the role of four regulatory hormones in controlling testicular function in a delay model

Pergamon Mathl. Comput. Modelling Vol. 25, No. 5, pp. 101-116, 1997

Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

CPII: SO895-7177(97)00033-2 08957177/97 $17.00 + 0.00

The Role of Four Regulatory Hormones in Controlling Testicular Function

in a Delay Model

P. DAS AND A. B. ROY Department of Mathematics, Jadavpur University

Calcutta 700 032, India

(Received April 1995; revised and accepted June 1996)

Abstract-Cur proposed mathematical model for the secretion in hypothalamus-pituitary-go- nadal axis introduces four regulatory hormones viz. GnFtH, FSH, Testosterone, and Inhibin. Here, we have considered four dimensional delay differential equations with multiple negative feedback loops which accounts for the pulsatile release of these hormones. We have derived the conditions for local asymptotic stability of the steady state and have estimated the length of delay to preserve the stability. Regions for stability and oscillations of the system are given in K - 7 and m - 7 plane, and the role of Inhibin in regulating the male fertility status by altering the FSH level has been clearly shown by computer simulation of the model.

Keywords-Hormone, Negative feedback, Delay, Oscillation, Computer simulation.

1. INTRODUCTION

There axe many feedback mechanisms working in human body to maintain key physiological vari-

ables within normal limits. In these biochemical feedback networks, there are delays associated

with the time interval between sensing of some disturbances and mounting of an appropriate

physiological response.

A variety of oscillatory phenomena axe thought to arise because of negative feedbacks-from

high frequency neural activity to longer period circadian rhythms and endocrine oscillations [l].

The introduction of a time delay in feedback loops often alters the stability properties of a dy-

namical system and their effects have been found to depend very much on the type of delay [2-61.

Such effects were studied in certain dynamical respiratory and haematopoietic diseases by Mackey

and Glass [7-91.

In this paper, we have considered a model of the endocrine regulation network with multiple

feedback incorporating time delays. Reproductive phenomena bear a peculiarly important and

intricate relationship with central nervous system (CNS) mechanism. There are three active com-

ponents of the system-the hypothalamus, the anterior pituitary and the testis. Hypothalamus

is the key to cyclic reproductive activity. It secretes the gonadotrophic releasing hormone or

G&H in the portal capillary blood vessels of the anterior pituitary and regulating the synthesis

and secretion of Follicle stimulating hormone or FSH and luteinising hormone or LH from the

pituitary. FSH and LH have primarily different targets in the testis, LH acting on the Leydig cells and bring about the secretion of testosterone or T and other steroids. T is concentrated

in the semi-inferous tubules where it is necessary for maintenance of spermatogenesis, circulates in the blood. FSH stimulates the sertolli cell growth and activity leading to the production of

101

102 P. DAS AND A. B. ROY

Androgen Binding Protein-ABP and Inhibin and the initiation of spermatogenesis. Since the role of LH has been discussed in our previous work [lO], we have excluded it to make the model mathematically tractable. Our present aim is to study the pulsatile behaviour of testosterone secretion and to investigate the role of Inhibin or I in regulating male fertility status. With this view, we construct a model dealing with four regulatory hormones, GnRH, FSH, T, and I. It is accepted that Inhibin as the glycoprotein is called has an important role in regulating FSH secretion which is required for successful spermatogenesis [ll] and regular imbalance in T level causes dramatic personality changes. Experimental results show that administration of T causes an abrupt decline in gonadotrophic hormone in castrated males of all species though main sup- pressive effect is exerted in hypothalmic level by affecting GnRH. In the human male, much of the supporting evidences for Inhibin stem from the clinical and pathological situations 112,131, If the production of Inhibin is defective as in the case of some infertile male, I level becomes low in the blood and results in rise in FSH level of the patient 114,151 and vice-versa. Thus, Inhibin concentration has a negative feedback effect on the production of FSH. Again, any change in the FSH level alters the fertility status [16,17] of the individual. So the role of Inhibin is to regulate the FSH secretion, which in turn control the fertility status in male.

Smith [lg] proposed a compartmental model for the Testosterone production involving LHRH, LH, and Cartwright and Husain [19] proposed another model with R, L, and T as variables based on experimental results and showed periodicity in their production by computer simulation of the model. In our previous work [20], we have dealt with R, L, and T involving delays of finite size and represented the model by a system of delay differential equations with multiple negative feedbacks, and derived the conditions for asymptotic stability and instability of the system.

In our present work, we have considered the negative feedback effects of T-concentration on G&H and that of Inhibin on FSH as mathematically modelled in Section 2, and studied the asymptotic stability of the steady state without delay in Section 3, and with delay in Section 4. We also have estimated the length of delay to preserve stability by Nyquist criteria in Section 5. We derived the sufficient conditions for oscillations of the linearized system in Section 6. We have simulated the nonlinear model numerically on computer, results of which along with a table of values of parameters and discussions are given in Section 7.

2. MATHEMATICAL MODEL

In formulating the mathematical model of negative feedback regulation of testosterone secretion, the following events are considered. GnRH (R) is secreted from the hypothalamus and it

reaches the pituitary after a time 72. It stimulates the secretion of FSH(F) from the anterior pituitary. To make the model mathematically tractable, Testosterone secretion is considered to be stimulated after a time 74 by GnRH [21] directly, though in reality GnRH stimulates LH which in turn stimulates T. FSH stimulates the Inhibin secretion after a time 75. Inhibin and Testosterone concentrations are considered to have negative feedback effect on FSH and GnRH secretion after the time 73 and rr, respectively (see Figure 1).

The interactions among the hormones are all composed of complex enzyme reactions following cooperative kinetics rather than Michaelis Menten type. In forming the growth equations of hormones GnRH and FSH, we take into account the negative feedback effects of G&H by T and on FSH by I. The rate of growth of GnRH and FSH due to corresponding feedbacks are represented by the nonlinear term of the form A/(K + Xn(t - q)) where X(t) denotes the concentration of the negative feedback agent and m > 0 is called the Hill coefficient, which is related to the degree of cooperativity of ligand hormones with the specific binding protein and pi, i.e., transmission delay is the time taken by the hormones to be transported between the hypothalamus, pituitary, and the testis because of their spatial separation and the fact that the hormones are transported by the circulating blood.

Four Regulatory Hormones 103

;Et HYPOTHA;AMUS 1

LCELL TESTIS

I

CELL 1

I

Figure 1. A schematic feedback model involving four hormones GnRH, FSH, Testos- terone, and Inhibin.

Rate equations of growth of the reproductive hormone are written in the form of following

nonlinear delay differential equations:

k(t) = -c&(t) +

P(t) = -/P(t) +

A I--cl +Tm(t-71)’

&R(t - 72) + A3

K1 + Pqt - 73) ’ (2-l) 5!‘(t) = -yT(t) + A4R(t - 74),

i(t) = -61(t) + AsF(t - s),

where Al, AZ, Ad, A5 axe the respective rates of synthesis of R, F, T, and I hormones, and Q, p, y, b

are the decay rates of R, F, T, and I in the blood stream. It is assumed that each of these hormones

is cleared from the blood stream according to the first-order kinetics.

We associate the initial values of the form:

R(r) = +6(t), for - 7s 5 t < 0, where 7s = max(rs, 74),

F(t) = h(t), for - 75 2 t 5 0,

T(t) = dl (t), for - 71 I t 2 0, (2.2)

I(t) = 43(t), for - 7-3 5 t 5 0,

where #i(t) E C( [-q, O],R+) and &(O) > 0, for i = 1,3,5,6. We now want to study the stability and oscillatory behaviour of the solutions of our system of equations (2.1) with (2.2).

Dimensionless System

Let R ,2 T I t

5=--, a1 Q2’

.z=-, w=--, 7=-

a3 a4 a5’ (2.3)


:. (2.1) becomes (writing t for r)

i(t) = -aa5z(t) + &a5/(~3m)

Kl/cq + z”(t - 71)’

A2ala5 si(t) = --/3a5?&) + -

A3(.r5&2@)

a2 x0 - 72) + ICI/q + wm(t - 73) ’

~44~~5 i(t) = -yQz(t) + - cy3 Y(t - 74h

A5a2a5 ?i(t> = -6a5w(t) + - Q4 ?/(t - 75)>

where

03 = mfi, a4 = m 6, A3

a5=-j--&.

:. The dimensionless equations are

k(t) = -a&) + A

l+P(t-71)’

G(t) = -42$/(t) + z(t - 72) + A

1 + uP(t - 73) ’

(2.4

(2.5)

i(t) = -asz(t) + Kx(t - 74),

ti(t) = -arzu(t) + y(t - s),

where AQCY. A3P ~437 A36

a’ = A1A2’ a2 = AIA2 ’ a3 = A1A2’ u4 = AlA2’

A = A:As 1 A1 A4 (2.6)

A:A; @+1)/m’ and K = a’

3. EXISTENCE OF STEADY STATES AND LOCAL STABILITY WITHOUT DELAY

Let ,$i, &&‘C;) be a steady state of the system (2.5). Evidently, j = (a$/K), y = a$;,

where 2 and 6 are the positive solutions of the equations

ala38 *VI+’ + alas: - AK = 0, and (3-l) *ffl+1_**m

020420 xw +~12a46- A+$ =O. ( > (3.2)

Using Descartes’ rule of sign, we find that (3.1) has only one positive root E, (3.2) can have either one or three positive root(s) depending on the parameters. Thus, the existence of the steady state is proved.

Now, for uniqueness of the steady state since (3.1) has only one positive solution E, $ = (C&/K) has only one value.

For &, we consider (3.2) as

(3.3)

and consider the L.H.S. of (3.3) as

fi(G) = a2a4& - ii, (3.4)


and R.H.S. as A

f&l) = p. 1+&m

(3.5)

As shown in Figure 2, fr(&) increases from -I to co, while A(6) decreases from A to 0. Since

both fr (&) and f 2 w are strictly monotone (i.e., always decreasing or always increasing) there ( * )

is only one intersection. Thus, there is only one positive &, and hence, only one positive i.

Therefore, &(g, G, E, 2;) is unique for any m.

Figure 2. Showing the existence of only one positive root ; of equation (3.2).

Stability

We investigate the stability of the steady state .k(i, i, E, 6) by linearization.

x=x1-;, Y = Y’ - $7 Z = Z’ - I, w=w’-??I. (3.6)

Then the linearized form of the system (2.5) about the steady state & is (writing x, y, Z, w for

Z’, y’, z’, w’)

Adm-1

g? = -azy(t) + x(t - 72) - Am&“-l

(1 t&m) 2 4 - 73), (3.7)

Let

i- = -a3z(t) + Kz(t - q),

ti = -arw(t) + y(t - 75).

Am,??-1 (1 + “_)2 = 61,

The characteristic equation is

D(X, 7+) = X4 + AX3 + (B + 62e-X(r3+T5) + 61Ke-x(Tl+T“)) X2

+ ( C + (al + a3)&e -Wn+n) + &fqa2 + u4)e-wn+d

>

x (3.8)

+ D + u,u362e-(r3+*s5) +u2u4S1Ke-(r1+T4) + K6162e-(rl+r3+T4+T5) = 0.

106 P. DAS AND A. B. Rou

When delays are absent in the system, the characteristic equation becomes,

D(x, 0) = ii* + AX3 + (B + 62 + &K)ii2 + (c + (Ul + a3)62 + 61K(a2 + U4)) x

+D+UlU362+U2U461K+K6162 =O, (3.9)

where A = al + u2 + u3 +u4,

B =U1@2 +a2a3+ U3U4+U4U1+0103+0204,

c=UlU2U3 +UlU2U4+UlU3U4+U2U3U4, (3.10)

D = ala2a3a4.

The necessary and sufficient conditions for local asymptotic stability (without delay) are given by Routh-Hurwitz condition, which are in our case

DI = al + ua+ U3 +a4 > 0,

D2 = A(B+Sz +&K)-(C+(ul +a~)62 +6lK(u2 +a*))

= AB -C+(ul + u3)KS1 +S2(u2 +a*) > 0,

03 = {C+(ul + as)62 +S1K(u2 + ~4)) D2

- (al + a2 + a3 + a4)2 (ala2a3a4 - hTbla2a4 + 62UlU3 + x&62),

D4 = DD3 > 0.

(3.11)

Since all the Dl, D2, D3, and D4 are positive, D(X, 0) has roots whose real parts are all negative. Hence, the steady state is always asymptotically stable if delays are absent.

4. LOCAL STABILITY WITH DELAY

We are now ready to examine the effects of time delay on the asymptotic stability of the linearized system (3.7).

THEOREM 4.1. The trivial solution of the linearized system of equations (3.7) is asymptotically stable if

l+K max - ( 1 61 3

‘&$ u4 >

<1

' a1 (4.1)

for any delay ri 2 0 (i = 1,2,3,4,5).

PROOF. Let us consider the Lyapunov functional v(t) defined by

v(t) = V(t, z, Y, 2, w) = Izl + k/I + I4 + I4 + 61 lITI Iz(s)l ds

s

t t + Is(s)1 ds +

s Iw(s)l ds + K

t-72 t-n l:, Ids)l ds- (4.2)

4 Ids)I ds + LIT

J

Calculating the right derivative D+w of v along the solutions of (3.7), we get

D+v(t) I -[(a - (1 +K)) l+)l+ (U2 - 1) Iv(t)1 + (U3 - 61) Iz(t)l+ (U4 - 62) lw(t)ll, (4.3)

if max ((1 + K)/ol, l/oz, &/a3,62/a4) < 1.

From (4.2) and (4.3), we get

Idt)l + Iv@>l + Iz(t)l + IWI + (~1 - Cl+ WI 1’ Ms>l ds + (~1 - 1) I’ IY(s)I ds

+ (a3 - 61) Of Iz(s)l ds + (U4 - 62) I' b(s>I ds 5 $‘) < 00. J

(4.4)


From (4.4), and by using Barbalat’s Lemma [22] and (4.3), we conclude that [z(t)1 + Ip( +

Ht)l + WY -+ 07 as t + 00 if (4.1) holds.

:. $4, c, I, &) is locally asymptotically stable.

The condition (4.1) in the above theorem states that local asymptotic stability is ensured if

the dimensionless parameters (1 + K/or), (l/as), (&/as), (6 2 a4 never exceeds 1, whatever the / ) value of delay is. We see that the parameters depend mainly on nature of the feedback terms

m, K, and degradation rates.

5. ESTIMATION FOR THE LENGTH OF DELAY TO PRESERVE STABILITY

From (3.11), we see that in absence of delay 2 is always locally asymptotically stable. By

continuity, all eigenvalues will continue to have negative real parts for sufficiently small delays ri,

(i = 1,2 ,..., 5).

To get our estimate on the length of delay, we shall use the Nyquist criterion for solutions. For

this, we consider the system (3.7) and the space of real valued continuous functions defined on

[ri, oo] satisfying the initial conditions (2.2). Now, (3.7) can be written as (writing -ai = ai and

-& = S,!)

? = ais + +(t - 71)

9 = a!&t) + z(t - 72) + 6$(t - 5),

2 = a$z(t) + Kx(t - 7-4),

72 = a&w(t) + y(t - 75).

(5.1)

Let W?, WL WY, and @(I’) denote the Laplace transform of z(t), y(t), z(t), and w(t),

respectively. The Laplace transform of the system (5.1) yields

(P - a;) Z(P) = I~~~-~“E(P) + 61,ewPT1K1(P) + x(O),

(P - aa)g(P) = emPT2Z(P) + Ghe-P73Cj(P) + emPT2K2(P) + i)2empT3K3(P) + y(O),

(P - a$) Z(P) = Ke-p”S(P) + KemPT4K4(P) + z(O),

(5.21

(P - a>) C(P) = e-PTs&(P) + ewPT5Ks(P) + w(O),

0 Kl(P) = J esPtz(t) dt, K2(P) =

-71 J 0

eePtz(t) dt, -T?

J 0

J 0

K3(P) = emPtw(t) dt, K,(P) = emPtz(t) dt, and (5.3) -73 -74

J 0

K5(P) = eePty(t) dt. -75

F&arranging (5.2), we get

p4 + AP3 + (B + b2e-P(T3+Q5) + &K~-P(TI+T*)) p2

+ (C + (al + a3)62e -p(73+T5) + &K(az + a4)e -P(T1+74) p > (5.4)

+ (D + K61a2a4e-P(T1+T4) + 62a1a3e-p(73+1~) + K6162e-(‘l+T3’3+T4+T5’5) )I g(p)


= zo(P + a3)(P + ug)empT2 + j/o {(P + al)(P + aa)(P + ad) + (P + u4)KSle-P(T1+T4J}

- .zo(P + u4)61e-P(T1+T2) - wo 1 (P + al)(P + u3)S2emPT3 + K6152e-P(T1+T3+T4) > - (P + u3)(P + u&e -w?+T2)&(p)

+ {(P + al)(P +u,)(P +a4)emPTz + (P+ c~~)K6le_~(~~+~~+~~)} K2(P)

- (P+al)(P+ ug)(P+u4)S2e {

-pT3 +(P+ u4)K~~~2e-PtT1+T3+T44) K3(P) >

- (P+a4)Kbe -Pb'l+m+T4)&(P)

- (P+a)(P-t u3)62e 1

--p(73+75) +K&&e- p(,l+T2+T4+75) K,(p)

>

= 50 { P2 + (a~ + a4)P + a3a4} eepT2

+ y0 P3 + (al + a3 + u4)P2 + (ula4 + 01~3 + a3u4)P

+ (P + a4)KGle -P(fl+T4) +alaaad

>

(5.4) cont.

- zo(P + u4)61e-p(T1+r2) - 200 1 (P” + (al + a3)P + ula3) S2emPT3

+ K&&e- P(Tl+Q+d >

- {P” + (~3 + u4)P + u3a4) 61e-P(T1+T21Kl(P)

+ {P” + (al + a3 + u4)P2 + (ala4 + ala3 + a3a4)P + 6~~3~4) espr2

+ (P + a4)KGle -P(Tl+T2+74) K2(p) 1 -

[ { P3 + (al + a3 + u4)P2 + (ala4 + ala3 + a3u4)P + a1u304) 62espr3

+ (P + u4)K&&e -p(T1+T2+T4) 1 K3(P) _ (P + a4)K61e-P(TI+f”+f’)K4(P)

- P2 + (al + u3)P + u~a3~2e-p(73+7s) + KS1~2e-P(71+T3+74+T5) >

KS(P).

The inverse Laplace transform of ij(P) will have terms which exponentially increase with time,

if y(P) has poles with positive real parts. For _k($, k, 5,;) to be locally asymptotically stable, it

is necessary and sufficient that all poles of g(P) have negative real parts. We shall employ the

Nyquist criterion, which states that if P is the arc length along a curve encircling the right half

plane, the curve g(P) will encircle the origin a number of times equal to the difference between

the number of poles and the number of zeroes of y(P) in the right half plane. We see that the

conditions for local asymptotic stability of k(;, 5, E, G) are given by

Im F(ivo) > 0, (5.5.1)

ReF(iwo) = 0, (5.5.2)

where

F(P) = p* + APs + B + K&e-P(rl+T4) + 620-p(T3+TJ) p2 ( >

+ 1

C + (a2 + ,*)K6re-P(T1+T4) + (al + a3)62e-P(T3+‘S) >

P F-6)

D + KSla2u4e(TI+r4) + fi2alu3e--P(r3+76) + K6162e-P(TI+T3+T4+Ts5) 11 ,


and vo is the smallest positive root of the equation (5.5.2). Conditions (5.5.1) and (5.5.2), in our

case become

71: - Bv,2 + D = K&vi cos T~vo + SZV~ cosTzv0 - K&vo(az + u4) sinTlv0

- &&al + as) sin T2v0 - K&a@4 COs TlVO

- &ala3 cos T2vo - K&62 cos(T1 + Tz)vo, (5.7)

-A$ + CVO > -K&vi sinTlv0 - 62vi sinT2vo - KS~vo(u2 + ~4) cosT~vo

- &vo(Ul + as) COST2Vo + K&U@4 sin T~VI-J

+ &ala3 sinT2v0 + K&62 sin(T1 + t2)vo. (5.8)

To get our estimate on the length of delay we shall utilize the following conditions:

v4 - Bv2 + D = K&v2 cosTlv + 62v2 cosT2v - K&v(u~ + Q) sinTlv

- 62v(al + Us) sinT2v - K6lU2U4 COSTlv

- 62UlU3 cosT2v - K&62 cos(T1 + T2)v, (5.9)

-Au3 + Cv > -K&v2 sinTlv - 62v2 sinT2v - KSlv(uz + ~4) cosTlv

- 62v(al + ~3) cosT2v + K61azu4 sinTlv

+ 62~1~3 sinT2v + K&62 sin(T1 + T2)v. (5.10)

Recall that ,6 will be stable if the inequality (5.10) holds at v = VO, when 00 is the first positive

root of the equation (5.7). Our technique will be to find an upper bound v+ on VO, independent

of Tl and T2, and then to estimate Tl and T2 so that (5.10) holds, for all values of v, 0 5 v < vu+,

hence, in particular, at v = vo.

The unique positive solution of

v4-v2(B+K61+52)-v [K&(uz + u4)+62(ul+a3)]+[o - Kb lu2u4-62ulu3+Kbl62] = 0, (5.11)

denoted by v+, is always greater than or equal to vo. By some calculation one can determine

(5.12)

where

p+q+r= B + Kc%+ 62

2 ’

pq + qr + rp = B+K61+62 2

4 > - [D - K6l~2~4 - 62~1~3 + K&62], and (5.13)

dvGJ;:= K&(u2+u4)+62(ul+u3) G

8 = --*

2

Here, v+ is always independent of Tl and Tz. We now need an estimate on Tl and T2 so that (5.10)

holds, for all v such that 0 5 v 5 v+. We now write (5.10) as

Av2 < C + K&v - K6z2a4) sinTlv + (62v - e) sinT2v

K&62 (5.14)

+ K61 (U2 + ~4) cos Tlv + 62(ul + ~3) cos T2v - 21 sin(Tl + T2)v.

We note that when ri = 0, the solution of (5.9) is

v2 = (B + K6l + 62) f [(B + K6l + 62)2 - 4 (D + K&@&j + &alas + K6l62)] 1’2

=+Av’=+ j@%i, 2

(5.15)

110

where,

P. DAS AND A. B. ROY

(i) L = A(B + KS1 + 62),

(ii) &if = (D + K&u@4 + 62a1u3 + K&62),

(iii) N = KSl(a2 + ~4) + 62(ul + ~3) + C,

(iv) S = AB - C + K&(uI + ~3) + b2(a2 + u4).

Now, since from (3.11), we get

(5.16)

D3 = {c + (ul + u3)62 + hK(u2 + u4)) [AB - C + (ul + U3)f& + 62(U2 + U4)]

- A2(D - K61~2~4 + 62~1~3 + K&62) > 0, i.e., SN > A2M.

So, we get

Av2 < C + Kh(u2 + ~4) + ~%(a1 + u3), (5.17)

:. (5.14) for Ti = 0 at w = Now, will hold for small uo.

Substituting from (5.9) into (5.14), get

[A(K61v2 K61~2~4) - Kbl(~2 + u~)w”] COST~V

+ [K~Iu~u~v - K&w~ - AK61v(u2 + u4)] sinTrv

+ [A(62v2 - 62~1~3) - 62(~l + u~)v”] COST~V

+ [S~~IU~V - 62~~ - AS2v(ul + u3)] sinT2V

+ KSr&vsin(Tr + TZ)V - AK6162 cos(T~ + T2)w < (C - AB)v2 + AD,

=S [A(KS1w2 - K61~2~4) - KGl(a2 + u4)u2] (COST~ZJ - 1)

+ [K61~2~4~ - K&v3 - AK&v(u2 + u4)] sin TOW

+ [A&v2 - 62~1~3) - S~(UI + u~)w~] (COST~W - 1)

+ [62~1~3t1- 52~~ - AS~U(U~ + ~3)] sinTp

+ K&5221 sin(Tr + TZ)V - AK6162 cos((T~ + T2) - 1)~

(5.18)

< A(D l kK&U2U4 i-b2UlU3 i-K6162)

-v2{A(B+K61 +b2) - (C+K&(UI +a3)+62(u2 +u4))} 1 = A(D + Kbl~2~4 + 62~1~3 + K&62) - v2D2

=AM-v2D2<AMq

(say), (as D2 > 0 from (3.10)). We denote the left-hand side of (5.18) by [&(Tl, v) + 4!~2(T2, v)].

Using the estimates sinTiv I Tiv

sin(Tl + TZ)V I (TI + T2)w,

(1 - cosTiv) I fTfv2 (i = 1,2),

1 - cos(Tl + T2)w I (Tf + T;) v2,

and (5.19)

we have,

E K&v2 [ (

f lv2(ul + u2) - Au2u41 + A$) Tf + K61v2 (02 + A(u~ + ~4) - ~204 + 62) Tl]

f1v2(u2 + ~4) - Au~usl + AK&) T,2 (5.20)

+ 62w2 (v” + A(ul + ~3) - 010~ + K&) T2 1 < 77.


We note that for 0 I v I v+, we have

$l(Tl,V) +62(T2,v) I !k(Tl,V) +1C12(T2rV) L $1(T1,v+) +1cIz(T29v+),

here if

$1(~1,v+) + ti2(T2,v+) < rl7 then til(Tl,vo) < v, 42P2,vo) < 7,

Let T;’ and Tz denote the unique positive root of

&(Tl,v+) = arl, 1c, (T 2 2, v+) = P% wherecr+p=l,

T:=&[-i$+dz],

T;=&[-N2+dm]r

where

MI = K&v: f I’J:(uI + us) - Aa + AC?,) ,

NI = K~IV~ (V: -I- A(u~ -I- U4) - 0204 + 62) ,

M2 = 62~: f Iv:@2 + u4) - Awsl + AK&) ,

(5.21)

(5.22)

(5.23)

(5.24)

iV2 = 62~; (v: + A(UI + ~3) - ulu3 + K61) .

Then for Tl < Tc and T2 < T$, the Nyquist criterion holds. T;’ and T$ give the estimates for

the length of delay for which stability is preserved.

6. OSCILLATIONS

We want to study the oscillatory behaviour of the linearized system (3.7) involving five distinct

delays which are different. But, so far as the author’s knowledge goes, there are very few studies

on the analysis of oscillation of models with unequal delays. To make the study mathematically

tractable, all the delays are assumed to be equal and equal to the 1/5th of the sum of all the delays.

From physiological data, total delay is nearly 28-30 min [19], from the numerical simulation of the

linearized system, it is seen that the pulsatile or oscillatory behaviour is present, if the individual

unequal delay exceed 5mins (see Figure 7). So, without much loss of generality, we assume

71 = 72 = 73 = 74 = 75 = 7 (say), where ri > 5. Then the system (3.7) can be written as

? = -a1z - &z(t - T),

ti = -aa9 + z(t -T) - &?20(t - 7),

t = -u3z + Kz(t - 7), (6.1)

ti = -u4w + g(t - T), [Ui > 0, 7 > 0, 61,62 > 01.

We will find a set of sufficient conditions for all bounded solutions of the linearized system (6.1)

to be oscillatory (defined below) when the system has equal multidelays (see [23-251). Here we

adopt the following definition.

DEFINITION. A nontrivial vector z = {z(t), y(t), z(t), w(t)}T defined on [O,(Y) is said to be oscil-

latory, if and only if at least one component of z(t) has arbitrary large zeroes on [0, a). Let us

define max{61,62,1,K+l} =J (say) > 0,

max{--al, -a2, -u3, -u4} = -a (say) < 0.

THEOREM. Let us assume the following conditions:

(i) 8 > a and

(ii) e7a < &e. (6.2)

112 P. DASAND A. B.RoY

Then all the bounded solutions of (6.1) corresponding to continuous initial conditions on [-r, 0]

are oscillatory on [0, w).

PROOF. Suppose that there exists a solution z = {x(t)ry(t),z(t),w(t)}T of (6.1), which is bounded and nonoscillatory on [0, oo).

* It will then follow that there exists a t > 0 such that no component of x(t) has a zero for

t > i + T, and as a consequence, we will have

$ lz(t)l I -a1 INI + I-611 It(t - 711,

-$ IYWI I -a2 Iv(t)l + Ix@ - 711 + I-621 Mt - 711 3

$ Iz(t)l 5 --a3 lz(t)l + K lx(t - T)I ?

$ W)l I -a4 Iw(t)l + IYV - T)I 7 for t > i + 27.

(6.3)

Let

u(t) = Ixc(t)l + IYWI + IWI + IWI ’ 07 for t 2 i + 7. (6.4)

:. We get -&u(t) < --au(t) +h(t -T) for t 2 ! +27. Now, we consider the scalar delay differential equation

-$(t) = -aW(t) + h(t - T), for t 2 i + 27, with (6.5)

u(s) = u(s), SE i&T . [ 1 (6.6)

Using [26], we get

u(t) I v(t), fort> ;+27. (6.7)

We now claim that all bounded solution of (6.6) are oscillatory on [T + 27, oa). Suppose this is not the case, then the characteristic equation associated with (6.6) is given by

A = -a + ZeTX7, (6.8)

will have a nonpositive root, say, A** < 0 and it will follow from (i) A** # 0, then A** < 0, and hence, we have from

The local inequality contradicts (ii), and hence, our claim regarding the oscillatory nature of v on [0, oo) is valid. Since v has arbitrarily large zeroes by (6.7), which means that u(t) = Ix(t)1 +

Iv(t>l+ Iz(t)l + IwWl is oscillatory implying that if(t) is oscillatory, but this is absurd. Since Z(t)

is taken to be nonoscillatory vector. So, there cannot exist a bounded nonoscillatory solution of (6.1) when the conditions (i) and (ii) hold, and thus, the proof is complete.

7. NUMERICAL RESULTS AND DISCUSSIONS

The model (2.1) was simulated numerically on a micro computer (IBM). The program was developed by using the fourth order Runge Kutta Fehlberg method [27] for obtaining accurate numerical results and constructed following algorithm to incorporate delay terms [28]. While defining R.H.S. of four control equations (3.1) as four functions in the Runge Kutta Scheme, we have substituted the variable X(t) as

X(t) = x,

Four Regulatory Hormones

(4 (b)

Gn RH 3

t\ T

4 12 16

(4 (4

Figure 3. Computer simulation of the model portraying (a) asymptotic behaviour of GnRH, FSH, T, and Inhibin with m = 5, rr = 72 = 73 = 74 = 75 = 0, and (b) showing pulsatile behaviour of the hormones for m = 5, 71 = 8, T-J = 6, 73 = 8, 74 = 12, 75 = 10mins. Computer simulation of the model showing pulsatile behaviour of the hormones for m = 5, for equal delays (c) ri = 7mins, (d) ri = 8mins, and (e) ri = 9mins (i = 1,2,3,4,5).

X(t - 7) =x0, if t 5 7 when X0 is the initial value of X(t).

This is quite evident from (2.2).

= Xl, if t > 7.


E E c

‘-

P

30

g 20

F- fZ

FSH

;E z uo

LL

2 Inhibin

1

i

3 6 9 3 6 9 3 6 -9

TIME in hours -

Figure 4. Showing low Inhibin level results in high concentration of FSH.

144

ne 112

96

00

64

48

32

1”“““’ 123456789

(b;n Figure 5. Region of asymptotically stable (RS) and periodic solutions (RP) of the system (2.1) in (a) K - T plane and (b) m - T plane.

The values of X, Xc, Xr are called from the array X(t) formed by storing value of X(t) at each

step of time during the integration.

The numerical results from computer simulation have been given as Figures 3-5, which are

now discussed. In Figure 3a, the asymptotic behaviour of the concentrations of four hormones

viz. GnRH, FSH, Testosterone, and Inhibin are shown in absence of delay terms. (That is, the

concentrations approach a stable equilibrium value if there is no delay. This has been analytically

derived by using Routh Hurwitz criterion in Section 3.) Figures 3b-e show the oscillatory (pul-

satile) character of these hormones in presence of delay and the period of oscillation is nearly two

hours. The experimental value for normal adults is about 130min (291. So, the theoretical value

is in good agreement with the experimental results. Physiological observation regarding the cor-

relation of Inhibin and FSH concentration (see the Introduction) is clearly depicted in Figure 4.

This again confirms that our model is reasonably realistic.

Observations for different values of 7, m, and K have been summarized in Figures 5a and 5b,

which shows two clear regions of oscillation and stability in the relevant parameter viz. K - r

and m - T plane, respectively.

It is found from the numerical results, that if the Hill coefficient (m) associated with the binding

of the hormones with plasma proteins is less than or equal to 3, pulsatile behaviour of R, F, T,

and I are absent for any value of the total delay. If m exceeds 3, there is an upper bound of

the total delay for which the pulsatile behaviour is absent. There is a nonlinear interrelationship

(hyperbolic) between the Hill coefficient (m) and the total delay y (see Figure 5) for presence of

pulsatile behaviour. From the available physiological data obtained from different experimental

Four Regulatory Hormones

Table 1. Values of parameters and constants.

115

in ng/ml/min in min-’ in min- l Fig. no. m 71 72 ~3 74 75

in min.

A1 = 1 a1 = 1 (Y = 0.1 3a 5 0 0 0 0 0

a2 = 0.01 p = 0.015 3b 5 8 6 8 12 10

a3 = 0.01 y = 0.021 3c 5 7 7 7 7 7 I

a4 = 0.01 6=0.1 3d 5 8 8 8 8 8

a5 = 0.05 3e 5 9 9 9 9 9

4 5 8 6 8 12 10

results, it appears from Figure 5, that if the total delay does not exceed 32 mins the corresponding

value of m is greater than 9, for which there is stable steady state. Oscillation appear if the total

delay exceed 32mins where the Hill coefficient lies between 3 and 9; but 9 as the value of m, i.e.,

cooperativity is not realistic.

The values of parameters were chosen partly to be physiologically reasonable, partly to fit

existing human data [8,30,31]. The figures for the delays are based on the data [32], the values

of (Y, 0, y, and S were taken from [30,33-361.

For drawing the figures, values of concentrations of GnRH, FSH, T, and Inhibin have been

suitably magnified.

1.

2.

3.

4.

5.

6.

7. 8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

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the role of four regulatory hormones in controlling testicular function in a delay model

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