Walid Djema
email: walid.djema@inria.fr

Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Catherine Bonnet
email: catherine.bonnet@inria.fr

Analysis of a Nonlinear Delay Differential-Difference Biological Model

Keywords: Lyapunov Functionals, Exponential Stability, Basin of attraction

Recently, a new model describing the cell dynamics in hematopoiesis was proposed. It can be described as a delay differential-difference model. Under some conditions on the biological parameters, it admits two equilibrium points. The first one is the 0-equilibrium and the second one, which does not always exist, is a strictly positive point. We propose a Lyapunov functional construction in order to investigate the stability properties of both equilibria. For the 0 equilibrium, we establish the global exponential stability when the positive equilibrium does not exist. For the positive equilibrium, we establish its local exponential stability, estimate the decay rate of solutions and provide a subset of its basin of attraction.

INTRODUCTION

Different research communities have been interested in the phenomenon of hematopoiesis over the last years. The complexity of the mechanism of blood cells formation and the multiple diseases that affect this process, including leukemia, made mathematical modeling of these phenomena essential in order to understand both the healthy and unhealthy hematopoiesis. A significant work has been done by [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF], improving the initial model proposed by [START_REF] Mackey | Unified hypothesis of the origin of plastic anemia and periodic hematopoiesis[END_REF]. Considerable works have followed thereafter covering both modeling and stability analysis of the dynamics of healthy and cancerous cells; see for instance: Özbay et al. (2008), Özbay et al. (2012), [START_REF] Avila | Analysis of a new model of cell population dynamics in acute myeloid leukemia[END_REF], Fridman et al. (2015), [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], [START_REF] Djema | Stability of Immature Cell Dynamics in Healthy and Unhealthy Hematopoiesis[END_REF] and recently [START_REF] Adimy | Age-Structured and Delay Differential-Difference Model Of Hematopoietic Stem Cell Dynamics[END_REF].

The present contribution is based on the latter work, where a new way to model the fast self-renewal dynamics of hematopoietic stem cells is introduced. The new model takes into account a subpopulation of cells that remains constantly active in the proliferating compartment. Its main interest is that it may explain the dynamics of cancer cells in the case of acute myeloid leukemia, where a high level production of abnormal immature white blood cells is observed.

The resulting model is a nonlinear delay differentialdifference system. It is worth mentioning that it can have bounded or unbounded solutions. The first equilibrium of interest is the 0-equilibrium which always exists. Biologically, it means the extinction of all the blood cells. This situation becomes interesting in the case where the studied model represents the dynamics of cancer cells that we want to eradicate. Under some conditions, a unique strictly positive steady state will exist in addition to the 0equilibrium. Biologically, the positive equilibrium reflects the survival of blood cells and its stability analysis offers a way to maintain the density of cells at a desired level and avoid the unbounded or periodic behaviors observed in some hematopoietic diseases.

From a mathematical point of view, it is well known that theory of Lyapunov functions offers some strong advantages when investigating the stability properties of time-delay systems, but unfortunately the construction of suitable Lyapunov functionals may be difficult. Some tools making possible to construct Lyapunov functionals for some nonlinear differential-difference equations are available (see, [START_REF] Pepe | On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equation[END_REF], [START_REF] Karafyllis | Stability results for systems described by coupled retarded functional differential equations and functional difference equations[END_REF], [START_REF] Gu | Lyapunov-Krasovskii Functional for Uniform Stability of Coupled Differential-Functional Equations[END_REF], and the references therein), or, when it is possible, one may formulate the problem in the neutral time-delay framework and take advantage of the existing literature in this field (see, for more information, [START_REF] Mazenc | Lyapunov Technique and Backstepping for Nonlinear Neutral Systems[END_REF] for nonlinear systems and [START_REF] Fridman | New LyapunovKrasovskii functionals for stability of linear retarded and neutral type systems[END_REF] for linear systems), but that does not apply when the trajectories are not differentiable. On the other hand, it is worth mentioning that even if an equilibrium is known to be asymptotically stable, we still need some explicit strict Lyapunov functionals in order to establish some robustness results [START_REF] Malisoff | Constructions of Strict Lyapunov Functions, Serie : Communications and Control Engineering[END_REF]. Besides the difficulties related to the construction of suitable functionals for nonlinear systems, the stability analysis may be more complicated due to some mathematical or practical considerations. Indeed, we wish to point out that it is more difficult to investigate the stability properties of piecewise solutions (see, for instance, [START_REF] Michel | Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Systems & Control: Foundations & Applications[END_REF]) than the uniformly continuous solutions. For instance, the Barbalat's lemma requires uniform continuity of solutions which is not the case of the studied model describing hematopoiesis. We also emphasize that, generally speaking, some robustness issues are expected when the studied models are dealing with living organisms, in contrast with other phenomena that are almost perfectly described by physical laws. It is worth mentioning that these obstacles are overcome by the knowledge of a strict Lyapunov functional. In light of previous comments, in this paper, we develop a Lyapunov approach to analyze the delay differentialdifference model of hematopoiesis. We construct two types of functionals that can be used to perform a robustness analysis.

The work is organized as follows: in the remainder of the introduction, the studied model is briefly introduced. In Section 2, we prove the global exponential stability of the 0-equilibrium via a linear functional, then, in Section 3, we use some quadratic functionals to analyze the local stability of the positive equilibrium and we provide a subset of its basin of attraction.

The reader is invited to refer to Sections 1-3 in [START_REF] Adimy | Age-Structured and Delay Differential-Difference Model Of Hematopoietic Stem Cell Dynamics[END_REF] to get the accurate presentation of the biological model represented in Figure 1, and analyzed in the present paper. The delay differential-difference system which interests us is given by: ẋ

(t) = -(δ + β(x(t))) x(t) + 2Le -γτ u(t -τ ), u(t) = β(x(t))x(t) + 2Ke -γτ u(t -τ ), t > 0, ( 1
)
where we consider that the parameters δ, K, L = 1 -K, γ and τ are strictly positive real numbers and K ∈ (0, 1).

x represents the total density of resting cells and u is the density of the new proliferating cells. The function β is continuous, decreasing and lim x→∞ β(x) = 0. A unique piecewise continuous solution (x(t), u(t)) exists for all t ≥ 0, when the system (1) is associated with the initial conditions x(0) ∈ R and ϕ u ∈ PC ([-τ, 0], R) [START_REF] Gu | Lyapunov-Krasovskii Functional for Uniform Stability of Coupled Differential-Functional Equations[END_REF]). Moreover, system (1) is positive (i.e. the solutions of system (1) associated with positive initial conditions are positive). In this paper we consider only the positive solutions of (1).

We consider that the nonlinearity β is locally Lipschitz and its expression is given by

β(x) = β(0) 1 + bx n (2)
where b, n and β(0) are strictly positive real numbers. (See, [START_REF] Mackey | Unified hypothesis of the origin of plastic anemia and periodic hematopoiesis[END_REF] and [START_REF] Adimy | Age-Structured and Delay Differential-Difference Model Of Hematopoietic Stem Cell Dynamics[END_REF] for some numerical values and their interpretation). To ease the notation, let us define the following constants:

K = 1 2 e γτ , (3)
µ = β(0) δ , (4)
K = (µ + 1)K -µ.
(5) Notice that K and µ are strictly positive.

STABILITY ANALYSIS OF THE 0-EQUILIBRIUM

We start by providing stability and instability results for the trivial equilibrium E 0 = (0, 0) of system (1). Biologically, E 0 reflects the extinction of all blood cells. In the case where the model (1) describes the dynamics of unhealthy hematopoietic stem cells, the stability analysis of E 0 may give a way to eradicate the cancerous cells. Our approach is slightly different from the one used in [START_REF] Adimy | Age-Structured and Delay Differential-Difference Model Of Hematopoietic Stem Cell Dynamics[END_REF] in the sense that we prove the exponential stability, with an estimate of the rate of convergence, via a construction of a strict Lyapunov functional. Theorem 1. For all K ∈ (0, K), (6) i) if the condition s := K -K > 0, (7) is satisfied, the origin of the system (1) is globally exponentially stable. ii) if s := K -K < 0, (8) then no positive trajectory converges to the origin of system (1). Remark 1. In Section 3, we shall prove that if ( 7) is satisfied, the origin is the unique equilibrium point of system (1). This explains why the stability results in Theorem 1 are global.

Proof. Let us introduce the following functional:

M(x(t), u t ) = x(t) + µ + 1 µ + t t-τ u(l)dl, (9) with
= - s 2(K -K)µ . ( 10
)
Notice that, since K ∈ (0, K), then < 0, when ( 7) is satisfied and > 0 when (8) holds.

We start by proving ii). Let us proceed by contradiction. We assume that the condition ( 8) is satisfied and a positive solution (x(t), u(t)) converges to the origin.

Since > 0, the functional M is positive on the positive orthant. Moreover, its derivative along the trajectories of (1) is

Ṁ(t) = -δ + µ + 1 µ + -1 β(x(t)) x(t) + - µ + 1 µ -+ µ + 1 µ + K K + 1 -K K u(t -τ ). ( 11
)
On the other hand, from (4), we notice that

-δ + µ + 1 µ + -1 β(0) = β(0) > 0. ( 12
)
Since x(t) converges to zero, and β is continuous and decreasing, we conclude that there exists a time instant t 1 > 0, such that for all t ≥ t 1 ,

-δ + µ + 1 µ + -1 β(x(t)) ≥ β(0) 2 > 0. ( 13
)
It follows from ( 11) and ( 13) that for all t ≥ t 1 ,

Ṁ(t) ≥ β(0) 2 x(t) + 1 µK -(µ + 1)K + µ + K + µ(K -K) u(t -τ ).
Since s = (µ+1)K -µ-K, the previous inequality rewrites as

Ṁ(t) ≥ β(0) 2 x(t) + 1 µK -s + µ(K -K) u(t -τ ) = β(0) 2 x(t) - s 2µK u(t -τ ),
where the last equality is a consequence of (10). Since s < 0 and > 0, we conclude that for all t ≥ t 1 , Ṁ(t) ≥ 0.

It follows that M(x(t), u t ) ≥ M(x(t 1 ), u t1 ) > 0. ( 14) Therefore, M does not converge to zero. On the other hand, M converges to zero since (x(t), u(t)) converges to the origin. This yields a contradiction.

Next, let us prove i). We consider the case where the inequality ( 7) is satisfied. An immediate consequence is that < 0.

First, to ensure that the functional M is positive on the positive orthant, let us check that µ+1 µ + > 0. From the definitions of and s, we get

µ + 1 µ + = 2 (µ + 1)K -µK -2K -s 2(K -K)µ . (15)
Since K ∈ (0, 1), we deduce that

(µ + 1)K -µK > (µ + 1)K -µ = K. It follows that µ + 1 µ + > 2(K -K) -s 2(K -K)µ = s 2(K -K)µ > 0. ( 16
)
Consequently, the functional M is positive on the positive orthant. Moreover, its derivative along the trajectories of (1) satisfies

Ṁ(t) = -δ + µ + 1 µ + -1 β(x(t)) x(t) - s 2µK u(t -τ ) ≤ -δ + µ + 1 µ -1 β(0) x(t) + β(x(t))x(t) - s 2µK u(t -τ ) = β(x(t))x(t) - s 2µK u(t -τ ), (17)
where the last equality is a consequence of (4).

By integrating (17), we get, for all t ≥ 0,

M(x(t), u t ) ≤M(x(0), ϕ u ) + t 0 β(x(m))x(m)dm - s 2µK t 0 u(m -τ )dm. ( 18
)
Since < 0 and s > 0, it follows from ( 9) and ( 18) that for all t ≥ 0, the trajectory x(t) is upper bounded by a strictly positive constant x b .

In order to complete the proof, let us introduce the following functional: (19) where ψ and ψ are positive real numbers to be selected later. From ( 16), observe that if we choose

W(x(t), u t ) =M(x(t), u t ) -ψ t t-τ u(l)dl + ψ t t-τ t m e l-t u(l)dldm,
ψ = min s 4(K -K)µ , s 2µK , ( 20
)
then the functional W is positive on the positive orthant and its derivative along the trajectories of ( 1) is:

Ẇ(t) = Ṁ(t) -ψ [u(t) -u(t -τ )] + ψ - t t-τ e l-t u(l)dl + τ u(t) -ψ t t-τ t m
e l-t u(l)dldm.

Selecting ψ = ψ τ , we obtain

Ẇ(t) = Ṁ(t) + ψu(t -τ ) -ψ t t-τ e l-t u(l)dl -ψ t t-τ t m e l-t u(l)dldm ≤ Ṁ(t) + ψu(t -τ ) -ψe -τ t t-τ u(l)dl -ψ t t-τ t m e l-t u(l)dldm. (21)
Then from ( 17) and ( 21), we deduce that

Ẇ(t) ≤ β(x(t))x(t) - s 2µK -ψ u(t -τ ) -ψe -τ t t-τ u(l)dl -ψ t t-τ t m e l-t u(l)dldm ≤ β(x(t))x(t) -ψe -τ t t-τ u(l)dl -ψ t t-τ t m e l-t u(l)dldm, (22)
where the last inequality is a consequence of (20).

Combining the facts that < 0, β is a continuous decreasing function and x(t) is upper bounded by x b > 0, for all t ≥ 0, we deduce that

Ẇ(t) ≤ β(x b )x(t) -ψe -τ t t-τ u(l)dl -ψ t t-τ t m e l-t u(l)dldm. (23)
Finally, we conclude that there exists a strictly positive s, such that Ẇ(t) ≤ -sW(x(t), u t ).

(24) We deduce that W(x(t), u t ) ≤ e -st W(x(0), ϕ u ).

(25) Therefore,

x(t) ≤ e -st W(x(0), ϕ u ). ( 26)

It follows that x converges exponentially to zero. Consequently, form the second equation in (1), we prove that u converges exponentially to zero, since the condition (6) implies that 2Ke -γτ < 1. Remark 2. Theorem 1 gives a necessary and sufficient condition for the exponential stability of the trivial steady state when K ∈ (0, K) (Eq. ( 6)). In the case where K ≥ 1, it follows that Theorem 1 is available for all K ∈ (0, 1). However, if K < 1, then the conditions ( 7) and ( 8) give stability/instability conditions only for K ∈ (0, K). In order to complete the analysis, we show that if K ≤ 1, then no positive trajectory converges to the origin for all K ∈ (K, 1), by similar arguments as in ii) in Theorem 1, where we consider the functional M, defined in ( 11) with any strictly positive constant. Moreover, notice that K > K implies that 2Ke -γτ > 1. Therefore, using the second equation in (1), we observe that in this case system (1) has unbounded solutions.

ANALYSIS OF THE POSITIVE EQUILIBRIUM

Contrary to the trivial equilibrium, E 0 = (0, 0), studied in the previous part, the strictly positive equilibrium E = (x e , u e ) does not always exist. In this section, first, we recall the conditions for its existence then we investigate its stability properties by a Lyapunov approach.

In order to analyze the stability of the positive equilibrium, the authors in [START_REF] Adimy | Age-Structured and Delay Differential-Difference Model Of Hematopoietic Stem Cell Dynamics[END_REF] develop a method which consists in writing the model (1) as a special case of a neutral differential system and then investigating the local stability properties of the linear approximation of the neutral system by a frequency approach. Notice that the latter procedure requires the solutions of the system (1) to be differentiable.

In the present contribution, without extra assumptions on the system (1), which may admit discontinuous solutions, we prove under suitable sufficient conditions that E is locally exponentially stable by a direct Lyapunov approach, which allows us to determine an approximation of its basin of attraction.

Existence of a strictly positive equilibrium point

If a positive equilibrium E = (x e , u e ) exists, then it satisfies:

(δ + β(x e ))x e = 2Le -γτ u e , β(x e )x e = (1 -2Ke γτ )u e . ( 27
)
From the second equation in ( 27), it follows that

u e = K K -K β(x e )x e . ( 28
)
From the previous equation, observe that the existence of E implies necessarily that K < K. By substituting u e in the first equation in ( 27), and since x e is not zero, we get

1 -K K -K β(x e ) = δ.
Since δ > 0 and K < K, we deduce that K < 1.

(29) Since β is continuous, decreasing and lim a→∞ β(a) = 0, it follows from (28) that the existence and uniqueness of E are guaranteed by

1 -K K -K > 1 µ .
Since 1 -K > 0, K -K > 0 and K = (µ + 1)K -µ, the previous inequality is equivalent to K < K.

(30)

We conclude that the condition K ∈ (K, K) is necessary for the existence of E. This statement justifies what we have mentioned in Remark 1. We can easily check that if (29) holds, then K < K, and consequently E may exist. Next, since K may be positive or negative, and using the fact that K > 0, we conclude that we have proved the following: Proposition 2. A unique strictly positive steady state E = (x e , u e ) exists if and only if the conditions:

0 < K < K < µ µ + 1 , (31)
or, 0 < K < K < K < 1, (32) are satisfied. Remark 3. In the previous statement, the uniqueness concerns the existence of the positive equilibrium E. We recall that the 0-equilibrium of system (1) always exists.

Stability properties of the positive equilibrium point

In this section, we assume that the positive equilibrium E exists. First, the changes of coordinates x = x -x e and ũ

= u -u e give        ẋ(t) = -(δ + β(x(t) + x e )) (x(t) + x e )
+2Le -γτ (ũ(t -τ ) + u e ), ũ(t) = -u e + β(x(t) + x e )(x(t) + x e ) +2Ke -γτ (ũ(t -τ ) + u e ), t > 0,

or, equivalently,

       ẋ(t) = -δ x(t) -β(x(t) + x e )(x(t) + x e )
+2Le -γτ ũ(t -τ ) -δx e + 2Le -γτ u e , ũ(t) = β(x(t) + x e )(x(t) + x e ) + 2Ke -γτ ũ(t -τ ) +(2Ke -γτ -1)u e , t > 0.

(34) Analyzing the stability properties of the origin of (34) seems to be a difficult task. Therefore, we write the model ( 34) in an equivalent form that eases its analysis. (2) . We deduce that for all t > 0, the system (34) is equivalent to

      
ẋ(t) = -δ x(t) -β(x e )x e -θx(t) -I(x(t)) +2Le -γτ ũ(t -τ ) -δx e + 2Le -γτ u e , ũ(t) = β(x e )x e + θx(t) + I(x(t))

+2Ke -γτ ũ(t -τ ) + (2Ke -γτ -1)u e .

(38)

Using ( 27), we get for all t > 0, ẋ(t) = -(δ + θ) x(t) -I(x(t)) + 2Le -γτ ũ(t -τ ), ũ(t) = θx(t) + I(x(t)) + 2Ke -γτ ũ(t -τ ).

(39) Observe that, due to the shifting, the trajectories are no longer positive (x > -x e , ũ > -u e , where x e > 0 and u e > 0). To analyze the model (39), we introduce the following function and functionals

Q(a) = 1 2 a 2 , (40)
Θ(ũ t ) =

Fig. 1 .

 1 Fig. 1. Schematic representation of blood cells formation.

  Using the Taylor formula, we can write β(z + x e )(z + x e ) = β(x e )x e + θz + I(z), (35) with θ = β(x e ) + β (x e )x e ,

  , let us state the following result: Claim 4. For all x e > 0 and x > -x e , there exist two strictly positive constants z 1 and z 2 , such that|I(x)| ≤ z 1 Q(x) and |I(x)| ≤ z 2 |x|,(43)where,z 1 = 2c 4 β(0), (44) z 2 = β(0) max c 4 , 2 max{b, b -1 } + |c 1 | , (45) with c 1 = 1 h -nbx n e h 2 , c 2 = x e , c 3 = 1nb(n -1)(1 + b(2x e ) n ) (|c 2 | + |c 3 |) max b, b b(2x e ) n ) |c 3 | h 2 .

This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissement d'Avenir" program, through the "iCODE Institute project" funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.

Proof. The proof of this simple result is omitted here.

Next, let us define the following constants which depend on the biological parameters:

K = 2Ke -γτ 2 + 2K|θ|e -γτ -1, (47)

and

Now we are ready to prove the following result: Theorem 5. If the conditions 0 < w < w, (50) are satisfied, then the positive equilibrium E of the nonlinear system (1) is locally exponentially stable. Moreover, an open subset of the basin of attraction of E can be determined.

Proof. The derivative of the functional Q(x(t)) along the trajectories of (39) satisfies

where the last inequality is a consequence of (43). Using the inequality x(t)ũ(t -τ

On the other hand, we prove that Q(ũ(t)) satisfies,

where the last inequality is a consequence of the inequalities

By grouping the terms we obtain:

where

, a 2 = Ke -γτ z 1 , and ξ and K are the constants defined respectively in ( 46) and ( 47). Using (53), we deduce that the derivative of the functional Θ, defined in (41), along the trajectories of (39) satisfies

Now, let us assume that the condition (50) is satisfied.

A direct consequence is that K < 0. Furthermore, we introduce the strictly positive constant

By combining ( 52) and ( 54), we conclude that, since K = 0, the derivative of the functional: V (x(t), ũt ) = Q(x(t)) + ωΘ(ũ t ), (56) along the trajectories of (39), satisfies the inequality,

with ξ > 0 and K < 0. From (55), we conclude that

where q = min {ξ (ω -ω) , -K (ω -ω)} > 0. On the other hand, the derivative of the functional Λ, introduced in (42), along the trajectories of (39), satisfies

where the last inequality is a consequence of (53). Now, by combining ( 59) and ( 58), we straightforwardly conclude that the derivative of the functional

where p = q 2τ max{ξ,K+1} , along the trajectories of (39), satisfies

Since |x(t)| ≤ 2U(x(t), ũt ), we deduce that

Therefore, for all initial conditions x(0) and φu , satisfying

the derivative of the functional U satisfies:

with r = min 1, pe -τ ω , q 4 . By virtue of the functional U, we conclude that x converges exponentially to its origin with a decay rate smaller or equal to r 2 . Using the second equation in (39), one can prove that ũ converges exponentially to its origin, since the condition K < 0 implies that 2Ke -γτ < 1. Notice that the set of all initial conditions satisfying the condition (63) provides an estimate of the basin of attraction of the origin of the shifted system (39). Therefore, the set

is a subset of the basin of attraction of the strictly positive equilibrium E of the positive system (1).

CONCLUSION

We studied the exponential stability of the 0-equilibrium of a differential-difference biological model by constructing a suitable linear functional. The local exponential stability of a positive equilibrium was performed via quadratic functionals and a subset of its basin of attraction was determined. Future work will concern the analysis of more refined models that take into account the coexistence of healthy and unhealthy cells.


Walid Djema

Frédéric Mazenc

Catherine Bonnet

Stability of Immature Cell Dynamics in Healthy and Unhealthy Hematopoiesis

Keywords:

A nonlinear system with distributed delays is studied. The model takes into account the fast self-renewal dynamics observed in Acute Myeloid Leukemia (AML). Stability and instability conditions are derived for the zero equilibrium of the AML model. Notice that, biologically, the zero equilibrium means the eradication of all malignant cells, which is the aim of the anti-AML therapy. The novelty of this work is that we consider time-varying biological parameters reflecting the fact that the differentiating and self-renewing parameters become time-variant both under the effect of the disease and the drugs. A simpler case of the studied model is well suited to describe the healthy hematopoiesis. In this case, we focus on the stability of the positive steady state since it reflects the surviving of all healthy blood cells. Via a construction of a novel Lyapunov functional, we derive sufficient conditions for the local exponential stability of the favourable equilibrium and we propose an estimate of its basin of attraction.

I. INTRODUCTION

Time-delay systems (TDS) appear in many modelling areas such as traffic flow, chemistry, communication networks, economics and biology (see [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF], [9], and references therein). The stability analysis of these models is a key issue and it is well known that the Lyapunov theory makes it possible, in some cases, to estimate rates of convergence of solutions, to establish ISS robustness results and to determine approximations of the basins of attraction for locally asymptotically stable equilibrium points (see [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF], [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF], [START_REF] Mazenc | Lyapunov-Krasovskii Functionals and Application to Input Delay Compensation for Linear Time-Invariant Systems[END_REF] for more information). However, the search for a suitable Lyapunov functional for nonlinear TDS is generally quite difficult, since no systematic methods apply [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF], and this is the case of the studied model of hematopoiesis.

In a healthy bone marrow, the hematopoeitic stem cells (HSC) develop into white blood cells, red blood cells, platelets and many other blood cells. The HSC are immature cells able to produce cells with the same maturity level and to differentiate into more mature and specific cells. In the case of acute myelogenous leukemia (AML), an overproduction of abnormal blasts is observed. These malignant cells are unable to develop further into mature white blood cells and prevent production of adequate numbers of other types of healthy blood cells. The mathematical model that describes healthy or unhealthy hematopoiesis is given by nonlinear transport Walid Djema, Frédéric Mazenc and Catherine Bonnet are with Inria, Université Paris-Saclay, L2S-CentraleSupélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France.

walid.djema@inria.fr, frederic.mazenc@l2s.centralesupelec.fr, catherine.bonnet@inria.fr.

equations, which are transformed to a nonlinear system with distributed delays using the method of characteristics. The new results in this contribution are outlined according to the following organization of the paper:

In Section II, we complete the analysis of the zero equilibrium of the multi-stage AML model analyzed recently in [START_REF] Avila | Analysis of a new model of cell population dynamics in acute myeloid leukemia[END_REF] and [START_REF] Fridman | Stability of the cell dynamics in Acute Myeloid Leukemia[END_REF], by considering the case of time-varying differentiating, self-renewing and fast self-renewing rates.

In Section III, we focus on the model without fast selfrenewal dynamics and with constant biological parameters. The latter case is well suited to describe the process of healthy hematopoiesis. This simpler model was proposed by Adimy in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] and partially analyzed in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF] and [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF]. Recall that the positive equilibrium is the most interesting from a practical point of view because it models the normal process of hematopoiesis, where all the generations of healthy cells survive. Moreover, several diseases may disrupt strongly the hematopoiesis process and a stability analysis of the linear approximation is insufficient. Indeed, as mentioned in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], estimates of the domain of attraction of the positive equilibrium point and of the exponential decay rate for the nonlinear full-order system have not been found and these are still important open problems. In the present paper, first, exponential stability of the positive steady state is established and decay rate of solutions are approximated. Moreover, an open subset of its basin of attraction is provided via a construction of a novel Lyapunov functional. This region depends on the different biological parameters involved in the biological process of hematopoiesis.

II. STABILITY OF THE MODEL WITH FAST SELF-RENEWAL

DYNAMICS

A schematic representation of the cell dynamics is given in Figure 1. For all i ∈ I n = {1, ..., n}, by x i we denote the total population density of resting cells and by xi the fast self-renewing cells, which have n ≥ 1 stages of maturity. The constant δ i is the death rate of quiescent cells. The function g i (a) is defined by g i (a) = e -γia f i (a) where f i is a density function describing the mitosis and is such that τi 0 f i (a)da = 1, and γ i is the death rate of the proliferating cells (apoptosis). Proliferating cells can divide between the moment they enter the proliferating phase and a maximal age τ i > 0. The reintroduction functions β i (x i ) and βi (x i ) from the resting to the proliferating phases are considered to be nonlinear, continuous, decreasing functions, and lim a→+∞ βi (a) = lim a→+∞ β i (a) = 0. The total population densities x i and xi are described by the following time-delay system, for all i ∈ I n = {1, . . . , n},

     ẋi (t) = -(δ i + β i (x i (t))x i (t) + L i (t) i (w i , wi ) +2K i-1 (t) i-1 (w i-1 , wi-1 ) ẋi (t) = -wi (x i (t)) + Li (t) i (w i , wi ), t ≥ 0, (1)
where, with an abuse of notation,

i (w i , wi ) = τi 0 g i (a)[w i (x i (t -a)) + wi (x i (t -a))
]da, where a is the age of cells. We assume that the functions L i (•) and Li (•) are of class C 0 , for all i ∈ I n , such that L i (t

) := 2σ i (t)(1 - K i (t)), Li (t) := 2(1 -σ i (t))(1 -K i (t)), K 0 = 0,
for all t ≥ 0. The rates K i (t) represent the proportion of cells that differentiate, while the functions 1 -σ i (t) characterize the probability of fast self-renewal at the time instant t. Moreover, we introduce for later use the parameters

C i = τi 0 g i (a)da, for all i ∈ I n . (2)
Originally, the model described by (1) was introduced in [START_REF] Avila | Analysis of a new model of cell population dynamics in acute myeloid leukemia[END_REF] with constant parameters K i , L i , Li and σ i , for all i ∈ I n . Since AML is characterized by a blockade in the differentiation process, which becomes also a target for some drugs already used in the treatment of this disease, we consider in this paper the time-varying system [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF].

Since each proliferating cell, which does not die by apoptosis and completes its mitosis, gives birth to two daughter cells, we deduce that K i (t) ∈ [0, 1] for all t ≥ 0. In fact, we assume that K(t) ∈ [K i min , K i max ] ⊂ (0, 1), for all t ≥ 0, meaning that even for an unhealthy hematopoiesis, the differentiating process will not completely vanish. More precisely, in the AML case, due to the blockade in the differentiation process, we expect that lim t→∞ K(t) = K i min = 0. Similarly, we consider the case where for all t ≥ 0,

σ i (t) ∈ [σ i min , σ i max ] ⊆ [0, 1].
From medical practice we know that, in the AML case, a blockade in the differentiation process (i.e. lim t→∞ K i (t) = K i min ) and a high rate of fast selfrenewing (i.e. lim t→∞ σ i (t) = σ i min ) are simultaneously observed. This fact yields to lim t→∞ Li (t) = Li max , in the AML case, with Li max = 2(1 -

σ i min )(1 -K i min ).
The situation where σ i = 0 describes the worst leukemic case in which all the proliferating cells, that complete their mitosis, will choose the fast self-renewing process. On the other hand, the case where σ i = 1, and all the parameters involved in the model are constant, is well suited to the healthy hematopoiesis as discussed in Section III.

Moreover, we define for all i ∈ I n ,

Li (t) := L i (t) + Li (t) = 2 (1 -K i (t)) , t ≥ 0. Notice that L(t) ∈ [ Li min , Li max ] ⊂ (0, 2), for all t > 0, where Li max = 2(1 -K i min ).
It is easy to prove that system (1) is a positive system (i.e. for nonnegative initial conditions, the solutions are nonnegative). Throughout this section, we employ the positivity of the system to construct a suitable Lyapunov-Krasovskii functional in order to investigate the stability properties of the overall system. We start with the following Proposition: Proposition 1: If the condition

C i Li max < 1, (3)
for all i ∈ I n , is not satisfied, then there exists a system in the family of systems (1) whose origin is not globally asymptotically stable.

Proof: We prove Proposition 1 by contradiction. Let us assume that (3) is not satisfied for a given j ∈ I n and that all the positive solutions of the system (1), with for all i ∈ I n , Li (t) = Li max , for all t ≥ 0, converges to the origin.

Observe that

ẋi (t) ≥ -wi (x i (t))+ Li max t t-τi g i (t-a) wi (x i (a))da (4)
Let us introduce the functionals

h i (x it ) = t t-τi t m g i (m -a + τ i ) wi (x i (a))dadm. (5) Simple calculations give ḣi (t) = C i wi (x i (t)) - t t-τi g i (t -a) wi (x i (a))da. (6)
From ( 4) and ( 6), it follows that the derivatives of the functionals H i (x it ) = xi (t) + Li max h i (x it ), along the trajectories of (1), satisfy

Ḣi (t) ≥ (C i Li max -1) wi (x i (t)). (7)
Since there exists j ∈ I n such that C j Lj max > 1, we get Ḣj (t) ≥ 0. It follows that, H j (x jt ) ≥ H j (0) > 0. Therefore, H j (x jt ) does not converge to zero. But we consider a solution that converges to the origin, which implies that H j (x jt ) converges to zero. This yields a contradiction. Remark 1: If σ i min = 0, we obtain Li max = Li max . Therefore, from Proposition 1, we observe that if the condition C i Li max < 1 is not satisfied for all i ∈ I n , then there exists a system (1) such that its origin is not globally asymptotically stable. Consequently, we can not establish global exponential stability of the origin of system (1) if the condition C i Li max > 1 holds for some i ∈ I n . Now, let us state and prove the following result: Theorem 1: For all i ∈ I n , if the conditions

C i Li max < 1, (8)
are satisfied, then all the positive trajectories of (1) converge exponentially to the origin.

Proof: First, we observe that when the conditions (8) are satisfied, then for all λ i ∈ 1, 1+Ci Li max 2Ci Li max , the following inequalities are verified

1 -λ i C i Li max > 1 -C i Li max 2 > 0. (9)
Next, let us introduce, for all i ∈ I n , the functionals defined along the trajectories of system (1), by

v i (x it , xit ) = t t-τi t m e -ρi(t-m-τi) g i (m -a + τ i ) × [w i (x i (a)) + wi (x i (a))] dadm, (10)
where ρ i are strictly positive constants to be selected later. The derivatives of the functionals [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], along the trajectories of (1), satisfy

vi (t) ≤ -ρ i v i (x it , xit ) -i (w i , wi ) + [w i (x i (t)) + wi (x i (t))] e ρiτi C i . ( 11
)
Let us introduce the following functional for the first compartment of unhealthy cells

V 1 (x 1t , x1t ) = x 1 (t) + x1 (t) + λ 1 L1 max v 1 (x 1t , x1t ), (12)
where λ 1 satisfies (9). Using [START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF], it follows that

V1 (t) ≤ -δ 1 + 1 -λ 1 e ρ1τ1 L1 max C 1 β 1 (x 1 (t)) x 1 (t) -1 -λ 1 e ρ1τ1 L1 max C 1 w1 (x 1 (t)) -λ 1 ρ 1 L1 max v 1 (x 1t , x1t ) + L1 (t) -λ 1 L1 max 1 (w 1 , w1 ). (13)
Since λ 1 satisfies (9). It follows that by selecting any

ρ 1 ∈ 0, 1 τ 1 ln 1 + λ 1 C 1 L1 max 2λ 1 C 1 L1 max , (14)
we obtain 1 -λ 1 e ρ1τ1 L1 max C 1 > 1-λ1 L1 max C1 2 > 0. Moreover, since L1 (t) ≤ L1 max , we deduce that (13) gives

V1 (t) ≤ -[δ 1 + θ 1 β 1 (0)] x 1 (t) -θ 1 w(x 1 (t)) -λ 1 L1 max ρ 1 v 1 (x 1t , x1t ) -λ 1 1 (w 1 , w1 ), (15)
where the constants θ 1 = 1 -λ 1 e ρ1τ1 L1 max C 1 and λ 1 = (λ 1 -1) L1 max are strictly positive. Since the right hand side of (15) is always nonpositive, we deduce that by integrating (15), we obtain for all t ≥ 0,

V 1 (x 1t , x1t ) ≤ V 1 (ϕ x1 , ϕ x1 ). ( 16
)
This means in particular that x(t) is bounded by a constant x1b > 0. Since the function β1 is decreasing, it follows that, for all t ≥ 0, the inequality (15) gives

V1 (t) ≤ -[δ 1 + θ 1 β 1 (0)] x 1 (t) -θ 1 β1 (x 1b )x 1 (t) -λ 1 L1 max ρ 1 v 1 (x 1t , x1t ) -λ 1 1 (w 1 , w1 ). ( 17
)
We conclude that for all t ≥ 0,

V1 (t) ≤ -s1 V 1 (x 1t , x1t ) -λ 1 1 (w 1 , w1 ), (18)
where, s1 = min

δ 1 + θ 1 β 1 (0), θ 1 β1 (x 1b ), λ 1 ρ 1 L1 max .
By virtue of the functional V 1 , we conclude that the origin of the subsystem i = 1 is globally exponentially stable, with a decay rate smaller than s1 . Next, in the rest of the proof, we consider the case where the conditions (8) are satisfied for any number of compartments (i ∈ I n ). Let us introduce the following functional that takes into account the first and the second generations of cells:

V 2 (X 2t ) =x 2 (t) + x2 (t) + λ 2 L2 max v 2 (x 2t , x2t ) + 2K 1 max λ 1 V 1 (x 1t , x1t ), (19)
with X 2 = (x 1 , x1 , x 2 , x2 ). Similarly to the case i = 1, we select ρ 2 such that the derivative of the functional (19) along the trajectories of (1) satisfies

V2 (t) ≤ -[δ 2 + θ 2 β 2 (0)] x 2 (t) -θ 2 w2 (x 2 (t)) -λ 2 L2 max ρ 2 v 2 (x 2t , x2t ) + 2K 1 (t) 1 (w 1 , w1 ) -λ 2 2 (w 2 , w2 ) + 2K 1 max λ 1 V1 (t). (20)
Combining ( 20) and ( 18), we deduce that for all t ≥ 0,

V2 (t) ≤ -[δ 2 + θ 2 β 2 (0)] x 2 (t) -θ 2 β2 (x 2b )x 2 (t) -λ 2 L2 max ρ 2 v 2 (x 2t , x2t ) -λ 2 2 (w 2 , w2 ) - 2s 1 K 1 max λ 1 V 1 (x 1t , x1t ) -2(K 1 max -K 1 (t)) 1 (w 1 , w1 ). (21)
Since K 1 (t) ≤ K 1 max , we straightforwardly deduce that there exists s2 > 0 such that, for all t ≥ 0,

V2 (t) ≤ -s 2 V 2 (X 2t ) -λ 2 2 (w 2 , w2 ). (22)
Next, by induction, one can readily determine constants a i > 0, b i > 0 and sn > 0, such that the derivative of the functional

V n (X nt ) = n i=1 [x i (t) + a i xi (t) + b i v i (x it , xit )] , (23)
with X n = (x 1 , x1 , ..., x n , xn ), along the trajectories of (1), satisfies

Vn (t) ≤ -s n V n (X nt ), with sn > 0. (24)
From the inequality (24) and the properties of the functional V n , we conclude that the origin of the system (1) is globally exponentially stable with a decay rate smaller than sn .

III. STABILITY OF THE MODEL DESCRIBING HEALTHY

HEMATOPOIESIS

In this section we focus on a simpler form of system (1) which fits well with the healthy hematopoiesis. The model of interest is in the form ẋi

(t) = -(δ i + β i (x i (t))) x i (t) + 2L i i (w i ) + 2K i-1 i-1 (w i-1 ), (25)
with i (w i ) = t t-τi g i (t -a)w i (x i (a))da, to ease the notation of the distributed delayed terms. Observe that the system (1) is equivalent to the system (25) when xi ≡ 0 and σ i ≡ 1, for all i ∈ I n , as illustrated in Figure 2. In other terms, we consider that the fast self-renewal process is observed only in leukemic cell dynamics (System (1)), but not in the healthy hematopoiesis (System (25)). Compared to Section II, the analysis of system (25) is different since the survival of all the healthy cells is the purpose of normal hematopoiesis. More precisely, we focus on the strictly positive steady state X e = (x e 1 , ..., x e n ), x e i > 0, of system (25), since it reflects the surviving of all the generations of blood cells. All the biological parameters involved in (25) are strictly positive constants with the same definitions introduced in Section II. The rates of differentiation, K i , and selfrenewing, L i , are positive constants such that K i ∈ (0, 1) and L i = 1 -K i , for all i ∈ I n . Furthermore, we introduce for later use the following parameters, for all i ∈ I n ,

α i = 2L i C i -1, (26)
where C i are the constants defined in (2). The model described by (25) was first proposed in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF], widely analyzed by an input-output approach in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF] and recently by a Lyapunov approach in [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF]. Using a new Lyapunov approach, we extend the results by establishing the local exponential stability under the same conditions that are sufficient for the local asymptotic stability, proposed in [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], and by determining a subset of the basin of attraction of the strictly positive equilibrium of the nonlinear system. Now, let us recall some useful results: i) The solutions of system (25) associated with positive initial conditions are positive. Throughout our work, we consider only positive solutions of (25).

ii) We assume that α i > 0 is satisfied for i ∈ I n . A unique strictly positive equilibrium X e may exist and the following proposition gives a necessary and sufficient condition that ensures its existence.

Proposition 2: Under the previous assumptions, the system (25) admits a strictly positive equilibrium X e = (x e 1 , ..., x e n ) if and only if

β 1 (0) > δ 1 /α 1 . ( 27
)
The reader is invited to refer to [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], for the proofs of results given above. In the remainder of this work, we assume that α i > 0 and that a unique positive steady state X e exists.

We begin the analysis by performing the change of coordinates xi = x i -x e i . From (25), it follows that ẋi

(t) = -δ i [x i (t) + x e i ] -w i (x i (t) + x e i ) + 2L i i (w i ) + 2K i-1 i-1 (w i-1 ), (28)
with i (w i ) = t t-τi g i (t -a)w i (x i (a) + x e i ) da. We are looking for a new representation of (28) that eases the analysis of its origin. Observe that,

w i (x i + x e i ) = w i (x e i ) + µ i xi + R i (x i ), ( 29
)
where

µ i = ∂wi ∂xi xi=x e i = β i (x e i ) + β i (x e i )
x e i , and

R i (x i ) = x e i +xi x e i [x i + x e i -l]w (2) i (l)dl. ( 30
)
As an immediate consequence,

ẋi (t) = -β i * xi (t) + 2L i µ i t t-τi g i (t -a)x i (a)da -R i (x i (t)) + 2L i t t-τi g i (t -a)R i (x i (a))da + 2K i-1 µ i-1 t t-τi-1 g i-1 (t -a)x i-1 (a)da + 2K i-1 t t-τi-1 g i-1 (t -a)R i-1 (x i-1 (a))da, (31)
with β i * = δ i +µ i . If the solutions of system (31) converge to the origin, then the positive solutions of system (25) converge to the positive steady state X e . Due to the shifting, we observe that the solutions of system (31) are not positive. Consequently, contrary to Section II, we can not take advantage from linear functionals constructions. This observation motivates the introduction of the following quadratic function and functionals:

Q(a) = 1 2 a 2 , (32)
Ω i (ϕ it ) = t t-τi t l g i (l -a + τ i )Q(ϕ i (a))dadl, (33)
Λ i (ϕ it ) = t t-τi e l-t t l g i (l -a + τ i )Q(ϕ i (a))dadl, (34)
and the following constants

ς i = β i * -2L i |µ i |C i = δ i + µ i -2L i |µ i |C i . (35)
We observe the similarity between the functionals Ω i and Λ i , inspired from the construction of Kolmanovskii and Myshkis [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF]. For technical reasons, we use a weighted combination between them to compensate the distributed delayed terms and estimate the exponential decay rate.

A. Exponential stability of the linear approximation

We extend the results in [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF] by designing a novel Lyapunov-Krasovskii functional implying local exponential stability. Let us prove the following result:

Theorem 2: The conditions

ς i > 0, (36)
for all i ∈ I n , ensure that the origin of (31) is locally exponentially stable.

Proof: The linear approximation of the system (31) at its origin is given by

żi (t) = -β i * z i (t) + 2L i µ i t t-τi g i (t -a)z i (a)da + 2K i-1 µ i-1 t t-τi-1 g i-1 (t -a)z i-1 (a)da. (37)
Let us define for all i ∈ I n the following functionals

S i (z it ) = 1 2 Q(z i (t)) + L i |µ i |Ω i (z it ), (38)
and

N 1 (z 1t ) = S 1 (z 1t ) + ς 1 2C 1 Λ 1 (z 1t ), (39)
and for all i ∈ {2, ..., n},

N i (z it , z i-1t ) = S i (z it ) + ς i 2C i Λ i (z it ) + ψ i Λ i-1 (z i-1t ), (40)
where ψ i is an appropriate strictly positive constant to be determined later for all i ∈ {2, ..., n}. First, observe for later use that the derivatives of the functionals Ω i and Λ i , for all i ∈ I n , along the trajectories of (37) satisfy,

Ωi (t) = C i Q(z i (t)) - t t-τi g i (t -a)Q(z i (a))da, (41) Λi (t) ≤ -Λ i (z it ) -e -τi t t-τi g i (t -a)Q(z i (a))da + C i Q(z i (t)), (42)
where the last inequality is a consequence of the definition of the constants C i given by (2). Next, For the sake of clarity, we decompose the proof of Theorem 2 in two steps: we start by the first generation of cells (i = 1), then we generalize to any number of compartments i ∈ I n .

1) Lyapunov-Krasovskii functional for the first compartment: Using the inequality |z

1 (t)z 1 (a)| ≤ 1 2 |z 1 (t)| 2 + 1 2 |z 1 (a)| 2
, we prove that the derivative along the trajectories of (37) of Q(z 1 (t)), where Q is the function defined in (32), satisfies the inequality

1 2 Q(t) ≤ [-β 1 * + L 1 |µ 1 |C 1 ] Q(z 1 (t)) + L 1 |µ 1 | t t-τ1 g 1 (t -a)Q(z 1 (a))da. (43)
Using (41), ( 42) and (43), we prove that the derivative of the functional N 1 , defined in (39), along the trajectories of (37), satisfies

Ṅ1 (t) ≤ - ς 1 2 Q(z 1 (t)) - ς 1 2C 1 Λ 1 (z 1t ). (44)
On the other hand, from the definition of N 1 , we readily check that

N 1 (z 1t ) ≤ 1 2 Q(z 1 (t)) + s 1 Λ 1 (z 1t ), (45)
with s 1 = ς1+2L1C1|µ1|e τ 1

2C1

. From (44) and (45), we get

Ṅ1 (t) ≤ -ς1 N 1 (z 1t ) - ς 1 4 Q(z 1 (t)), (46)
for all ς1 ∈ 0, min ς1 2 , ς1 ς1+2L1C1|µ1|e τ 1 . We conclude from (46) that the origin of the subsystem (i = 1) is exponentially stable.

2) Lyapunov-Krasovskii functional for the overall system: Now, using the inequalities:

|z i (t)z i (a)| ≤ Q(z i (t)) + Q(z i (a), for all i ∈ I n , and |z i (t)z i-1 (a)| ≤ ξ i Q(z i (t)) + 1 ξi Q(z i-1 (a)
), with ξ i > 0, for all i > 1, we prove that the derivatives of the functions Q(z i (t)), for all i > 1, along the trajectories of (37), satisfy

1 2 Q(t) ≤ [-β i * + L i |µ i |C i ] Q(z i (t)) + L i |µ i | t t-τi g i (t -a)Q(z i (a))da + ξ i K i-1 |µ i-1 |C i-1 Q(z i (t)) + K i-1 |µ i-1 | ξ i t t-τi-1 g i-1 (t -a)Q(z i-1 (a))da.
We observe that, for i > 1, the derivatives of the functionals N i , defined in (40), along the trajectories of (37), satisfy

Ṅi (t) ≤ [-β i * + 2L i |µ i |C i ]Q(z i (t)) + ξ i K i-1 |µ i-1 |C i-1 Q(z i (t)) + K i-1 |µ i-1 | ξ i t t-τi-1 g i-1 (t -a)Q(z i-1 (a))da - ς i 2C i Λ i (z it ) + ς i 2 Q(z i (t)) + ψ i Λi-1 (t). (47)
Let us select, for all i ∈ {2, ..., n},

ψ i = K i-1 |µ i-1 |e τi-1 ξ i , and
ξ i = ς i 4K i-1 |µ i-1 |C i-1 . (48)
Then, the inequality (47) gives

Ṅi (t) ≤ - ς i 4 Q(z i (t)) - ς i 2C i Λ i (z it ) -ψ i Λ i-1 (z i-1t ) + ψ i C i-1 Q(z i-1 (t)). (49)
Similarly to the case i = 1, (44)-(46), we easily determine a strictly positive real number ςi such that

Ṅi (t) ≤ -ςi N i (z it , z i-1t ) - ς i 8 Q(z i (t)) + θ i ς i-1 8 Q(z i-1 (t)), (50)
with θ i = 8ψiCi-1 ςi-1 . Finally, we introduce the functional:

W (Z t ) = p 1 N 1 (z 1t ) + n i=2 p i N i (z it , z i-1t ), (51)
with p i = 2 n-i n k=i+1 θ k , and p n = 1. From ( 46) and (50), we conclude that

Ẇ (t) ≤ -p 1 ς1 N 1 (z 1t ) - n i=2 p i ςi N i (z it , z i-1t ) - n i=1 p i ς i 8 Q(z i (t)) + n i=2 p i θ i ς i-1 8 Q(z i-1 (t)). (52)
Noticing that p i θ i = 1 2 p i-1 , we easily determine a strictly positive constant ς such that

Ẇ (t) ≤ -ςW (Z t ), (53)
To summarize, by the virtue of the functionals N i , for all i ∈ I n , and the inequality (53) we conclude that the origin of the system (37) is exponentially stable.

B. Exponential stability of the nonlinear system

The next step consists in determining an approximation of the basin of attraction of the strictly positive equilibrium of the nonlinear system (31). To achieve this, we consider the case where

β i (x i ) = β i (0) 1 + b i x ni i , (54)
with β i (0) > 0, b i > 0 and n i > 2. This typical choice of the functions β i was justified by Mackey in [START_REF] Mackey | Unified hypothesis of the origin of plastic anemia and periodic hematopoiesis[END_REF], then assumed in [START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF], [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] and [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF]. After long but simple calculations, using the expression of β i in (54), we prove that there exist constants ŝi > 0, for all i ∈ I n , such that

R i (x i ) ≤ ŝi Q(x i ). ( 55
)
where R i are the functions in (30). Let us consider the functionals S i and N i defined in (38), ( 39) and (40). In order to ease the notation, we denote

I i (x it ) = t t-τi g i (t -a)Q(x i (a))da. (56)
Finally, we define the constants

ki = ς i 8ŝ i , ki = ς i 4C i L i ŝi e τi and N i = min k2 i , k2 i . (57)
Now we are ready to state and prove the following theorem:

Theorem 3: Let the system (31) be such that

ς i > 0 (58)
for all i ∈ I n . Therefore all the solutions of (31) with initial conditions φi ∈ C([-τ i , 0], R) satisfying

N i ( φi , φi-1 ) < N i , (59)
converge exponentially to the origin.

Notice that for all i ∈ I n , ki and ki are only dependent on the constant biological parameters of the model. Moreover, if the conditions (58) are satisfied, then N i > 0, for all i ∈ I n .

Proof: For the sake of brevity, we give only the outlines of the proof. First, using (55), we check that the derivative of the function Q(x 1 (t)), introduced in (32), along the trajectories of the nonlinear system (31) satisfies

Q(t) ≤2 [-β 1 * + L 1 |µ 1 |C 1 ] Q(x 1 (t)) + 2L 1 (|µ 1 | + ŝ1 |x 1 (t)|) I 1 (x 1t ) + ŝ1 |x 1 (t)|Q(x 1 (t)).
(60) Then, using (60) in conjunction with the fact that |x 1 (t)| ≤ 2 N 1 (x 1t ), we determine the following upper bound for the derivative of the functional N 1 , introduced in (39), along the trajectories of the nonlinear system (31),

Ṅ1 (t) ≤ -ς 1 N 1 (x 1t ) + ŝ1 N 1 (x 1t ) - ς 1 4 Q 1 (x 1 (t)) + 2L 1 ŝ1 N 1 (x 1t ) - ς 1 e -τ1 2C 1 I 1 (x 1t ) - ς 1 8 Q(x 1 (t)).
Now, observe that if the condition (59) is satisfied then

Ṅ1 (t) ≤ -ς1 N 1 (x 1t ) - ς 1 8 Q(x 1 (t)). (61)
This allows us to conclude that the origin of the system (31) is exponentially stable with a decay rate smaller than ς1 when i = 1, for all initial conditions satisfying (59). This result is generalized to the overall system, i ∈ I n , by similar arguments as in Theorem 2.

IV. CONCLUSIONS

We proposed explicit constructions of suitable strict Lyapunov functionals for nonlinear systems with distributed delays. In contrast with the methods already used, the Lyapunov approach allowed us to address more general statements, as the case of time-varying parameters and the approximation of the basin of attraction of the positive steady state. Furthermore, compared to the previous works where the cell division probability density function (mitosis) was taken as a Dirac mass function in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF], and assumed to be a truncated exponential term in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], our results were derived without any extra assumption on the mitosis function. For lack of space, we have not been able to illustrate some simulations that validate our theoretical results. Finally, it is worth mentioning that we are working on the identification of the parameters involved in the mathematical models.

Fig. 1 .

 1 Fig. 1. Schematic representation of the age-structured model describing unhealthy hematopoeisis.

Fig. 2 .

 2 Fig. 2. Schematic representation of the model of healthy hematopoeisis.

This work is supported by a public grant overseen by the French National Research Agency (ANR) through the "iCODE Institute project" funded by the IDEX Paris-Saclay.


Ramon Puigjaner
email: putxi@uib.cat

Progressing toward Digital Equity

Keywords: Digital equity, Building the infrastructure, Education levels, ITU WSIS, UN SDG, IFIP

In the computer domain, the objective to be reached in order to give opportunities equality to all persons is defined as Digital Equity. However, different international organizations are working to reach this goal. The analysis of the key points for reaching this objective and how international organizations (International Telecommunications Union, United Nations, International Federation for Information Processing) are working for this goal will be presented in the paper.

Introduction

ICT is a set of technologies that well used should allow the progress and the wellbeing of people everywhere in the world. However as there are important differences in the current situation of people, depending on its general level of use in each country, the use of ICT to reach the progress and the wellbeing cannot be the same in all countries. To evaluate this influence of the ICT we will use the concept of Digital Equity.

Digital equity [START_REF]International Society for Technology Education: Digital Equity Toolkit -Working Draft[END_REF] is the social-justice goal of ensuring that everyone in our society has equal access to technology tools, computers and the Internet. Even more, it is when all individuals have the knowledge and skills to access and use technology tools, computers and the Internet. A simple definition of digital equity can be a state in which both the digital divide and the participation gap are bridged.

Digital equity ensures that everyone has equal opportunities to use the tools and resources needed to fully participate as a citizen in today's digitally-powered world. Lacking these opportunities causes people to encounter educational, economic and social limitations that negatively impact their quality of life. The progress towards the Digital Equity, as it is established in the IFIP strategic action, should be achieved by:  Promoting accessibility to ICT,  Promoting good practices,  Promoting and enhancing appropriate access to knowledge and experiences,  Organizing and contributing to activities aimed at achieving the goals of the World Summit on the Information Society (ITU-WSIS) [START_REF]World Summit in Information Society: WSIS Action lines[END_REF],

 Organizing and contributing to activities aimed at achieving the UN Sustainable Development Goals (SDGs) [START_REF]United Nations: Transforming our World -The 2030 Agenda for Sustainable Development goals[END_REF], as we will analyze in Section 4, To analyze in depth the points suggested in the previous paragraphs, this paper will be organized as follows: in section 2 the concept of Digital Equity will be detailed; section 3 will be devoted to the relations of Digital Equity with the ITU-WSIS Action Lines; the relation of ITU-WSIS Action Lines with the UN Sustainable Development Goals will be the topic of Section 4. In Section 5 it is analyze the differences of Digital Equity in developed and developing countries and how IFIP [START_REF]International Federation for Information Processing: Structure of IFIP[END_REF] can contribute to reach these goals. Finally Section 6 gives some conclusions about the coming future.

2

Concept of Digital Equity

What is Digital Equity?

As it has been stated in the previous section, Digital equity is the social-justice goal of ensuring that everyone in our society has equal access to technology tools, computers and the Internet. Even more, it is when all individuals have the knowledge and skills to access and use technology tools, computers and the Internet. Digital equity can be defined as the state in which both the digital divide and the participation gap are both bridged. Digital equity ensures that everyone has equal opportunities to use the tools and resources needed to fully participate as a citizen in today's digitally-powered world. Lacking these opportunities causes people to encounter educational, economic and social limitations that negatively impact their quality of life.

Technology is so commonplace in our lives that it may be hard to believe there are still many people with limited access to and knowledge of the resources that are available online today. Children still stand in lines at libraries for a brief stint on a computer and parents have difficulty completing online employment applications; a clear and important difference between the new digital generation and the previous ones. While cell phones may have helped to close the gap to some degree, there are still important activities that are not well-suited for small mobile devices.

2.2

The Five Dimensions of Digital Equity

These dimensions have been chosen as fundamental categories by educators and professionals working in the field. If you are just beginning to learn about this field then these categories should help you address your basic needs:  Access to technology resources (hardware, software, wiring and connectivity): possibility to have access to the technological resources allowing us to access the information existing in the network.  Access to high quality digital content; if the information available in the network is not of good quality, people will not be attracted to access it.

 Access to high quality, culturally relevant content; the available information should not only be of high quality but adapted to the context in which each community is leaving.  Educators skilled in using these resources effectively for teaching and learning; it Is obvious that for approaching the digital equity it is necessary to educate users but in many environments there are not enough people with appropriate skills for learning end users in all the needed categories (e.g. basic user, advanced user, expert in installation and maintenance, hardware and software developer and builder, ICT research).  Opportunities for learners and educators to create their own content; a way to increase the capacity of learners and educators is to offer them some tools to create contents accessible to the appropriate end users.

Digital Divide Impacts

It is difficult to have progress toward Digital Equity homogeneously in its five dimensions and, in consequence the risk to fall in Digital Divide exists, at least in someone of the five dimensions. How to avoid this inconvenient situation? What are the initial steps to progress toward Digital Equity?  First it is necessary to allow that people can obtain the education allowing to conveniently using computers and networks in a large sense that is that users need to acquire a convenient knowledge for a correct use of computers and networks depending on the people desired education level: basic user, advanced user, expert in installation and maintenance, hardware and software developer and builder, ICT research, etc..  And second people should be able to easily and deeply use both computers and networks for accessing and using all kind of information. Without infrastructure the other dimensions have no sense. But the investment in a new infrastructure has to be done only if the possibility of maintaining it is ensured; otherwise there is an important risk of wasting the funds of this investment.

In summary, the first steps to reach Digital Equity should be education and infrastructure. However the progress towards Digital Equity cannot be reached through individual efforts but by the coordinate effect of actions promoted by governments and governmental organizations. The global organizations to consider will be the International Telecommunication Union (ITU), the United Nations (UN) and the International Federation for Information Processing (IFIP).

Relation between Digital Equity and the ITU-WSIS Action Lines

WSIS Action Lines

The Action Lines of the World Summit in Information Society (WSIS), proposed by the ITU are: AL1. The role of public governance authorities and all stakeholders in the promotion of ICTs for development AL2. Information and communication infrastructure AL3. Access to information and knowledge AL4. Capacity building AL5. Building confidence and security in the use of ICTs AL6. Enabling environment AL7. ICT Applications: benefits in all aspects of life AL7.1 E-Government AL7.2 E-Business AL7.3 E-Learning AL7.4 E-Health AL7.5 E-Employment AL7.6 E-Environment AL7.7 E-Agriculture AL7.8 E-Science AL8. Cultural diversity and identity, linguistic diversity and local content AL9. Media AL10. Ethical dimensions of the Information Society AL11. International and regional cooperation

Relation between WSIS and Digital Equity

The action lines proposed by the ITU have as global goal the implementation of the Digital Equity in order to avoid the Digital Divide. The description of these Action Lines can be found in [START_REF]World Summit in Information Society: WSIS Action lines[END_REF]. Maybe the idea of Digital Equity with respect to the WSIS Action Lines is not explicitly expressed, but after reading the documentation contained in [START_REF]World Summit in Information Society: WSIS Action lines[END_REF], the conclusion is that Digital Equity should be considered as a transversal goal, common to all action lines. The goal of each one of these Action Lines works, in some way, to complete one or several dimensions of Digital Equity. Some examples of this fact are presented in the coming sections.

Example 1. Action Line 2: Information and communication infrastructure.

The vision of this action line, as it is defined in [START_REF]World Summit in Information Society: WSIS Action lines[END_REF], is: Infrastructure is the cornerstone in achieving goals such as digital inclusion, enabling universal, sustainable, ubiquitous and affordable access to ICTs by all, taking into account relevant experiences from developing countries and countries with economies in transition.

Digital Equity should be supported by a convenient infrastructure. So the infrastructure suggested in this Action Line has to be able to allow all citizens the correct access and use of ICT, covering one of the five dimensions of the Digital Equity.

Example 2: Action Line 3: Access to information and knowledge.

The vision of this action line, as it is defined in [START_REF]World Summit in Information Society: WSIS Action lines[END_REF], is: For the post-2015, it is envisioned inclusive information and knowledge societies to facilitate access and use of information and exchange of knowledge among all people, including those coming from previously marginalized groups and regions in addition to persons with impairments with a significant portion of knowledge flows and innovations that advance human rights and the attainment of development goals.

To succeed with the Digital Equity the government has to create and offer information and knowledge to all citizens or to establish the appropriate mechanisms allowing the creation and offering of information and knowledge accessible for all citizens. Otherwise there is the risk of provoking the Digital Divide between the citizens of the same country or between the citizens of the country with the citizens of other countries.

Example 3: Action Line 8: Cultural diversity and identity, linguistic diversity and local content.

The vision of this action line, as it is defined in [START_REF]World Summit in Information Society: WSIS Action lines[END_REF], is: The vision on inclusive Knowledge Societies is that of a more culturally and linguistically diverse digital world, where around 40% of all existing languages are present in cyberspace and where development takes into account local, national and regional contexts, builds on the knowledge generated by all communities, promotes innovation and creativity, and allows all human beings to practice their own culture and enjoy that of others free from fear. It is a world where marginalized groups, including indigenous peoples, and those coming from migrations, diasporas and from language minorities, enjoy increased recognition and equity; artists, cultural professionals and practitioners are empowered to create, produce, disseminate enjoy and preserve a broad range of cultural goods, services and activities; and where intangible expressions inherited from past generations are kept alive for future generations.

If the global goal is to attain the Digital Equity it is necessary to respect the cultural diversity and identity giving the convenient importance to the linguistic diversity and local content. Otherwise there will be people without interest in accessing the information existing in the network and avoiding the attainment of the Digital Equity.

ITU-WSIS Action Lines and the UN Sustainable Development Goals

UN Sustainable Development Goals (SDG)

The recently defined by the United Nations Sustainable Development Goals are:

SDG1. End poverty in all its forms everywhere SDG2. End hunger, achieve food security and improved nutrition and promote sustainable agriculture SDG3. Ensure healthy lives and promote well-being for all at all ages SDG4. Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all SDG5. Achieve gender equality and empower all women and girls SDG6. Ensure availability and sustainable management of water for all SDG7. Ensure access to affordable, reliable, sustainable and modern energy for all SDG8. Promote sustained, inclusive and sustainable economic growth, full and productive employment and decent work for all SDG9. Build resilient infrastructure, promote inclusive and sustainable industrialization and foster innovation SDG10. Reduce inequality within and among countries SDG11. Make cities and human settlements inclusive, safe, resilient and sustainable SDG12. Ensure sustainable consumption and production patterns SDG13. Take urgent action to combat climate change and its impacts SDG14. Conserve and sustainably use the oceans, seas and marine resources for sustainable development SDG15. Protect, restore and promote sustainable use of terrestrial ecosystems, sustainably manage forests, combat desertification, and halt and reverse land degradation and halt biodiversity loss SDG16. Promote peaceful and inclusive societies for sustainable development, provide access to justice for all and build effective, accountable and inclusive institutions at all levels SDG17. Strengthen the means of implementation and revitalize the global partnership for sustainable development Each one of these goals is divided in [START_REF]United Nations: Transforming our World -The 2030 Agenda for Sustainable Development goals[END_REF] in a number of sub-goals (83) whose analysis allows a better understanding with respect to WSIS ALs and to the IFIP TCs.

Relations between the UN SDG and the WSIS Action Lines

WSIS has prepared a matrix indicating the effect of each one of the Action Lines on the different Sustainable Development Goals [START_REF]World Summit in Information Society: WSIS-SDG Matrix Linking WSIS Action Lines with Sustainable Development Goals[END_REF]. Table 1 presents these relations. If all WSIS Action Lines have some impact in the UN SDGs, the actions conducting to the Digital Equity will contribute at some level to approach the UN SDGs. So we can say that Digital Equity establishes the UN SDG in the specific domain of ICT that is WSIS Action Lines represent the application of SDG in the ICT domain. In some sense the dimensions of Digital Equity are included in one or several SDG and actions going toward Digital Equity contribute to the UN SDGs.

Digital Equity in Different Groups or Environments

Is Digital Equity in identical situation in all countries? Is Digital Equity in identical situation for the different genders?

It would be interesting to see the situation of Digital Equity with respect to different criteria like developed versus developing countries, like gender issue, etc. but let us concentrate in this article in the differences between developed and developing countries.

Situation of Digital Equity in developed and developing countries

Digital Equity is neither yet reached in developed nor in developing countries. However the level of Digital Equity is quite different in developed and in developing countries. We can analyze these differences comparing the state of the five dimensions of Digital Equity in both types of countries.

 Access to technology resources (hardware, software, wiring and connectivity): possibility to have access to the technological resources allowing us to access the information existing in the network: Obviously the situation of the technological resources in developed countries is more comfortable than in developing ones, mainly because the extension of the network coverage and its capacity in developed countries are much more higher, faster and reliable than in developing ones. Also because the availability of devices in developed countries is greater, with higher novelty of models and at prices economically cheaper taking into account the economic level of the country. And finally the possibility for the end users of receiving the appropriate education is also higher; and this is true at all levels of education: basic user, advanced user, expert in installation and maintenance, hardware and software developer and builder, ICT research, etc.; it is easy to find the necessary education at reasonable prices for all type of education. Also in many countries there are strong differences in the access possibilities for the different genders.  Access to high quality digital content; if the information available in the network is not of good quality, people will not be attracted to access it. Developed countries have an infrastructure allowing the access to all information existing in Internet by both the quality of the physical connection and the freedom for accessing all kind of information. Developing countries can have limitations by one or both of these aspects; in some cases the network has not enough capacity for accessing heavy information, and in other cases the access is limited by political reasons.  Access to high quality, culturally relevant content; the available information should not only be of high quality but adapted to the context in which each community is living. Most of the web pages are written in English (or at least is the greatest minority) making difficult the access of people not fluent in reading this language. Also the topics contained in these pages are thought with Anglo-Saxon parameters making the access by people of other cultures of low interest. Consequence: not all pages are equally interesting for all people in the world, increasing the disadvantages of people of developing countries.  Educators skilled in using these resources for effectively teaching and learning; it Is obvious that for approaching the digital equity it is necessary to educate users but in many environments there are not enough people with appropriate skills for learning end users in all the needed categories. Education is a key point for reaching Digital Equity. However to deliver education it is necessary to have educators of all the needed profiles. The number and quality of the education centres dedicated to the educators are higher in developed than in developing countries.  Opportunities for learners and educators to create their own content; a way to increase the capacity of learners and educators is to offer them some tool to create contents accessible to the appropriate end users. The hardware and software infrastructure is much more solid in developed than in developing countries.

Contribution of IFIP to Digital Equity

The International Federation for Information Processing is a society created in 1960 under the auspices of UNESCO grouping computer societies under the base of one member society per country. From the technical point of view IFIP is composed of a number of Technical Committees (TC) each one dedicated to a specific aspect of computer science, engineering and applications. Each TC is composed of a number of Working Groups (WG), each one devoted to a specific aspect of the TC domain. Currently there are 13 TCs and 120 WGs grouping thousands of computer professionals. IFIP has decided that Digital Equity is one of its Strategic Activity Lines. In general we can consider that the work of all these bodies contribute to the Digital Equity. However there are some of them are more specifically oriented to tackle the Digital Divide and empowering Digital Equity.

From the previous introduction let us review the different TCs and analyze what is their involvement in some aspect of the Digital Equity. The IFIP TCs are:  TC1 on Foundations of Computer Science, whose aims are: to support the development of theoretical computer science as a fundamental science; and to support the development and exploration of fundamental concepts, models, theories, systems, and and the understanding of laws, limits, and possibilities of information processing. Its influence on Digital Equity is generic but not directly implied.  TC2 on Software, Theory and Practice, whose aim is to obtain a deeper understanding of programming concepts in order to improve the quality of software by studying all aspects of the software development process, both theoretical and practical. So, building good software products has a positive influence on Digital Equity mainly promoting good practices in the creation of software products.  TC3 on Education whose aims are: to provide an international forum for educators to discuss research and practice in: teaching informatics; and educational uses of communication and information technologies (ICT); to establish models for informatics curricula, training programs, and teaching methodologies; to consider the relationship of informatics in other curriculum areas; to promote the ongoing education of ICT professionals and those in the workforce whose employment involves the use of information and communication technologies; to examine the impact of information and communication technologies on the whole educational environment: teaching and learning; administration and management of the educational enterprise; and local, national and regional policy-making and collaboration. As education is a key factor for reaching Digital Equity, we can see that the aims of this TC are fully in line with the dimensions of Digital Equity.  TC5 on Information Technology Applications whose aim is to promote research and development of fundamental concepts, models, and theories to support applications of ICT. So, it has a generic interest for Digital Equity; only the promotion of applications appropriate for developing countries can help to attain Digital Equity.  TC6 on Communication Systems, whose aims are to promote the international exchange of information related to communication systems; to bridge gaps existing between users, telecommunication operators, service providers and computer and equipment manufacturers; and to establish working contacts with international bodies concerned with data communication, such as ITU, ETSI, ISO, IEEE, IETF, ITC and ATM Forum. The aims of this TC are fully in line with a dimension of the Digital Equity (creation of a convenient infrastructure). In particular in this TC there is a WG devoted to developing countries so dedicated to bridge the gap between developed and developing countries with respect to Digital Equity. This is the WG6.9 on Communications Systems for Developing Countries, whose aims are: to identify and study technical problems related to the access to, understanding of and application of network and telecommunications technology in developing countries or regions; to encourage cross-fertilization of concepts and techniques among developing countries, and between developing countries and developed countries; to promote activities oriented to the diffusion of the methods and techniques for accessing computer networks in developing countries or regions.  TC7 on System Modelling and Optimization, whose aims are: to provide an international clearing house for computational (as well as related theoretical) aspects of optimization problems; to promote the development of necessary theory to meet the needs of complex optimization problems and cooperate with the International Mathematics Union; and to foster interdisciplinary activity on optimization problems spanning the areas such as Economics, Biomedicine, Meteorology, etc., in cooperation with associated international bodies. So, it has not a specific interest for Digital Equity.  TC8 on Information Systems whose aim is to promote and encourage interactions among professionals from practice and research and advancement of investigation of concepts, methods, techniques, tools, and issues related to information systems in organizations. So, it has a generic interest for Digital Equity specially proposing good practices for the information systems.  TC9 on ICT and Society whose aims are: to develop understanding of how ICT progress is associated with change in society; and to influence the shaping of socially responsible and ethical policies and professional practices. The aims of this TC are fully in line with several dimensions of the Digital Equity.

In particular it has a WG devoted to Developing countries dedicated to bridge the gap between developed and developing countries. It is the WG9.4 on Social Implications of Computers in Developing Countries, whose aims are: to collect, exchange and disseminate experiences of ICT implementation in developing countries; to develop a consciousness amongst professionals, policy makers and public on social implications of ICT in developing nations; to develop criteria, theory, methods, and guidelines for design and implementation of culturally adapted information systems; and to create a greater interest in professionals from industrialized countries to focus on issues of special relevance to developing countries.  TC10 on Computer Systems Technology, whose aim is the promotion of the Stateof-the-Art and the coordination of information exchange on concepts, methodologies, and tools in the stages in the life cycle of computer systems. So, it has a generic but not specific interest for Digital Equity.  TC11 on Security and Privacy Protection in Information Processing Systems whose aim is to increase the trustworthiness and general confidence in information processing and to act as a forum for security and privacy protection experts and others professionally active in the field. Obviously the topics concerned by this TC are of high importance for reaching Digital Equity, ensuring the needed security and privacy protection to the information and to the end users.  TC12 on Artificial Intelligence whose aims are: to foster the development and understanding of Artificial Intelligence and its applications worldwide; to promote interdisciplinary exchanges between Artificial Intelligence and other fields of information processing; and to contribute to the overall aims and objectives and further development of IFIP as the international body for Information Processing. The aims of this TC are not of specific interest for Digital Equity.  TC13 on Human-Computer Interaction whose aims are :to encourage empirical research (using valid and reliable methodology, with studies of the methods themselves where necessary); to promote the use of knowledge and methods from the human sciences in both design and evaluation of computer systems; to promote better understanding of the relation between formal design methods and system usability and acceptability; to develop guidelines, models and methods by which designers may be able to provide better human-oriented computer systems; and to cooperate with other groups, inside and outside IFIP, so as to promote userorientation and "humani-zation" in system design. Obviously the topics concerned by this TC are important for reaching Digital Equity, because a good interface to the searched information helps a lot to its usability by the end users.  TC14 on Entertainment Computing whose aims are: to enhance algorithmic research on board and card games; to promote a new type of entertainment using information technologies; to encourage hardware technology research and development to facilitate implementing entertainment systems; and to encourage nontraditional human interface technologies for entertainment. Maybe the topics concerned by this TC are marginal for reaching Digital Equity, but it works on topics making the access to information more attractive to the end users.

So IFIP has convenient bodies to tackle the problems derived from and related to Digital Equity in general and specifically in developing countries. The recent creation of a Standing Committee in Digital Equity will increase and coordinate the action to promote the Digital Equity.

Conclusions

We have analyzed the concept of Digital Equity and the different aspect it includes. Also we have seen how several international organizations; the Action Lines of the World Summit in Information Society, the United Nations Sustainable Development Goals and the Technical Committees of the International Federation for Information Processing have a strong overlapping with the goals proposed by Digital Equity. Maybe these international organizations have not an exact coincidence on their goals, but an important overlap between them has been clearly stated.

Table 1 .

 1 Effect of each one of the Action Lines on the SDG

	Sustainable Development Goal	WSIS Action Lines