WHAT'S BEHIND

Standards and, more broadly, organizational rules are a key topic in the organizational literature. Since these are meant to be tools to improve the efficiency of individual and collective contributions to the achievement of an organization's goals, organizational scholars have been interested for a long time in the study of the process of their development as well as of their actual impact (see for example [START_REF] Selznick | TVA and the Grass Roots[END_REF][START_REF] Gouldner | Patterns of industrial bureaucracy[END_REF]. And indeed, analogies between some of their questionings and results of empirical studies -conducted in areas other than in the healthcare sector -and those of science studies on medical guidelines may be identified. First, the results of organizational studies contradict some assumptions (or threats) on the power of standards or organizational norms to make organizations function uniformly and they even underline that their adaptation to the specificities of every organization is a prerequisite for their successful diffusion [START_REF] Segrestin | L'entreprise à l'épreuve des normes de marché[END_REF]Brunsson & Jacobsson, 2000). This is compatible with well-known results about the incapacity of guidelines to hinder practice variations and about the necessity of local negotiations and adaptation processes to everyday care practices and institutionalized patterns [START_REF] Timmermans | Standardization in Action: Achieving Local Universality through Medical Protocols[END_REF][START_REF] Zuiderent-Jerak | Preventing Implementation: Exploring Interventions with Standardization in Healthcare[END_REF]. Second, organizational studies conclude that the introduction of standards does not always benefit the categories of actors who are in charge of the supervisory activities or, say, the regulators [START_REF] Segrestin | La normalisation de la qualité et l'évolution de la relation de production[END_REF][START_REF] Cochoy | Comment l'écrit travaille l'organisation: le cas des normes ISO 9000[END_REF][START_REF] Castel | Quand les normes de pratiques deviennent une ressource pour les médecins[END_REF]. This of course may echo back to empirical studies which underplay the risks of "deprofessionalization" through the development of medical standards [START_REF] Berg | Guidelines, Professionals and the Production of Objectivity: Standardization and the Professionalism of Insurance Medicine[END_REF][START_REF] Timmermans | Evidence-Based Medicine and the Reconfiguration of Medical Knowledge[END_REF][START_REF] George | From clinical counting to evidence-based medicine[END_REF]. Third, organizational-oriented studies [START_REF] Segrestin | L'entreprise à l'épreuve des normes de marché[END_REF]Brunsson & Jacobsson, 2000) as well as STS studies [START_REF] Berg | Rationalizing medical work. Decision support techniques and medical practices[END_REF][START_REF] Bourret | BRCA patients and clinical collectives[END_REF][START_REF] Cambrosio | Regulatory objectivity and the generation and management of evidence in medicine[END_REF] have analyzed standards as potentially useful coordinative devices. However, beside these convergences, the organizational approach and some of its key concepts may enlighten aspects of the dynamics of medical guidelines, which to my knowledge have received only scant attention thus far. Drawing from the seminal works by Weber, organizational scholars showed that collective action is problematic, notably because actors pursue their own and often non-convergent interests, and that power is a given dynamic in relations between actors placed in a context of strategic interdependence [START_REF] Emerson | Power-Dependence Relations[END_REF][START_REF] Crozier | The Bureaucratic Phenomenon[END_REF][START_REF] Crozier | Actors and systems: the politics of collective action[END_REF][START_REF] Pfeffer | and Row)[END_REF]. This approach leads to an emphasis on the 'strategic interactions among a set of actors placed in a given field of action and mutually dependent for the solution of some common "problem"' [START_REF] Crozier | Organizations and collective action[END_REF]. Taking the French oncology sector as a case study, I will show that guidelines are used strategically by individual physicians and groups of physicians. This framework has thus two heuristic consequences. First, it reveals another force propelling the development of guidelines. While some studies made convincing arguments about the rise of guidelines as a result of analyses by some medical researchers that the practices of physicians hardly correspond to medical science [START_REF] Berg | Turning a Practice Into a Science: Reconceptualizing Postwar Medical Practice[END_REF]Daly, 1995) or as one of the manifestations of a new type of objectivity, 'regulatory objectivity' [START_REF] Cambrosio | Regulatory objectivity and the generation and management of evidence in medicine[END_REF], in the medical domain, the case of French oncology shows evidence that the proliferation of medical guidelines is also the result of an attempt by some physicians to improve their position relative to competing groups. Second, it points to other kinds of changes which guidelines may involve, beyond medical practices and coordination: the evolution of the structure of power relationships inside the medical profession. This point reinforces the argument made by science studies that, paradoxically, standardization does not eliminate the diversity of practices among physicians, even if it may sometimes transform them.

Material and Methods

This paper is based on fieldwork undertaken since 1999 which consists of several case studies aimed at understanding some developments of the organization of cancer care in France, and in particular the trend to rationalize medical work. Public teaching hospitals were created in 1958. The Hospital Reform Act was passed to modernize the French hospital system by linking regional public hospitals to university medical schools (see Jamous & Peloille, 1970). Public teaching hospitals have become the keystone of the French healthcare system, since they are expected to offer the best and most advanced treatments, to train physicians and to conduct medical research for all pathologies. Last but not least, general hospitals and private clinics participate in cancer care by providing diagnostics and/or treatments in surgery, radiotherapy and/or chemotherapy. Fieldwork was carried out in six different settings: at the national level, within the FNCLCC, which is the main producer of cancer guidelines in France, and in five French administrative regions.

• Fieldwork at the national level (1999)(2000)(2001)(2002) focused on the genesis and the main goals assigned to the federal guidelines project. Data was obtained from daily informal exchanges and semi-directive interviews with the main federal actors in charge of the project and with directors. I also had access to working documents and the proceedings of meetings where the scope of the project was discussed, decided and (re)defined.

•

At the regional level, interviews were conducted with representatives of regulatory bodies, directors of hospitals and physicians. Nearly 300 semi-directive interviews have been conducted so far (1999)(2000)(2001)(2002)(2003)(2004). In a specific region, I was able for three years (2002)(2003)(2004) to observe collective decisions-making processes about how guidelines were negotiated and validated. I had also the opportunity to observe the therapeutic decision-making processes (N 450) of four medical staffs. It was an opportunity to observe the manner in which guidelines were referred to when physicians had to define a therapeutic strategy for actual patients.

A non-unified process led by professional organizations

A professional initiative

Early attempts to institutionalize the production and use of guidelines in the French healthcare system can be placed at the turn of the 1990s. In 1987, a national agency was created to develop medical evaluation, but its actual results had been rather modest. The 1991 Health Law underlined once again the importance of evaluation (including the evaluation of medical practices) in the healthcare sector, but this did not lead to any significant change either. In fact, the first national standards procedures occurred in 1993. These norms, however, aimed at forbidding dangerous and costly practices rather than promoting the best ones [START_REF] Kerleau | L'hétérogénéité des pratiques médicales, enjeu des politiques de maîtrise des dépenses de santé[END_REF]. Furthermore, cancer care was not influenced by these public guidelines. The first French recommendation regarding cancer was published in 1993 by a medical society (the French Society of Mammography and Breast Pathology) and dealt with quality criteria in mass screening for breast cancer -i.e. a diagnostic procedure. But from then on, cancer centers took the lead in developing cancer guidelines in France. In 1993, the board of the FNCLCC mandated some of their peers to launch a project aimed at 'harmonizing clinical practices between cancer centers, concerning diagnostic, classification, treatment and follow-up procedures' (Annual Report of the FNCLCC, 1994; author's translation). The selected methodology to elaborate these guidelines was initially a consensus conference. But a trip to the United States, where the project leaders visited some prestigious U.S. medical institutions, changed their mind: guidelines should rather be based on a critical review of medical literature in order to limit the impact of subjective opinions by the most reputed physicians and consequently to 'objectify' these guidelines. This project was thus explicitly conceived as a part of the emerging evidence-based medicine. From this point onwards, the scope of the project also changed its aim to not only defining the best practices for CLCCs but for the entire sector. Methodologists were recruited by the Federation. They were public health physicians who were trained to review the literature and to evaluate whether the final guidelines were congruent with this literature or not. They were meant to argue against physicians if the latter were prone to develop guidelines according to their own experience or first results of the latest clinical trials rather than to a systematic review of the literature.

In addition to the methodologists, the project's leaders planned from the start to involve as many cancer center physicians as possible in the elaboration of guidelines. From their point of view, this strategy presented some assets: first (and quite obviously), these physicians (about 1000 physicians) represented a skilled labor force; secondly, their participation was conceived as a way to train them to learn and practice evidence-based medicine and to facilitate the acceptation and appropriation of guidelines. By the end of 1994, 300 cancer center physicians had already participated either in the task groups (which elaborated the clinical practice guidelines) or in the feedback process (consisting of comments on the first versions of the guidelines). Six hundred physicians had taken part in the project by 1998. The first documents, called 'Standards, Options and Recommendations' (SOR), were produced in 1995, consisting of long monographs (around 200 pages) and of summaries published by the main French cancer journal Bulletin du cancer 1 .

Collaboration and competition

Beginning with the publication of the first SORs, the National Federation has involved more and more people working outside the CLCCs. At first, in 1995, an advisory board was created in addition to the executive committee, which was exclusively composed of 5 directors. In 1996, the executive committee welcomed one more CLCC physician, 2 representatives of medical societies (pediatric oncology and gynecologic oncology) and 4 physicians who practiced outside CLCC. The advisory board grew to 74 members. To finish, it was decided in 1997 to create a scientific board, exclusively composed of people working outside CLCC.

Until recently, more than half of physicians who have been involved in the elaboration of guidelines or in the feed-back process worked outside CLCC. Since 1998, the FNCLCC has published 29 clinical practice guidelines in collaboration with a medical society (see table 1). Despite this alliance with other actors or institutions, the FNCLCC still manages the process and keeps the label 'Standards, Options and Recommendations' and the (legal) responsibility for its guidelines. It is responsible for 81 out of 148 French clinical practice guidelines (54%) which have been published so far (see graph 1). Furthermore, FNCLCC's guidelines have been published in the British Journal of Cancer, one of the most reputed journals in Europe. FNCLCC is the only French organization whose cancer guidelines are diffused outside France. The SORs are approved by the State and medical audits conducted by the Social Insurance used them as reference. Other French producers of guidelines are the medical societies. They have published 46 sets of guidelines since 1997. Like the FNCLCC, their guidelines concern the management of specific cancers. Among them; the French Association of Urology (14 published guidelines -9%), the French Society of Hematology (9 guidelines) and the French Federation of Digestive Oncology (6 guidelines) are the most productive. Other medical societies published only 1 guideline during this period. Guidelines by the National Agency are of a different kind: they correspond either to recommendations for prevention and screening procedures, the evaluation of new techniques, or they answer precise questions (e.g. "Can initially non-resectable hepatic metastases be made resectable?").

FNCLCC with medical societies 21%

Agencies 16%

Medical Societies 30% FNCLCC alone 33%

Graph 1: French cancer guidelines' producers

Diversity of the implementation processes at the regional level

A recent study funded by the French National Cancer Institute confirmed the results of two previous surveys conducted by the FNCLCC: SOR guidelines are generally considered by physicians as the most reliable ones in oncology. However, among these physicians, those who worked in CLCCs and physicians who devoted more than half of their activity to cancer patients, were the ones who declared more familiarity with them and utilized them more frequently. Nonetheless, they are not considered as tools for everyday practice, except by a few practitioners who work inside teaching hospitals or cancer centers. Cancer centers thus created voluntary and cooperative networks for the elaboration and implementation of regional treatment protocols adapted to the characteristics and resources of the local healthcare system in order to be relevant and handy for everyday practice. This entails initiating and monitoring the discussion process at the regional level through which federal review monographs are in fact transformed into decision-making algorithms and into specific recommendations of one among several scientifically appropriate treatments. In some regions, the cancer center created a network without involving the regional teaching hospital. As a result, these teaching hospitals organized opposing networks. Besides these regional networks, some physicians who were specialized in the treatment of specific organs -and mainly urologists, gastroenterologists, visceral surgeon, lung specialists, and hematologists -declared that they used the guidelines of medical societies rather than the SORs.

Development of guidelines as a collective strategy for intra-professional legitimacy

The specific dynamics of this movement -its initial impulse and the diversity of involved organizations at the national and regional levels -is tributary to the specific organization of cancer care in France. In particular, the rivalry and competition between specialties (pathology-centered specialists vs. organ-centered specialists) and between hospitals influenced this development.

The reinstatement of cancer centers as dominant actors through the development of guidelines

The traditional collective identity of French Cancer Centers and the evolution of their position in the French oncology sector explain why their physicians first became involved in the development and diffusion of guidelines. Indeed, these guidelines have been considered as a tool to reinstate themselves as scientific leaders in the sector and as partners for other physicians. This project was the key-stone of an ambitious reform program conducted by the FNCLCC. This program, which consisted also of the development of research activities, an accreditation program and the negotiation of new professional statutes for employees, was a direct response to increasing competition for the CLCCs and to critiques by other stakeholders of the sector [START_REF] Castel | Institutional Change as an Interactive Process: the Modernization of the French Cancer Centers[END_REF].

In 1993, the instigator of the project was a pediatric oncologist, the director of a cancer center located in a very competitive region. His motives to develop guidelines were fourfold. First, the implementation of state-of-the-art practices in every hospital was a goal that he shared with his American peers (see [START_REF] Daly | Evidence-based medicine and the search for a science of clinical care[END_REF][START_REF] Berg | Turning a Practice Into a Science: Reconceptualizing Postwar Medical Practice[END_REF] and he could have been qualified, using Marks' expression, as a 'therapeutic reformer' [START_REF] Marks | The Progress of Experiment: Science and Therapeutic Reform in the United States, 1900-1990[END_REF]. Indeed, he could not abide by the heterogeneity among treatments which were prescribed by physicians in his center, since they could imply the unequal treatment of patients. He was all the more sensitive to this issue in that the international community of pediatric oncologists had already been involved for many years in the question of the harmonization of their practices [START_REF] Castel | La dimension politique de la rationalisation des pratiques médicales[END_REF]. Secondly, and furthermore, he analyzed that this heterogeneity was an obstacle to the regional legitimacy of his center. Indeed, because of this heterogeneity, physicians of other hospitals could put in doubt the quality of the advice that CLCC physicians offered them.

There have been two initial reasons to launch this project. The first one is an inner one. We, as pediatric oncologists, treated lymphoma with a specific protocol and we saved 75% of the children. In another service, physicians treated adults with lymphoma with another protocol and saved only 25% of patients. Thus, the poor patient was 25% likely to survive, if he was over 16, whereas children under 16 were 75% likely to survive. And I did not judge it justifiable that patients be treated differently from one service to another. (...) The second reason is an anecdote. Dr X [a head of service in a big public general hospital] called me once and told me: 'I called your center to ask if it was possible to prescribe birth-control pills to a woman with breast cancer and I had three different answers.' Then, he told me: 'As long as you won't be able to agree with each other -or at least to argue why you disagree -we will not consider you as the leading hospital.' (Interview with the director of the cancer center, 1999; author's translation)

Thirdly, this director hoped the development of guidelines could be used to reshape (and improve) the relationships between his center and other general hospitals. His center should appear to other hospitals as a facilitator to treat the more frequent cancers and as a scientific reference which could help them to treat more complex and rare cancers or to which they could even refer patients for enrolling into clinical trials or for specific care. Last but not least, he expressed in the same discourse that guidelines might be an argument to convince regulatory bodies to pay for new, expensive treatments 3 . Other directors were soon convinced of the strategic opportunities of this project for CLCCs. At the national level, they were well aware that it might contribute to re-establishing their legitimacy towards the public authorities and the French medical community. Indeed, at a time when concern for quality in the healthcare system was rising, CLCCs would be the hospitals who set the example. The SORs were preeminently seen by the directors as a way to set the rules in the oncology sector. Indeed, teaching hospitals were particularly threatening to them. From the 1980s on, teaching hospitals began to claim that cancer centers were not useful anymore since they had the same missions in the field of cancer-care. The former further argued that their physicians, specialized in organ treatments and at the forefront of clinical research, were more qualified to treat (and cure) cancer and to teach other physicians the best way to do it. In this context of struggle for legitimacy, guidelines were conceived by directors as a way to re-affirm the usefulness of CLCCs along-side these teaching hospitals. But they were also a tool to defend a medical approach and the organization of cancer care which CLCCs have developed [START_REF] Pinell | The birth of a scourge: the anti-cancer war in France 1890-1940[END_REF] and are very proud of: an organization based on the participation of every medical specialist in the decision-making process related to a patient's treatment strategy -also called a multidisciplinary approach -whereas teaching hospitals tended to promote another model, in which mastership of the initial decision is given to the organ specialist (and mainly to the surgeon). It therefore comes as no surprise that one of the first guidelines elaborated by the FNCLCC was entitled 'Standards, Options and Recommendations concerning good practices in the multidisciplinary organization of cancer care' [START_REF] Chardot | Standards, options and recommendations for the multidisciplinary organization of oncology[END_REF]. At the regional level, the directors' expectations were congruent with those of the project leader's: they intended to use the SORs to restore the leadership of their center and to improve its relationships with some general (private and public) hospitals. But they also had specific strategic uses of the SORs depending on the local organizational context. For instance, for some directors who had decided to enhance collaboration with the regional teaching hospital, the SORs helped to initiate a dialogue between their physicians and those who worked in the teaching hospital. In one region, the elaboration of a common language and common therapeutic attitudes subsequently allowed a new division of cancer care between these two hospitals to be established. In other regions, the directors hoped the diffusion of guidelines would facilitate the treatment of common cancers by general hospitals, allowing CLCC physicians to concentrate upon complex tasks. On the contrary, in some regions, the SORs were a tool to restrict the entrance of newcomers to the sector.

When the organization of cancer care was discussed at the regional level [with regulatory bodies and hospital representatives] two years ago, we had a clear idea about which hospitals we wanted to work with and with which we did not. We came to the discussions with the SORs, saying: 'These are the best practices in oncology and we cannot sell off oncology. Thus, only hospitals which are able to comply with the SORs may care for cancer patients.' If you refer to some rules which you do not have written down and which describe what has to be done, it defuses a bomb. Then, we succeeded in selling the ideas of a multidisciplinary organization of cancer care and a unique regional medical record… It is not all achieved, but… People have made headway. We may notice that some hospitals have given up the idea of developing oncology, others have grown bigger.

(Interview with a vice-director of a CLCC, 2001; author's translation)

Whatever the different uses of the SORs, they all have in common that these guidelines represented for CLCC directors a tool to reorganizing cancer care in their regions. This was a major motive for adopting the project.

Counter-reactions

As we noted above, public authorities delegated cancer guidelines to the FNCLCC and approved the SORs, thus reinstating CLCCs as legitimate organizations in the French oncology sector. Then, medical societies stepped in and became involved in the development of cancer guidelines. Some of them collaborated with the FNCLCC and others produced their own guidelines. These different attitudes were partially linked with the specific relationships that oncologists had developed previously with the different organ specialties.

The medical societies which have been the most prolific producers of guidelines are dominated by organ specialists who traditionally intend to master the care of the corresponding cancers at the expense of oncologists, considering the latter as merely "subcontractors". Indeed, urologists, gastroenterologists and hematologists perceived oncologists as competitors. For instance, urologists usually consider surgery to be the best treatment against localized and locally advanced prostate cancers, even if some studies concluded that surgery and radiotherapy were equivalent treatments in terms of overall survival. For them, radiotherapy must remain only an option when surgery is not possible. The respective place of these two treatments for prostate cancers remains a controversial issue [START_REF] Hakenberg | Treatment of locally advanced prostate cancer. The case for radical prostatectomy[END_REF]. In the same vein, gastroenterologists consider themselves to be more qualified than oncologists to prescribe drugs for digestive cancers since they better understand the working of the corresponding organs and thus are able to adapt protocols to the organs' reactions (and weaknesses). However, even if they more or less openly criticize the FNCLCC for trying to lead the development of guidelines at the national level, all these societies cannot help making reference to the SORs, thus attesting to their importance in the sector.

It is true that there have been conflicts between oncologists and urologists. Things are changing leading to fewer conflicts. But this remains very much "physician-dependent". (…) But there is still a problem: the FNCLCC still tends to think that cancer centers embody cancer care. We answer that they are only one actor amongst others. (…) The methodology that the French Association of Urology is using to produce guidelines is becoming more and more credible. We are trying to elaborate them according to the level of evidence, like the SORs. (Interview with a member of the board of the French Association of Urology, 2003; author's translation)

On the contrary, the first medical societies which accepted to participate in the SOR project happened to be the ones which were composed either of organ specialists who were used to working with oncologists and/or of many oncologists. In particular, since the 1970s, the French Society of Pediatric Oncology has been composed of physicians of cancer centers and teaching hospitals who have developed common research and treatment protocols [START_REF] Castel | La dimension politique de la rationalisation des pratiques médicales[END_REF]. Inside the French Society of Gynecologic Oncology, gynecologists and oncologists have collaborated for a long time, since breast cancer is one of the first cancers, of which treatment has consisted in a multidisciplinary approach -combining surgery, radiotherapy and medical treatment - [START_REF] Pickstone | Contested Cumulations: Configurations of Cancer Treatments through the Twentieth Century[END_REF]. However, even if the organization of cancer care influences the development and implementation of guidelines, it does not determine it. Specialists, once competitors, may learn how to work together around the development of guidelines. For instance, the French Society of Dermatology finally accepted to co-sign two guidelines with the FNCLCC, although they had been reluctant to for a long time. The French Federation of Digestive Oncology also joined the SOR project three years ago, after it had published 6 guidelines by itself. One hypothesis for this evolution relies on the institutionalization of the SOR project in the French oncology sector, so that it is difficult for an (isolated) medical society to continue this activity in parallel. Another hypothesis is the regulatory capacity of evidence-based medicine, entailing an inherent reflexivity among participants and resulting in the production of conventions and concerted programs of action [START_REF] Cambrosio | Regulatory objectivity and the generation and management of evidence in medicine[END_REF].

Individual authority and guidelines

In the French oncology sector, physicians tend to hold a good opinion of guidelines because of the professional issues these guidelines have addressed [START_REF] Castel | Quand les normes de pratiques deviennent une ressource pour les médecins[END_REF]. On the one hand, they are generally considered as a tool to lessen medical uncertainty, since they may attenuate the difficulty to make a therapeutic decision in a context characterized by an increasing number of randomized controlled trials and since an error may have lethal consequences. On the other hand, they limit the risk of a patient's and a manager's mistrust towards the quality of medical practices. Furthermore, the fact that these guidelines are exclusively elaborated by professionals (including those which have been approved by public authorities) and that they are not used by public authorities to punish physicians, facilitate this acceptation. But I will argue in this section that participation in the development and implementation of guidelines may also be the result of an individual strategy by physicians, in order to increase their specialization in cancer care and their visibility in the medical community, and thus, to have greater control of patients' trajectories in Strauss' sense (1985). Intra-professional legitimacy is also at stake for the individual practitioner, which will partially explain why some become involved in the elaboration and implementation of guidelines, and others do not.

The interests of cancer center physicians in the SOR project

The SOR project would probably have failed if the physicians who directed CLCCs had not convinced "rank-and-file" oncologists to participate. As we saw above, nearly two thirds of the medical community enrolled within a few years. The CLCC physicians we interviewed between 1999 and 2002 criticized neither the developmental process of the SORs nor the quality of the final documents. To the contrary, they generally expressed pride in their participation. For instance, in 1995, seven inter-CLCC scientific groups, among the most significant ones (gynecologic tumors, breast cancers, digestive cancers, genetic oncology, infectious diseases in oncology, statistics, radiology), stated their participation in the SORs prominently in their annual report. Three reasons may explain this success. The first is an institutional one. On the one hand, CLCC physicians were aware that the oncology sector was becoming increasingly competitive and, thus, increasingly threatening for them. They believed that the SORs could contribute to re-establishing cancer centers as leaders in the French oncology sector.

The SORs are a wonderful tool and only CLCCs are able to produce it -thanks to our multidisciplinary culture. We receive great credibility when we publish a SOR. (…) When we write such texts, we become incontestable in the sector. (Interview with a radiotherapist, 2000; author's translation)

On the other hand, since clinical research is tightly linked with the birth of oncology as a medical specialty [START_REF] Cambrosio | Rationalisation et médecine des preuves en oncologie : quelques remarques à propos des pratiques biomédicales (Commentaire)[END_REF][START_REF] Pickstone | Contested Cumulations: Configurations of Cancer Treatments through the Twentieth Century[END_REF], CLCC physicians were more prone to participate in a project that claimed its insertion in the Evidence-Based Medicine movement. Moreover, the traditional multidisciplinary organization inside CLCCs has facilitated the acceptation and the organization of this project by their physicians. They were more likely to accept a formalized (collective and multidisciplinary) regulation of their practices.

The guidelines are part of a global evolution, but this does not represent a revolution for us. For other hospitals, it may be a cultural revolution (…): the systematic reference to a document for therapeutic decision-making and the multidisciplinary organization of care are not innate; it requires a specific organization and structure. (Interview with a CLCC surgeon, 2002; author's translation)

The second reason for this success related to the management of this project, which was not intended to be a top-down project. On the one hand, the whole medical community was invited to get involved. It was not limited to opinion leaders. Furthermore, every guideline had to be validated by every CLCC medical committee (which is composed of all physicians who work inside the center) before it was published. On the other hand, reformers presented this project to the physicians as a way to improve their practices rather than as a coercive one to control them. Medical evaluations were intended to evaluate the conformity of medical practices with the SORs but no sanction was considered.

Thirdly, physicians consider that the SORs help them in their daily practice. On the one hand, their involvement in the development and dissemination process contributes to their hyper specialization (and authority) on certain cancers. Indeed, belonging to a national task force which is responsible for a SOR enhances or consolidates a physician's reputation in the treatment of the corresponding cancer (similar to publishing the results of randomized clinical trials in leading medical journals). In the same vein, the management of a task force at the regional level in order to adapt and implement the SORs contributes to the regional authority of this physician: not only can this physician make clear during discussions that he is wellversed in the literature and that he is aware of the latest clinical trials but, at the same time, he demonstrates to his regional colleagues that he is inclined to help them -and so exhibiting a kind of altruism.

[A medical leader] is someone who works a lot and who works well. We may evaluate this through his care for the patients we refer to him. But he is also someone who participates in task groups and in protocols. And he is someone who knows how to share his knowledge. The number of publications is one criterion but it is not the only one. What counts more is showing one's intention of working (…) and on the way one is trying to be acknowledged as a leader. There is also the human touch. (Interview with an oncologist in a general hospital, 2003; author's translation)

On the other hand, due to a significant increase in the number of articles related to randomized clinical trials at the turn of the 1990s, it was difficult even for the CLCC physicians to be well informed on current treatments for every cancer. Most of them became specialized in certain kinds of cancers. Thus, the SORs were consulted as a resource to find scientific information when it came to taking care of patients who suffered from cancers of which they were less familiar. In particular, cancer center physicians were interested in the SORs for their activity of outside counseling. Indeed, when they had to give advice to peers on cancers in which they are not specialized, the SORs constituted a precious help.

I am an oncologist, but I do not know 10% of all the decisions to take in oncology. We are hyper-specialized. For instance, I am specialized in prostate cancer. But when it departs from your hyper-specialization, it becomes very complicated. Guidelines will be referred to by physicians when the case departs from their specialty. (Interview with a radiotherapist, 2000; author's translation)

Opportunist uses of guidelines

Other physicians may also perceive the strategic interest of their participation in the regional implementation process. Organ specialists who want to specialize in oncology or oncologists who work in general public or private hospitals and who have thus to be trusted by surgeons may choose to participate in the regional implementation of guidelines in order to acquire local legitimacy. Indeed, physicians have to convince their colleagues who may refer patients to them, that they are competent and, in particular, that they are able to prescribe the correct treatments. There is no available study which compares overall survival rates among different hospitals in France. A doctor's reputation rests therefore on other (social) mechanisms. The participation in clinical research which may allow a physician to acquire a distinctive reputation through publications or communications is a frequently used strategy. But one may analyze the current development of oncology networks in France, which generally elaborate collegial guidelines and promote the creation of multidisciplinary (and sometimes multihospitals) medical staffs in charge of making therapeutic decisions, as another medical response to restore medical authority. Physicians participate in the regional elaboration of guidelines or in multidisciplinary medical staffs to increase their knowledge, to appear competent when talking with their peers and to acquire a kind of quality distinction.

A regional task group aiming at implementing the SORs allows physicians who work in general healthcare organizations to have privileged and personal access to the knowledge and experience of their academic 4 counterparts. Guidelines facilitate the sharing of specialized knowledge, so that "rank-and-file physicians" depend less on the personal advice of their illustrious colleagues for the most frequent types of cancers. We may then apply Segrestin's analysis of ISO norms (1996) to guidelines: indeed, the 'logic' of both kinds of norms consists of sharing knowledge which was only mastered by experts before. This participation is a resource for physicians in their efforts to gain greater control over the management of patient "trajectories" in both Strauss' senses [START_REF] Strauss | The Social Organization of Medical Work[END_REF]: trajectory of illness and organization of medical work. This allows them to intervene more in the work of their colleagues. First, they may acquire a greater legitimacy over physicians who have not specialized in oncology. Some find this label useful in convincing their patients and peers (general practitioners, surgeons, diagnostic physicians…), who may assign patients to them, that they are as competent as, say, academic physicians. Their authority relies then more on local, collective norms rather than on their individual competence. Indeed, some of their nonspecialized counterparts reported more trust (and thus refer more frequently) to those of their peers who participated in this collective process. We may thus understand why only oncologists and organ specialists who want to specialize in oncology participate in such networks. On the opposite, organ specialists (gastroenterologists, surgeons, etc.) for whom cancer patients do not represent a major part of their activity do not get involved in networks.

The network is useful to attest my competencies. I decided to adhere to it as soon as it began. (…) Surgeons, when they send me a patient, know that some regional standards exist. Thus, I am writing in the medical record: 'standard of [the regional network]'. It legitimizes my decision. They know that every patient is treated in the same way.

(Interview with a medical oncologist, 2003; author's translation)

Secondly, during task group discussions, they have the opportunity to hinder the academic physicians' monopoly over cancer care. The first social mechanism is negotiation, as Timmermans and Berg showed before ( 2003): they negotiate with academic physicians so that guidelines do not stipulate treatments that are too complex and that cannot be prescribed by general facilities (e.g. clinical trials, intensive chemotherapies, up-to-date diagnostic exams…). The second mechanism is trust: during discussions, they get a chance to convince their (specialized) colleagues that they are competent and may even demonstrate some expertise. Indeed, some academic physicians that were interviewed recognized that before this process, they tended to mistrust their colleagues who worked in general healthcare organizations, believing that they were not competent enough. Nowadays, they have changed their mind. As a result, academic physicians are more prone to let these physicians take care of patients who suffer "standard" cancers (such as colon cancer, breast cancer…) or even to re-assign those patients to them.

This allowed me to meet some physicians who worked in general hospitals and whom I had not known before. It helped me to avoid certain prejudices. In the medical world, it is terrible how fast we label people! 'The private sector does not think anything but money!' 'The public sector does not do anything at all!' But, when we came to know each other, we realized that people share a common interest: the patient! And we also realized that everybody worked hard. (Interview with an academic lung specialist, responsible for a regional task group, 2003; author's translation)

Last but not least, guidelines constitute a robust argument for physicians, so that they may try to restrict the autonomy of their peers and to influence the management of patients. For instance, during multidisciplinary staff meetings, physicians may argue that a therapeutic decision is not 'evidence-based' and thus try to change it, even if this patient is not under their responsibility. Physicians who want to propose a "non-conform" treatment have to justify it. Another example relied in the changing relationships between surgeons and radiotherapists in a region where protocols had been broadly diffused. Surgeons grumbled about the growing interference of radiotherapists in their autonomy. Indeed, radiotherapists dared to ask them to perform a new operation when surgical margins were insufficient related to the diffused guidelines. A last frequent example relied in the relationships between diagnostic specialists and clinicians: the latter were more prone to use guidelines to frame their colleagues' activity in terms of delay and quality of their exams reports.

[The regional protocols] make clear that I am not maniac or obsessional when I am asking them for some quality criteria: these criteria are written in the guidelines! For example, I must have exams' results in pathology or in radiology fifteen days later. As for pathologists, when we ask them to write some details, it is not because we are maniacs, it is only because it has to be that way. (Interview with a private surgeon, 2003; author's translation)

Of course, this does not mean that physicians are compelled to obey their colleagues if they rely on guidelines. Other types of authority still exist in the sector. For example, physicians may invoke the uniqueness of every therapeutic relationship to adapt a protocol. And authority of academic physicians may still rely on charisma or personal prestige. Indeed, some academic physicians refuse to participate in networks which implement guidelines. They would rather organize frequent meetings when they give personal advice on specific cases which are brought by non-specialized physicians. Nonetheless, our observations made clear that rational-legal authority [START_REF] Weber | Economy and Society[END_REF] is becoming dominant in the French oncology sector. Physicians who are able to justify therapeutic decisions and advice thanks to these guidelines and, more broadly, to the medical literature, are becoming more and more legitimate among their peers. Furthermore, guidelines facilitate an increasing 'lateral control regime' [START_REF] Lazega | Rule Enforcement Among Peers: A Lateral Control Regime[END_REF] among physicians. Contrary to previous periods when analysts noticed that physicians had limited opportunities to meet and to evaluate each other [START_REF] Freidson | Processes of Control in a Company of Equals[END_REF][START_REF] Crozier | The Bureaucratic Phenomenon[END_REF], guidelines represent a device which increases interdependency among physicians and a reference frame to judge one's colleagues.

Conclusion

Medical standardization may not only alter relationship dynamics between physicians and non-physicians (specifically patients, hospital managers and regulatory bodies), but it may also alter relationships between physicians themselves. In line with [START_REF] Timmermans | The Gold Standard. The Challenge of Evidence-Based Medicine and Standardization in Health Care[END_REF] and Cambrosio and colleagues (2006), this case study shows that standardization implies an increase of relationships among physicians (through task groups, for instance) and the development of shared conventions, which facilitate these inter-personal exchanges. As it is in the industrial sector [START_REF] Segrestin | L'entreprise à l'épreuve des normes de marché[END_REF][START_REF] Cochoy | Comment l'écrit travaille l'organisation: le cas des normes ISO 9000[END_REF]Brunsson & Jacobsson, 2000), standardization in the medical field allows for an improvement of relationships between actors through mechanisms of collective learning.

The study of standardization in the sector of French oncology shows also that it is heuristic to take political considerations into account when studying the development of standardized guidelines. This conclusion appears to be congruent with some of those of [START_REF] Timmermans | The Gold Standard. The Challenge of Evidence-Based Medicine and Standardization in Health Care[END_REF] who proposed "a study of the politics of standardization in practice". However, their conception of power differs from the one used in this paper. Although they did not explicitly set forth a definition of power, their analyzes and reference to The Birth of the Clinic by [START_REF] Foucault | The Birth of the Clinic: An Archaeology of Perception[END_REF] underlay a substantive and abstract conception of power. In the cases they studied, standards either embodied institutionalized power relationships (which impeded the establishment of standards, like in the "CPR" case) or they changed the dynamics of these relationships; however, the authors did not analyze why either happened as it did. By contrast, the sociology of organization, to which I refer, considers an interactive and strategic concept of power, which is a basic ingredient of exchange relations between actors placed in a context of strategic interdependence [START_REF] Crozier | The Bureaucratic Phenomenon[END_REF][START_REF] Chazel | Pouvoir[END_REF][START_REF] Boudon | Dictionnaire critique de la sociologie[END_REF]. Power can be conceptualized as the unequal and negotiated exchange of capacities of action in which every actor looks to simultaneously constrain other members of an organization in order to satisfy expectations and avoid being constrained by other members [START_REF] Crozier | Actors and systems: the politics of collective action[END_REF]Friedberg, 1997). Strategy and resources are, therefore, two key ideas related to this concept of power. In the case of French oncology, I established that standards could indeed be strategic resources used by professional actors at the expense of other professionals to improve their position in regards to the following two issues: 1) maintaining a sufficient volume of activity and 2) increasing control over therapeutic decisions. CLCCs and cancer center physicians were the ones who benefited the most from these standards. But, in this theoretical perspective, even actors placed in a favorable situation could not eliminate the capacity of other actors to resist. This explains why negotiations took place at every level of analysis: national, regional and local. The perspective of the sociology of organization, which places concrete exchange and bargaining relations at the core of its analysis and in which social control is continually challenged and (re)produced, helps to identify other reasons why, as well-illustrated by STS studies [START_REF] Timmermans | Standardization in Action: Achieving Local Universality through Medical Protocols[END_REF][START_REF] Zuiderent-Jerak | Preventing Implementation: Exploring Interventions with Standardization in Healthcare[END_REF], standardization does not prevent local specificities and variations but may even enhance them.

Notes

1 These guidelines are called 'Standards, Options and Recommendations' (SOR) since they are classified in three categories. A clinical attitude is called a 'Standard' when there is unanimity concerning its benefits, its inappropriateness or its potential danger. It is called an 'Option' when a majority of physicians agrees with the benefits, inappropriateness or dangers of a specific attitude. 'Recommendations' represent a choice by experts among different options. Each category is then explicated by a level of evidence, depending on the available scientific.

  Center for the Sociology of Organizations; CNRS; Paris Colloque : "The Institutions of Objectivity in Medicine: Informal and Formal Modalities of Regulation", McGill University (Montreal), 19-22 April 2007

  Four main types of healthcare organizations are in charge of cancer care in France. Created in the 1920s, French comprehensive cancer centers (Centres de Lutte Contre le Cancer; CLCCs) are 20 autonomous private hospitals publicly funded. Each one is directed by a physician. The CLCCs manage research activities and treat patients at the same time in their respective regions. The National Federation of CLCCs (FNCLCC), whose board is composed of the 20 directors, was created in 1964: conceived at first as an employers' association, its role has enlarged to comprise the facilitation of collaborative scientific projects between CLCCs.

Table 1 :

 1 Participation of medical societies in FNCLCC's guidelines

	Medical society	Number of guidelines
	French Dermatology Society	2
	French Society of Pediatric
	Oncology	6
	French Federation of Digestive
	Oncology	2
	French Society of Gynecologic
	Oncology	4
	Society for the Study and the
	treatment of Pain	5
	French Society of Pneumology	2
	Study Group of Lymphomas	2
	Others	2	6
			29

  Physicians who work in other hospitals want us to let them develop their business. (…) Our answer is the thesaurus [i.e. the treatments protocols], outside consultations (…); it is an openness which leads us to advise a treatment for patients we do not necessarily see. It is an evolution you know well. (Discourse in front of the medical Board of the cancer center, 1993; author's translation)

  At the time [i.e. in 1993, at the beginning of the project], the Social Security agency was about to elaborate guidelines… Then, what was our usefulness if, as oncologists, we were not able to produce cancer guidelines? We thus decided together: 'We will produce guidelines by ourselves and respect them, before we are forced to do so by someone else.' (…) Furthermore, our oncologists used to say that they were the best ones. They had to prove it [in producing guidelines]! (Interview with a director, 2002; author's translation)

  If someone wants to practice traditional surgery, [the work on guidelines] is not that important. But if someone wants to specialize in oncology, then… It is changing so fast! Nobody is able to attend all scientific meetings; nobody is able to read the whole published literature. (…) If someone wants to "hyperspecialize", he has to know up-todate treatments and nobody can do it on their own. (Interview with a private surgeon, 2003; author's translation)

I am grateful to Martha Zuber for her editing assistance.

This corresponds to medical societies who participated in only one guideline: French Sarcoma Group, Frenchspeaking Society of Enteral Nutrition, French Society of Pathology, French Association of Urology, Frenchspeaking Society of Brain-surgery, French Society of Nuclear

Medicine. 3 In the same vein,Berg and colleagues (2000) showed that guidelines had been a tool for insurance physicians in the Netherlands to restore the legitimacy of their practices, which were criticized by non-

physicians.4 By "academic physician", I mean physicians who work in hospitals whose missions are research and teaching, and thus teaching hospital and cancer center physicians.


Jean Avan

Nicolas Grosjean

Thierry Huillet

DID THE EVER DEAD OUTNUMBER THE LIVING AND WHEN? A BIRTH-AND-DEATH APPROACH

Keywords: population growth, Malthusian, constant vs time-dependent birth/death rates, time-inhomogeneous Markov chain, age-structured models

. We start developing simple deterministic Malthusian growth models of the problem (with birth and death rates either constant or time-dependent) before running into both linear birth and death Markov chain models and age-structured models.

Introduction

Starting from a question raised in (of all places) Huffington Post, asking whether the total number of dead people ever outgrew the total number of living people at a certain time, and when the crossover (if any) occurred, leads us to analyze some simplified models of population growth based on "mean-field" approximations, i.e. dealing with the notion of mean birth/death rates.

We shall successively consider a simple deterministic growth model for the whole population (Section 2); a linear birth/death Markov chain model for the growth of the whole population (Section 3); a deterministic age-structured model (Lotka-McKendrick-von Foerster) in Section 4, for which we shall explicitly (as far as possible) derive general solutions for the population sizes (in all 3 cases), the extinction probability, time to extinction and joint law for the number of people ever born and ever dead (for the Markovian process). Some explicitly solvable cases will be considered.

Simple birth and death models for population growth (deterministic)

We get first interested in the simple deterministic evolution of a population whose individuals are potentially immortal, non-interacting and living in a bath with unlimited resources.

Constant rates (Malthusian model).

Let the mean size of some population at time t = 0 be x (0) = x 0 and let x (t) be its mean size at time t.

Let (λ b , λ d ) be the birth and death rates per individual, assumed constant in the first place. With . = d/dt, consider the evolution [START_REF] Bharucha-Reid | Elements of the theory of Markov processes and their applications[END_REF] .

x (t) = (λ b -λ d ) x (t) =:

.

x b (t) -.

x d (t) , where .

x (t) is the growth rate at time t of the population, whereas .

x b (t) = λ b x (t) and .

x d (t) = λ d x (t) are the birth and death rates at time t of this population (the rates at which newborns and dead people are created, respectively).

Integrating, we have x (t) = x b (t) -x d (t), with x (0) = x b (0) . The quantities x b (t) and x d (t) are the mean number of individuals ever born and ever dead between times 0 and t and therefore x (t) is the current population size (x b (t) includes the initial individuals). Obviously, with λ := λ b -λ d x (t) = x (0) e λt and therefore (if

λ b = λ d ) x b (t) = x (0) + λ b λ b -λ d (x (t) -x (0)) x d (t) = λ d λ b -λ d (x (t) -x (0)) .
Under this model, given that some individual is alive at time t, its lifetime τ d obeys P (τ d > t + τ | τ d > t) = P (τ d > τ ) = e -λ d τ , independent of t (the memory-less property of the exponential distribution). The individuals are immortal in that there is no upper bound for their lifetime although very long lifetimes are unlikely to occur, i.e. with exponentially small probabilities. We assume the supercritical condition: λ = λ b -λ d > 0 so that x (t) grows exponentially fast. Then, as t → ∞ [START_REF] Cohen | Ergodic theorems in demography[END_REF] x b (t)

x (t) → λ b λ b -λ d and x d (t) x (t) → λ d λ b -λ d .
Remark: The cases λ b = λ d or λ b -λ d < 0 correspond to the critical and subcritical situations, respectively. In the first case, the population size x (t) remains constant (while x b (t) and x d (t) grow linearly) whereas in the second case, the population size goes to 0 exponentially fast. Unless otherwise specified we shall stick to the supercritical case in the sequel.

• Suppose we ask the questions:

-how many people have ever lived in the past and -do the ever dead outnumber the living and when?

The answer to the first question is x b (t).

The answer to the second question is positive iff

λ b > λ d and λ d λ b -λ d > 1 or else if λ d < λ b < 2λ d .
Then the time when the overshooting occurred is t * defined by

x d (t * ) x(t * ) = 1. Thus t * = - 1 λ log 2 - λ b λ d > 0,
which is independent of x (0). At time t * , the number of people who have ever lived in the past is thus

x b (t * ) = x (0) 1 + λ b λ b -λ d e λt * -1 = x (0) 2λ d 2λ d -λ b > x (0) .
Note also that the population size at t * is

x (t * ) = x (0) e λt * = x (0) λ d 2λ d -λ b > x (0) , so twice less than x b (t * ) consistently with x (t * ) = x b (t * ) -x d (t * ) and x d (t * ) = x (t * ) .
Suppose that at some (large) terminal time t f we know:

-x b (t f ) the number of people who ever lived before time t f .

-the initial population size x (0) at time t = 0.

-the current population size x (t f ) .

Then

λ = λ b -λ d = 1 t f log x (t f ) x (0) and λ b λ b -λ d ∼ x b (t f )
x (t f ) and both λ b and λ d are known. As a result,

t * = - 1 λ log 2 - λ b λ d ∼ t f log (1 + x (t f ) / (x b (t f ) -2x (t f ))) log (x (t f ) /x (0))
can be estimated from the known data.

Example:

The number of people who ever lived on earth is estimated to x b (t f ) = 105.10 9 . Suppose 10.000 years ago the initial population was of 10 6 units. Then x (0) = 10 6 and t f = 10 4 . The current world population is around x (t f ) = 7.10 9 . Then λ b ∼ 15.10 -3 , λ d ∼ 14.10 -3 and t * ∼ 75 years.

These rough estimates based on most simple Malthus growth models are very sensitive to the initial condition. ✷ 2.2. Time-dependent rates. Because the assumption of constant birth and death rates is not a realistic hypothesis, we now allow time-dependent rates, so both λ b and λ d are now assumed to depend on current time t. We shall assume that λ b (t) is bounded above by λ + b < ∞ and below by λ - b > 0 and similarly for λ d (t) with upper and lower bounds λ ± d . Unless specified otherwise, we assume supercriticality throughout, that is λ - b > λ - d . We shall also assume that both λ b (t) and λ d (t) are nonincreasing functions of t.

1 Let Λ b (t) = t 0 dsλ b (s) , Λ d (t) = t 0 dsλ d (s) be primitives of λ b (t) and λ d (t) .
1 Under this hypothesis, both the expected lifetime and the mean reproduction time of each individual increase as time passes by, reflecting an improvement of the general survival conditions.

Under this model, given some individual is alive at time t, its lifetime τ d obeys

P (τ d > t + τ | τ d > t) = e -(Λ d (t+τ )-Λ d (t)) so P (τ d ∈ dt | τ d > t) = λ d (t) dt.
Similarly, its reproduction time

τ b obeys P (τ b > t + τ | τ b > t) = e -(Λ b (t+τ )-Λ b (t)) so P (τ b ∈ dt | τ b > t) = λ b (t) dt.

With

• ≡ d/dt, we thus have to consider the new evolution

.

x (t) = (λ b (t) -λ d (t)) x (t) =: . x b (t) - . x d (t) ,
where .

x (t) is the growth rate at time t of the population, whereas .

x b (t) = λ b (t) x (t) and .

x d (t) = λ d (t) x (t) are again the birth and death rates at time t of this population.

We immediately have x (t) = x b (t)-x d (t), with x (0) = x b (0) . The quantities x b (t) and x d (t) are the mean number of individuals ever born and ever dead between times 0 and t (x b (t) including the initial individuals) and therefore x (t) is the current population size. Clearly, with Λ (t) = Λ b (t) -Λ d (t)

x (t) = x (0) e Λ(t)
and therefore

x b (t) = x (0) 1 + t 0 dsλ b (s) e Λ(s) x d (t) = x (0) t 0 dsλ d (s) e Λ(s) . Note x (t) = x (0) 1 + t 0 ds (λ b (s) -λ d (s)) e Λ(s)
We assume that Λ (t) is non-decreasing with t so that x (t) grows and also that, with

λ - b > λ - d > 0 λ b (t) → λ - b > 0 and λ d (t) → λ - d > 0 as t → ∞. Then, as t → ∞ (4) x b (t) x (t) = 1 + t 0 dsλ b (s) e Λ(s) 1 + t 0 ds (λ b (s) -λ d (s)) e Λ(s) → λ - b λ - b -λ - d and x d (t) x (t) = t 0 dsλ d (s) e Λ(s) 1 + t 0 ds (λ b (s) -λ d (s)) e Λ(s) → λ - d λ - b -λ - d . Indeed, by L'Hospital rule, if f (t) g(t) → t→∞ c ⇒ F (t) G(t) → t→∞ c where (F, G) are the primi- tives of (f, g) . Applying L'Hospital rule to f = λ b (t) e Λ(t) and g = (λ b (t) -λ d (t)) e Λ(t)
gives the result. Suppose

λ - b > λ - d . Whenever λ - d λ - b -λ - d > 1 or else λ - b < 2λ - d , there is a unique t * such that x d (t * )
x(t * ) = 1.

Remark: Assume λ - b > λ - d > 0.
We have the detailed balance equations:

d dt x b (t) x d (t) = λ b (t) -λ b (t) λ d (t) -λ d (t) x b (t) x d (t) , with x b (0) x d (0) = x b (0) 0 .
In vector form, this is also .

x (t) = B (t) x (t), x (0) where x (t) 0 (componentwise). The matrix B (t) has eigenvalues (0,

λ b (t) -λ d (t)) , with (0, λ b (t) -λ d (t)) → 0, λ - b -λ - d as t → ∞. λ 1 (t) = λ b (t) -λ d (t)
is its dominant eigenvalue. Letting X (t) be a 2 × 2 matrix with X (0) = I, and . X (t) = B (t) X (t) , we have x (t) = X (t) x (0) and

e -t 0 λ1(s)ds X (t) → xy ′ x ′ y as t → ∞, with B (∞) x = θx, y ′ B (∞) = θy ′ , θ = λ - b -λ - d := λ 1 (∞) the dominant eigenvalue of B (∞).
The left and right eigenvectors of B (∞) associated to θ are found to be x ′ = λ - b , λ - d and y ′ = (1, -1) . The latter convergence result holds because, with y ′ independent of t, y ′ B (t) = λ 1 (t) y ′ for all t > 0 and B (t) 1 = 0 (1 is in the kernel of B (t) for all t), see [START_REF] Cohen | Ergodic theorems in demography[END_REF]. We are thus led to

e -t 0 λ1(s)ds X (t) = e -Λ(t) X (t) → 1 λ - b -λ - d λ - b -λ - b λ - d -λ - d as t → ∞ and e -Λ(t) x (t) → x (0) λ - b -λ - d λ - b λ - d as t → ∞,
which is consistent with the previous analysis making use of L'Hospital rule, recalling

x (t) = x b (t) -x d (t) = x (0) e Λ(t) . Note 1 t t 0 λ 1 (s) ds → θ as t → ∞.
Examples: (d -c)

(i) (homographic
Λ (t) = (a -c) t + δ log (t + 1) and x (t) = x (0) (t + 1) δ e (a-c)t . Note δ ≷ 0, depending on b ≷ d. Thus x b (t) = x (0) 1 + t 0 ds (as + b) (s + 1) δ-1 e (a-c)s x d (t) = x (0) t 0 ds (cs + d) (s + 1) δ-1 e (a-c)s and x d (t) x(t)
→ t→∞ c a-c . When a > c, log x (t) /t → a -c > 0 and the population grows at exponential rate.

If in addition a < 2c, there is a unique t * such that x d (t * ) = x (t * ) :

t * 0 ds (cs + d) (s + 1) δ-1 e (a-c)s = (t * + 1) δ e (a-c)t * . (ii) Gompertz (a critical case λ - b = a = c = λ - d ):
We briefly illustrate here on an example that population growth under criticality also has a rich structure. Let

λ b (t) = a + (b -a) e -αt and λ d (t) = c + (d -c) e -αt . Assume α > 0, a = c and b -d > α. Then Λ b (t) = at + b -a α 1 -e -αt and Λ b (t) = ct + d -c α 1 -e -αt Λ (t) = Λ b (t) -Λ d (t) = b -d α 1 -e -αt . Because λ - b = a = c = λ - d , x d (t) and x (t) will never meet. We have x (t) = x (0) e Λ(t) = x (0) e (b-d)(1-e -αt )/α → x (0) e (b-d)/α as t → ∞ : due to b-d > α,
after an initial early time of exponential growth, the population size stabilizes to a limit. This model bears some resemblance with the time-homogeneous logistic (or Verhulst) growth model with alternation of fast and slow growth and an inflection point between the two regimes. ✷

Birth and death Markov chain (stochasticity)

The deterministic dynamics discussed for (x (t) , x b (t) , x d (t)) arise as the mean values of some continuous-time stochastic integral-valued birth and death Markov chain for (N (t) , N b (t) , N d (t)) which we would like now to discuss; N b (t) , N d (t) are now respectively the number of people who ever lived and died at time t to the origin, while

N (t) = N b (t) -N d (t)
is the number of people currently alive at t. Due to stochasticity, some new effects are expected to pop in, especially the possibility of extinction, even in the supercritical regime.

P (N (t + dt) = n + 1 | N (t) = n) = λ b (t) ndt + o (dt) P (N (t + dt) = n -1 | N (t) = n) = λ d (t) ndt + o (dt) , with state {0} therefore absorbing. Let φ t (z) = EE z N (t) | N (0) =: E z N (t) be the pgf of N (t) averaged over N (0) . Let φ 0 (z) = E z N (0) be the pgf of N (0) . Then, defining g (t, z) = (λ b (t) -λ d (t)) (z -1) + λ b (t) (z -1)
2 , φ t (z) obeys the PDE (see [START_REF] Bharucha-Reid | Elements of the theory of Markov processes and their applications[END_REF])

∂ t φ t (z) = g (t, z) ∂ z φ t (z) , with φ t=0 (z) = φ 0 (z)
or equivalently, solves the non-linear Bernoulli ODE problem

. ϕ t (z) = -g (t, ϕ t (z)) , with ϕ t=0 (z) = z and φ t (z) = φ 0 ϕ -1 t (z) . Setting ψ (t) = e Λ(t) and η (t) = t 0 dsλ b (s) e Λ(t)-Λ(s) , both going to ∞ as t → ∞, assuming λ b (t) → λ - b > 0 and λ d (t) → λ - d > 0 as t → ∞ and λ - b = λ - d , we easily get the solution (6) φ t (z) = φ 0 1 -(η (t) -ψ (t)) (z -1) 1 -η (t) (z -1) ,
where inside φ 0 (•) we recognize ϕ t (z) as an homographic pgf, solution to the ODE problem. We choose for initial condition a 'Bernoulli thinned' geometric random variable with homographic pgf 2 φ 0 (z) = q (q 0 + p 0 z) 1 -p (q 0 + p 0 z) = q + p 0 q (z -1) q -p 0 p (z -1) ;

this considerably simplifies [START_REF] Hoppensteadt | Mathematical methods of population biology[END_REF], since the composition of two homographic pgfs is again an homographic pgf. With this initial condition, we finally get

φ t (z) = q -(qη (t) -p 0 qψ (t)) (z -1) q -(qη (t) + p 0 pψ (t)) (z -1)
.

With x (0) = φ ′ 0 (1) = p 0 /q > 0 (with ′ the derivative with respect to z), averaging over N (0), we obtain

x (t) = E (N (t)) = EE (N (t) | N (0)) = φ ′ t (1) = φ ′ 0 (1) ψ (t) = x (0) e Λ(t) Var (N (t)) = φ ′′ t (1) + φ ′ t (1) -φ ′ t (1) 2 = x (0) 2 (2p -1) ψ (t) 2 + x (0) ψ (t) (1 + 2η (t)) .
Remark: For pgfs of the homographic form [START_REF] Hoppensteadt | Mathematical methods of population biology[END_REF], it is easy to see that, averaging over N (0), for n ≥ 1 3

P (N (t) = n) = EP (N (t) = n | N (0)) = [z n ] φ t (z) = C (t) n-1 (φ t (0) C (t) + D (t)) ,
where, with φ ′ t (0

) = P (N (t) = 1) = φ ′ 0 (0) ψ (t) (1 + η (t)) -2 C (t) := 1 - φ ′ t (0) 1 -φ t (0)
and

D (t) := φ ′ t (0) 1 -φ t (0)
-φ t (0) .

3.1.2.

The extinction probability and time till extinction. The extinction probability reads

P (N (t) = 0) = φ t (0) = 1 - p 0 ψ (t) q + qη (t) + p 0 pψ (t)
.

Because P (N (t) = 0) is also P (τ e ≤ t) where τ e is the extinction time (the event τ e = ∞ corresponds to non-extinction), we obtain

P (τ e > t) = p 0 ψ (t) q + qη (t) + p 0 pψ (t) . Whenever λ - b > λ - d > 0 (supercritical regime), η (t) /ψ (t) → κ > 1 (as t → ∞) because η (t) /ψ (t) = t 0 dsλ b (s) e -Λ(s) > t 0 dsλ (s) e -Λ(s) = 1 -e -Λ(t) → 1. 2 E z N(0) := φ 0 (z) = φ G (φ B (z)) is the composition of a geometric pgf φ G (z) = (qz) / (1 -pz) (p + q = 1) with a Bernoulli pgf φ B (z) = q 0 + p 0 z (p 0 + q 0 = 1). So N (0) d = G i=1 B i
where the B i s are iid Bernoulli, independent of G with E (N (0)) = p 0 /q and Var(N (0)) = p 0 (q 0 q + p 0 p) /q 2 = q 0 E (N (0)) + pE (N (0)) 2 .

3 [z n ] φ t (z) is the coefficient of z n in the series expansion of φ t (z) .

As it can be checked, were both λ b (t) and λ d (t) be identified with their limiting values λ - b and λ - d , then:

η (t) /ψ (t) → 1/ (1 -ρ) = λ - b / λ - b -λ - d where ρ := λ - d /λ - b (< 1 in the supercritical regime). Thus, with κ = 1/ (1 -ρ) > 1 (7) P (τ e > t) → P (τ e = ∞) = p 0 qκ + p 0 p as t → ∞.
The extinction probability reads

ρ e = P (τ e < ∞) = 1 -P (τ e = ∞) = q (κ -p 0 ) qκ + p 0 p = φ 0 (ρ) , N (t) vanishes at time τ e | τ e < ∞.
Note ρ e → 0 as x (0) → ∞ (or q → 0) and

φ t (z) → ρ e = φ 0 (ρ) as t → ∞.
We have

τ e = inf (t > 0 : N d (t) ≥ N b (t))
the first time the ever dead outnumber the ever living. In the randomized setup, even in the supercritical regime of mean exponential growth for

x (t) = x b (t)-x d (t),
there is a "no-luck" possibility that this event occurs and this corresponds to global extinction of the population at time τ e .

These results answer the somehow related question on if and when the ever dead (N d (t)) outnumber the ever living (N b (t)) in the supercritical linear birth and death Markov chain model; but so far the question on whether the ever dead (N d (t)) outnumber the living (N (t) = N b (t) -N d (t)) has not been addressed.

N (0) = N b (0) .
With n = n b -n d , the joint transition probabilities are obtained as

(8) P (N b (t + dt) = n b + 1, N d (t + dt) = n d | N (t) = n) = λ b (t) ndt + o (dt) P (N b (t + dt) = n b , N d (t + dt) = n d + 1 | N (t) = n) = λ d (t) ndt + o (dt)
allowing to derive the evolution equation of

EP (N b (t) = n b , N d (t) = n d | N (0)).
Introducing the joint pgf of N b (t) and

N d (t) as Φ t (z b , z d ) := E z N b (t) b z N d (t) d
, we obtain its time evolution as (9)

∂ t Φ t = (λ b (t) (z b -1) + λ d (t) (z d -1)) (z b ∂ z b -z d ∂ z d ) Φ t , with initial condition Φ 0 (z b , z d ) = φ 0 (z b ), noting N d (0) = 0. 3.2.2. Correlations. Consistently, we have x b (t) = ∂ z b Φ t (1, 1) and x d (t) = ∂ z d Φ t (1, 1) .
Taking the derivative of ( 9) with respect to z b and also with respect to z d and evaluating the results at z b = z d = 1 we are lead to

.

x b (t) = λ b (t) (x b (t) -x d (t)) = λ b (t) x (t), x b (0) = x (0) and to . x d (t) = λ d (t) x (t), x d (0) = 0. With E N b (t) 2 := ∂ 2 z b z b Φ t (1, 1) , E N d (t) 2 := ∂ 2 z d z d Φ t (1, 1) , E (N b (t) N d (t)) := ∂ 2 z b z d Φ t (1, 1)
we also have

d dt     E N b (t) 2 E N d (t) 2 E (N b (t) N d (t))     =   2λ b (t) 0 -2λ b (t) 0 -2λ d (t) 2λ d (t) λ d (t) -λ b (t) λ b (t) -λ d (t)       E N b (t) 2 E N d (t) 2 E (N b (t) N d (t))     , with initial condition     E N b (0) 2 E N d (0) 2 E (N b (0) N d (0))     =    E N (0) 2 0 0    .
In vector form, this is also .

x (t) = C (t) x (t), x (0) where x (t) 0 (componentwise).

The matrix C (t) tends to a limit C (∞) as t → ∞.

It has eigenvalues (0, λ b (t) -λ d (t) , 2 (λ b (t) -λ d (t))) , with (0, λ b (t) -λ d (t) , 2 (λ b (t) -λ d (t))) → 0, λ - b -λ - d , 2 λ - b -λ - d , as t → ∞. We identify now λ 1 (t) = 2 (λ b (t) -λ d (t))
as its dominant eigenvalue (the one for which lim t→∞ -2) . The latter convergence result holds because, with y ′ independent of t, y ′ C (t) = λ 1 (t) y ′ for all t > 0 and C (t) 1 = 0 (1 is in the kernel of C (t) for all t).

1 t t 0 λ (s) ds is largest). Letting X (t) be a 3 × 3 matrix with X (0) = I and . X (t) = C (t) X (t) , we have x (t) = X (t) x (0) and e -t 0 λ1(s)ds X (t) → xy ′ x ′ y as t → ∞, with C (∞) x = θx, y ′ C (∞) = θy ′ , θ = 2 λ - b -λ - d := λ 1 (∞)
Indeed,

X (t) = x (t) y ′ solves . X (t) = C (t) X (t)
and from Theorem 1 in [START_REF] Levinson | The asymptotic nature of solutions of linear systems of differential equations[END_REF], assuming 4

∞ . λ b (t) dt < ∞ and ∞ . λ d (t) dt < ∞, then x (t) ≃ xe t T λ1 (
s)ds (for large t ≫ T ). We are thus led to [START_REF] Levinson | The asymptotic nature of solutions of linear systems of differential equations[END_REF] e

-t 0 λ1(s)ds Cov (N b (t) , N d (t)) → C b,d -C b C d > 0 as t → ∞ 4
The homographic birth and death rates satisfy these conditions. More generally, the conditions hold for any non increasing rate function λ of the form λ (t) = a + (ba) P (T > t) where b > a > 0 and P (T > t) is the tail probability distribution of any positive random variable T.

where

C b,d = E N b (0) 2 λ - b λ - d / λ - b -λ - d 2 , C b = E (N b (0)) λ - b / λ - b -λ - d and C d = E (N b (0)) λ - d / λ - b -λ - d so C b,d -C b C d =Var(N b (0)) λ - b λ - d / λ - b -λ - d 2 >
0, translating an asymptotic positive correlation of (N b (t) , N d (t)) . Note

1 t t 0 λ 1 (s) ds → θ as t → ∞.
∂ t Φ t = λ b (t) ζξ 2 -(λ b (t) + λ d (t)) ξ + λ d (t) ζ ∂ ξ Φ t , Φ 0 (ξ, ζ) = φ 0 (ξζ) .
Φ t is therefore obtained as a function of a suitable combination of ξ and time (solvable by the method of characteristics), and separately of ζ in a way determined by initial conditions.

Fixing ζ to a certain value, one may then seek for an homographic ansatz

(11) Φ t (ξ, ζ) = Φ 0 a (t) ξ + b (t) c (t) ξ + d (t) , ζ = φ 0 a (t) ξ + b (t) c (t) ξ + d (t) ζ , with a (0) = d (0) = 1 and b (0) = c (0) = 0.
Plugging this ansatz into the evolution equation for Φ and denoting respectively

α (t) = λ b (t) ζ, β (t) = -(λ b (t) + λ d (t)) , γ (t) = λ d (t) ζ, we need to solve . ξ (t) = α (t) ξ (t) 2 + β (t) ξ (t) + γ (t) with initial condition ξ (0) = ξ
and the guess ξ (t) = a(t)ξ+b(t) c(t)ξ+d(t) . We get

(t) = ←-exp t 0 β/2 -α γ -β/2 (s) ds.
Although this system can rarely be solved explicitly by quadrature for all t, this in principle solves Φ t (ξ, ζ) hence Φ t (z b , z d ) when back to the original variables.

Remark: We note from the above differential equation [START_REF] Ohkubo | McGregor-like formula in a simple time-inhomogeneous birth-death process[END_REF]

(t) = ± 1 2 (λ b (t) + λ d (t)) 2 -4ζ 2 λ b (t) λ d (t). With A 1 = -1/2 -ζ 0 1/2 and A 2 = -1/2 0 ζ 1/2 , we have A (t) = λ b (t) A 1 + λ d (t) A 2 with [A 1 , A 2 ] = 0; In fact A 1 , A 2 generate the full su(2) algebra hence it is consistent to define (14) X (t) = exp (α 1 (t) L 1 + α 2 (t) L 2 + α 3 (t) L 3 ) ,
for some α i (t) obeying a Wei-Norman type non-linear differential equation [START_REF] Huillet | Minimal realizations of the matrix transition Lie group for bilinear control systems: explicit results[END_REF] and L i Pauli matrices. For recent further developments, see [START_REF] Ohkubo | McGregor-like formula in a simple time-inhomogeneous birth-death process[END_REF].

With i = 1, 2, let x i (t) = X (t) x i (0) with x 1 (0) ′ = (1, 0) ⇒ x 1 (t) ′ = (a, b) and x 2 (0) ′ = (0, 1) ⇒ x 2 (t) ′ = (c, d) .
We set

A (t) = A (∞) + A (t) A (∞) = λ - b A 1 + λ - d A 2 A (t) = λ b (t) -λ - b A 1 + λ d (t) -λ - d A 2 .
Of course, the matrix A (∞) has two real opposite eigenvalues

µ ± = ± 1 2 λ - b + λ - d 2 -4ζ 2 λ - b λ - d with µ ± (t) → µ ± as t → ∞.
From Theorem 1 in [START_REF] Levinson | The asymptotic nature of solutions of linear systems of differential equations[END_REF], assuming

∞ . λ b (t) dt < ∞ and ∞ .
λ d (t) dt < ∞, with T the (large) time after which the process enters its asymptotic regime, we have

(15) x i (t) ∼ exp t T µ + (s) ds φ + (1) φ + (1) φ + (2) φ + (2) x i (T ) , as t → ∞
where φ + is the right column eigenvector of A (∞) associated to µ + , i.e. A (∞)

φ + = µ + φ + . The row vector φ ′ + is φ ′ + := φ + (1) , φ + (2) = 1, µ + + λ - b + λ - d /2 / -ζλ - b . We conclude that, as t → ∞, both |x i (t)| → ∞. Recalling det(X) = a (t) d (t) - b (t) c (t) = |x 1 (t)| |x 2 (t)| sin x 1 (t) , x 2 (t) = 1
for all t, it follows that the angle x 1 (t) , x 2 (t) goes to 0 as t → ∞ at rate 2 t T µ + (s) ds > 0 : the vectors x 1 (t) , x 2 (t) become parallel (or anti-parallel) with the same time evolution ∼ exp t T µ + (s) ds which guarantees that they stay homothetical at large t ≫ T (as seen from the asymptotic evaluation (15)). Therefore both a (t) /c (t) and b (t) /d (t) tend to the same limit (homothetical ratio x 1 /x 2 ) as t → ∞, namely (a (T ) + b (T )) / (c (T ) + d (T )). This common limit exists and, from the above scaling argument [START_REF] Webb | Theory of Age Dependent Population Dynamics[END_REF], it can be written in the form say h ζ 2 /ζ, for some (unknown) function h encoding the finite size effects till time T ; it depends on ζ and ζ 2 . Note that, consistently, both a (t) /b (t) and c (t) /d (t) also have a common limit (angular azimut of both x 1 , x 2 ) which is

φ + (1) /φ + (2) = -ζλ - b / µ + + λ - b + λ - d /2 = u ζ 2 /ζ, where u ζ 2 = 2µ + -λ - b + λ - d / 2λ - d . So the limit structure of X (t) is known and as t → ∞ (16) Φ t (ξ, ζ) → φ 0 h ζ 2 .
The dependence on ξ has disappeared in the limit. Coming back to the original variables, recalling ξ ≡ z b /z d and ζ ≡ √ z b z d , as t → ∞ yields the limiting shape of Φ t (z b , z d ) . We get

Φ t (z b , z d ) = E z N b (t) b z N d (t) d → t→∞ E z N b (∞) b z N d (∞) d = φ 0 (h (z b z d )) .

3.2.5.

A particular solvable case. It turns out that in several case there is an explicit solution to [START_REF] Ohkubo | McGregor-like formula in a simple time-inhomogeneous birth-death process[END_REF], illustrating the general conclusions just discussed. Let us give an example.

• Explicit solution. Let ρ > 0 be a fixed number. When λ d (t) = ρλ b (t) , ( 12) can be solved explicitly while observing

β/2 -α γ -β/2 = λ b (t) -(1+ρ) 2 -ζ ζρ (1+ρ) 2
and diagonalizing the latter matrix. With Λ b (t) = t 0 λ b (s) ds and putting

θ ζ 2 = 1 2 (1 + ρ) 2 -4ρζ 2 ,
we indeed get ad -bc = 1 where

a = - (1 + ρ) 2θ ζ 2 sinh θ ζ 2 Λ b (t) + cosh θ ζ 2 Λ b (t) b = ζρ θ ζ 2 sinh θ ζ 2 Λ b (t) c = - ζ θ ζ 2 sinh θ ζ 2 Λ b (t) d = (1 + ρ) 2θ ζ 2 sinh θ ζ 2 Λ b (t) + cosh θ ζ 2 Λ b (t) . so that Φ t (ξ, ζ) = Φ 0 aξ + b cξ + d , ζ = φ 0 aξ + b cξ + d ζ .
Recalling ξζ = z b and ζ 2 = z b z d , coming back to the original variables, we get

Φ t (z b , z d ) = E z N b (t) b z N d (t) d = φ 0 A t (z b z d ) z b + B t (z b z d ) z b z d C t (z b z d ) z b + D t (z b z d )
where

A t (z b z d ) = - 1 + ρ 2θ (z b z d ) sinh (θ (z b z d ) Λ b (t)) + cosh (θ (z b z d ) Λ b (t)) B t (z b z d ) = ρ θ (z b z d ) sinh (θ (z b z d ) Λ b (t)) C t (z b z d ) = - 1 θ (z b z d ) sinh (θ (z b z d ) Λ b (t)) D t (z b z d ) = 1 + ρ 2θ (z b z d ) sinh (θ (z b z d ) Λ b (t)) + cosh (θ (z b z d ) Λ b (t))
each depend on t and z b z d .

• t → ∞ asymptotics.

This fully solvable case enables us to obtain exact asymptotics for the generating function Φ t (z b , z d ) in the supercritical case ρ < 1. From the explicit formulae for (a, b, c, d) we get:

lim t→∞ a (t) c (t) = 1 + ρ -2θ ζ 2 2ζ = 2ζρ 1 + ρ + 2θ ζ 2 = lim t→∞ b (t) d (t)
identified indeed by definition of θ ζ 2 . This illustrates the fact that the vectors x i (t), i = 1, 2 become parallel up to exp (-2Λ b (t)) when t gets large. Note that one still have of course ad

-bc = 1 ⇔ (ad) / (bc) -1 = 1/ (bc) = O (exp (-2Λ b (t))). Indeed, it appears that a/c -b/d = 1/ (cd) = O (exp (-2Λ b (t)
)) → 0 (without being strictly = 0). Hence we get that the generating function yields

Φ t (z b , z d ) → φ 0 1 + ρ -2θ 2 , θ = 1 2 (1 + ρ) 2 -4ρz b z d .
Dependence on ξ has disappeared. Observing that as t → ∞

(17) Φ t (z b , z d ) = E z N b (t) b z N d (t) d → ρ e E (z b z d ) N d (∞) | extinct ,
because either extinction occurred (with probability ρ e ) and

N b (∞) = N d (∞) < ∞ or not in which case N b (∞) = N d (∞) = ∞, we conclude (18) ρ e E z N d (∞) | extinct = φ 0   1 + ρ -(1 + ρ) 2 -4ρz 2   .
Recalling ρ e = φ 0 (ρ), we obtain the pgf of the cumulative population size (ever born or ever dead)

N b (∞) = N d (∞) < ∞,
given extinction occurred, as the compound expression

E z N d (∞) | extinct = φ 0 (h (z)) /φ 0 (h (1))
where

h (z) = 1 + ρ -(1 + ρ) 2 -4ρz 2 In particular (with h (1) = ρ = λ - d /λ - b < 1) E (N d (∞) | extinct) = φ ′ 0 (h (1)) φ 0 (h (1)) h ′ (1) = φ ′ 0 (ρ) φ 0 (ρ) ρ 1 -ρ .
Note that, would one consider a subcritical case ρ > 1, then h (1) = 1 leading to

ρ e = φ 0 (h (1)) = 1 (almost sure extinction) with E z N d (∞) = φ 0 (h (z)) char- acterizing the law of N b (∞) = N d (∞) < ∞ at extinction. Note E (N d (∞)) = E (N b (0)) ρ ρ-1 . In the critical case ρ = 1, h (1) = 1 (ρ e = 1) and E z N d (∞) = φ 0 1 - √ 1 -z with E (N d (∞)) = ∞.
Remark: This solvable case includes the situation where both λ b (t) and λ d (t) are independent of time. It suffices to particularize the latter formula while setting

Λ b (t) = λ b t. ✷ If z b = z, z d = z -1 , Φ t z, z -1 = E z N (t) = φ 0 A t (1) z + B t (1) C t (1) z + D t (1)
which is the previous homographic expression for E z N (t) in the particular case when λ d (t) = ρλ b (t).

If

z b = z, z d = 1 or z b = 1, z d = z, we get the marginals Φ t (z, 1) = E z N b (t) = φ 0 z A t (z) + B t (z) C t (z) z + D t (z) or Φ t (1, z) = E z N d (t) = φ 0 A t (z) + B t (z) z C t (z) + D t (z) .
The processes (N b (t) , N d (t)) both are monotone non-decreasing.

If z b = z -1 , z d = z 2 , we are interested in the Laurent series

(19) ψ t (z) := E z 2N d (t)-N b (t) = Φ t z -1 , z 2 = φ 0 A t (z) z -1 + B t (z) z C t (z) z -1 + D t (z) .
The quantity ψ t (z) = Φ t z -1 , z 2 turns out to be useful in the understanding of the statistical properties of the first time that ∆ (t) := 2N d (t) -N b (t) hits 0, starting from ∆ (0) = -N b (0) = -N (0) < 0. We now address this point.

3.2.6. First time N d (t) ≥ N (t) = N b (t) -N d (t) (∆ (t) ≥ 0). The process ∆ (t)
takes values in Z. It moves up by 2 units when a death event occurs and down by 1 unit when a birth event occurs. Starting from ∆ (0) = -N b (0) < 0, the process ∆ (t) will enter the nonnegative region ∆ (t) ≥ 0 for the first time at some random crossing time τ ∆(0) where the dependence on the initial condition has been emphasized.

There are two types of crossing events:

-type-1: ∆ (t) enters the nonnegative region ∆ (t) ≥ 0 as a result of ∆ (t) = -1 followed instantaneously by a death event leading to ∆ (t + ) = +1.

-type-2: ∆ (t) enters the nonnegative region ∆ (t) ≥ 0 as a result of ∆ (t) = -2 followed instantaneously by a death event leading to ∆ (t + ) = 0.

Consider the process ∆ 1 (t) which is ∆ (t) conditioned on t = 0 being a type-1 first crossing time (∆ (0) = -1 and ∆ (0 + ) = +1). Let λ 1 (t) be the rate at which some type-1 crossing occurs at t for ∆ 1 . We let

λ 1 (s) := ∞ 0 e -st λ 1 (t) dt
be the Laplace-Stieltjes transform (LST) of λ 1 (t) .

Let τ 1 denote the (random) time between 2 consecutive type-1 crossing times for ∆ 1 . By standard renewal arguments [START_REF] Feller | An introduction to probability theory and its applications[END_REF] (20)

λ 1 (s) = 1/ 1 -f 1 (s) ,
where f 1 (s) is the LST of the law of τ 1 .

Similarly, consider the process ∆ 2 (t) which is ∆ (t) conditioned on t = 0 being a type-2 first crossing time (∆ (0) = -2 and ∆ (0 + ) = 0). Let λ 2 (t) be the rate at which some type-2 crossing occurs at t for ∆ 2 . Let τ 2 denote the random time between 2 consecutive type-2 crossing times for ∆ 2 . Then λ 2 (s

) = 1/ 1 -f 2 (s)
where λ 1 (s) is the LST of λ 2 (t) and f 2 (s) the LST of the law of τ 2 .

Clearly, between any two consecutive crossings of any type, there can be none, one or more crossings of the other type.

The laws of the first return times (τ 1 , τ 2 ) of (∆ 1 , ∆ 2 ) are thus known once the rates (λ 1 (t) , λ 2 (t)) are.

Let now λ 1,∆(0) (t) (respectively λ 2,∆(0) (t)) be the rate at which some type-1 (respectively type-2) crossing occurs at t when ∆ is now simply conditioned on any ∆ (0) < 0.

Let also τ 1,∆(0) (respectively τ 2,∆(0) ) be the first time that some type-1 (respectively type-2) crossing occurs for ∆ given ∆ (0) < 0.

Again by renewal arguments for this now delayed renewal process:

(21) λ 1,∆(0) (s) = f 1,∆(0) (s) / 1 -f 1 (s) and λ 2,∆(0) (s) = f 2,∆(0) (s) / 1 -f 2 (s)
where λ 1,∆(0) (s) (respectively λ 2,∆(0) (s)) is the LST of λ 1,∆(0) (t) (respectively λ 2,∆(0) (t)) and f 1,∆(0) (s) (respectively f 2,∆(0) (s)) is the LST of the law of τ 1,∆(0) (respectively τ 2,∆(0) ). Finally, we clearly have

(22) τ ∆(0) = inf τ 1,∆(0) , τ 2,∆(0) .
The knowledge of all these quantities (in particular the law of τ ∆(0) ) is therefore conditioned on the knowledge of the rates. Conditioned on the initial condition ∆ (0) < 0, we have for instance

(23) λ 1,∆(0) (t) dt = z -1 ψ t (z) • λ d (t) • E ∆(0) (N d (t) + 1 | ∆ (t) = -1) dt λ 2,∆(0) (t) dt = z -2 ψ t (z) • λ d (t) • E ∆(0) (N d (t) + 2 | ∆ (t) = -2) dt,
corresponding respectively, through a jump of size 2 of ∆ (t), to the death events ∆ (t + dt) = 1 given ∆ (t) = -1 and ∆ (t + dt) = 0 given ∆ (t) = -2. In (23), ψ t (z) = Φ t z -1 , z 2 is given by (19).

The term z -1 ψ t (z) (respectively z -2 ψ t (z)) is the probability that ∆ (t) = -1 (respectively ∆ (t) = -2).

The term

E (N d (t) + 1 | ∆ (t) = -1) (respectively E (N d (t) + 2 | ∆ (t) = -2)) is the expected value of N (t) = N b (t) -N d (t) given ∆ (t) = -1 (respectively of N (t) given ∆ (t) = -2).
These last two informations can be extracted from the joint probability generating function of (N d (t) , ∆ (t

) := 2N d (t) -N b (t) | ∆ (0)) which is known from Φ t (z b , z d ) = E z N b (t) b z N d (t) d .
Therefore, the law of τ ∆(0) follows in principle, although the actual computational task left remains huge.

Age-structured models

We finally address a different point of view of the birth/death problems, once again deterministic but now based on age-structured models (see e.g. [START_REF] Charlesworth | Evolution in age-structured populations[END_REF], [START_REF] Cushing | An introduction to structured population dynamics[END_REF], [START_REF] Hoppensteadt | Mathematical methods of population biology[END_REF] and [START_REF] Webb | Theory of Age Dependent Population Dynamics[END_REF]).

Let ρ = ρ (a, t) be the density of individuals of age a at time t. Then ρ obeys (24)

∂ t ρ + ∂ a ρ = -λ d (a, t) ρ,
with boundary conditions ρ (a, 0) = ρ 0 (a) and ρ (0, t) = λ b (a, t) ρ (a, t) da.

The quantities λ b (a, t) and λ d (a, t) are the birth and death rates at age and time (a, t) and they are the inputs of the model together with ρ 0 (a) . The equation (24) with its boundary conditions constitute the Lotka-McKendrick-Von Foerster model, [START_REF] Mckendrick | Applications of mathematics to medical problems[END_REF].

The total population mean size at t is x (t) = ρ (a, t) da.

With

• ≡ d/dt, integrating (24) with respect to age and observing ρ (∞, t) = 0, we get .

x (t) = λ b (a, t) ρ (a, t) da -λ d (a, t) ρ (a, t) da =: . x b (t) - . x d (t)
where .

x b (t) and .

x d (t) are the birth and death rates at time t. We note that .

x b (t) = ρ (0, t), the rate at which new individuals (of age a = 0) are injected in the system.

Integrating, we have

x (t) = x b (t) -x d (t), with x (0) = x b (0) in view of λ d (a, 0) = 0.
x b (t) and x d (t) are the mean number of individuals ever born and ever dead between times 0 and t (x b (t) including, as before, the initial individuals). The answer to the question of whether and when the ever dead outnumbered the living is transferred to the existence of the time t * when some overshooting occurred, defined by x d (t * )

x(t * ) = 1. As in Section 2, this requires the computation of both x d (t) and x (t) and we now address this question.

With a ∧ t = min {a, t}, let

(25) Λ d (a, t) = a∧t 0 λ d (a -s, t -s) ds.
Integration of (24), using the method of characteristics yields its solution as

(26) ρ (a, t) = ρ 0 (a -t) e -Λ d (a,t) if t < a = . x b (t -a) e -Λ d (a,t) if t > a.
Recall ρ 0 (a) = ρ (a, 0) and .

x b (t) = ρ (0, t) as from the boundary conditions of (24). We shall distinguish 3 cases.

x b (t) = λ b (t) x (t) and . x d (t) = λ d (t) x (t) . Supposing x (0) = ρ 0 (a) da < ∞, we get x (t) = x (0) e Λ b (t)-Λ d (t) ,
where Λ b , Λ d are the primitives of λ b , λ d . Furthermore,

Λ d (a, t) = a∧t 0 λ d (t -s) ds = Λ d (t) if t < a = Λ d (t) -Λ d (t -a) if t > a.
We thus obtain

x b (t) = x (0) + t 0 λ b (s) x (s) ds = x (0) 1 + t 0 λ b (s) e Λ b (s)-Λ d (s) ds x d (t) = t 0 λ d (s) x (s) ds = x (0) t 0 λ d (s) e Λ b (s)-Λ d (s) ds
and for the age-structure

ρ (a, t) = ρ 0 (a -t) e -Λ d (t) if t < a = . x b (t -a) e -(Λ d (t)-Λ b (t-a)) if t > a, which is completely determined because . x b (t) = x (0) λ d (t) e Λ b (t)-Λ d (t) is.
Λ d (a, t) = a∧t 0 λ d (a -s) ds = Λ d (a) -Λ d (a -t) if t < a = Λ d (a) if t > a.
5 Not very realistic since it assumes that individuals may reproduce at any age and are potentially immortal eg no cutoff to λ b (a) for a < a crit or a > a crit , and λ d (a) does not increase with a.

6 which amounts to assuming that no modification of the demographical dynamics ever occurs.

We thus obtain

ρ (a, t) = ρ 0 (a -t) e -(Λ d (a)-Λ d (a-t)) if t < a = . x b (t -a) e -Λ d (a) if t > a
which is completely determined as well once .

x b (t) = ρ (0, t) is. We have Inverting the Laplace transform yields .

x b (t). The solution of ( 24) is completely determined from (26). Inverting this Laplace transform yields x (t).

4.3. The complete case. This is the realistic case when both λ b (a, t) and λ d (a, t) fully depend on (a, t) . Looking at .

x (t) = λ b (a, t) ρ (a, t) da -λ d (a, t) ρ (a, t) da =:

.

x b (t) -.

x So we need to solve the following inverse problem: find the unknown functions λ b (a, t) and λ d (a, t) such that λ b (a, t) ρ * (a, t) da = .

x b (t) and λ d (a, t) ρ * (a, t) da = .

x d (t) which is a linear problem (not in the convolution class). Then (x b (t) , x d (t) , x (t)) will constitute a solution consistent with (24) and the rates λ b (a, t) , λ d (a, t) .

We briefly sketch another point of view:

Suppose we were given λ d (a, t) and .

x b (t) = ρ (0, t) in the first place.

Then ρ (a, t) is completely known from Equations ( 25) and (26), together with x (t) = ρ (a, t) da and .

x d (t) = λ d (a, t) ρ (a, t) da. Also x b (t) is known from x b (0) = x (0) = ρ (a, 0) da. where the unknown function is λ b (a, t) . Suppose this inverse problem was solved. Then (x b (t) , x d (t) , x (t)) constitute a solution consistent with (24) and the rates λ b (a, t) , λ d (a, t) .

1 d and λ - 1 b

 11 Similarly, its reproduction time τ b obeys P (τ b > t + τ | τ b > t) = P (τ b > τ ) = e -λ b τ . After each reproduction time, each individual is replaced by two individuals by binary splitting. Then λ -are the mean lifetime and reproduction times.

  After each reproduction time, each individual is again replaced by two individuals, giving birth to a new individual by binary splitting.

  rates) With b > a > 0 and d > c > 0, assume λ b (t) = at + b t + 1 and λ d (t) = ct + d t + 1 Thus λ - b = a, λ + b = b and λ - d = c, λ + d = d. Then Λ b (t) = at + (b -a) log (t + 1) and Λ d (t) = ct + (d -c) log (t + 1) and with δ = (b -a) -

3. 1 .

 1 The current population size N (t).3.1.1. Probability generating function (pgf ) of N (t). Let N (t) ∈ {0, 1, 2, ...} with N (0) d ∼ π 0 for some given initial probability distribution π 0 . Define the transition matrix of a (linear) birth and death Markov chain[START_REF] Karlin | Linear growth birth and death processes[END_REF] as[START_REF] Feller | An introduction to probability theory and its applications[END_REF]

  the dominant eigenvalue of C (∞). The left and right eigenvectors of C (∞) associated to θ are found to be x ′ = λ - b 2 , λ - d 2 , λ - b λ - d and y ′ = (1, 1,

3. 2 . 3 .

 23 Joint pgf revisited. Introducing the change of variables from (z b , z d ) to ξ ≡ z b /z d and ζ ≡ √ z b z d , the right-hand-side of (9) only contains ∂ ξ . The evolution equation for Φ in the new variables indeed reads

.

  ad -bc) γ.Fixing the CP 3 gauge choice of any homographic form aξ+b cξ+d (invariant under a, b, c, d → λa, λb, λc, λd) by fixing ad -bc =

4. 1 .

 1 Case: λ b (a, t) and λ d (a, t) independent of age a 5 . Suppose λ b (a, t) = λ b (t) and λ d (a, t) = λ d (t) for some given time-dependent rate functions λ b (t) , λ d (t) .

  Then

.

4. 2 .

 2 Case: λ b (a, t) and λ d (a, t) independent of time t 6 . Suppose λ b (a, t) = λ b (a) and λ d (a, t) = λ d (a) for some given age-dependent rate functions λ b (a) , λ d (a) . Then

. 0 λ 0 e 0 e

 000 x b (t) = λ b (a) ρ (a, t) da= t b (a) . x b (t -a) e -Λ d (a) da + ∞ t λ b (a) ρ 0 (a -t) e -(Λ d (a)-Λ d (a-t)) da, the sum of two convolution terms. Taking the Laplace transforms, with α (z) := ∞ -za λ b (a) e -Λ d (a) da and β (z) := ∞ -za ρ 0 (a) e Λ d (a) da, we get the Laplace-transform of .x b (t) asx . b (z) = x . b (z) α (z) + β (z) α (z) , leading to x . b (z) = α (z) β (z) 1 -α (z) .

0 .

 0 We also havex (t) = ρ (a, t) da = t x b (t -a) e -Λ d (a) da + ∞ t ρ 0 (a -t) e -(Λ d (a)-Λ d (a-t)) dathe sum of two convolution terms. Taking the Laplace transforms, withα 0 (z) := ∞ 0 e -za e -Λ d (a) da, we get x (z) = x . b (z) α 0 (z) + β (z) α 0 (z) . So (27) x (z) = α 0 (z) β (z) 1 -α (z) = α 0 (z) α (z)x . b (z) .

  Now.x b (t) = λ b (a, t) ρ (a, t) da where now λ b (a, t) has to be determined. We thus need to solve the inverse problem. x b (t) = t 0 λ b (a, t) . x b (t -a) e -Λ d (a,t) da + ∞ t λ b (a, t) ρ 0 (a -t) e -Λ d (a,t) da,

  d (t) suggests to introduce the mean birth and death rates (averaging over age) Then, with x (t) = ρ (a, t) daWe are back to a case where λ * b , λ * d are independent of age although we know that there is an age dependency for both λ b (a, t) and λ d (a, t). (t) and x (t) are known from Subsection 4.1. Furthermore ρ (a, t) = ρ * (a, t) withρ * (a, t) = ρ 0 (a -t) e -Λ * d (t) if t < a = . x b (t -a) e -(Λ * d (t)-Λ * b (t-a)) if t > ais the known age-dependent density.

	So	. x (t) = (λ * b (t) -λ * d (t)) x (t) =: x (t) = x (0) e Λ * b (t)-Λ * d (t) ,
	where Λ * b , Λ * d are the primitives of λ * b , λ * d and
	. x b (t) = x (0) λ * b (t) e Λ * b (t)-Λ * d (t) ,	. x d (t) = x (0) λ * d (t) e Λ * b (t)-Λ * d (t) .
	Suppose we are given (λ * b (t) , λ * d (t)) in the first place. Then	. x b (t),
	λ * b (t) =	λ b (a, t) ρ (a, t) da ρ (a, t) da	and λ * d (t) =	λ d (a, t) ρ (a, t) da ρ (a, t) da	.

.

x b (t) -. x d (t) . . x d

Acknowledgments: T.H. acknowledges partial support from the labex MME-DII (Modèles Mathématiques et Économiques de la Dynamique, de l' Incertitude et des Intéractions), ANR11-LBX-0023-01.