MATHEMATICAL ANALYSIS AND GLOBAL DYNAMICS FOR A TIME-DELAYED CHRONIC MYELOID LEUKEMIA MODEL WITH TREATMENT

. In this paper, we investigate a time-delayed model describing the dynamics of the hematopoietic stem cell population with treatment. First, we give some property results of the solutions. Second, we analyze the asymptotic behavior of the model, and study the local asymptotic stability of each equilibrium: trivial and positive ones. Next, a necessary and suﬃcient condition is given for the trivial steady state to be globally asymptotically stable. Moreover, the uniform persistence is obtained in the case of instability. Finally, we prove that this system can exhibits a periodic solutions around the positive equilibrium through a Hopf bifurcation. types of leukemia, including: 47, 49] Leukemia models ordinary diﬀerential equations (ODE), [5, 10, 26, 39] for models with partial diﬀerential equations (PDE). Recently, numerous papers are dealing with optimal therapy modeling as in [37, 38, 40, 44]. The stochastic and impulsive diﬀerential equations are also used to describes the dynamics of leukemia stems cells as in [12, 17]. We can also ﬁnd competition models of healthy and leukemic (stem) cells as in [8, 11, 48]. For more interesting models for leukemia, we can also see

. Mathematical models provide a convenient and inexpensive mechanism for studying biological processes and interventions for which experimental data may be scarce or expensive. Cancer research is among the disciplines that have begun to use mathematical modeling to analyze several processes related to cancer diseases, including immunity and treatment, as in [6,9,46].
Along this paper, based on the model of Radulescu et al. [41], we will analyze a mathematical model on chronic myeloid leukemia with treatment called imatinib, the latter was already introduced in the literature, we cite as an example [13]. The sensitivity of leukemic stem cells to treatment with imatinib is the subject of various speculations, which is why additional studies are necessary to make this phenomenon more clear (see [15,23,24,43]). The effect of imatinib on stem cells, even if it is very weak, is taken into consideration, this effect is observed on the reduction of the proliferation of leukemic stem cells (see [18,[32][33][34][35][36]).
We will present some results about existence and uniqueness of solutions of stem cell dynamics model. In our work we show the stability of the positive steady state by establishing Hopf bifurcation which is also used in [3,27,30,31].
The paper is organized as follows. In Section 2, we develop the mathematical model to be studied and we discuss basic properties of solutions, including their existence, uniqueness, boundedness, and positivity. In Section 3, steady state solutions will be given. In Section 4, we investigate local and global stability of the trivial steady state. Section 5 is devoted to the uniform persistence of the model. Stability of the positive equilibrium and Hopf bifurcation with illustrative numerical simulations are provided in Section 6. In the last Section, we give some conclusions.

The model
Let Q(t) denotes the density of leukemic stem cell population. In the proposed model, the cell is divided into two daughter cells, either by a symmetrical division with the fraction η 1 > 0, or by an asymmetrical division with the fraction η 2 > 0. The others, are supposed to renew with the fraction (1 − η 1 − η 2 ). The parameter τ describes the duration of the cell division, which is supposed to be the same for all types of division. For a more clear understanding of different types of cell division, we give a schematic representation in Figure 1.
D(t) denotes the amount of the drug that reaches the bloodstream, and P (t) denotes the amount of drug in the plasmatic compartment.
The evolution of the population is described by the following system, for t > 0, +k 0 η 1 e −γτ Q(t − τ ) − r(P (t))Q(t), D(t) = −κD(t) + K, P (t) = −vP (t) + κD(t). (2.1) The system (2.1) is completed by the following initial conditions, Q(t) = ϕ(t) ≥ 0, t ∈ [−τ, 0], D(0) = D 0 ≥ 0 and P (0) = P 0 ≥ 0. (2.2) In this model, β(Q) is a positive and decreasing function of Q. It denotes the rate of self-renewal. As example, β can be considered as a Hill function β(Q) = β 0 θ n θ n + Q n , n > 1.  The fraction value of asymmetric division (none) η 2 The fraction value of symmetric division (none) β The rate of self renewal (day −1 ) γ The mortality rate (day −1 ) k 0 The rate of differentiation and of asymmetric division (day −1 ) τ The duration of the cell division(day) K The constant dose of administrated drug (mg/day) κ The first order absorption rate (day −1 ) v Clearance ratio, volume of drug distribution (hour −1 ) β 0 Maximal self renewal rate (day −1 ) θ The value for which the function β(Q) attains half of its maximum value (cells.kg −1 ) m A Hil coefficient (none) P 0 The half of the maximum activity concentration x 0 The number of infected cells (cells.kg −1 ) R 0 The number of cells resistent to treatment (cells.kg −1 ) The function r(P ) is supposed to be increasing and bounded. It describes the treatment's effect, which represents the reduction rate of leukemic stem cell proliferation. For instance, r(P ) is given by Both β and r are supposed to be continuously differentiable. All the parameters are nonnegative constants and they are described in the Table 1 . Throughout this paper, we assume that (ϕ, D 0 , P 0 ) ∈ C + × R + × R + . The existence and uniqueness of nonnegative solutions of (2.1)-(2.2) can be obtained by using the theory of functional differential equations as in [21] (see also [20] and [28]).
Proposition 2.1. All solutions of system (2.1) with nonnegative initial conditions are nonnegative.
Proof. Let (Q, D, P ) be a solution of (2.1) associated to the initial condition (ϕ, D 0 , P 0 ) ∈ C + × R + × R + . We prove the nonnegativity on the interval [0, τ ], and we apply the same reasoning by steps on each interval [kτ, (k + 1)τ ], for k = 1, 2 . . .. For t ∈ [0, τ ], we have t − τ ∈ [−τ, 0]. Then, the system (2.1) transforms to (2. 3) The idea is to extend the analogous result known for ODE to the delay differential equations (DDE) as established in the Theorem 3.4 in [45]. More precisely, the positivity of DDE follows also as ODE with the classical sufficient conditions on the nonlinearity. We have the following implications and This yields Q(t) ≥ 0, D(t) ≥ 0 and P (t) ≥ 0 for t ∈ [0, τ ]. One can just repeat the argument by steps on [kτ, (k + 1)τ ], k ∈ N. We conclude that Q, D and P are nonnegative on [0, +∞).
Next, we show the boundedness of solutions of (2.1).
Proposition 2.2. All solutions of system (2.1) are bounded.
Proof. Let (Q, D, P ) be the solution of (2.1) associated to the initial condition (ϕ, D 0 , P 0 ) ∈ C + × R + × R + . First, we focus on the components D and P . From the differential equation of D we have, for t > 0, Then, D never blows up in finite time since lim sup t→+∞ D(t) < +∞. We set sup t≥0 D(t) = M . Then, for t > 0, This implies that for t > 0, we have Focus now on the component Q, and assume that Since β is a decreasing function and lim x→+∞ β(x) = 0, there exists a unique Q 0 > 0 such that We can chek that (2.5) In fact, let y ∈ [0, Q). We distinguish two cases. If y ≤ Q 0 , then If y > Q 0 , then Hence, (2.5) holds for all y ∈ [0, Q). Assume, by contradiction, that lim sup t→+∞ Q(t) = +∞, where Q(t) is a solution of (2.1). Then, there existst > τ such that From (2.1), we havė From (2.5), we obtain that Consequently, we getQ This leads to a contradiction. As a consequence, lim sup t→+∞ Q(t) < +∞, that is Q is bounded.

Existence of steady states
In this section, we focus on the existence of steady states for system (2.1). Let (Q , D , P ) be a steady state of (2.1). Then, it satisfies As a result, we have either Q = 0 or The point (0, D , P ) is always an equilibrium of (2.1). In another side, since β is a decreasing function on [0, +∞[, then the existence of a positive steady state Q is equivalent to This condition is satisfied for where In fact, we have The following result summarizes the existence of the steady states.

Stability of the trivial steady state
In this section, we focus on the local stability and the global stability of the trivial steady state. We will show that the solution of the first component of (2.1) disappears when the trivial equilibrium is the only steady state.

Local asymptotic stability
The purpose of this part is to show the local asymptotic stability by studying the characteristic equation of the linearized system of (2.1).
Theorem 4.1. If τ > τ max , then the unique trivial steady state of system (2.1) is locally asymptotically stable. If τ < τ max , then it is unstable.
Proof. The linearization of (2.1) corresponding to (0, D , P ) is given by The characteristic equation is given by the following formula Thus, (4.1) becomes Since κ > 0 and v > 0, the stability is determined by the sign of the real part of λ ∈ C satisfying D 0 (λ, τ ) = 0.
If we consider D 0 as a real function, the derivative of D 0 (λ, τ ), with respect to λ, is Consequently, D 0 (λ, τ ) is an increasing function with respect to λ, and satisfies Then, there exists λ 0 a unique real solution of the equation D 0 (λ, τ ) = 0. Moreover, we have If τ < τ max , then we can check that D 0 (0, τ ) < 0, which implies that λ 0 > 0. Hence, the trivial steady state is unstable.

Global asymptotic stability of the trivial steady state
In this section, we investigate the global asymptotic stability of the trivial steady state. First, we focus on the subsystem of (2.1) All the parameters of (4.5) are positive. The set of the initial conditions is included in Ω := {(D, P ) ∈ R 2 + , D ≥ 0, P ≥ 0}. By the resolution of the subsystem (4.5) we obtain Since lim t→+∞ e −κt = lim t→+∞ e −vt = 0, the solution of the system (4.5) tends towards the unique equilibrium of the model (K/κ, K/v). Then we conclude that the unique steady state (D , P ) of system (4.5) is globally asymptotically stable. Next, we use the global convergence of D, P to D , P , respectively, to obtain the global asymptotic stability of the equilibrium (0, D , P ) of system (2.1). In fact, with the choice of > 0 small enough, there exists a sufficiently large T > 0, such that P (t) ≥ P − for all t ≥ T . Since s → r(s) is an increasing function, then for all t ≥ T we haveQ (4.7) We will use a comparison principle. Then, we consider the following problem (4.8) Without loss of generality, we can assume that the system (4.8) holds for all t ≥ 0 by takingφ(δ) = Q(T + δ) with δ ∈ [−τ, 0]. We note that zero is also a steady state for the system (4.8). The global stability of this equilibrium is given in the following result.
Then, the zero equilibrium of (4.8) is globally asymptotically stable.
Proof. Consider the following continuous function The derivative of V along the solution trajectory t → Q (t) of (4.8) is calculated as followṡ Note that β is a decreasing function. If 1 − 2e −γτ ≥ 0, then we havė From our hypothesis, we can observe easily that ν( ) is positive. Consequently, the both situations imply that the trivial steady state of (4.8) is globally asymptotically stable.
In the next theorem, we give a necessary and sufficient conditions for the trivial equilibrium to be globally asymptotically stable (see Fig. 2).
Theorem 4.3. If τ > τ max , where τ max is given by (3.4), then the unique trivial equilibrium of (2.1) is globally asymptotically stable. Moreover, we can observe that if we suppose the condition (3.3), then there exists a small > 0 such that (4.9) holds. This allows us to say that under (3.3) and using (4.10) the state Q(t) converges to zero. Recalling that (0, D , P ) is locally asymptotically stable. Then, it is globally asymptotically stable. This completes the proof.

Uniform persistence
Persistence of the solutions of system (2.1) ensures survival of the leukemic cells since solutions do not converge towards the trivial steady state.
With this choice of , we are going to show that (5.1) holds true. On the contrary, suppose that (5.1) does not hold. Then, there exists a sufficiently large T ,˜ > T˜ such that Q(t) ≤ for all t ≥ T ,˜ . From (2.1), for t ≥ T ,˜ we haveQ We notice that We have again that, for t ≥ T ,˜ , Since, Q(t) ≤ , for any t ≥ T ,˜ , and β is an decreasing function, we have Then, for t ≥ T ,˜ −e −λT ,˜ Q(T ,˜ ) The conclusion is that This leads to a contradiction.
The Theorem 5.1 shows the persistence of solution but not the strong one. The uniform (strong) persistence means that there exists a positive constant 1 > 0 such that The Theorem 5.1 shows only that lim sup t→+∞ |Q(t)| ≥ for some small > 0 (weak persistence). Based on the theory in ( [19], Thm. 11), it follows in our case that the uniform weak persistence implies the uniform (strong) persistence (see also [1,2] for a same corresponding proof). Then, for any positive initial condition, there exists a positive constant 1 > 0 such that From above, we see that the condition (3.3), associated to the existence of the unique positive steady state or not, implies that it plays the role of a threshold not only for the eradication but also for the persistence of cells.

Local asymptotic stability and hopf bifurcation
We focus now on the positive steady state of (2.1). In order to study the local asymptotic stability of the positive steady state, we write the characteristic equation given by the following formula and Then, the characteristic equation becomes We recall that a steady state is locally asymptotically stable if all roots of the associated characteristic equation have negative real parts, and unstable if at least one root with positive real part exists. Since κ and v are positive, we will consider the equation First, we will check that λ = 0 is not a root of D(λ, τ ).
Proof. Assume that λ = 0 is a root of the characteristic equation. Then, we obtain In fact, the existence of the positive steady state implies that 1 − 2e −γτ < 0 and β is a decreasing function. This completes the proof. Now, we study the sign of the root of D(λ, τ ), given by (6.1). In the next lemma, we show the stability at τ = 0.
This yields to .
Lemma 6.5. The functions S k given by (6.10) satisfy, for all k ∈ N and τ ∈ [0,τ ), Therefore, provided that no root of S k is a local extremum, the number of the positive of S k , for k ∈ N, on the interval τ ∈ [0,τ ) is even. From the previous lemma, we conclude that if a fixed S k has no root on [0,τ ), then all functions S j , with j > k, also have no roots on [0,τ ). Consequently, we have the following proposition.
Now, let suppose that S 0 has at least one positive root, on the interval [0,τ ). We denote byτ ∈ (0,τ ) the smallest root of S 0 . Then, (Q , D , P ) is locally asymptotically stable for all τ ∈ [0,τ ) and loses its stability when τ =τ . A finite number of stability switch may occurs as τ increases and passes through roots of the functions S k (See Fig. 4).
Next, we will prove that (Q , D , P ) can be destabilized through a Hopf bifurcation as τ increases. We need to guarantee the transversality condition of the Hopf bifurcation theorem. We start by proving that if an imaginary characteristic root iω exists, then it is simple.
Consequently, for τ > 0, Since λ is a purely imaginary root, we lead to a contradiction.
From [7], we have the following result.
Proposition 6.8. The transversality condition is reduced to Proof. Since ∂h ∂z (z(τ ),τ ) = 1, then it follows from [7] that sign dRe(λ(τ )) dτ The following theorem states the stability of the positive equilibrium and the existence of periodic solutions through the Hopf bifurcation.
Proposition 6.9. Assume that (3.3) and (6.5) hold true. If S 0 (τ ) has at least one positive root on the interval (0,τ ), then the positive steady state (Q , D , P ) is locally asymptotically stable for τ ∈ [0,τ ), whereτ is the smallest root of S 0 (τ ) on (0,τ ), and (Q , D , P ) loses its stability when τ =τ . A finite number of stability switch may occur as τ passes through the roots of the functions S k . Moreover, if then a Hopf bifurcation occurs at (Q , D , P ) for τ =τ .
The following theorem summarizes the result dealing with the asymptotic stability of the positive steady state (Q , D , P ) of the system (2.1) (See Fig. 5).

Conclusions
In this work we have considered a model of chronic myeloid leukemia under medical treatment. A mathematical analysis is proposed to draw conclusions.
The study of the existence of equilibria makes it possible to distinguish two stationary solutions, the trivial equilibrium corresponding to the extinction of the called stem cells which exists for any value of the cell cycle τ , and the nontrivial equilibrium which exists only for τ ∈ (0, τ max ).
The study of stability allows to conclude that under certain conditions on the parameters of the model, we can deduce the stability of the nontrivial equilibrium when it exists with the instability of the trivial equilibrium, and when the trivial is alone it is even overall stable. This involves the disappearance of the leukemic cells.
More precisely, from the results obtained in this work, in particular from Proposition 3.1 and Theorem 4.1, the amplitude τ of the cell cycle, which can be compared to a certain value τ max , which depends on the parameters of the model, influences either the onset and viability of the disease for τ < τ max corresponding to the existence and stability of the nontrivial equilibrium E 1 , or the reduction of the disease in the case where τ < τ max . From (3.4), we see that τ max depends on the parameter of medical treatment r(P ), then one could do the choice of the medical control protocol to move from a situation of the installation of the disease (for τ < τ max ) to a situation of control and significant reduction of the disease (for τ > τ max ).
Otherwise, other questions which were not considered in this work, can be raised and discussed in future works, for example considering leukemic progenitor cells in addition to stem cells or/and resistance to treatment.