A note on global attractivity of the periodic solution for a model of hematopoiesis

A note on global attractivity of the periodic solution for a model of hematopoiesis Teresa Faria Departamento de Matem´atica and CMAF-CIO, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal e-mail: [email protected] Jos´e J. Oliveira1 CMAT and Departamento de Matem´atica e Aplica¸c˜oes, Escola de Ciˆencias, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal e-mail: jjoliv[email protected] Abstract For the generalized periodic model of hematopoiesis x0(t) = −a(t)x(t) + m X i=1 bi(t) 1 + x(t−τi(t))n, with 0 < n ≤1, sufficient conditions for the global attractivity of its positive periodic solution are given, improving previous results in the literature. The effectiveness of the present criterion is illustrated by a numerical example. Keywords: Hematopoiesis, Global attractivity, Yorke condition, Periodic solution Mathematics Subject Classification: 34K13, 34K25, 92D25 1 Introduction The delay differential equation (DDE) x0(t) = −γx(t) + β 1 + x(t−τ)n, t ≥0,(1.1) where γ, β, τ, n ∈(0,∞), was introduced by Mackey and Glass [8] and since then has proven to be a suitable model to describe the hematopoiesis (the process of production, multiplication, and specialization of blood cells in bone marrow). In (1.1), x(t) is the density of the mature cells in the circulation, τis the time delay between the production of immature cells in the bone marrow and their maturation for release in the bloodstream, γis a destruction rate, and βis the maximal production rate. More details can be found in [8]. Since Mackey and Glass’ publication, model (1.1) has been studied by many researchers (see [1, 4, 5, 6, 7] and references therein) from different points of view. As all coefficients are positive, equation (1.1) has a unique steady state k, which is the positive solution of the equation γk =β 1 + kn. The stability of the equilibrium point khas been studied by several authors. For the case 0 < n ≤1, E. Liz et al. [6] proved that kis a global attractor of (1.1) (in the set of positive solutions) without further constraints. As the environment plays an important role in many biological and ecological dynamical systems, the model is often more realistic if periodic parameters are considered, taking into account the periodicity of the environment. In fact, periodic versions of (1.1) have attracted the attention of several authors (see [1, 5, 10] and references therein). 1Corresponding author. 1 For ω > 0, and a, b :R→(0,∞), τ:R→[0,∞)ω-periodic continuous functions, the model x0(t) = −a(t)x(t) + b(t) 1 + x(t−τ(t))n, t ≥0,(1.2) has a positive ω-periodic solution ex(t) (see [10]) but, to the best of our knowledge, for 0 < n ≤1, to prove or disprove that all positive solutions of (1.2) converge to the positive periodic solution ex(t) remains an open problem [1]. A first contribution to solve this problem was given by G. Liu et al. in [5], where the authors considered the following generalized model of hematopoiesis: x0(t) = −a(t)x(t) + m X i=1 bi(t) 1 + x(t−τi(t))n, t ≥0,(1.3) with m∈N,n∈(0,1], and a, bi:R→(0,∞), τi:R→[0,∞)ω-periodic continuous functions i= 1, . . . , m. For this equation, G. Liu et al. proved the existence of a unique positive ω-periodic solution ex(t) [5, Theorem 2.1], and established a criterion for its global attractivity in the set of positive solutions [5, Theorem 3.1]. Motivated by the open problem referred to above (see [1, Problem 4]), in this note we give an improvement of the global stability criterion presented in [5, Theorem 3.1]. For this purpose, we insert (1.3) in a more general framework studied in our previous works [2, 3]. An illustrative numerical example in Section 4 shows that the requirements to apply such criterion are easy to check. 2 Preliminaries and Notations First, we set some notations. Let C:= C([−τ, 0]; R) be the Banach space of continuous functions from [−τ, 0] to Requipped with the sup norm, kϕk= max −τ≤θ≤0|ϕ(θ)|. For delay functions τi: [0,∞)→ [0,∞) continuous and bounded (1 ≤i≤m), define τ(t) = max 1≤i≤m{τi(t)}, τ = sup t≥0{τ(t)}, and consider a DDE of the form y0(t) = −a(t)y(t) + m X i=1 fi(t, yi t), t ≥0,(2.1) where a: [0,∞)→[0,∞) is continuous, fi(t, yi t) = fit, y|[t−τi(t),t]and fi(t, ϕ), i= 1, . . . , m, are functions defined for t≥0 and ϕ∈C([−τi(t),0]; R) in the following way: for t≥0 and ϕ∈C([−τi(t),0]; R), we take the extension ϕ∗∈Cof ϕwhich is ϕ(−τi(t)) on [−τ, −τi(t)], and define fias the restriction of some continuous function Fi: [0,∞)×C→R, with Fi(t, ϕ∗) := fit, ϕ∗ |[−τi(t),0] =fi(t, ϕ). For a DDE (2.1) in C, initial conditions have the form yt0=ϕfor ϕ∈C, where, as usual, yt0is defined by yt0(θ) = y(t0+θ) for θ∈[−τ, 0]. Assuming f(t, 0) = 0 for t≥0, in [2, 3] the authors considered the model (2.1) (with and without impulses) and gave sufficient conditions for the stability and global attractivity of the trivial solution. Due to the interpretation of the models under consideration, one frequently restricts the set of admissible initial conditions, as well as the set of solutions, so that the concept of attractivity only applies to a subset of initial conditions S∗⊆C. A set S∗is said to be an admissible set of initial conditions for (2.1) if ϕ∈S∗⇒yt(·, t0, ϕ)∈S∗,for t≥t0≥0, where y(t, t0, ϕ) denotes the solution of (2.1) such that yt0=ϕ. The definition of global attractivity is recalled below. 2 Definition 2.1. We say that a solution y∗(t)of a DDE (2.1) is globally attractive in an admissible set S∗⊆Cif y∗ t∈S∗and y∗(t)−y(t)→0,as t→ ∞, for all solutions y(t)with initial condition in S∗. For the non-impulsive situation, the main assumptions in [2] can be summarized as follows: (H1) Z∞ 0 a(u)du =∞; (H2) there are piecewise continuous functions λ1,i, λ2,i : [0,∞)→[0,∞) such that −λ1,i(t)Mi t(ϕ)≤fit, ϕ|[−τi(t),0] ≤λ2,i(t)Mi t(−ϕ), t ≥0, ϕ ∈S∗, i = 1, . . . , m, where Mi t(ϕ) = max (0,sup θ∈[−τi(t),0] ϕ(θ))is the so-called Yorke’s functional; (H3) there is T > 0 with T−τ(T)>0 such that α∗ 1α∗ 2<1,(2.2) where the coefficients α∗ j:= α∗ j(T) are given by α∗ j= sup t≥TZt t−τ(t) m X i=1 λj,i(s) e−Rt sa(u)du ds, j = 1,2. Note that the above hypothesis (H2) implies that fi(t, 0) = 0 for all t≥0, thus y(t) = 0 is an equilibrium point of (2.1). The following stability result holds: Theorem 2.1. [2] Assume (H1)-(H3), with S∗⊆Can admissible set of initial conditions for (2.1). If 0∈S∗, then the zero equilibrium point is globally attractive (in S∗). Recall that, for a function z(t) defined for t≥0, we say that z(t) is oscillatory if it is not eventually zero and it has arbitrarily large zeros; otherwise, it is called non-oscillatory. Remark 2.1. In [2], (H3) was used only to treat the case of oscillatory solutions. Moreover, from the proofs of Lemmas 2.2, 2.4 and Theorems 2.1, 2.2 in [2], the conclusion of Theorem 2.1 is obtained under (H1), (H3) and a weaker version of (H2), as follows: for solutions y(t) of (2.1) with initial condition in S∗, (i) if y(t) is non-oscillatory, the solution-segment ytsatisfies (H2∗)for i= 1, . . . , m and large t≥0, fit, y|[t−τi(t),t]≤0 if y|[t−τi(t),t]≥0 and fit, y|[t−τi(t),t]≥0 if y|[t−τi(t),t]≤0; (ii) if y(t) is oscillatory, (H2) is satisfied for large twith ϕ|[−τi(t),0] replaced by y|[t−τi(t),t]. We now go back to the generalized model of hematopoiesis (1.3). For the functions a,biand τi in (1.3), we shall denote τ(t) = max 1≤i≤m{τi(t)}, τ = max t∈[0,ω]{τ(t)},¯a= max t∈[0,ω]{a(t)},and bi= min t∈[0,ω]{bi(t)}, and consider it in the phase space C=C([−τ, 0]; R). By biological reasons, only positive solutions of (1.3) are meaningful and therefore hereafter we consider the set of initial conditions as S={ϕ∈C:ϕ(θ)≥0 for −τ≤θ < 0, ϕ(0) >0}. 3 It is easy to show that Sis an admissible set for (1.3), i.e., xt(·, t0, ϕ)∈Sfor t≥t0≥0 and ϕ∈S, where x(t, t0, ϕ) denotes the solution of (1.3) with initial condition xt0=ϕ. In this note, we study the global attractivity of an ω-periodic solution ex(t) of (1.3) in the set S, whose existence was established in [5]. We shall use some uniform lower bound estimates proven in [5, Lemmas 3.1, 3.2, and 3.3]. Lemma 2.2. [5] The positive ω-periodic solution of (1.3),ex(t), satisfies ex(t)≥x1exp −sup t∈[0,ω]Zt t−τ a(u)du!=: X1,for all t∈R,(2.3) where x1is the unique positive solution of the equation ¯ax = m X i=1 bi 1 + xn. Lemma 2.3. [5] If x(t)is a positive solution of (1.3) such that x(t)−ex(t)is oscillatory, then there is Tx> T such that x(t)≥X1,for t≥Tx.(2.4) Remark 2.2. In fact, in the proof of [5, Lemma 3.1], there is an oversight in the definition of X1, and the correct definition should be as above in (2.3). 3 Global Attractivity In this section, we prove the following criterion for the global attractivity of the positive ω-periodic solution of (1.3), which improves the stability criterion in [5, Theorem 3.1]. Theorem 3.1. If there is T > 0such that nXn−1 1 (1 + Xn 1)2sup t≥TZt t−τ(t) m X i=1 bi(s) e−Rt sa(u)du ds < 1,(3.1) then the positive ω-periodic solution ex(t)of (1.3) is globally attractive (in the set of all positive solutions). Proof. By the change of variables y(t) = x(t)−ex(t), model (1.3) is reduced to y0(t) = −a(t)y(t) + m X i=1 bi(t)1 1+(y(t−τi(t)) + ex(t−τi(t)))n−1 1 + ex(t−τi(t))n, t ≥0.(3.2) Clearly, zero is an equilibrium point and the equation (3.2) has the form of the DDE (2.1) with fi(t, ϕ) = bi(t)1 1+(ϕ(−τi(t)) + ex(t−τi(t)))n−1 1 + ex(t−τi(t))n,(3.3) for all t≥0, ϕ∈C, and i= 1, . . . , m. As only positive solutions of (1.3) are admissible, naturally e S={ϕ∈C:ϕ(θ)≥ −ex(θ) for −τ≤θ < 0, ϕ(0) >−ex(0)}(3.4) is the set of admissible initial conditions for (3.2). We need to show that the zero equilibrium of (3.2) is globally attractive in e Sand, in order to do so, the cases of oscillatory and non-oscillatory solutions are considered separately. 4 On the one hand, as the positive continuous function a(t) is ω-periodic, then (H1) holds, and, on the other hand, it is clear that hypothesis (H2∗) holds for all ϕ∈e S. Consequently, from [2, Lemma 2.2] we conclude that all non-oscillatory solutions of (3.2) converge to zero as t→ ∞. Now we consider y(t) an oscillatory solution of (3.2) with an initial condition in e S. Naturally, y(t) = x(t)−ex(t) for some x(t) positive solution of (1.3) and by Lemma 2.3 there is Tx> T such that (2.4) holds. From Lemma 2.4 and Theorem 2.2 in [2] (see Remark 2.1), it is enough to show that hypothesis (H2) holds for ϕ∈e Sreplaced by yt, for all t>Tx. For t>Txand i∈ {1, . . . , m}, the Lagrange’s Theorem allows us to conclude that there is ξ=ξ(t, y) between y(t−τi(t)) + ex(t−τi(t)) = x(t−τi(t)) and ex(t−τi(t)) such that fi(t, yt) = bi(t)1 1+(y(t−τi(t)) + ex(t−τi(t)))n−1 1 + ex(t−τi(t))n =−nξn−1bi(t) (1 + ξn)2y(t−τi(t)). From Lemmas 2.2 and 2.3, we know that ex(t)≥X1and x(t)≥X1for all t > Tx. Consequently, ξ=ξ(t, y)≥X1for all t≥Txand, as σ7→ nσn−1 (1+σn)2is a non-increasing function on (0,∞), we have fi(t, yt) = nξn−1 (1 + ξn)2bi(t)−y(t−τi(t))≤nξn−1 (1 + ξn)2bi(t)Mi t(−yt)≤nXn−1 1 (1 + Xn 1)2bi(t)Mi t(−yt). Analogously, we have fi(t, yt) = −nξn−1 (1 + ξn)2bi(t)y(t−τi(t)) ≥ − nξn−1 (1 + ξn)2bi(t)Mi t(yt)≥ − nXn−1 1 (1 + Xn 1)2bi(t)Mi t(yt). Thus, (H2) holds, with ϕ=yt, for t > 0 large and λ1,i(t) = λ2,i(t) = nXn−1 1 (1 + Xn 1)2bi(t). Finally, condition (3.1) is equivalent to (2.2) with α∗ 1=α∗ 2, and we conclude that y(t)→0 as t→ ∞. The proof is complete. Remark 3.1. In [5, Theorem 3.1], G. Liu et al. prove that the positive ω-periodic solution ex(t) of (1.3) is a global attractor of all positive solutions if nXn−1 1 1 + Xn 1 eA(ω) eA(ω)−1Zω 0 m X i=1 bi(s)ds ≤1,(3.5) where A(ω) = Zω 0 a(u)du. We claim that (3.1) is a weaker condition than (3.5). In fact, for t > 0 5 and considering N(t)∈Nsuch that τ(t)∈(N(t)−1)ω, N(t)ω, we have Zt t−τ(t) m X i=1 bi(s) e−Rt sa(u)du ds ≤ N(t) X j=1 Zt−(j−1)ω t−jω m X i=1 bi(s) e−Rt sa(u)du ds = N(t) X j=1 e−(j−1)A(ω)Zt−(j−1)ω t−jω m X i=1 bi(s) e−Rt−(j−1)ω sa(u)du ds =  N(t) X j=1 e−A(ω)j−1 Zt t−ω m X i=1 bi(s) e−Rt sa(u)du ds =e−N(t)A(ω)−1 e−A(ω)−1Zt t−ω m X i=1 bi(s) e−Rt sa(u)du ds <eA(ω) eA(ω)−1Zt t−ω m X i=1 bi(s) e−Rt sa(u)du ds ≤eA(ω) eA(ω)−1Zω 0 m X i=1 bi(s)ds, which shows that condition (3.5) implies condition (3.1). Therefore, Theorem 3.1 improves the stability criterion in [5]. Moreover, the example given below shows that condition (3.1) is strictly less restrictive than condition (3.5). 4 Numerical example In this section, a numerical example is given to illustrate the effectiveness of the new result presented in Theorem 3.1. Here, we have used the Matlab software [9], to plot the numerical simulation of the solutions. Letting n= 1, m= 2, a(t) = 1+ 1 2cos(2πt), b1(t) = 3 81 + 1 2cos(2πt),b2(t) = 3 81 + 1 2sin(2πt), and τ1(t) = τ2(t) = 1 2(1 + sin(2πt)) in the hematopoiesis model (1.3), we have de DDE x0(t) = −1 + 1 2cos(2πt)x(t) + 6 + 3 2(cos(2πt) + sin(2πt)) 8(1 + x(t−1 2(1 + sin(2πt)))), t ≥0.(4.1) Eq. (4.1) is 1-periodic and, by easy computations, we can see that X1=√2−1 2e−1and Zt t−τ(t) (b1(s) + b2(s)) e−Rt sa(u)du ds ≤Z1 0 (b1(s) + b2(s))ds =3 4, thus 3 4 1 (1+X1)2≈0.64757 <1. Consequently, from Theorem 3.1, the positive 1-periodic solution ex(t) of (4.1) attracts all positive solutions. See the numerical simulations for three solutions shown in Figure 1. However, since nXn−1 1 1 + Xn 1 eA(ω) eA(ω)−1Zω 0 m X i=1 bi(s)ds =2 e 2 e +√2−1 e e−1 3 4≈1.10248 >1, condition (3.5) is not satisfied, thus the result of G. Liu et al. [5] cannot be applied to this example. Figure 1 illustrates our example by plotting the graph of three solutions of (4.1) with the following initial conditions in S: x0=ϕ1,where ϕ1(θ) = cos(θ),for θ∈[−1,0]; x0=ϕ2,where ϕ2(θ)=0.5 eθ,for θ∈[−1,0]; x0=ϕ3,where ϕ3(θ)=0.1−sin(πθ),for θ∈[−1,0]. 6 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time t solution x(t) Figure 1: Behavior of three positive solutions of (4.1). Acknowledgments This work was partially supported by Funda¸c˜ao para a Ciˆencia e a Tecnologia under Project UID/MAT/04561/2019 (Teresa Faria) and UID/MAT/00013/2013 (Jos´e J. Oliveira). References [1] L.Berezansky, E.Braverman, and L.Idels, Mackey-Glass model of hematopoiesis with monotone feedback revisited, Appl. Math. Comp. 219 (2013), 4892-4907. [2] T. Faria and J.J. Oliveira, On stability for impulsive delay differential equations and applications to a periodic Lasota-Wazewska model, Disc. Cont. Dyn. Systems Series B, 21 (2016), 2451-2472. [3] T. Faria and J.J. 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