J.-P Salmon
email: salmon@loria.fr

L Wendling
email: wendling@loria.fr

ARG based on arcs and segments to improve the symbol recognition by genetic algorithm

Keywords: graphic recognition, vector representation, inexact graph matching, technical drawings

In this paper a genetic matching scheme is extended to take into account primitive arcs and complex description in the pattern recognition process. Classical ways only focus on segments and are sensitive to over segmentation effect. Our approach allows to improve the recognition by handling more accurate description and also to decrease processing time by limiting the number of vertices to match. Experimental studies using real data attest the robustness of our approach.

Introduction

Symbol recognition is especially well suited to using structural pattern recognition techniques, unlike usual statistical classification techniques. When dealing with specific families of symbols, techniques similar to OCR could be used; this is the case for symbols having all a loop [START_REF] Okazaki | An Automatic Circuit Diagram Reader with Loop-Structure-Based Symbol Recognition[END_REF] or for music recognition. However these techniques have their own limitations, in terms of computational complexity and of discrimination power, and in the present case we have a family of symbols which is much too large and diverse for these techniques to be suitable. Actually, graph matching techniques are especially adapted to the specificities of symbol recognition [START_REF] Lladós | Symbol Recognition: Current Advances and Perspectives[END_REF]. Generally, methods match symbols to be recognized with model graphs using graph distance computations. In its basic principle, it is sensitive to errors and noise; as we usually cannot assume that segmentation is perfect nor reliable, this means that the graphs to be processed can also have a number of extra or missing vertices. To deal with this problem, errortolerant subgraph isomorphism algorithms have been proposed [START_REF] Bunke | Error Correcting Graph Matching: On the Influence of the Underlying Cost Function[END_REF][START_REF] Messmer | A New Algorithm for Error-Tolerant Subgraph Isomorphism Detection[END_REF]. Another problem with graph matching is the computational complexity of subgraph isomorphism methods. To deal with this, even if a lot of efforts have therefore been devoted to optimizing the matching process through continuous optimization [START_REF] Kuner | Efficient Techniques to Solve the Subgraph Isomorphism Problem for Pattern Recognition in Line Images[END_REF] or constraint propagation techniques to perform discrete [START_REF] Habacha | Structural Recognition of Disturbed Symbols Using Discrete Relaxation[END_REF][START_REF] Wilson | Structural Matching by Discrete Relaxation[END_REF] or probabilistic relaxation [START_REF] Christmas | Structural Matching in Computer Vision Using Probabilistic Relaxation[END_REF]. Arcs and circles are basic elements contained in technical drawings (engineering, electrical networks, mechanical...) and often relate to parts of symbols (for instance diode or logical nor representations) [START_REF] Kasturi | A system for interpretation of line drawings[END_REF]. Considering raster data, powerful operator dedicated to their recognition should be required for both analysis and understanding of documents [START_REF] Cordella | Symbol recognition in documents: a collection of techniques[END_REF][START_REF] Messmer | Automatic learning and recognition of graphical symbols in engineering drawings[END_REF][START_REF] Tombre | Interpretation of engineering drawings[END_REF]. In the early years, many approaches have been developed (or extended) to extract such primitives [START_REF] Song | Effective multiresolution arc segmentation: Algorithms and performance evaluation[END_REF][START_REF] Wenyin | Extended summary of the arc segmentation contest[END_REF] as arcfitting methods [START_REF] Rosin | Segmentation of edges into lines and arcs[END_REF][START_REF] Elliman | An algorithm for arc segmentation in engineering drawings[END_REF], using Hough transform [START_REF] Conker | A dual plane variation of the hough transform for detecting nonconcentric circles of different radius[END_REF][START_REF] Leavers | The dynamic generalized hough transform: Its relationship to the probabilistic hough transforms and an application to the concurrent detection of circles and ellipses[END_REF] and stepwise arc extension methods [START_REF] Dori | Vector-based arc segmentation in the machine drawing understanding system environment[END_REF][START_REF] Liu | Incremental arc segmentation algorithm and its evaluation[END_REF][START_REF] Song | Effective multiresolution arc segmentation: Algorithms and performance evaluation[END_REF]. We can also denote that several methods were embedded in comparative studies using GREC contest [START_REF]GREC'01 contest. Proceedings of 4th IAPR International Workshop on Graphics Recognition[END_REF][START_REF]GREC'03 contest. Proceedings of 5th IAPR International Workshop on Graphics Recognition[END_REF]. For instance the last edition has shown than RANVEC method [START_REF] Hilaire | Ranvec and the arc segmentation contest: Second presentation[END_REF] supersedes most of them in many cases. This approach has been used here to ensure an accurate description from raster image.

In this paper we propose to integrate primitive arcs and segments in Attributed Relational Graph. Second step consists in matching such ARGS with a genetic algorithm to improve the recognition while keeping a faster processing.

Graph Description

ARG Structure

The vertices of our Attributed Relational Graph represent the graphic pritimives and the edges characterize the topological relations between them. First, we consider the contour representation provides in [START_REF] Khoo | Evaluation of genetic operators and solution representations for shape recognition by genetic algorithm[END_REF] to handle with filled symbols (see figure Fig. 1). In the figure Fig. 1 the associated planar graph is entirely connected, that is each vertex is linked to two others. The variability of symbols encountered in graphic documents, requires to extend such process to take into account symbols split into strings of points. Obviously vertex constraints are weaker: a vertex can be connected at least to one other (see 2).

This kind of representation does not allow to really take into account symbols composed of several connected components (see fig. Fig. 3).

The noise presents can remove some jonctions in image corresponding to remove some edges in the ARG. To fix this problem, lookup areas has been defined around ends of graphical primitives to locate the nearest graphical pritimives. In this way, if a segment is isolated from the others then a particular relation is created to link it. The lookup areas is spread-out in order to connected all the graphical primitives. Thus, all the vertices of the ARG are connected at less with an other. This graph provides a more accurate description of symbols. Nevertheless symbols may have arcs in their composition. Generally they are approximated by a set of segments depending on the scale ( Fig. 4). The ability to handle with arcs has been studied to improve the accuracy during the recognition step.

Attributes of ARG (features of arcs and segments)

A set of sharing features is required in order to be able to compare arcs and segments when vectorization errors remain. Unary features are usually process within the recognition step to characterize graphic primitives, as the length, the sum of internal angles in relation to adjacent primitives and the distance between the middle of primitive and the middle of the associated chord (equal to 0 in case of segment) (see Fig. 5). Binary relations between primitives are also used as angle between two primitives, distance between middle of primitives, angles between farer points of primitives (see Fig. 6). Determining an angle between a segment and an arc has no sense. So we consider the angle between the segment and an adjoining segment of the arc (see Fig. 7).

All these features are rotation and translation invariants. The acute angle between two primitives is invariant to scale factor as well as the sum of internal angles. Li [START_REF] Li | Matching: Invariant to Translations, Rotations and Scale Changes[END_REF] proposes to integrate a logarithm function to ensure scale factor invariance of sizes and distances. Other ways focus on normalized height and distance following maximal values computing on the shape under consideration. Nevertheless these approaches are sensitive to segmentation errors. Here the fitness function, proposed Khoo and Suganthan [START_REF] Khoo | Evaluation of genetic operators and solution representations for shape recognition by genetic algorithm[END_REF], is adapted to efficiently take into account the scale factor. Such description allows to efficiently differentiate the shapes provided figure Fig. 8 while preserving a suitable recognition despite unexpected vectorization errors.

Genetic Algorithm

In previous part, we presented a description composed of segments and arcs, which is more robust and accurate to describe complex symbols, especially for scale sensitive symbols. Such description is structured into an ARG.

Nevertheless considering real data, noises due to the scan can disturb original signal while adding unexpected artifacts and defomations into the ARG representation. Therefore, the recognition method should be able to match the sub-graphs and to be error-tolerant. That the reason why, we chose the genetic algorithm.

Fitness function

For any feature of the vertex, the matching cost is calculated as follows:

C i,fi,u = |a fi,u -a i,u | (1)
with f i the vertex matched with the vertex i following the feature u. Let us consider a vertex i, the global cost A i,fi is calculated as follows:

A i,fi =      U u=1 tanh(k u (t u -C i,fi,u )), f i = N ull 0 Otherwise (2)
with U the number of features. This sum relies on the global matching cost obtained from vertex i. The use of function tanh allows to normalize values between -1 and 1 and to act on the importance of each feature. The parameter k relies on the slope of tanh and t is a threshold assuming the matching cost is too high to consider symbols are similar. We introduce a close process to handle with edges (see eq. 3 and 4 next page) with i, j adjacent vertices and b an edge feature.

C i,j,f i ,f j ,b = |a f i ,f j ,b -a i,j,b | (3)
D i,j,f i ,f j = B b=1 tanh(k b (t b -C i,j,f i ,f j ,b )), fi = N ull, fj = N ull, 0 Otherwise (4)
The fitness function became:

F = S i=1 A i,fi + λ S i=1 S j=i+1 D i,j,fi,fj (5)
with S the number of the vertices. A value of fitness greater than 0 implies vertices are matched, equal to zero relies on unknown information (NULL) and lower than 0 implies a bad match. Parameter λ weights the importance of vertex features with respect to edge features during the initialization step.

Matching

In our application a gene is associated to the matching between the vertex of a scene and those of the model. Strings of genes are randomly set in the initialization step. More the scene is complex (following the number of connected components and the number of primitives)and more the string is long. Following the generation, crossovers are carried out with the strings and new genes are randomly added. The crossovers consist in randomly swapping genes between two strings. When the number of generations is completed, only genes having the best quality according to the fitness function are kept. A number of generations is preset to guarantee an acceptable response time. A way to provide optimal values of the algorithm parameters is proposed in experimental section § 4.

We have studied two approaches of matching. First, we compared a scene to each model, one by one, using the genetic algorithm. Also, for each model, a similarity value and the best matching (vertex to vertex),is obtained. The algorithm returns the model having the greatest similarity value.

In a second way, one scene is compared to all the models in a unique step and the most similar parts of models are kept. Also, for each vertex of the scene, the vertex of the model having the greatest similarity value is selected. Such method is interesting to detect parts (or symbols) included in the composition of other symbols (see figure Fig. 9). Obviously this method relies on the number of symbols contained in the database and is sensitive to scale. Nonetheless the method is able to process these two cases. Let us consider a set of learning samples.The study leaded to find optimal values for parameters k and t has been performed as follows: Let us consider p discrete values for k and n queries. We keep the value of k which maximizes the percentage of right matching. Then the value of t is determined in a same way from k. For example the figure Fig. 10 shows the recognition rates obtained for k and t using n = 1000 queries. Optimal values should be set a 0.1 for k and 5 for t.

Variability of the results

The accuracy of the method relies on the number of generations. The influence of such parameter was studied to improve the robustness of our approach by refining the stopping criterion of the basic algorithm. For instance let us consider n queries (for example n = 1000 see figure Fig. 11.a), we can remark a minimal of generations allowing to achieve a maximal recognition rate (here 300).

It is well know that when one uses stochastic optimization algorithms results obtained in a single experiment do not guarantee repeatability and are not significant. That the reason why, we have also determine how many queries are enough to keep a stable recognition rate by calculated a cumulated mean as follows: rate(n) = ((n -1) * rate(n -1) + δ(n)) n with: rate(n) the mean recognition rate after n queries and δ(n) is equal to 1 if the query n is successful,0 otherwise. Figure Fig. 11.b shows that recognition rates are relatively stable about 100 queries.

Experimental study

In this sub-section, all recognition rates are computed with the results of previous studies.

First we have tested our approach with symbols made of several strings of points using GREC database (distortion 2). We have considered 7 classes merged with 8 other classes in the dataset of 74 symbols (following the class, the number of images is different) to show the robustness of the approach. Figure To show the interest of handling with primitive arc, another test has been performed using a database composed of 9 classes of 11 symbols by class having arcs. A random model of deformation has been applied to the extremities of the primitives (see figure Fig. 13). The basic approach [START_REF] Khoo | Evaluation of genetic operators and solution representations for shape recognition by genetic algorithm[END_REF] gives rise to a mean recognition rate of 52.1% (with classical vectorization in a set of segments). Handling with arcs in the calculation of fitness allows to reach a mean recognition rate of 90.4% (see figure Fig. 14).

Conclusion and future work

In this paper, we introduced an Attributed Relational Graph able to describe the symbols composed of several strings of points and several connected components. Likewise, we used the segments and the arcs to describe more accurately the symbols and also to avoid the mistakes of matching. Common invariant features to geometric transformations have been proposed to take into account both primitives in the ARG. We demonstrated than a genetic algorithm based on this ARG provides better results. In our future work, we want to take into account more complexe graphical primtives.

Fig. 1 .

 1 Fig. 1. A contour vector and its graph.

Fig. 2 .

 2 Fig. 2. A Symbol composed of several strings and its associated graph.

Fig. 3 .

 3 Fig. 3. A symbol composed of 3 connected components and its associated graph.

Fig. 4 .

 4 Fig. 4. Problem of vectorization : increase in the number of segments due to scale factor.

Fig. 5 .

 5 Fig. 5. Common vertex features: length, sum of internal angles, distance between the primtives and its chord.

Fig. 6 .

 6 Fig. 6. Edge features: the angle, distance between center points, the maximum distance between end points

Fig. 7 .

 7 Fig. 7. Angle between a segment and a adjoining segment of an arc.

Fig. 8 .

 8 Fig. 8. Similar description with ambiguities between an arc and two unexpected segments.

Fig. 9 .

 9 Fig. 9. a) scene composed from 2 models b) first model (5 matched vertices) c) second model (6 matched vertices).

4 Experimental study 4 . 1

 41 Parameters k and t setup

Fig. 10 .

 10 Fig. 10. Recognition rates following t and k.

Fig. 11 .

 11 Fig. 11. Recognition rates following the number of generation (left) and cumulated recognition rates (right).

Fig. 12 .

 12 Fig. 12. Recognition rates.

Fig. 13 .

 13 Fig. 13. Examples of distorted symbols.

Fig. 14 .

 14 Fig. 14. Recognition rates obtained.


Marie Puren

Charles Riondet

Laurent Romary

Dorian Seillier

Lionel Tadjou

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

A standardized data model

The scenarios are described using the TEI format (Text Encoding Initiative). All the information displayed within the SSK proceed from TEI files :

• A scenario is a list of events (<tei:listEvents>) • Each step in a scenario is an event (<tei:event>)

• TEI files are all available on GitHub (https:// github.com/ParthenosWP4/SSK) • The documentation is also available (https:// parthenoswp4.github.io/SSK/SSKspec.html)

THE SSK NEEDS YOU !

Start contributing to the SSK by adding new scenarios on GitHub and soon directly on the interface.

Make your Arts and Humanities research go standard.

SSK by example

Marie Puren, Charles Riondet, Laurent Romary, Dorian Seillier, Lionel Tadjou

  https://ssk-application.parthenos.d4science.org/ssk/#/ ssk@inria.fr The Standardization Survival Kit The « research scenarios » The Standardization Survival Kit is a collection of research use cases (the « scenarios ») illustrating digital best practices in Digital Humanities and Heritage research. Gain efficiency, interoperability and sustainability by using standards ! Guidelines and tools are easier to understand and use when presented through examples. The SSK provides a variety of standardized procedures and resources in a meaningful context. The research scenarios help you identify the most useful material to help you carry out your research project, step by step. Browse research scenarios corresponding to your research project, follow the steps and find the most fitted resources & standards ! PARTHENOS is funded by Hoziron2020 of the European Commission. The project started on 1 May 2015 and runs for 48 months.


Walid Djema

Catherine Bonnet

Frédéric Mazenc

Jean Clairambault

Introducing Cell-Plasticity Mechanisms into a Class of Cell Population Dynamical Systems

Keywords: Delay, Nonlinear, Cellular plasticity

Cells have amazing features that allow them to guide their development paths and determine their individual and collective fates. Dedifferentiation and transdifferentiation (cell plasticity) are littleunderstood phenomena that allow cells to regress from an advanced differentiated state to a less differentiated one, including the case where cells lose their specific function and become stem cells. In this paper, we introduce cell plasticity into a class of mathematical models we are interested in. We explore a new model involving a dedifferentiation function in the case of two cell maturity stages (stem cells and progeny). We highlight the role that dedifferentiation may have in the survival of cancer cells during therapy. The latter hypothesis appears to be in line with some medical observations.

I. Introduction

1) Stem cells (SCs):

SCs are undifferentiated cells characterized by their ability to self-renew and their multipotency, which is the ability to differentiate into more specialized cells. A SC that engages in the division process (the cell cycle) undergoes a series of transformations until becoming two daughter cells. Most of the cells are in a non-proliferative phase, known as G 0 or resting phase. A resting cell prepares its entrance into the cell cycle, which consists of four consecutive discrete subphases G 1 , S, G 2 and M . The M phase is mitosis (nuclear division) that ends with cytokinesis, which is the final step of production of two daughter cells ( [START_REF] Cooper | The Cell: A Molecular Approach[END_REF], Chap. 17). In view of the broad range of options available for SCs, including: self-renewal, differentiation, lineage specification, programmed cell death (apoptosis), and quiescence G 0 , determining the fate of a given SC becomes a key challenge [START_REF] Enver | Stem cell states, fates, and the rules of attraction[END_REF]. In particular, it is agreed that a better understanding of the processes guiding the fates of SCs may elucidate the causes of cancer and the rise of Cancer Stem Cells (CSCs) (see, e.g., [START_REF] Reya | Stem cells, cancer, and cancer stem cells[END_REF], [START_REF] Enver | Stem cell states, fates, and the rules of attraction[END_REF]).

The mathematical modeling and analysis become crucial in the query of understanding cancer [START_REF] Murray | Mathematical Biology: I: An Introduction[END_REF]. In general, the ultimate goal behind theoretical studies of biological systems is to firmly grasp their healthy or unhealthy Walid Djema* corresponding author, walid.djema@inria.fr.

Walid Djema, Catherine Bonnet and Frédéric Mazenc are with Inria, University Paris-Saclay, L2S (CNRS), CentraleSupélec, Gifsur-Yvette, France. Jean Clairambault is with Inria, University of Sorbonne, UPMC, Laboratory Jacques-Louis Lions, Paris, France. walid.djema@inria.fr, frederic.mazenc@inria.fr catherine.bonnet@inria.fr, jean.clairambault@inria.fr behaviors [START_REF] Mackey | What has mathematics done for biology?[END_REF], including mechanisms governing cell population dynamics (as for blood cell dynamics [START_REF] Apostu | Understanding cyclical thrombocytopenia: A mathematical modeling approach[END_REF], [START_REF] Dale | Understanding, treating and avoiding hematological disease: Better medicine through mathematics?[END_REF]). The modeling and analysis approaches are in fact further complicated by some new biological evidences, that enhance cell features and expand their fate possibilities, such as dedifferentiation 1 (see [START_REF] Yamada | Concise review: dedifferentiation meets cancer development: proof of concept for epigenetic cancer[END_REF], [START_REF] Cai | Dedifferentiation: a new approach in stem cell research[END_REF], [START_REF] Friedman-Morvinski | Dedifferentiation and reprogramming: origins of cancer stem cells[END_REF] [START_REF] Leder | The therapeutic implications of plasticity of the cancer stem cell phenotype[END_REF], [START_REF] Jilkine | Effect of dedifferentiation on time to mutation acquisition in stem cell-driven cancers[END_REF]).

2) Cell plasticity: It has been believed that once a cell differentiates into a particular cell type that has a distinctive function (e.g. when an hematopoietic SC differentiates into one type of white blood cells), it permanently loses the potential for diverse functions and stably maintains its identity [START_REF] Yamada | Concise review: dedifferentiation meets cancer development: proof of concept for epigenetic cancer[END_REF]. The discovery of dedifferentiation contradicted this biological postulate, and thus opening the way to a much richer and complex cellular behavior. In addition, it was also proved that adult SCs may first reside in one place and then contribute to another tissue [START_REF] Tata | Cellular plasticity: 1712 to the present day[END_REF]. This phenomenon is known as transdifferentiation 2 . In fact, when the process of cell generation and continuous replenishment is perturbed (e.g. tissue injury or hemorrhage), the homeostatic mechanisms are invoked to allow adequate regeneration of damaged tissues [START_REF] Tata | Cellular plasticity: 1712 to the present day[END_REF]. Cell plasticity is already used to produce SCs in regenerative medicine 3 [START_REF] Enver | Stem cell states, fates, and the rules of attraction[END_REF], thus avoiding the ethical issues that usually arise regarding the use of embryonic SCs in research [START_REF] Cai | Dedifferentiation: a new approach in stem cell research[END_REF]. Cell plasticity provides also new hypotheses for the origin of cancer and some insights into its treatment (see, e.g., [START_REF] Doherty | Cancer stem cell plasticity drives therapeutic resistance[END_REF], [START_REF] Friedman-Morvinski | Dedifferentiation and reprogramming: origins of cancer stem cells[END_REF], [START_REF] Leder | The therapeutic implications of plasticity of the cancer stem cell phenotype[END_REF], [START_REF] Jilkine | Effect of dedifferentiation on time to mutation acquisition in stem cell-driven cancers[END_REF]).

3) Objectives and organization of the work: We aim to deepen the analysis of cell dynamics, by taking into account the process of cell dedifferentiation in living organisms, in the class of systems as in [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF]. More precisely, we consider a general model, that includes a resting and proliferating phases as in [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], [START_REF] Burns | On the existence of a G 0 phase in the cell cycle[END_REF], and which takes into account several discrete cellmaturity stages, as in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF]. Then, we extend the model through the addition of some cell plasticity mechanisms. In fact, since this is the first time cellular plasticity is 1 «Dedifferentiation is an important biological phenomenon whereby cells regress from a specialized function to a simpler state reminiscent of stem cells» [START_REF] Cai | Dedifferentiation: a new approach in stem cell research[END_REF] 2 «Transdifferentiation is defined as the conversion of one cell type to another. It belongs to a wider class of cell type transformations called metaplasias which also includes cases in which SCs of one tissue type switch to a completely different SC» [START_REF] Shen | Transdifferentiation, metaplasia and tissue regeneration[END_REF] 3 «Regenerative medicine aims to replace damaged cells in the human body through a new source of healthy transplanted cells or by endogenous repair» [START_REF] Doherty | Cancer stem cell plasticity drives therapeutic resistance[END_REF]. Indeed, by artificially reprogramming adult cells (using Yamanaka Factors [START_REF] Takahashi | Induction of pluripotent stem cells from adult human fibroblasts by defined factors[END_REF]), a new category of cells that expresses embryonic SCs characteristics is obtained: also known as induced Pluripotent Stem Cells (iPSCs).

introduced in this class of systems, we begin in Section II with a general framework which serves as a basis for plasticity functions. Next, in Section III, we select a typical form of dedifferentiation mechanisms, where an unhealthy (differentiated) subpopulation of progeny cells regresses to a SC (cancerous) state. We highlight one of the expected cellular-plasticity effects, which is the possible survival of unhealthy cells as a result of progeny dedifferentiation. In fact, a rich behavior of the system is observed when the dedifferentiated cells are directly active in the proliferating phase, which is the common behavior of unhealthy cells when dedifferentiation is associated with cancer [START_REF] Tung | Epigenetic mechanisms of tumorigenicity manifesting in SCs[END_REF] (e.g. due to some genetic/epigenetic mutations that increase the selfrenewal activity of abnormal cells [START_REF] Ley | DNMT3A mutations in acute myeloid leukemia[END_REF], [START_REF] Tung | Epigenetic mechanisms of tumorigenicity manifesting in SCs[END_REF]). A stability result is given in Section III-B, followed by a numerical example, that we discuss in Section IV.

Summary and key points

In our study, we provide a general model for cell population dynamics that may describe many types of cancers. We particularly emphasize the case of hematopoietic SCs and early blood progeny (see some similar models in [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], [START_REF] Bélair | Agestructured and two-delay models for erythropoiesis[END_REF], [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF], [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], [START_REF] Djema | Analysis of a model of dormancy in cancer as a state of coexistence between tumor and healthy stem cells[END_REF]). In fact, hematopoiesis provides a common paradigm for understanding all the mammalian SCs, in particular when it comes to study cell differentiation.

We are interested in modeling a dedifferentiation scenario where a portion (i.e. a mutated subpopulation) of a given generation of differentiated progenitor cells (that belongs for instance to a specific blood lineage hierarchy as in Fig. 2, blue hierarchy), regresses into an hematopoietic stem cell proliferating state.

In the general case, anti-cancer therapy aims to eradicate all the malignant cancer cells. Therefore, we focus on the stability properties of the origin of our model, since it reflects extinction of malignant cells.

Our approach could incite modelers to systematically take into account cellular plasticity features when modeling cell population dynamics. Moreover, our result supports the idea that unhealthy cells protect themselves during anti-cancer therapy through dedifferentiation. We provide a stability condition which could help medical experts restrain -or inhibitdedifferentiation, in order to ensure cell eradication. This procedure seems in line with some medical experiments, where anti-dedifferentiation mechanisms are already used (see Section IV).

II. Modeling cell populations and cellular plasticity: a general framework

We discuss in this section a refined model (illustrated in Fig. 12) that highlights, qualitatively, the impact of dedifferentiation on the behavior of cancer cells. Firstly, we focus on the general form of the model in Fig. 1, in which we have n distinguishable maturity levels (i ∈ I n = {1, . . . , n}, such that i = 1 is the SC compartment). Models with quiescent and proliferating phases date back to some pioneer works such as Burns & Tannock [START_REF] Burns | On the existence of a G 0 phase in the cell cycle[END_REF] and Mackey [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], which have been improved by Adimy et al. [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF], and which we extend now -as previously mentionedby adding some cell plasticity functions. {G1, S, G2, M }. In the general case, we can consider that all the involved biological parameters (rate of differentiation σi ∈ (0, 1), rate of self-renew λi = 1 -σi, apoptosis rate dpi, death rate of resting cells dri) are time-varying. On the right, the flux due to the cell-plasticity functions, ξi and ωi for all i ∈ In, are represented as input or output functions. In fact, both ξi and ωi have a part related to dedifferentiation and another part related to transdifferentiation. The former one depends on the (modeled) state variables, which are the total densities of the resting cells xi as explained in Section II, while the parts quantifying trandifferentiation (denoted ξ † i and ω † i , in equations ( 2), ( 5)) are assumed to depend only on time, since they are biologically related to distant (or, external) cell tissues and lineages (i.e. which are not explicitly modeled in the specific studied cell-maturity hierarchy, see also Fig. 2).

The dynamics of resting cells, r i (t, a), of the i-th generation (i ∈ I n ), of age a > 0, at time t ≥ 0, and the dynamics of proliferating cells, p i (t, a), of age a ∈ [0, τ i ], at time t ≥ 0, are governed by the agestructured (McKendrick) PDEs:

∂p i ∂t + ∂p i ∂a = -[dpi(t) + hi(a)] pi(t, a), ∂r i
∂t

+ ∂r i ∂a = -[dri(t) + βi (xi(t))] ri(t, a) -ξi (t) , (1)
where, for all i ∈ I n , x i (t) = +∞ 0 r i (t, a)da, and, for all i ∈ I n and t ≥ 0, d ri (t) is the death rate of the resting cells, while d pi (t) is the death rate of proliferating cells. The reintroduction function from the resting phase to the proliferative phase, β i , is decreasing and lim →∞ β i ( ) = 0. Moreover, we consider that for all t ≥ 0,

ξi(t) = ξi    x1(t), . . . , xi-1(t) out. dediff , xi(t), out. transdiff ξ † i (t)    .
(2)

The system (1) is associated with some initial conditions (i.e. initial age distributions when t = 0) which are L 1 -functions defined by:

pi(0, a) = p 0 i (a), 0 ≤ a ≤ τi, ri(0, a) = r 0 i (a)
, for all a ∈ [0, +∞).

(

) 3
Moreover, we assume from biological considerations that for all t ≥ 0, lim a→∞ r i (t, a) = 0.

About the functions ξi ξi ξi As illustrated in Fig. 2, the function ξi quantifies the output (cell loss) either by dedifferentiation or by transdifferentiation. As formulated in (2), the function ξi depends on xi and may be affected by some of, or all, the generations of cells which are less mature than the i-th generation, i.e. the cell densities x1, . . . , xi-1. This is -by definition-justified by the fact that dedifferentiation of the i-th generation addresses a need (either healthy or cancerous) that arises in a less mature generation within the same hierarchy (i.e. at least one of the densities x1, . . . , xi-1). In addition, the loss by transdifferentiation, ξ † i , aims to address a need from more distant tissues, i.e. ξ † i does not depend on the modeled cell populations xi, but only on time (as a representation of any external event that occurs beyond the studied hierarchy).

Finally, the renewal conditions, which give the birth rate at the initial age a = 0, are introduced through the following boundary conditions:

     pi(t, 0) = βi(xi(t))xi(t) + ωi(t), ri(t, 0) = 2σi-1(t) τ i-1 0 hi-1(a)pi-1(t, a)da +2 (1 -σi(t)) τ i 0 hi(a)pi(t, a)da, ( 4
)
where σ i (t) represents the time-varying rate of differentiation and, consequently, λ i (t) = 1 -σ i (t) is the rate of self-renewal of the i-th cell generation. In addition, we consider with an abuse of notation that for all t ≥ 0,

ωi(t) = ωi    xi(t), xi+1(t), . . . , xn(t) in. dediff , in. transdiff ω † i (t)    .
(5)

About the functions ωi ωi ωi

The function ωi appears in the boundary conditions (4), since it represents the new cell birth resulting from the dedifferentiation of more mature cell generations, together with the transdifferentiation of other cell lineages. The functions ωi are implicitly related to the functions ξi in (2), since cells that join the i-th generation by dediffentiation (the input ωi) are, for instance, the sum of the dedifferentiated cells from more mature generations (the outputs ξi+1, ξi+2, . . .). Therefore, for all i ∈ In, the function ωi depends on xi and may depend on (all or some of) the more mature generations in the same hierarchy, xi+1, . . . , xn. Moreover, the time-varying term ω † i in (5) quantifies the incoming by trandifferentiation from distant cell lineages or types, outside the hierarchy of interest.

In summary, we say that generally in McKendrick type models, the removal terms (e.g. death rates) appear in the PDE system (model (1), in our case), while new births appear in the boundary conditions (given by (4), in our case). The PDE system (1)-( 4)-( 3) provides a general framework to describe the cell dynamics within a given hierarchy formed by n discrete-maturity stages. However, we still need to specify the nature and the operating mode of the dedifferentiation and the transdifferentiation functions (ξ i and ω i , for all i ∈ I n ), in order to determine the behavior of the overall system. For that purpose, we need to focus on a typical explicit dedifferentiation mechanism. More precisely, we start in this study with an unhealthy case in which a portion of the j-th cell generation (j ∈ I n , j > 1) dedifferentiates and joins the SC compartment (i = 1), as presented in the sequel.

Fig. 2:

The blue lineage hierarchy is the one on which we are focusing, and which is modeled by our system. More precisely, each i-th stage in the blue hierarchy corresponds to the i-th generation in the model illustrated in Fig. 1. In contrast, the gray generations/lineages are those which cannot be explicitly modeled in our mathematical framework. Next, for the i-th cell generation, the function wi quantifies the input by dedifferentiation from the upcoming more mature generations that belong to the studied blue lineage, together with the input by transdifferentiation (ω † i ), which comes from some nonmodeled distant-cell types. In a similar spirit, ξi quantifies the loss in cell density of the i-th generation, for instance in response to some requirements from less mature generations that belong to the blue hierarchy (i.e. this is the loss due to dedifferentiation), or from other types of cells which are in gray, i.e. this is the loss/output by transdifferentiation (ξ † i ).

III. A specific dedifferentiation process

We focus now on a typical case, in which a proportion κ ∈ [0, 1] of the total density of resting cells of the jth generation x j becomes diseased, e.g. as a result of a series of abnormal mutations (see, e.g., [START_REF] Xie | Age-related mutations associated with clonal hematopoietic expansion and malignancies[END_REF] for mutations inducing leukemia). Then, we consider that only the subpopulation of mutated cells can undergo dedifferentiation, and we denote by D the characteristic pattern that describes the dedifferentiation mechanism from the j-th generation into the SC-proliferating compartment.

A. A model involving one dedifferentiation feedback

Let j ∈ [2, n] be the generation of cells that includes a malignant (mutated) subpopulation, that we assume capable of generating cancer stem cells (CSCs) by dedifferentiation (see Fig. 3 in [START_REF] Friedman-Morvinski | Dedifferentiation and reprogramming: origins of cancer stem cells[END_REF]). For the sake of simplicity, we consider that the transdifferentiation mechanisms are negligible compared to dedifferentiation within the same hierarchy (i.e. we put ξ † i (t) = ω † i (t) = 0, for all i ∈ I n and t ≥ 0). We notice that -with an abuse of notationthe functions ξ i and ω i can be rewritten in this case as: where we can select the function D as follows, D( ) = tanh ( p ) , where, p > 0, for all ≥ 0.

(

) 7
Let us say few words about the cell-plasticity functions ξ i and ω i in [START_REF] Cai | Dedifferentiation: a new approach in stem cell research[END_REF]. Firstly, we are limiting ourselves to the case of constant κ, where κ ∈ [0, 1]. The function D depends on the total density of mutated cells, which is given by κx j , since this subpopulation is the unique one that may undergo dedifferentiation. The function D is selected such that lim →+∞ D( ) = 1, meaning that the aggressiveness of mutated cells increases (i.e. mutated cells are more likely to dedifferentiate into CSCs) with respect to the total density of mutated cells κx j . In fact, the features of the tangent hyperbolic function D, as defined in [START_REF] Cooper | The Cell: A Molecular Approach[END_REF], lead to the following qualitative behavior: x ξ j ≡ 0 and ω 1 ≡ 0, if x j = 0 or κ = 0, i.e. dedifferentiation does not exist without abnormal mutations. y when the j-th cell generation forms a relatively small population over all the n cell generations (i.e. a low cell density x j ), or when the portion of unhealthy cells is minimal within the genetic diversity landscape (i.e. a low mutated portion of cells, quantified by κ), then the gain of the dedifferentiation process is minimal. Indeed, the subpopulation of mutated cells is not expected to entirely dedifferentiate and form aggressive CSCs, in this case, because it is too small to impose itself upon all the other cells. We actually expect that immunological mechanisms are sufficient in this case to control the small population of mutated cells [START_REF] Schreiber | Cancer immunoediting: integrating immunity's roles in cancer suppression and promotion[END_REF]. Let us notice that, in our model, the gain of the dedifferentiation process is represented by the quantity κD(κx j ), since ξ j (x j ) = κD(κx j )x j . So, for low x j and low κ, the amplitude of κD(κx j ) is relatively small, then it increases in order to approach its highest value, when the total size of the population x j is too large, i.e. the maximum possible gain of dedifferentiation is κ (since lim xj →∞ κD(κx j ) = κ).

Remark 1: The selected dedifferentiation functions (6) depend only on the mutated subpopulation of cells belonging to the j-th maturity generation. The latter choice can be argued as an unhealthy dedifferentiation process triggered by some abnormal mutations that occur in the j-th maturity stage. However, different choices may be considered within the general framework introduced in Section II. For instance, a healthy dedifferentiation process can be formulated by considering that ξ i and ω i depend on the total density of resting SCs, x 1 . In such a configuration, SCs have the ability to trigger by themselves the dedifferentiation of more mature cells, when needed (e.g. after hemorrhage or injury, when the body is in a hurry to regenerate itself). However, this healthy situation is beyond the scope of the current work.

Finally, we rewrite the model ( 1) describing the dynamics of resting and proliferating cells, in the form:

                     ∂p i ∂t + ∂p i ∂a + [dpi(t) + hi(a)] pi(t, a) = 0, ∂r i
∂t + ∂r i

∂a + [dri(t) + βi (xi(t))] ri(t, a) = 0, for all i ≥ 1 where i = j, and,

∂p j ∂t + ∂p j ∂a + [dpj(t) + hj(a)] pj(t, a) = 0, ∂r j ∂t + ∂r j ∂a + [drj(t) + βj (xj(t))] rj(t, a) +κD (κxj(t)) xj(t) = 0. (8)
We emphasize the case of (4) described throughout the current section, in which the renewal conditions are introduced through the following boundary conditions:

                       p1(t, 0) = β1 (x1(t)) x1(t) + κDj (κxj(t)) xj(t), pi(t, 0) = βi (xi(t)) xi(t), for all i ∈ {2, . . . , n} , r1(t, 0) = 2 (1 -σ1(t)) τ 1
0 h1(a)p1(t, a)da, and for all i > 1 :

ri(t, 0) = 2 (1 -σi(t)) τ i 0 hi(a)pi(t, a)da +2σi-1(t) τ i-1 0 hi-1(a)pi-1(t, a)da. (9)
Finally, we associate to ( 8)-( 9) some suitable initial conditions similar to (3). Next, using the method of characteristics (see similar applications in [START_REF] Bélair | Agestructured and two-delay models for erythropoiesis[END_REF] and [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF]), we can reduce the model ( 8)-( 9)-( 3) into a nonlinear system with finite distributed delays and time-varying parameters. For the sake of clarity, we illustrate these statements in the case involving two cell-maturity generations, in which all the biological parameters are constant and a dedifferentiation mechanism is established from the 2nd generation into the SC one, as shown in Fig. 3.

For instance, we consider that a progeny subpopulation κx 2 , where κ ∈ [0, 1], has some abnormal mutations (e.g. DMNT3A increasing the self-renewal activity), that may trigger dedifferentiation into CSCs [START_REF] Cozzio | Similar MLL-associated leukemias arising from self-renewing stem cells and short-lived myeloid progenitors[END_REF], as it appears to be the case in leukemia [START_REF] Tung | Epigenetic mechanisms of tumorigenicity manifesting in SCs[END_REF], [START_REF] Ley | DNMT3A mutations in acute myeloid leukemia[END_REF]. The time-delay version of the model of interest is given by: [START_REF] Dale | Understanding, treating and avoiding hematological disease: Better medicine through mathematics?[END_REF] where f i ( ) = h i ( )e -0 hi(m)dm and g i ( ) = f i ( )e -dpi , (see [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF] for a similar application, but also [START_REF] Bélair | Agestructured and two-delay models for erythropoiesis[END_REF]). In addition, we can notice that the system (10) is positive.

                       ẋ1(t) = -[dr1 + β1(x1(t))] x1(t) +2 (1 -σ1) τ 1 0 g1(a)β1 (x1(t -a)) x1(t -a)da +2 (1 -σ1) τ 1 0 g1(a)κx2(t -a)D (κx2(t -a)) da, ẋ2(t) = -[dr2 -β2 (x2(t)) -κD (κx2(t))] x2(t) +2 (1 -σ2) τ 2 0 g2(a)β2 (x2(t -a)) x2(t -a)da +2σ1 τ 1 0 g1(a)β1 (x1(t -a)) x1(t -a)da +2σ1 τ 1 0 g1(a)κx2(t -a)D (κx2(t -a)) da,

B. Sufficient global asymptotic stability conditions

Now, we want to determine stability conditions of the origin of the model [START_REF] Dale | Understanding, treating and avoiding hematological disease: Better medicine through mathematics?[END_REF]. Convergence of the model trajectories to zero is equivalent to total cell eradication, e.g. thanks to anti-cancer infusions therapy. Firstly, let us define for later use the constants:

s i = δ i -α i β i (0), α i = 2λ i C i -1, for all i ∈ I n , and, a = s2 -κD 2σ1λ1C 2 1 β1(0) s1 + 2σ1C1 -1 , (11)
where, D = sup ≥0 D( ). Notice that, D = 1, when, D( ) = tanh( p ), p > 0. It is worth mentioning that for a model without dedifferentiation (i.e. when κ = 0), the conditions s i > 0, for all i ∈ I n are necessary and sufficient conditions for global exponential stability of the origin of the corresponding system (see [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF] and [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF]). We state in this paper -without details of the proof-the following stability result:

Theorem 1: Let us assume that si > 0 for all i ∈ In.

If, in addition, the conditions,

1 -2σ1C1 - 2σ 1 λ 1 C 2 1 β 1 (0) s 1 < 0, and, a > 0, or, 1 -2σ1C1 - 2σ 1 λ 1 C 2 1 β 1 (0) s 1 ≥ 0, ( 12
)
are satisfied, then the origin of the model ( 10) is globally asymptotically stable. 3: Red cells are malignant (mutated) cells. White cells are ordinary stem cells (SCs), while the gray cell compartment is the one of (progeny) differentiated cells, that includes the subpopulation (in red) κx2. A mechanism of dedifferentiation of malignant cells (in red, from i = 2 to i = 1) is established through the function D. In hematopoiesis, a translocation effect after some epigenetic mutations in the progeny compartment (here i = 2) may lead to a dedifferentiation of progeny ( [START_REF] Cozzio | Similar MLL-associated leukemias arising from self-renewing stem cells and short-lived myeloid progenitors[END_REF]) and the rise of cancer stem cells (CSCs), that trigger a quick progression of leukemia [START_REF] Tung | Epigenetic mechanisms of tumorigenicity manifesting in SCs[END_REF]. For instance, a DNMT3A mutation may provide abnormal cells with a selfrenewal activity as important as the one of SCs [START_REF] Ley | DNMT3A mutations in acute myeloid leukemia[END_REF], [START_REF] Tung | Epigenetic mechanisms of tumorigenicity manifesting in SCs[END_REF].

Remark 2: In fact, the model (10) may admit a positive steady state even if s i > 0, which is not the case when κ = 0. Therefore, we notice that then conditions s i > 0, for all i ∈ I n are no longer sufficient to ensure that the 0-equilibrium is globally stable when dedifferentiation exists. In light of Theorem 1, we deduce that zero is stable if we ensure that an upper-bound on κ is satisfied, i.e. if dedifferentiation does not cross the threshold defined by the condition a > 0 in [START_REF] Djema | Analysis of a model of dormancy in cancer as a state of coexistence between tumor and healthy stem cells[END_REF]. Medical practice supports this observation ( [START_REF] Iwasa | Radiosensitizing effect of YM155, a novel small-molecule survivin suppressant, in non-small cell lung cancer cell lines[END_REF], [START_REF] Rhodes | Mathematical Modeling of the Role of Survivin[END_REF]), as discussed in the sequel.

IV. A numerical illustration and some concluding comments

Medical practice shows that sometimes if dedifferentiation is neglected, cancer cells may survive to therapy. We consider the concrete situation studied in [START_REF] Iwasa | Radiosensitizing effect of YM155, a novel small-molecule survivin suppressant, in non-small cell lung cancer cell lines[END_REF], [START_REF] Rhodes | Mathematical Modeling of the Role of Survivin[END_REF].

A glimpse into the medical experience

In [START_REF] Iwasa | Radiosensitizing effect of YM155, a novel small-molecule survivin suppressant, in non-small cell lung cancer cell lines[END_REF], some experiences have been conducted on human non-small cell lung cancer. Their results suggested that non-stem cancer cells which were targeted through radiotherapy, have protected themselves by dedifferentiation processes (CSCs are particularly resistant to radiotherapy, see for example [START_REF] Bao | Glioma stem cells promote radioresistance by preferential activation of the DNA damage response[END_REF], [START_REF] Dahan | Ionizing radiations sustain glioblastoma cell dedifferentiation to a stem-like phenotype through survivin: possible involvement in radioresistance[END_REF]). In [START_REF] Rhodes | Mathematical Modeling of the Role of Survivin[END_REF], an ODE-model was proposed to fit the data of [START_REF] Iwasa | Radiosensitizing effect of YM155, a novel small-molecule survivin suppressant, in non-small cell lung cancer cell lines[END_REF]. In a second time, the experience was renewed by adding some survivin inhibitors (known as YM155 ), that undermined dedifferentiation of cancer cells. The therapy efficacy was substantially improved when YM155 was used along with radiotherapy [START_REF] Rhodes | Mathematical Modeling of the Role of Survivin[END_REF], [START_REF] Iwasa | Radiosensitizing effect of YM155, a novel small-molecule survivin suppressant, in non-small cell lung cancer cell lines[END_REF].

We check the qualitative behavior of our system through some situations that are reflective of those encountered in [START_REF] Iwasa | Radiosensitizing effect of YM155, a novel small-molecule survivin suppressant, in non-small cell lung cancer cell lines[END_REF], [START_REF] Rhodes | Mathematical Modeling of the Role of Survivin[END_REF]. For that purpose, we consider for instance the cell division probability densities of the forms f i (a) = mi e m i τ i -1 e mia , with m i > 0, i ∈ I n , along with the following functions and parameters: After simple calculations, we get: s 1 = 0.0518 and s 2 = 0.2218. Therefore, in the case where dedifferentiation does not exist (i.e. κ = 0), the origin of the studied model is globally exponentially stable [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF], [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF], as illustrated in Fig. 4-(a). Next, let us assume that at t = 20 days, the dedifferentiation mechanism is triggered (by setting κ = 0.8 at t = 20 days). In that case, we still have s 1 > 0 and s 2 > 0, but, however, we note that the sufficient stability conditions [START_REF] Djema | Analysis of a model of dormancy in cancer as a state of coexistence between tumor and healthy stem cells[END_REF], in Theorem 1, are not satisfied:

1 -2σ1C1 - 2σ1λ1C 2 1 β1(0) s1 = -4.45, and, a = s2 -κD 2σ1λ1C 2 1 β1(0) s1 + 2σ1C1 -1 = -3.33.
In simulation, we observe that for some initial conditions, the origin of the system (10) is not asymptotically stable, as illustrated in Fig. 4-(b). More importantly, we notice that the trajectories converge in this case to a strictly positive steady state, given by x * 1 = 1.47 and x * 2 = 0.66, that exists even if s i > 0 (contrary to the case κ = 0, where zero is the unique steady state when s i > 0). In addition, we can determine the gain of the dedifferentiation function, which converges to κD(κx * 2 ) = 0.12, when the trajectories of the system (10) approach the positive steady state. Thus, in this example, the unhealthy cells prevent themselves from total extinction thanks to their dedifferentiation ability (Fig. 4 (a)-(b)). Finally, we notice in Fig. 4-(c) that for some sufficiently Fig. 4: In (a), κ = 0, si > 0, the stability conditions are satisfied and we observe that the trajectories converge to zero. In (b) and (c), dedifferentiation processes are triggered at t = 20. In (b), the trajectories do not converge to zero (they converge to a small positive steady state), meaning that dedifferentiation prevents cells from total extinction. This observation supports the hypothesis claiming that cancer cells may protect themselves through dedifferentiation during anti-cancer therapy [START_REF] Doherty | Cancer stem cell plasticity drives therapeutic resistance[END_REF]. We suggest that therapeutics should take into account cell plasticity features, by including drugs that inhibit the dedifferentiation mechanisms, in order to satisfy the conditions given in Theorem 1. In other words, we want to design a therapy that avoids the case described in (b). Otherwise, malignant cells may regenerate themselves after therapy. In (c), dedifferentiation is also triggered at t = 20, but this time, we consider that the initial mutated subpopulation κxj is relatively too small. In this case, dedifferentiation is ineffective in ensuring unhealthy cell survival (mutated cells disappear over time without creating an aggressive cancer). Sometimes it is suggested that immmunological mechanisms (immunoediting [START_REF] Schreiber | Cancer immunoediting: integrating immunity's roles in cancer suppression and promotion[END_REF]) may prevent mutated cells from the outgrowth that characterizes CSCs (i.e. the immune system acts to restrain the gain of dedifferenciation mechanisms).

small initial conditions and κ = 0.8, the dedifferentiation process is not sufficient to avoid the total cell eradication.

Fig. 1 :

 1 Fig. 1: A cartoon representation of the discrete maturity model of interest, involving multiple dedifferentiation and transdifferentiation functions. Cells of the i-th maturitygeneration, where i ∈ In = {1, . . . , n} , n > 1, are in a resting phase G0, or in proliferation (cell-division cycle){G1, S, G2, M }. In the general case, we can consider that all the involved biological parameters (rate of differentiation σi ∈ (0, 1), rate of self-renew λi = 1 -σi, apoptosis rate dpi, death rate of resting cells dri) are time-varying. On the right, the flux due to the cell-plasticity functions, ξi and ωi for all i ∈ In, are represented as input or output functions. In fact, both ξi and ωi have a part related to dedifferentiation and another part related to transdifferentiation. The former one depends on the (modeled) state variables, which are the total densities of the resting cells xi as explained in Section II, while the parts quantifying trandifferentiation (denoted ξ † i and ω † i , in equations (2), (5)) are assumed to depend only on time, since they are biologically related to distant (or, external) cell tissues and lineages (i.e. which are not explicitly modeled in the specific studied cell-maturity hierarchy, see also Fig.2).

  , . . . , xi) = 0, for all i ∈ In, i = j, ξj(x1, . . . , xj) = ξj(xj) = κD(κxj)xj, ω1(x1, . . . , xn) = ω1(xj) = κD(κxj)xj, ωi(xi, . . . , xn) = 0, for all i ∈ {2, . . . , n} , (6)

  Fig.3: Red cells are malignant (mutated) cells. White cells are ordinary stem cells (SCs), while the gray cell compartment is the one of (progeny) differentiated cells, that includes the subpopulation (in red) κx2. A mechanism of dedifferentiation of malignant cells (in red, from i = 2 to i = 1) is established through the function D. In hematopoiesis, a translocation effect after some epigenetic mutations in the progeny compartment (here i = 2) may lead to a dedifferentiation of progeny ([START_REF] Cozzio | Similar MLL-associated leukemias arising from self-renewing stem cells and short-lived myeloid progenitors[END_REF]) and the rise of cancer stem cells (CSCs), that trigger a quick progression of leukemia[START_REF] Tung | Epigenetic mechanisms of tumorigenicity manifesting in SCs[END_REF]. For instance, a DNMT3A mutation may provide abnormal cells with a selfrenewal activity as important as the one of SCs[START_REF] Ley | DNMT3A mutations in acute myeloid leukemia[END_REF],[START_REF] Tung | Epigenetic mechanisms of tumorigenicity manifesting in SCs[END_REF].


THE IMPACT OF A COMMON AGRICULTURAL POLICY ON THE FRENCH RURAL TERRITORIES AND THE STAKEHOLDERS

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

The main purpose of this research is to understand how the European subsidies of the Common Agricultural Policy have an impact over both farmers and the territories they live in. Our study reveals that the French program of rural development tends to enhance the territorial projects. Farmers who really depend on the subsidies from the CAP adopt various strategies and attitudes in order to keep their activities and income. Farmers have mainly a short term view on the prosperity of their farm. They hardly plan their future ahead and their working habits. Farmers are encouraged by the program to develop new multi-use related activities and to focus on the landscaping of the french rural territories. The zoning of the program helps understand territories differently by putting forward a new reference system. Besides, the program is aimed at making the cohabitation easier between urban and rural territories and thereby leading the stakeholders towards a better comprehension of the overall needs and requirements of the society.

THE IMPACT OF A COMMON AGRICULTURAL POLICY ON THE FRENCH RURAL TERRITORIES AND THE STAKEHOLDERS A research study on 6 representative rural and agricultural territories in France Methodology

Research areas studied:

-a study on the effects of the rural development program when launched -a gathering of « counties » Data used for the 6 rural territories:

-a quantitative study of the rural development program

-interconnection with other programs (european, national, regional)

Interview guidelines according to the stakeholders interwiewed:

-the rural and agricultural administration officials and professionals -234 semi direct interviews : 37 per research area

Qualitative data analysis:

-theme analysis -text analysis with the software program (Sphinx Plus²)

Research issues

What are the different aspects of the CAP and how is this policy implemented on the territories? How do the stakeholders use for their own purpose the CAP subsidies?

What is the impact on the stakeholders and on both the rural and agricultural territories?

Which policy is the CAP opting for the french rural territories?

Main achievements

The CAP impact on the stakeholders:

• the farmers behave differently according to the subsidies they get

• the stakeholders adopt various strategies

• the farmers patterns change accordingly

• it raises a selection in the projects and among the subsidized ones

• the CAP is used differently according to the stakeholders.

The CAP impact on the territories:

• the urban expectations are met through:

-diversification (in-farm sales, rural tourism)

-amenities (landscape, environment)

• different aspects in agriculture:

-"agri-resource": maintenance, development of economic activities -"agri-heritage": maintenance of the existing landscapes and their appealing touristic aeres -"agri-nature": a respect for environment

• the CAP emphasizes on territories rather than on branches -tools: CTE / CAD, MAE.

The CAP at stake today


W Djema

F Mazenc

C Bonnet

J Clairambault

E Fridman

Stability Analysis of a Nonlinear System with Infinite Distributed Delays Describing Cell Dynamics

Keywords: Infinite Distributed Delay, Lyapunov-Krasovskii Functionals (LKFs), Exponential Stability

We want to reconcile some earlier modeling ways of the cell cycle in one common framework. Accordingly, we consider a model that contains a compartment where cells may be quiescent for an unlimited time, along with a proliferating phase in which most of the cells may divide, or die, while few of them may be arrested during their cycle for unlimited time. In fact, the cell-cycle arrest may occur for many reasons (DNA damages detected at some checkpoints, insufficient resources for cell grow, drug infusions). We actually extend some early models involving finite distributed delays (taken from [2], [1]) to the case of infinite distributed delays and time-varying parameters. Our main result relies on the construction of a novel Lyapunov-Krasovskii functional, suitable for the analysis of the origin of the system involving infinite distributed delays and time-varying parameters.

I. Introduction

Cells that engage in the division process (cell cycle, [START_REF] Hartwell | Cell cycle control and cancer[END_REF]) usually undergo a series of active transformations and a mechanism of nuclear division (mitosis), that ends with the division of each mother cell into two daughter cells (cytokinesis, [START_REF] Morgan | The Cell Cycle: Principles of Control[END_REF]). The cell cycle leads to the growth of tissues in all multicellular organisms. A small category of adult cells is known as stem cells (SCs): these are undifferentiated cells characterized by their extensive ability to self-renew and their multipotency, which is the ability to differentiate into more specialized cells [START_REF] Tuch | Stem cells-a clinical update[END_REF]. When cells become more and more mature, they gradually lose their self-renewal feature. Even in fast-renewing tissues, most of the cells are in fact in a non-proliferating state known as the resting or quiescent phase, G 0 [START_REF] Morgan | The Cell Cycle: Principles of Control[END_REF]. A resting cell prepares its entrance into the cycle, which is formed by four consecutive discrete subphases G 1 , S, G 2 and M (see [START_REF] Cooper | The Cell: A Molecular Approach[END_REF], Chap. 17), where the M -phase stands for mitosis, or nuclear division. Several checkpoints are used by the body to control cells that undergo division. These checkpoints gather a set of regulatory proteins that Walid Djema* corresponding author, walid.djema@inria.fr.

Walid Djema, Frédéric Mazenc and Catherine Bonnet are with Inria, University Paris-Saclay, L2S (CNRS), CentraleSupélec, Gifsur-Yvette, France. Jean Clairambault is with Inria, University of Sorbonne, UPMC, Laboratory Jacques-Louis Lions, Paris, France. Emilia Fridman is with the Department of Electrical Engineering and Systems at the School of Electrical Engineering, Tel-Aviv, Israel. walid.djema@inria.fr, frederic.mazenc@inria.fr catherine.bonnet@inria.fr, jean.clairambault@inria.fr emilia@eng.tau.ac.il monitor the progression of cells through different stages in the cycle. If some requirements are not fulfilled at specific moments, the regulatory proteins network takes over the issue by preventing the cell progression through the cycle. Consequently, abnormal cells (e.g. the ones with damaged DNA) cannot move forward in the division process, causing cell arrest. These cells are then obliged to undergo a process of DNA repair, or they are doomed to die by apoptosis.

There are in fact three main checkpoints: G 1 /S, G 2 /M , and the metaphase checkpoints. In the G 1 /S checkpoint, the cell ensures that it has enough materials to replicate its DNA. Thus, unhealthy or malnourished cells are stuck at this point. Similarly, in the G 2 /M checkpoint, the cell ensures that it has enough cytoplasm and phospholipids for two (daughter) cells, otherwise, it is prevented from performing mitosis. Lastly, at the metaphase checkpoint, the division is stopped if chromosomes are not all well-aligned at the spindle equator before anaphase. The length of the cycle is approximately 24 hours for fast-dividing mammalian cells. However, even when cells are not arrested, this duration may vary from one type of tissue to another, and even from one cell to another within the same tissue. Some authors focused on the length of G 1 and its possible applications ( [START_REF] Lange | Cdks and cyclins link G1 length and differentiation of embryonic, neural and hematopoietic stem cells[END_REF], [START_REF] Salomoni | Cell cycle control of mammalian neural stem cells: putting a speed limit on G1[END_REF]); e.g., the manipulation of the G 1 length in neural SCs is discussed in [START_REF] Salomoni | Cell cycle control of mammalian neural stem cells: putting a speed limit on G1[END_REF], including also its impacts on the differentiation of neural precursors. We refer to Fig. 1 in [START_REF] Lange | Cdks and cyclins link G1 length and differentiation of embryonic, neural and hematopoietic stem cells[END_REF] that gives the lengths of the cell cycle (in particular the length of G 1 ) of some important categories of cells, including blood lineages, gut lineage and neural lineage. Note that, in the classification given in [START_REF] Lange | Cdks and cyclins link G1 length and differentiation of embryonic, neural and hematopoietic stem cells[END_REF], the cycle length of the common myeloid progenitors (CMP) -in the hematopoietic system-appears as unknown or undetermined. We also emphasize that cancer dormancy is sometimes explained through cellular dormancy, i.e. G 0 and G 1 -arrest [START_REF] Aguirre-Ghiso | Models, mechanisms and clinical evidence for cancer dormancy[END_REF], which might be also associated to the cell arresting power of some chemotherapies [START_REF] Rader | Dual CDK4/CDK6 inhibition induces cell-cycle arrest and senescence in neuroblastoma[END_REF].

Our objective is to extend some modeling aspects in earlier models describing population cell dynamics. For that purpose, in this paper, we simply slightly modify the model presented in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF], [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF], where all the cells have been assumed to divide or die before a finite-time during their cycle. So, here we consider that a minority of cells can be arrested for an infinite time within the division-cycle. Then, we generalize the stability analysis of the trivial steady state of the resulting system, from a finite to an infinite proliferating phase.

II. Trends and models of the cell cycle

We pointed out that the cell-cycle length can be different within the total population of cells, and that few cells may be arrested for an indefinite duration. In fact, some models are already taken into account a proliferative phase of infinite duration, while other models assume that all the cells are obliged to divide before a maximum finite age (see Fig. 1). Basically, the following questions may arise in this case: If some models take into account the cell arrest, while other models consider that all the cells must divide or die before a finite maximum age, can we expect equivalent representations in both frameworks? More precisely, does the minority of arrested cells cause a change in the asymptotic behavior of the model that has been assumed to have a finite proliferating phase?

Let us begin here with a brief review of some existing models of the cell cycle. Firstly, it is worth mentioning that some models in the literature were introduced in order to investigate the behavior of cells, undergoing proliferation, at their specific checkpoints, i.e. the interest was on the transitions from a given sub-phase of the cell cycle to the next one. Accordingly, some models consider some explicitly separated sub-phases of the cell cycle (see, e.g., [START_REF] Billy | Age-structured cell population model to study the influence of growth factors on cell cycle dynamics[END_REF], [START_REF] Avila | A new model of cell dynamics in Acute Myeloid Leukemia involving distributed delays[END_REF]). The latter perspective is in fact beyond the scope of this work. Indeed, here we consider a model that represents the transition between the resting phase G 0 (which is not explicitly modeled in [START_REF] Billy | Age-structured cell population model to study the influence of growth factors on cell cycle dynamics[END_REF]) and the proliferating compartment (i.e. the sum of G 1 , S, G 2 and M ), together with their respective lengths. We recall that cell population models with separated quiescent and proliferating compartments date back to some pioneers works such as Burns & Tannock [START_REF] Burns | On the existence of a G 0 phase in the cell cycle[END_REF] and Mackey [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], which have been more recently improved by many authors, including Adimy et al. [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF], [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF], Özbay et al. [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], Djema et al. [START_REF] Djema | Analysis of a model of dormancy in cancer as a state of coexistence between tumor and healthy stem cells[END_REF].

In Fig. 1, we point out some differences between two main modeling approaches: on the one hand, we observe the configurations (A) and (B) where the G 0state is taking into account, and on the other hand, the configuration (C) where G 0 is assimilated to the phase G 1 . These two trends may be -in some sense -reconciled by adopting the representation (D) (see the arguments given in Fig. 1). More precisely, the configuration (D) improves the one illustrated in (C) by clearly separating the G 0 compartment from the cell cycle. In addition, contrary to (A) and (B), the representation (D) extends the representation of the cell cycle by considering it of unlimited length. Thus, the role of the growth phase G 1 and the gap phase G 2 (which are of variable duration in healthy and unhealthy cases, and which may halt some dividing cells) is taken into account. The configuration in (D)-Fig. 1 is the one that we want to study. Our first objective is to extend the LKF constructions we provided in [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF] to the case of models involving infinite distributed delays, modeling the infinite proliferating phase.

In order to better understand the cell dynamics in healthy or unhealthy situations, mathematical modeling and analyses are generally required. In particular, since the aim of anti-cancer therapies is the eradication of all malignant cells, the focus is put on investigating the stability properties of the zero solution of the model (here it is about the model in (D)-Fig. 1). Indeed, the convergence of the model trajectories to zero means total-cells extinction, which reflects to desired fate for unhealthy and cancer cells. It is well known that theory of Lyapunov functions offers efficient tools to study nonlinear systems ( [START_REF] Khalil | Nonlinear systems, 3rd[END_REF], [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF]), possibly involving delays ( [START_REF] Gu | Stability of timedelay systems[END_REF]). However, the drawback of this approach is that no systematic method exists to find a suitable Lyapunov function for a given nonlinear system (see [START_REF] Kolmanovskii | Introduction to the theory and applications of functional differential equations[END_REF], [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF], [START_REF] Khalil | Nonlinear systems, 3rd[END_REF] and [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF]).

At this juncture, we mention that the version of the biological model of interest, involving finite distributed delays (introduced and analyzed in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF], [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF]), has been already investigated through the construction of strict Lyapunov-Krasvoskii functionals (LKFs) in [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF] (for its two steady states: the trivial and the strictly positive ones). However, extending the LKF construction given in [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF], to investigate the stability properties of the origin of the model involving infinite distributed delays, is not a trivial task.

III. A nonlinear cell population model involving infinite distributed delays

The model of interest is the one illustrated in Fig. 2, in which we have n distinguishable maturity levels (i ∈ I n = {1, . . . , n}, where i = 1 is the SC-compartment). Some modeling aspects are simplified in this work since they deserve a separated study. More precisely, cell plasticity features are neglected in this paper (refer to [START_REF] Djema | Introducing Cell Plasticity Mechanisms into a Class of Cell Population Dynamical Systems[END_REF]).

We study the following nonlinear system with infinite distributed delays:

ẋi (t) = -d ri (t)x i (t) -β i (x i (t))x i (t) + 2λ i (t) +∞ 0 g i (t, a)β i (x i (t -a))x i (t -a)da + 2σ i-1 (t) +∞ 0 g i-1 (t, a)β i-1 (x i-1 (t -a)) × x i-1 (t -a)da, (1)
where

f i (a) = h i (a)e - a 0 hi(m)dm , g i (t, a) = f i (a)e - a 0 dpi(m+t-a)dm , ( 2
)
and h i is the cell-division rate. In [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF], h i fulfilled the condition1 τi 0 h i (a)da = +∞, and it has been considered as a continuous non-decreasing function satisfying lim a→τi h i (a) = +∞. Now, in our work, we consider that the (continuous) cumulative distribution function h i is defined over an infinite support [0, +∞), is nondecreasing and it satisfies lim a→+∞ h i (a) = 1. Indeed, we consider

(A) (B) (C) (D) finite ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

finite finite finite finite

Res$ng phase Prolifera$ng phase Res$ng phase Prolifera$ng phase Res$ng phase Prolifera$ng phase Prolifera$ng phase Fig. 1. Cartoon approximate representation of some cell cycle models from the literature. Configuration (A) is the one in which cells may be quiescent during their entire life, or they can enter to a proliferating stage of finite length (as in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF], [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF], [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF]). In (B), the cell cycle is splited into four sub-phases G 1 , S, G 2 and M , each one has a finite duration (as in [START_REF] Avila | A new model of cell dynamics in Acute Myeloid Leukemia involving distributed delays[END_REF]), i.e. in (B) the length of the proliferating phase is finite since all the sub-phases were assumed to be finite. Next, the situation in (C) (as in [START_REF] Billy | Age-structured cell population model to study the influence of growth factors on cell cycle dynamics[END_REF]) is completeley different, since in this approach the cell cycle is described as an infinite support without an explicit resting phase G 0 (that can be associated with G 1 ). Thus, in (C), the lengths of the sub-phases G 1 , S, G 2 and M are infinite. Finally, we notice that (D) represents the case in which we consider that, at least, G 1 or G 2 may be of infinite lengths in the cell cycle, which is compacted in a single infinite phase, along with a separated infinite resting compartment G 0 . In summary, one notices that the configuration (D) separates quiescence from proliferation (as in (A)-(B)), while proliferation is of infinite length (as in (C)). Some slightly different models are considered for instance in [START_REF] Brikci | Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle[END_REF], where a molecular structured population (involving age-and-cyclin structured-PDEs) have been studied, considering that the length of the cell cycle is infinite.

that when the age a increases, the cells which do not die by apoptosis have an increasing probability to divide, and in fact the probability of division goes to 1 when a goes to infinity. The latter description captures the fact that a majority of cells may divide (if they do not die) and few of them may be arrested within the cycle. Next, we assume that [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF], and λ i (t) = 1-σ i (t). Moreover, we consider that for all t ≥ 0, the death rates satisfy:

λ i (t) ∈ [λ i , λ i ] ⊂ (0, 1), σ i (t) ∈ [σ i , σ i ] ⊂ (0,
d ri (t) ∈ [d ri , d ri ] ⊂ (0, ∞), and d pi (t) ∈ [d pi , d pi ] ⊂ (0, ∞). A direct consequence is that e -adpi ≤ e - a 0
dpi(m+t-a)dm ≤ e -ad pi .

(

Then, for all t ≥ 0, and for all a ∈ [0, +∞),

f i (a)e -adpi ≤ g i (t, a) ≤ f i (a)e -ad pi . ( 4
)
The mitosis function f i satisfies +∞ 0 f i (a)da = 1. This basically means that cells which do not die by apoptosis are intended to divide, but in infinite time.

For all i ∈ I n and t ≥ 0, d ri (t) is the death rate of the resting cells, while d pi (t) is the death rate of proliferating Fig. 2. A cartoon representation of the discrete maturity model of interest. Cells of the i-th maturity-generation, where i ∈ In = {1, . . . , n} , n > 1, are in a resting phase G 0 , or in proliferation (celldivision cycle) {G 1 , S, G 2 , M } of infinite support (i.e. few cells may be arrested in the cell cycle for unlimited time). In the general case, we can consider that all the involved biological parameters (rate of differentiation σ i ∈ (0, 1), rate of self-renew λ i = 1 -σ i , apoptosis rate d pi , death rate of resting cells d ri ) are time-varying. On the right, the flux due to the cell-plasticity functions (ξ i and ω i for all i ∈ In) is represented. However, in the current paper, all these plasticity feature are ignored here: these issues are in fact discussed in a separated study in [START_REF] Djema | Introducing Cell Plasticity Mechanisms into a Class of Cell Population Dynamical Systems[END_REF]. Thus, the model on which we focus in the present paper extends the one in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF] to the case of time-varying parameters and to the situation in which a minority of cells may be arrested for an infinite time in the cell cycle.

cells. The reintroduction function β i is decreasing and lim →∞ β i ( ) = 0. The function ξ i describes the cell loss by cell plasticity mechanisms, that we assume in this work to be negligible when compared to d ri . Next, σ i (t) represents the time-varying rate of differentiation and, consequently, λ i (t) = 1 -σ i (t) is the rate of self-renewal of the i-th cell generation. Similarly to ξ i , we consider that ω i , which quantifies the input from cell-plasticity features, is negligible in this work. We invite the interested reader to refer to [START_REF] Djema | Introducing Cell Plasticity Mechanisms into a Class of Cell Population Dynamical Systems[END_REF] for more detail on plasticity functions related to cancer. Finally we notice that in our work, apart from the aforementioned extension to the case of unlimited cell-cycle length, we are also considering that all the biological parameters (differentiation, selfrenewing and apoptosis rates), involved in our model, are time-varying, which is not the case of the similar earlier models ([20], [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF], [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF]).

Remark 1: The system (1) is positive, i.e. for positive initial conditions the trajectories are positive for all t ≥ 0 (see for instance a similar proof in the case of finite distributed delays in Proposition 1, [START_REF] Djema | Lyapunov stability analysis of a model describing hematopoiesis[END_REF]). Throughout the work, we consider only positive solutions of (1). The positivity of system (1) makes it possible to use some unusual non-quadratic Lyapunov functionals, which can be approximated at the origin by linear functionals [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF].

IV. Stability analysis of the 0-equilibrium

In this section, we analyze the stability properties of the origin of the model (1), since the aim of anticancer therapy is the eradication of unhealthy cells. More precisely, we prove in this section the following result: Theorem 1:

Let C i = +∞ 0 f i ( )e -d pi d .
The system (1) admits the 0-equilibrium as a globally exponentially stable equilibrium point if the inequalities:

d ri -(2λ i C i -1)β i (0) > 0, (5)
are satisfied for all i ∈ I n = {1, . . . , n}. If

d r1 -(2λ 1 C 1 -1)β 1 (0) < 0, (6)
then no positive solution converges to 0-equilibrium. Corollary 1: (i) Assume that the conditions

2λ i C i -1 < 0, (7)
are fulfilled for all i ∈ I n = {1, . . . , n}, then the trivial steady state is globally exponentially stable.

(ii) Assume that the conditions

2λ i -1 < 0, (8)
are fulfilled for all i ∈ I n = {1, . . . , n}, then the trivial steady state is globally exponentially stable. Remark 2: It is clear that if the inequalities (8) are satisfied, then the conditions (7) are also satisfied (since C i < 1, as d pi > 0). Similarly, we see that if the conditions (7) hold true then those in (5) are also verified, since d ri > 0. In fact, the conditions (5) are the direct generalization of the conditions 2L i C i -1 > 0 in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF] (see also [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF] and [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF]) to the case of infinite distributed delays and time-varying parameters. Biologically, one readily check that the conditions (8) are extreme conditions that ensures exponential eradication of unhealthy cells even if apoptosis is stalled to zero (i.e. d pi = 0 and C i = 1).

Remark 3: The condition ( 5) is a direct generalization of the well-known necessary and sufficient stability condition for the 0-equilibrium of the class of systems with finite distributed delays and constant parameters (introduced in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF]). More precisely, the same kind of stability condition has been provided in [START_REF] Adimy | Discretematurity structured model of cell differentiation with applications to myelogenous leukemia[END_REF] for local asymptotic stability of the origin, and in [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF] for global exponential stability of the origin, of the model with finite distributed delays. However, proving an equivalent result in the case of infinite delays requires a novel Lyapunov-Krasovskii functional, slightly different from the one already used in [START_REF] Djema | Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis[END_REF].

Proof: The proofs of Theorem 1 and Corollary 1 are not provided in this brief version (refer to [START_REF] Djema | Understanding Cell Dynamics in Cancer from Control and Mathematical Biology Standpoints: Particular Insights into the Modeling and Analysis Aspects in Hematopoietic Systems and Leukemia[END_REF]).

This assumption describes the fact that all the proliferating cells which do not die by apoptosis during the cell cycle, are obliged to divide before they reach the maximal age τ i .