A clause is a disjunction of literals. It is represented as a list. A clause is satisfied if and only if at least one of its literals is satisfied,

Definition clause := list lit.
We provide some operations to manipulate clauses. The function lit_is_in checks if a literal is in a clause.

Definition lit_is_in: lit -> clause -> bool.
The function lit_insert adds a literal inside a clause.

Definition lit_insert: lit -> clause -> clause.
The function lit_rm removes from c2 all the literals that occur in c1.

Definition lit_rm (c1 c2: clause): clause.

The function clause_merge appends all the literals in c1 and c2.

Definition clause_merge (c1 c2: clause): clause.

Clauses

The problem to solve is a conjunction of clauses. The conjunction is represented as a list. To solve the problem, we need to satisfy each clause, i.e. we need to satisfy at least one literal in each clause.

Definition clauses:= list (nat * clause).

The integer associated to each clause represents the length of the clause, i.e. the number of literals in the clause. This integer is used to sort the list. Clauses with few literals are on top of the list. The function clause_insert adds a clause in the list of clauses.

Definition clause_insert: clause -> clauses -> clauses.
The function clause_merge appends two lists of clauses. The key function for manipulating clauses is the function clauses_update. It is used to update the list of clauses removing from the list all the clauses that contain the literal l and removing from each clause in the list all the literals that occur in c. This function is called to update the list of constraints when a new fact is known. The literal l is this new fact we know to hold (for example the cell (1, 2) contains 3). The list of literals c contains the facts we know not to hold (for example the cell (1, 2) does not contain 1, the cell (1, 2) does not contain 2, the cell (1, 2) does not contain 4, . . . ).

Generating problem constraints

In order to generate constraints, we need some list. The list indexes contains all the possible index for a position from 0 to size -1.

Definition indexes := progression size 0.

The list cross contains all the position (x,y) with 0 ≤ x < h and 0 ≤ y < w.

Definition cross := let p := progression h 0 in let q := progression w 0 in fold_right (fun x l => (map (fun y => (Pos x y)) q) ++ l) nil p.

The list cross1 contains all the pairs (x,y) with 0 ≤ x < size and 1 ≤ y ≤ size.

Definition cross1 := let p := indexes in let q := ref_list in fold_right (fun x l => (map (fun y => (x, y)) q) ++ l) nil p.

The function gen_row generates the constraints that the row number i contains the value z.

Definition gen_row i z := fold_right (fun y l => lit_insert (L (Pos i y) z) l) nil indexes.

The function gen_column generates the constraints that the column number i contains the value z.

Definition gen_column i z := fold_right (fun x l => lit_insert (L (Pos x i) z) l) nil indexes.

The function gen_rect generates the constraints that the subrectangle number i contains the value z.

Definition gen_rect i z := let x := h * div i h in let y := w * mod i h in fold_right (fun p l => lit_insert (L (shift p x y) z) l) nil cross.

where div and mod are the division and the modulo respectively. The function gen_cell generates the constraints that the cell at position p contains a value.

Definition gen_cell p := fold_right (fun z l => lit_insert (L p z) l) nil ref_list.

The function all_cell generates the constraint that all the cells of the grid contain a value.

Definition all_cell := let c0 := cross2 in (fold_right (fun p l => let res := gen_cell p in clause_insert res l) nil c0).

Generating complement literals

To speed up the search, when we have a new fact, we not only remove all the clauses that contains the corresponding literals but we also simplify clauses removing the literals that cannot hold. Computing from a known fact the list of fact that cannot hold is done by the function anti_literals.

Generating the initial constraints

To generate the initial constraints, we just accumulate the clauses that each number appears in each row, each number appears in each column, each number appears in each subrectangle and finally each cell contains a value. These constraints are redundant but most importantly a list that satisfies these constraints is a Sudoku. Note that the initial constraint only ensures that the list ref_list is included in each row, each column and subrectangle. It is sufficient because the list ref_list is composed of unique elements.

Inductive ulist : list A -> Prop := ulist_nil: ulist nil | ulist_cons: forall a l, ~In a l -> ulist l -> ulist (a :: l) .

Theorem ref_list_ulist : ulist ref_list.

So we can apply a derived version of the pigeon-hole principle.

Theorem ulist_eq_permutation: forall (l1 l2 : list A), ulist l1 -> incl l1 l2 -> length l1 = length l2 -> permutation l1 l2.

where incl is the inclusion predicate. Before starting the search, we first have to simplify the list of clauses with all the information that is present in the initial grid. This is done by the function gen_init_clauses_aux that walks through the initial grid s and updates the list of clauses c. The argument p contains the actual position of the element of the list s we are considering. The main function is gen_init_clauses.

Definition gen_init_clauses s := gen_init_clauses_aux s (Pos 0 0) init_c.

Finding one solution

To organize the search for a solution, we have to deal with Coq's restriction that only structural recursive schemas can be used in function definitions.

The first function we write is the function try_one that given the current state of the Sudoku and the list of constraints cs tries to satisfies the clause c. For this, it picks a literal of c, supposes it holds and calls the continuation f to check if the result is satisfiable. If so, it returns the solution, otherwise it tries the next literal. The function find_one_aux is the one that organizes the recursion. It takes the current state of the Sudoku s and the corresponding list of clauses cs. It looks at the top of the list of clauses. As this list is ordered, clauses with few literals come first. If the list contains the empty clause, this clause is unsatisfiable so there is no solution. Otherwise it takes the first clause and tries to satisfy it using try_one. The argument n is a dummy argument added to ensure structural termination. As at each call of try_one we remove at least one clause from the list of clauses, taking the initial list of clauses can safely by given as dummy argument. The main function is then Definition find_one s := let cs := gen_init_clauses s in find_one_aux cs s cs.

To state the correctness of this algorithm, we need to define the predicate that indicates if a grid is a refinement of another grid With this predicate, the correctness is expressed as

Theorem find_one_correct: forall s, length s = size * size -> match find_one s with None => forall s1, refine s s1 -> ~sudoku s1 | Some s1 => refine s s1 /\ sudoku s1 end.

Finding all solutions

It is trivial to modify the previous algorithm so that it returns not the first solution but all the solutions. The function try_all accumulates all the solutions instead of stopping at the first one. where the function merges:list nat -> list (list nat) -> list (list nat) inserts a solution in a list of solutions.

  Definition next (p: pos) := let (x,y) := pos in if eq_nat (S y) size then Pos (S x) 0 else Pos x (S y).

Fixpoint

  subst (n: nat) (v: A) (l: list A) : list A := match l with nil => nil | a :: l1 => match n with O => L :: l1 | S n1 => a :: subst n1 L l1 end end. Definition update p L (l: list nat) := subst (pos2n p) L l.

  Definition clauses_merge: clauses -> clauses -> clauses.

Fixpoint

  clauses_update (l: lit) (c: list lit) (cs: clauses) {struct cs}: clauses := match cs with nil => nil | (n , c1) :: cs1 => if lit_is_in l c1 then clauses_update l c cs1 else let res := lit_rm c c1 in clause_insert res (clauses_update l c cs1) end.

  Definition anti_literals (l: lit) := let c := l :: nil in let (p, v) := l in let (x, y) := p in clause_merge (lit_rm c (gen_row x v)) (clause_merge (lit_rm c (gen_column y v)) (clause_merge (lit_rm c (gen_rect ((div x h) * h + (div y w)) v)) (lit_rm c (gen_cell p)))).

  Definition init_c := let c1 := cross1 in fold_right (fun iz l => let res := gen_row (fst iz) (snd iz) in clause_insert res l) (fold_right (fun iz l => let res := gen_column (fst iz) (snd iz) in clause_insert res l) (fold_right (fun iz l => let res := gen_rect (fst iz) (snd iz) in clause_insert res l) all_cell c1) c1) c1.

Fixpoint

  gen_init_clauses_aux (s: list nat) (p: pos) (c: clauses): clauses := match s with nil => c | a :: s1 => let p1 := next p in let ll := L p a in if (In_dec eq_nat a ref_list) then let c1 := clauses_update ll (anti_literals ll) c in gen_init_clauses_aux s1 p1 c1 else gen_init_clauses_aux s1 p1 c end.

Fixpoint

  try_one (s: list nat) (c: clause) (cs: clauses) (f: list nat -> clauses -> option (list nat)) {struct c}: option (list nat) := match c with nil => None | (L p v) as k:: c1 => let s1 := update p v s in let cs1 := clauses_update k (anti_literals k) cs in match f s1 cs1 with None => try_one s c1 cs f | Some c1 => Some c1 end end.

Fixpoint

  find_one_aux (n: clauses) (s: list nat) (cs: clauses): option (list nat) := match cs with nil => Some s | (_, nil) :: _ => None | (_, p) :: cs1 => match n with nil => None | _ :: n1 => try_one s p cs1 (find_one_aux n1) end end.

  Definition refine s1 s2 := length s1 = size * size /\ length s2 = size * size /\ forall p, valid_pos p -> In (get p s1) ref_list -> get p s1 = get p s2.

Fixpoint

  try_all (s: list nat) (c: clause) (cs: clauses) (f: list nat -> clauses -> list (list nat)): list (list nat) := match c with nil => nil | (L p v) as k:: l1 => let s1 := update p v s in let cs1 := clauses_update k (anti_literals k) cs in merges (f s1 cs1) (try_all s l1 cs f) end.

Fixpoint

  find_all_aux (n: clauses) (s: list nat) (cs: clauses): list (list nat) := match cs with nil => s :: nil | (_, nil) :: _ => nil | (_, p) :: cs1 => match n with nil => nil | _ :: n1 => try_all s p cs1 (find_all_aux n1) end end.

The main function is then Definition find_all s := let cs := gen_init_clauses s in find_all_aux cs s cs.

The corresponding correctness statement is the following.

Theorem find_all_correct: forall s s1, refine s s1 -> (sudoku s1 <-> In s1 (find_all s)).

Running example

To use the Sudoku contribution, we first need to load it.

Require Import Sudoku.

Suppose we want to solve 3x3 Sudoku, we first create the initial position taking the convention that zeros correspond to empty cells.

Definition test := 5 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 4 :: 0 :: 8 :: 1 :: 0 :: 0 :: 0 :: 0 :: 0 :: 9 :: 3 :: 0 :: 0 :: 0 :: 0 :: 0 :: 2 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 2 :: 0 :: 3 :: 9 :: 0 :: 0 :: 7 :: 0 :: 0 :: 0 :: 0 :: 0 :: 2 :: 3 :: 0 :: 0 :: 0 :: 6 :: 0 :: 7 :: 0 :: 3 :: 6 :: 5 :: 1 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 5 :: 0 :: 8 :: 0 :: 0 :: 0 :: 0 :: 1 :: 0 :: 7 :: 0 :: 6 :: 0 :: 0 :: nil.

Doing

Eval compute in find_one 3 3 test.

returns 5 :: 8 :: 6 :: 2 :: 3 :: 7 :: 9 :: 1 :: 4 :: 7 :: 4 :: 2 :: 8 :: 1 :: 9 :: 3 :: 5 :: 6 :: 1 :: 9 :: 3 :: 4 :: 6 :: 5 :: 7 :: 8 :: 2 :: 6 :: 5 :: 7 :: 9 :: 8 :: 1 :: 2 :: 4 :: 3 :: 9 :: 1 :: 4 :: 7 :: 2 :: 3 :: 5 :: 6 :: 8 :: 2 :: 3 :: 8 :: 5 :: 4 :: 6 :: 1 :: 7 :: 9 :: 3 :: 6 :: 5 :: 1 :: 9 :: 8 :: 4 :: 2 :: 7 :: 4 :: 7 :: 9 :: 6 :: 5 :: 2 :: 8 :: 3 :: 1 :: 8 :: 2 :: 1 :: 3 :: 7 :: 4 :: 6 :: 9 :: 5 :: nil

Doing

Eval compute in length (find_all 3 3 test).

returns 1. We are then sure that there is a unique solution to this Sudoku The source code is available at ftp://ftp-sop.inria.fr/marelle/Laurent.Thery/Sudoku.zip


Kai Gerlinger

Charles Perrin

Where the hydro projections from Explore 2070 and FLOW MS that different?

Conclusions

• We updated the hydro projections of Mosel, Sarre and French Rhine

• We took into account uncertainties

• We undertook an automatic calibration of LARSIM

• Mean flows should slightly increase (except IPSL RCP 8.5)

• High flows should increase in near future, more uncertainty later

• Low flows should decrease in near future, more uncertainty later but potential very high decrease

• Uncertainties: GCMs > RCPs > Sampling > Hydro models

• Prior studies: with CMIP5, we globally are consistent with FLOW MS or Explore 2070, but:

• Explore 2070 corresponds rather to an intermediate state between MOSARH 21 near and far futures • FLOW MS cannot be linked to a specific RCP



  /2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) Aims of the MOSARH21 project Main goal: Producing future discharge estimations for the French tributaries of the Rhine Sub-goals:  Adopting a rigorous testing protocol for hydrological models  Temporal robustness  Quantifying uncertainties of the different steps of the modelling chain  Several elements used for each step  Comparing future discharge estimations of MOSARH21 with former projects results  CMIP3 vs CMIP5 p. 3 MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) /2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) Hydro models parameters calibration To better suit catchments characteristics {c, l, L, h, H, p} 1 {c, l, L, h, H, p} 2 /2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) On the temporal robustness of hydro models Application of a classical split-sample test, a necessity for CC applications (Thirel et al.Thirel, G., Andréassian, V., Perrin, C., 2015. On the need to test hydrological models under changing conditions. Hydrological Sciences Journal, 60:7-8, pages 1165-1173, DOI:10.1080/02626667.2015.1050027. p. 7 MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)LARSIM automatic calibrationFirst time an automatic calibration of LARSIM was done 7 parameters were selected and calibrated sequentially according to main processes:• Water balance (1 param)• Low flows (2 params)• Mid-flows (2 params)• High flows (2 params)For each step, a specific objective function was usedBased on the GR methodology p. 8MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)Evolution of performance of LARSIM during its calibration /2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) /2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)Temporal robustness of hydro models• Slight deterioration of performances on independent periods Models /2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)CC impact modelling chain p. 12 MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) /2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)Hydrological projections (evol. of mean flows in %) Low to medium increase of Qmean in near future except RCP2.6 Contrasted evolution in far future (see climate projections). GRSD behaves more extremely p. 16MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)Hydrological projections (evol. of high flows in %) Low to medium increase of high flows in near future  Contrasted evolution in far future (see climate projections). GRSD behaves more extremely p. 17

MOSARH21 24 /

 24 09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) Hydrological projections (evol. of low flows in %)  Low to medium decrease of low flows in near future, except RCP8.5 Contrasted evolution in far future. Poteltial very strong decrease for the most pessimistic scenarios p. 19 MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)

  Where the hydro projections from Explore 2070 and FLOW MS that different?

	Evol. of mean flows in % Evol. of High flows in % Evol. of low flows in %
	/!\ Beware, the Explore 2070 FLOW MS rather similar to RCP FLOW MS rather similar to
	future period is 2045-2065, not 8.5 CNRM and ADC projections
	2021-2050.
	Explore 2070 more uncertain, it Explore 2070 more pessimistic,
	FLOW MS rather similar to RCP is closer to far future of it is closer to far future of
	4.5 MOSARH 21 evolutions! MOSARH 21 evolutions!
	Explore 2070 differs, it is closer
	to far future of MOSARH 21
	evolutions!
	Prior studies Prior studies Prior studies	p. 20 p. 21 p. 22

MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr) MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)

Where the hydro projections from Explore 2070 and FLOW MS that different? MOSARH21 24/09/2020 / CHR meeting / Thirel Guillaume (guillaume.thirel@inrae.fr)

https://webgr.inrae.fr/en/mosarh21/