Some analytical and numerical results for a

nonlinear Volterra integro-differential equation

with periodic solution modeling hematopoiesis

Athena Makroglou+, Department of Mathematics,

University of Portsmouth,Portsmouth, Hampshire, PO1 3HE, England, UK.

athena.makroglou@port.ac.uk

and Yang Kuang, Department of Mathematics and Statistics,Arizona State University

Tempe, AZ 85287-1804, USA,kuang@asu.edu

February 2006

Abstract

We consider a nonlinear Volterra integro-differential equation withperiodic solution, a special case of which has been used in modelinghematopoiesis dynamics. Some analytical results concerning existenceand uniqueness of a positive periodic solution are presented togetherwith some numerical results based on the reformulation of the equa-tion as a system of ODEs and also from its direct solution using thetrapezoidal method.

+ Correspondence to: A. Makroglou.

1

1 Introduction

Volterra integro-differential equations (VIDEs) with periodic solutions haveapplications in many biological and biomedical areas, e.g. population biology(cf. [9], [15], [19], [21], [22], [34], [37]), epidemiology (cf. [1], [2], [13], [14],[20]), physiology (cf. [35]).

Various types of VIDEs with periodic solutions have been consideredin the literature for theoretical as well as practical purposes (existence anduniqueness of periodic solutions, stability). They include linear equations(cf. [3], [4]), equations of logistic and other nonlinear types (cf. [5], [6], [7],[11], [12], [16], [17], [18], [19], [22], [28], [30], [31], [35], [36], [37]).

Some of the equations treated are first order ones and some are ofsecond order (cf. [32]). In addition, some are ordinary VIDEs and some aredelay VIDEs.

Various approaches have been used in the proofs of existence and unique-ness of periodic solutions which can be categorised as follows (see also [16],p. 261):

(i) approaches that use the standard contraction principle and the simplebut effective fluctuation principle.

(ii) approaches which in some cases extrapolate known results that applyto equations without time delay.

(iii) approaches that apply the Horn’s asymptotic fixed-point theorem.

(iv) approaches that use the so called coincidence degree method. This lastmethod can only obtain existence statement.

All require assumptions that ensure some degree of uniform boundedness ofsolutions. Such requirements are readily met by well formulated models inapplications.

The numerical treatment of such equations has been considered by [3],who applied polynomial and mixed collocation methods to linear first and sec-ond order VIDEs with periodic solutions. Existence and uniqueness resultswere also given. Collocation methods have also been used for the numericalsolution of delay differential equations with periodic solution (cf. [10]).

In this paper we consider the following nonlinear VIDE with periodicsolution. It is derived from a model of hematopoiesis which has generalimplications in cell growth dynamics ([35]). It is a nonautonomous extension

2

of a delay differential equation (DDE) model proposed by M.C. Mackey andL. Glass (1977) ([25]), and subsequently studied in more forms by manyothers (see [24] and the many references cited by Pujo-Menjouet et al ([27])and Crauste ([8]).

y′(t) = −δ(t)y(t) −β(t)y(t)

1 + yn(t))

+ α(t)

∫ ∞

0

K(s)y(t − s)

1 + yn(t − s)ds, (1.1)

y(t) = φ(t), t ∈ (−∞, 0], (1.2)

where n > 0, δ(t), β(t), α(t) are continuous positive periodic functions on[0,∞) with a period ω and K(s) : [0,∞) → [0,∞) is a delay kernel, assumedto be piecewise continuous, nonincreasing eventually and normalized suchthat

∫ ∞

0K(s)ds = 1. It is also assumed that φ(s) ∈ C((−∞, 0], R+), φ(0) >

0. Here y(t) denotes the density of hematopoietic stem cells (see for example[25], [24], [29] and the recent paper [27] for an explanation of the modelingof the problem).

In particular, we will present theoretical results (existence and unique-ness) and also numerical results.

The organisation of the paper is as follows. An outline of a proof forexistence and uniqueness of periodic solutions of a generalisation of equation(1.1)-(1.2) is given in section 2. In section 3 we show how the VIDE (1.1)-(1.2) can be transformed to a system of ODEs. In section 4 we describemethods of numerical solution, and in section 5 we present numerical results.A discussion of the results and some ideas for further research are to be foundin section 6.

2 Existence and uniqueness of periodic solu-

tion

In this section, we present results on the existence and uniqueness (globalstability) of a periodic solution of a general equation that includes VIDE(1.1)-(1.2) as a special case. The method of proof is similar to that of [35].

3

We consider in this section

y′(t) = −δ(t)y(t) − β(t)f(y(t))

+ α(t)

∫ ∞

0

K(s)f(y(t − s))ds, (2.1)

y(t) = φ(t), 0 < φ(t) < M, t ∈ (−∞, 0], (2.2)

where M is a constant, f(x) is a differentiable nonnegative function boundedabove by M > 0, f ′(0) = 1, (f(x)/x)′ < 0 for x > 0, limx→∞ f(x)/x = 0 andf(0) = 0. Moreover, either there is a unique x∗ > 0, such that f ′(x∗) = 0 orf ′(x) > 0 for all x ≥ 0 (in this case, we will define x∗ = ∞). An example ofsuch a function is f(x) = x/(1 + xn), n ≥ 1.

To facilitate the proof, we introduce some notations.Notation-assumptions-preliminaries.

Letα0 = min

0≤t≤ω{α(t)}, α0 = max

0≤t≤ω{α(t)},

with β0, β0, δ0, and δ0 defined similarly.

The existence of a nonnegative periodic solution is implied by the dis-sipativity of the equation ([38]). So we will establish the dissipativity of(2.1)-(2.2) under appropriate (to be given below) conditions.

Notice thaty′(t) ≤ −δ0y(t) + α0M

implies thatlimt→∞

sup y(t) ≤ α0M/δ0.

This shows that all solutions are eventually uniformly bounded from aboveby y0 ≡ α0M/δ0. The difficulty is to show solutions are eventually uniformlybounded away from zero. We assume below that

α0 > δ0 + β0.

Hence there is a A > 0 such that g(A) = 0 where

g(x) = −(δ0 + β0) + α0f(x)/x, x > 0.

Definey0 = min{y0, A, x∗}/2.

4

Then, similar to the tedious but crucial proof of Lemma 2.4 in [35], we have

Lemma 1. Let y(t) be any positive solution of (2.1)-(2.2) and α0 >δ0 + β0, then there exists a T1 > 0 such that y(t) > y0 for t > T1.

The following simple but important Lemma is needed in the proof ofTheorem 1 below.

Lemma 2 ([35], p. 1026). Any ω-periodic solution y(t) of (2.1)-(2.2)is also an ω-periodic solution of the equation

y′(t) = −δ(t)y(t) − β(t)f(y(t)) + α(t)

∫ ω

0

H(s)f(y(t − s))ds, t ≥ 0, (2.3)

where

H(s) =

∞∑j=0

K(s + jω), s ∈ [0, ω], (2.4)

and vice versa.Let also

H = max{f ′(x), x ≥ y0}.

By adapting the rather technical and tedious arguments (such adap-tation is mostly straightforward and we omit the details) contained in theproofs of the Theorems 3.5 and 3.6 in [35], we can obtain the following result.

Theorem 1. Assume that∫ ∞

0

sK(s)ds < +∞

and that

(α0 + β0)H < δ0 ≤ δ0 + β0 < α0, x∗ < ∞,

or(α0 − β0)H < δ0 ≤ δ0 + β0 < α0, x

∗ = ∞.

Then,

5

(i) there exists a unique positive ω-periodic solution y∗(t) of (2.1)-(2.2),and

(ii) any positive solution y(t) of (2.1)-(2.2) satisfies

limt→∞

[y(t) − y∗(t)] = 0.

3 Reformulation of the VIDE as a system of

ODEs

Consider the VIDE (1.1). Setting t − s = u we rewrite it as

y′(t) = −δ(t)y(t) −β(t)y(t)

1 + yn(t)+ α(t)

∫ t

−∞

K(t − u)y(u)

1 + yn(u)du. (3.1)

The delay kernel is assumed to have the form

K(t) ≡ Km(t) =cm+1tme−ct

m!. (3.2)

We will use the so called ’linear chain trick’, see for example [33], p.248, originally in [23]. We set

zj(t) =

∫ t

−∞

Kj(t − u)y(u)

1 + yn(u)du,

j = 0, . . . , m, (3.3)

where Kj(t) = cj+1tje−ct

j!. Then

dzj(t)

dt=

∫ t

−∞

dKj(t − u)

dt

y(u)

1 + yn(u)du + Kj(0)

y(t)

1 + yn(t). (3.4)

After calculatingdKj(t−u)

dtwe easily find that

dzj(t)

dt= c[zj−1(t) − zj(t)],

j = 1 . . . , m, (3.5)

dz0(t)

dt= c[

y(t)

1 + yn(t)− z0(t)]. (3.6)

6

Using (3.3) in equation (3.1) we obtain

y′(t) = −δ(t)y(t) −β(t)y(t)

1 + yn(t)+ α(t)zm(t). (3.7)

So we have obtained an ODE system of m + 2 equations in y(t), zj(t), j =0, . . . , m.

4 Numerical methods

One approach for solving the VIDE (3.1) combined with the kernel (3.2) andthe initial conditions (1.2), is to solve the equivalent ODE system (3.5)-(3.7).

Initial conditions:

zj(0) =

∫ 0

−∞

Kj(−u)φ(u)

1 + φn(u)du, (4.1)

j = 1, . . . , m

z0(0) =

∫ 0

−∞

ce−c(−u) φ(u)

1 + φn(u)du. (4.2)

y(0) = φ(0). (4.3)

There is a wealth of methods suitable for solving ODE systems of equationswith periodic solution (cf. [26]). The numerical results presented in nextsection were obtained with one of the Matlab functions, the ODE45.

Solving the VIDE directly: The mixed collocation methods of [3]can be used. Also polynomial collocation methods. Here we describe the ap-plication of a simple method (trapezoidal) which is well known to be suitablefor integrals with periodic integrands. It is applied to the VIDE

y′(t) = G(t, y(t), z(t)), 0 ≤ t ≤ T, (4.4)

y(t) = φ(t), t ∈ (−∞, 0], (4.5)

where

z(t) =

∫ t

−∞

k(t, s, y(s))ds. (4.6)

7

We consider equidistant mesh ti = ih, i = 0, . . . , N , where h = T/N is theconstant stepsize.

Then the numerical method applied to the integrated form of (4.4) fromtj to tj+1 and (4.6) becomes

yj+1 = yj +h

2[G(tj, yj, zj) + G(tj+1, yj+1, zj+1)] (4.7)

zj+1 =

∫ 0

−∞

k(tj+1, s, φ(s))ds + h

j+1∑i=0

wik(tj+1, ti, yi), (4.8)

j = 0, . . . , N − 1,

where wj are the usual trapezoidal rule weights.Equations (4.7)-(4.8) are solved as a nonlinear algebraic system of equa-

tions in y1, . . . , yN ; z1, . . . zN .

5 Numerical results

Figures 6.1-6.4 show plots of y(t) using ([35], p. 1029)

α(t) = 20, β(t) =1

2| cos t| +

1

2, δ(t) = | sin t|,

and figures 5.5-5.8 show plots of y(t) using

β(t) =1

2| cos t| +

1

2, α(t) = 4β(t), δ(t) = | sin t|,

and φ(t) = et, t ∈ (−∞, 0]. The values of the parameters c, m, n used aregiven in the figures’ titles. ( c = 1, m = 2 and n = 1, n = 2).

In figures 6.1, 6.2, 6.7, 6.8 the t-interval is [0, 40π], and in figures 6.3,6.4, 6.5, 6.6 the t-interval is [0, 10π] so that more detail can be seen nearerto the origin.

The starting integrals were evaluated using a Gauss-Laguerre quadra-ture rule with 25 nodes.

The plots were obtained by using the Matlab ODE45 to solve the system(3.5)-(3.7) and the direct method based on the use of the trapezoidal method.The trapezoidal method was used with stepsize h = π/64.

8

6 Discussion

We considered a nonlinear VIDE with positive ω-periodic solution which hasapplications in hematopoiesis. The existence and uniqueness was proved fora more general nonlinear equation extending the work of [35]. The numericalresults obtained using α(t), β(t), δ(t) ω = π-periodic and positive functions,verified that the VIDE has a π-periodic solution. Printed results obtainedwith the Matlab function ODE45 and the direct trapezoidal method hadagreement in at least two significant decimal places.

Extensions can involve proof of existence and uniqueness using differ-ent approaches and introduction of numerical methods specially designed forproblems with periodic solution, extending ones that apply to linear VIDEsand to ODEs with periodic solution.

Acknowledgments

The research of Yang Kuang is supported in part by DMS-0077790 andDMS/NIGMS-0342388.

Keywords: Numerical solution, Volterra integro-differential equations,periodic solution, existence and uniqueness, hematopoiesis.

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0 4 8 12 16 20 24 28 32 36 401

5

9

13

17

21

25

29

33

t (in multiples of π) days

y(t)

c=1, n=1, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.1: α(t) = 20, β(t) = 1

2| cos t| + 1

2, δ(t) = | sin t|

13

0 4 8 12 16 20 24 28 32 36 401

2

3

4

5

6

7

8

9

t (in multiples of π) days

y(t)

c=1, n=2, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.2: α(t) = 20, β(t) = 1

2| cos t| + 1

2, δ(t) = | sin t|

0 1 2 3 4 5 6 7 8 9 101

5

9

13

17

21

25

29

33

t (in multiples of π) days

y(t)

c=1, n=1, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.3: α(t) = 20, β(t) = 1

2| cos t| + 1

2, δ(t) = | sin t|

14

0 1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

t in multiples of π) days

y(t)

c=1, n=2, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.4: α(t) = 20, β(t) = 1

2| cos t| + 1

2, δ(t) = | sin t|

0 1 2 3 4 5 6 7 8 9 10

1

1.5

2

2.5

3

3.5

t (in multiples of π) days

y(t)

c=1, n=1, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.5: β(t) = 1

2| cos t| + 1

2, α(t) = 4β(t), δ(t) = | sin t|

15

0 1 2 3 4 5 6 7 8 9 10

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

t (in multiples of π) days

y(t)

c=1, n=2, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.6: β(t) = 1

2| cos t| + 1

2, α(t) = 4β(t), δ(t) = | sin t|

0 4 8 12 16 20 24 28 32 36 40

1

1.5

2

2.5

3

3.5

t (in multiples of π) days

y(t)

c=1, n=1, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.7: β(t) = 1

2| cos t| + 1

2, α(t) = 4β(t), δ(t) = | sin t|

16

0 4 8 12 16 20 24 28 32 36 40

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

t (in multiples of π) days

y(t)

c=1, n=2, m=2

continuous line: ODE45dashed line: Trapezoidal

Figure 6.8: β(t) = 1

2| cos t| + 1

2, α(t) = 4β(t), δ(t) = | sin t|

17