| Mihaela Neamţu* arXiv:1709.08936v1 [math.DS] 26 Sep 2017 · Eva Kaslik1,2 | Mihaela Neamţu*1 1West University of Timişoara, Romania 2Institute e-Austria Timişoara, Romania Correspondence

arX

iv:1

709.

0893

6v1

[m

ath.

DS]

26

Sep

2017

Received <day> <Month>, <year>; Revised <day> <Month>, <year>; Accepted <day> <Month>, <year>

DOI: xxx/xxxx

ARTICLE TYPE

Stability and Hopf bifurcation analysis of a four-dimensional

hypothalamic-pituitary-adrenal axis model with distributed delays †

Eva Kaslik1,2 | Mihaela Neamţu*1

1West University of Timişoara, Romania2Institute e-Austria Timişoara, Romania

Correspondence

*Mihaela Neamţu, Email:[email protected]

Abstract

A four-dimensional mathematical model of the hypothalamus-pituitary-adrenal

(HPA) axis is investigated, incorporating the influence of the GR concentration and

general feedback functions. The inclusion of distributed time delays provides a more

realistic modeling approach, since the whole past history of the variables is taken

into account. The positivity of the solutions and the existence of a positively invariant

bounded region are proved. It is shown that the considered four-dimensional system

has at least one equilibrium state and a detailed local stability and Hopf bifurcation

analysis is given. Numerical results reveal the fact that an appropriate choice of the

system’s parameters leads to the coexistence of two asymptotically stable equilibria

in the non-delayed case. When the total average time delay of the system is large

enough, the coexistence of two stable limit cycles is revealed, which successfully

model the ultradian rhythm of the HPA axis both in a normal disease-free situation

and in a diseased hypocortisolim state, respectively. Numerical simulations reflect

the importance of the theoretical results.

KEYWORDS:

HPA axis, mathematical model, distributed time delay, stability, bistability, bifurcation, limit cycle,

numerical simulation

1 INTRODUCTION

The hypothalamus-pituitary-adrenal (HPA) axis is a neuroendocrine system which regulates a number of physiological processes(1, 2), playing an important role in stress response. It consists of the hypothalamus, pituitary and adrenal glands, as well directinfluences and positive and negative feedback interactions. Different types of stressors (e.g. infection, dehydration, anticipation,fear) activate the secretion of corticotropin-releasing hormone (CRH) in the hypothalamus, which induces the corticotropin(ACTH) production in the pituitary. ACTH travels by the bloodstream to the adrenal cortex, where it activates the release ofcortisol (CORT), which in turn down-regulates the production of both CRH and ACTH.

Dynamical systems have previously proved to be successful in studying metabolic and endocrine processes. Different typesof mathematical models of the HPA axis have been recently explored. Three dimensional systems of differential equations withor without time delays, with the state variables given by the hormone concentrations CRH, ACTH and CORT, have been usedto model the HPA axis in (3, 4, 5, 6, 7). The influence of the circadian rhythm in the mathematical model has been analyzed in(8). A more general three-dimensional model has been developed in (9), possessing a unique equilibrium state. If time delays

†This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project no. PN-II-RU-TE-2014-4-0270.

http://arxiv.org/abs/1709.08936v1

2 EVA KASLIK & MIHAELA NEAMŢU

are not taken into consideration, no oscillatory behavior has been observed (9, 10). Oscillatory solutions should be a featureof mathematical models of the HPA axis, as they correspond to the circadian / ultradian rhythm of hormone levels (11). Ageneralization of the "minimal model" (9) has been obtained in (12), including memory terms in the form of distributed delaysand fractional-order derivatives, which are shown to generate oscillatory solutions.

Due to the transportation of the hormones throughout the HPA axis, time delays should mandatorily be incorporated in theconsidered mathematical models. With the aim of reflecting the whole past history of the variables, general distributed delays areconsidered, proving to be more realistic and more accurate in real world applications than discrete time delays (13). Distributeddelay models appear in a wide range of applications such as hematopoiesis (14), population biology (15, 16, 17) or neuralnetworks (18, 19).

Four-dimensional models which incorporate the positive self-regulation of glucocorticoid receptors (GR) in the pituitary havebeen investigated in (20, 21, 22, 23, 24). In particular, in (24) we constructed a four-dimensional general model with distributedtime delays, which represents an extension of the minimal model of (9). In (20), it has been suggested that positive self-regulationof GR may trigger bistability in the dynamical structure of the HPA model, i.e. there exist two asymptotically stable equilibriumstates: one corresponding to the normal disease-free state with higher cortisol levels, and a second one with lower cortisol levelsrelated to a diseased state associated with hypocortisolism.

In this paper, an in-depth analysis is provided for the distributed-delay model introduced in (24), proving the positivity of thesolutions and the existence of a positively invariant bounded region. It is shown that the considered four-dimensional system hasat least one equilibrium state and a local stability and bifurcation analysis is provided. Numerical results reveal the fact that anappropriate choice of the system’s parameters leads to the coexistence of two asymptotically stable equilibria in the non-delayedcase. Moreover, when the total average time delay is large enough, it is shown that two stable limit cycles coexist, which appeardue to Hopf bifurcations, extending the results presented in (20, 24).

2 MATHEMATICAL MODEL OF HPA WITH DISTRIBUTED DELAYS

With the aim of formulating a mathematical model of the HPA axis, the following sequence of events is considered. Cognitiveand physical stressors stimulate CRH neurons in the paraventricular nucleus (PVN) of the hypothalamus to trigger the secretionof corticotropin-releasing hormone (CRH), which is released into the portal blood vessel of the hypophyseal stalk. CRH istransported to the anterior pituitary, where it stimulates the secretion of adrenocorticotropin hormone (ACTH), with an averagetime delay �1. ACTH then activates a complex signaling cascade in the adrenal cortex, stimulating the secretion of the stresshormone cortisol (CORT) with the average time delay �2. CORT exerts a negative feedback on the hypothalamus and thepituitary, suppressing the synthesis and release of CRH and ACTH, in an effort to return them to the baseline levels. On onehand, cortisol inhibits the secretion of CRH in the hypothalamus (25), with an average time delay �31. On the other hand, CORTbinds to glucocorticoid receptors (GR) in the pituitary and performs a negative feedback on the secretion of ACTH, with anaverage time delay �32. Moreover, the CORT-GR complex self-upregulates the GR production in the anterior pituitary, with anaverage time delay �34 .

Denoting the plasma concentrations of hormones CRH, ACTH and CORT by x1(t), x2(t), and x3(t) respectively, and theavailability of the glucocorticoid receptor GR in the anterior pituitary by x4(t), the following system of differential equationswith general distributed delays is considered:

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

x1(t) = k1f1

⎛⎜⎜⎝

t

∫−∞

x3(s)ℎ31(t − s)ds

⎞⎟⎟⎠−w1x1(t),

x2(t) = k2f2

⎛⎜⎜⎝x4(t)

t

∫−∞


⎞⎟⎟⎠

t

∫−∞

x1(s)ℎ1(t − s)ds −w2x2(t),

x3(t) = k3

t

∫−∞

x2(s)ℎ2(t − s)ds −w3x3(t),

x4(t) = k4

⎛⎜⎜⎝� + f3

⎛⎜⎜⎝x4(t)

t

∫−∞


⎞⎟⎟⎠

⎞⎟⎟⎠−w4x4(t).

(1)

EVA KASLIK & MIHAELA NEAMŢU 3

Here, the positive constants ki, i = 1, 4, relate the production rate of each variable to specific factors that regulate the rate ofrelease/synthesis (2). The basal production rate � and elimination constants w1, w2, w3, w4 are positive.

The function f1 represents the negative feedback of CORT on CRH levels in the paraventricular nucleus of the hypothalamuswhile the function f2 describes the negative feedback of the CORT-GR complex (at concentration x3(t)x4(t)) in the pituitary.The positive feedback function f3, describes the self-upregulation effect of the CORT-GR complex on GR production in theanterior pituitary. The following general assumptions will be considered:

• f1, f2 ∶ [0,∞) → (0, 1] are strictly decreasing, smooth and bounded on [0,∞);

• f3 ∶ [0,∞) → [0, 1) is strictly increasing, smooth and bounded on [0,∞);

• f1(0) = f2(0) = 1; f3(0) = 0.

As a special case, the feedback functions can be chosen as Hill functions, such as in (2, 9, 10, 20, 22), which verify the conditionsgiven above:

f1(u) = 1 − �u�1

c�11+ u�1

, f2(u) = 1 − �u�2

c�22+ u�2

, f3(u) =u�3

c�33+ u�3

(2)

with Hill coefficients �1, �2, �3 ≥ 1, �, � ∈ (0, 1], and microscopic dissociation constants c1, c2, c3 > 0.In system (1), the delay kernels ℎ1, ℎ2, ℎ31, ℎ32, ℎ34 ∶ [0,∞) → [0,∞) are probability density functions representing the

probability of occurrence of a particular time delay. These functions are bounded, piecewise continuous and satisfy∞

∫0

ℎ(s)ds = 1. (3)

The average time delay of a kernel ℎ(t) is

� =

∞

∫0

sℎ(s)ds < ∞.

In this paper, we focus our attention on two types of delay kernels:

• Dirac kernels: ℎ(s) = �(s − �), where � ≥ 0, equivalent to a discrete time delay:t

∫−∞

x(s)ℎ(t − s)ds =

∞

∫0

x(t − s)�(s − �)ds = x(t − �).

• Gamma kernels: ℎ(s) =sp−1e−s∕�

�pΓ(p), where p, � > 0, with the average delay � = p�.

In the mathematical modeling of real world phenomena, the exact distribution of time delays is generally unavailable, and hence,general kernels may provide better results (26, 27). The analysis of models which include particular classes of delay kernels(e.g. weak Gamma kernels with p = 1 or strong Gamma kernels with p = 2) may reveal the more realistic effect of distributeddelays on the system’s dynamics, compared to discrete delays.

Initial conditions associated with system (1) are of the form:

xi(s) = 'i(s), ∀ s ∈ (−∞, 0], i = 1, 2, 3, 4,

where 'i are bounded continuous functions defined on (−∞, 0], with values in [0,∞).

3 POSITIVELY INVARIANT SETS AND EQUILIBRIUM STATES

Lemma 1. Assume that g ∶ [0,∞) → [0,∞) is a continuously differentiable function such that there exist m1, m2 > 0 such

that g(0) ≤ m1

m2

and

g′(t) ≤ m1 − m2g(t), ∀ t ≥ 0.

Then, g(t) ≤ m1

m2

for any t ≥ 0.


Proof. From the hypothesis we easily obtain that the function G(t) = em2t(g(t) −

m1

m2

)is decreasing on [0,∞). Therefore,

as G(t) ≤ G(0) for any t ≥ 0, it follows that

g(t) ≤ m1

m2

+ e−m2t

(g(0) −

m1

m2

)≤ m1

m2

, ∀ t ≥ 0.

This completes the proof.

In the following, we denote:

k1

w1

= L1 ,k1k2

w1w2

= L2 ,k1k2k3

w1w2w3

= L3 ,k4

w4

= L4.

Proposition 1. The compact set

Ω =[0, L1

]×[0, L2

]×[0, L3

]×[0, (� + 1)L4

]⊂ ℝ

4+

and ℝ4+

are positively invariant sets for system (1).

Proof. Assume that (x1(t), x2(t), x3(t), x4(t)) denotes the solution of system (1) with the initial condition xi(s) = 'i(s),

s ∈ (−∞, 0], with i = 1, 4, where 'i are bounded positive continuous functions defined on (−∞, 0]. From the positivity of

the feedback functions it easily follows that

xi(t) ≥ −wixi(t), ∀ t > 0, i = 1, 4

and hence, the functions xi(t)ewit are increasing on (0,∞). Therefore:

xi(t) ≥ 'i(0)e−wit ≥ 0, ∀ t > 0, i = 1, 4.

Therefore, all positive initial conditions lead to positive solutions, i.e. ℝ4+

is positively invariant for system (1).

Moreover, assume ('1(s), '2(s), '3(s), '4(s)) ∈ Ω for any s ∈ (−∞, 0].

From the first equation of (1) and the boundedness of f1, it follows that

x1(t) ≤ k1 −w1x1(t), ∀ t > 0.

Using Lemma 1, as x1(0) ≤ L1, we have that x1(t) ≤ L1 for any t ≥ 0.

The second equation of (1), the boundedness of f2 and (3) provides

x2(t) ≤ k2L1 −w2x2(t), ∀ t > 0.

From Lemma 1 it follows that x2(t) ≤ L2 for any t ≥ 0.

From the third equation of (1) and (3) it follows that

x3(t) ≤ k3L2 −w3x3(t), ∀ t ≥ 0.

Lemma 1 leads to x3(t) ≤ L3 for any t ≥ 0.

The last equation of (1), the boundedness of f3 leads to

x4(t) ≤ k4(� + 1) −w4x4(t), ∀ t ≥ 0,

which, based on Lemma 1, provides the desired conclusion.

Remark 1. Due to the fact that x4(t) in the mathematical model (1) is a non-dimensional variable representing the availabilityof glucocorticoid receptors (2, 20), it is reasonable to demand that x4(t) ∈ [0, 1] for any t ∈ ℝ. Based on Proposition 1, thisis guaranteed if the following inequality is satisfied:

(� + 1)L4 ≤ 1.

The existence of an equilibrium point of system (1) is provided by the following:

Proposition 2. The equilibrium states of system (1) belong to the invariant set Ω and are of the form

E =

(L1f1(x0),

w3x0

k3, x0,

1

x0f−12

(x0

L3f1(x0)

)). (4)


where x0 ∈[0, L3

]is a solution of the equation

L4

(� + (f3◦f

−12)

(x

L3f1(x)

))=

1

xf−12

(x

L3f1(x)

). (5)

Proof. From Proposition 1 it follows that any equilibrium state of system (1) belongs to the set Ω. Moreover, An equilibrium

point of system (1) is a solution of the following algebraic system:

⎧⎪⎪⎨⎪⎪⎩

k1f1(x3) = w1x1,

k2f2(x3x4)x1 = w2x2,

k3x2 = w3x3,

k4(� + f3(x3x4)) = w4x4,

(6)

which is equivalent to

⎧⎪⎪⎨⎪⎪⎩

x1 = L1f1(x3),

x2 =w3x3

k3,

L3f2(x3x4)f1(x3) = x3,

L4(� + f3(x3x4)) = x4.

(7)

From the first two equations of (7) it follows that the first two components of an equilibrium state are uniquely determined

by the third component. The last two components of an equilibrium state represent a fixed point for the continuous function

F ∶ ℝ2→ ℝ

2 defined by

(u, v) → F (u, v) =(L3f1(u)f2(uv), L4(� + f3(uv))

)

From the boundedness properties of the functions fi, i ∈ {1, 2, 3} it easily follows that the function F maps the convex

compact set [0, L3] × [0, (�+1)L4] into itself. By Brouwer’s fixed-point theorem we obtain the existence of at least one fixed

point of the function F in the set [0, L3] × [0, (� + 1)L4]. Therefore, system (1) has at least one equilibrium state.

From system (7) we easily deduce (5), and hence we obtain the form of the equilibrium states given by (4).

Remark 2. In the case of the minimal model of the HPA-axis, it has been shown (9, 12) that there exists a unique equilibriumstate. For the extended four-dimensional model (1), Proposition 2 only shows the existence of at least one equilibrium state.The presence of the positive feedback function is often associated with the coexistence of several equilibrium states (20, 22).

4 LOCAL STABILITY ANALYSIS

In this section, necessary and sufficient conditions for the local asymptotic stability of an equilibrium point E are provided,choosing general delay kernels. Delay independent sufficient conditions are explored for the local asymptotic stability of theequilibrium point E, which may prove to be useful if the time delays in system (1) cannot be accurately estimated.

By linearizing the system (1) at an equilibrium point E, we obtain:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

y1(t) = k1f′1(x0)

t

∫−∞

y3(s)ℎ31(t − s)ds −w1y1(t),

y2(t) = k2f2(x0r0)

t

∫−∞

y1(s)ℎ1(t − s)ds +k1k2

w1

f1(x0)r0f′2(x0r0)

t

∫−∞

y3(s)ℎ32(t − s)ds+

+k1k2

w1

f1(x0)x0f′2(x0r0)y4(t) −w2y2(t),

y3(t) = k3

t

∫−∞

y2(s)ℎ2(t − s)ds −w3y3(t),

y4(t) = k4r0f′3(x0r0)

t

∫−∞

y3(s)ℎ34(t − s)ds + k4x0f′3(x0r0)y4(t) −w4y4(t).

(8)


where r0 =1

x0f−12

(x0

L3f1(x0)

).

The characteristic equation of the linearized system at the equilibrium point E is:

(z +w1)(z +w2)(z +w3)(z + w4) + a(w4 − w4)(z +w1)H2(z)H34(z)+ (9)

+ b(z + w4)H1(z)H2(z)H31(z) + a(z +w1)(z + w4)H2(z)H32(z) = 0,

where Hi(z) = ∫ ∞

0e−zsℎi(s)ds are the Laplace transforms of the kernels ℎi, i ∈ {1, 2, 31, 32, 34} and

a = −k1k2k3

w1

f1(x0)f′2(x0r0)r0 = −w2w3

x0r0f′2(x0r0)

f2(x0r0)> 0, (10)

b = −k1k2k3f′1(x0)f2(x0r0) = −w1w2w3

x0f′1(x0)

f1(x0)> 0, (11)

w4 = w4 − k4x0f′3(x0r0) < w4. (12)

For the theoretical analysis, we introduce the following set of inequalities:

(I0) w4 > 0;

(I1) (w1 + w4)(w2 + w4)(w3 + w4) ≥ (w4 −w1)(w4 −w4)(w1 +w2 +w3 + w4);

(I2) a(w1 +w4) + b ≤ (w1 +w2)(w2 +w3)(w1 +w3);

(I3)aw4

w4

+b

w1

< w2w3;

(I3)aw4

w4

+b

w1

≥ w2w3.

Theorem 1 (Local asymptotic stability).

1. If there is no time-delay and (I0), (I1) and (I2) are satisfied, the equilibrium pointE of system (1) is locally asymptotically

stable.

2. For any delay kernels ℎi(t), i ∈ {1, 2, 31, 32, 34}, if (I0) and (I3) hold, then the equilibrium pointE of system (1) is locally

asymptotically stable.

Proof. 1. In the absence of delays, the characteristic equation (9) is given by:

z4 + c1z3 + c2z

2 + c3z + c4 = 0, (13)

where

c1 = w1 +w2 +w3 + w4 > 0,

c2 = w1w2 +w2w3 +w1w3 + (w1 +w2 +w3)w4 + a > 0,

c3 = w1w2w3 + (w1w2 +w2w3 +w1w3)w4 + a(w1 +w4) + b > 0,

c4 = (w1w2w3 + b)w4 + aw1w4 > 0.

Based on the Routh-Hurwitz stability test, it suffices to prove that

c1c2c3 − c23− c2

1c4 > 0.

From this inequality it clearly follows that c1c2 − c3 > 0.

Denoting

S = (w1 +w2)(w1 +w3)(w2 +w3)

T = (w1 + w4)(w2 + w4)(w3 + w4)

we obtain

c1c2c3 − c23− c2

1c4 =(S − b − a(w1 +w4))(T + b + a(w1 +w4))+

+ a(w1 +w2 +w3 + w4)(T − (w4 −w1)(w4 −w4)(w1 +w2 +w3 + w4)))


Using inequalities (I0), (I1) and (I2) it is easy to see that c1c2c3 − c23− c2

1c4 > 0. The Routh-Hurwitz stability criterion

implies that the equilibrium point E is asymptotically stable.

2. In the presence of delays, the characteristic equation (9) can be expressed as

'(z) = (z),

where ' and are

'(z) = −(z +w1)(z +w2)(z +w3)(z + w4),

(z) = a(w4 − w4)(z +w1)H2(z)H34(z) + b(z + w4)H1(z)H2(z)H31(z) + a(z +w1)(z + w4)H2(z)H32(z).

The functions ' and are holomorphic in the right half-plane.

Considering z ∈ ℂ with ℜ(z) ≥ 0, the properties of the delay kernels (3) imply:

|Hi(z)| =|||||||

∞

∫0

e−zsℎi(s)ds

|||||||≤

∞

∫0

|e−zs|ℎi(s)ds =∞

∫0

e−ℜ(z)sℎi(s)ds ≤∞

∫0

ℎi(s)ds = 1,

for any i ∈ {1, 2, 31, 32, 34}. Therefore, based on inequalities (I0) and (I3), we have:

| (z)| ≤ a(w4 − w4)|z +w1||H2(z)||H34(z)| + b|z + w4||H1(z)||H2(z)||H31(z)|++ a|z +w1||z + w4||H2(z)||H32(z)|

≤ a(w4 − w4)|z +w1| + b|z + w4| + a|z +w1||z + w4|= |z +w1||z + w4|

(a(w4 − w4)

|z + w4| +b

|z +w1| + a)

≤ |z +w1||z + w4|(a(w4 − w4)

w4

+b

w1

+ a

)

< |z +w1||z + w4|w2w3

= |z +w1||z +w2||z +w3||z + w4| = |'(z)|.where the inequality |z +w| ≥ w, for any z ∈ ℂ with ℜ(z) ≥ 0 and w > 0, has been repeatedly used.

Hence, the inequality | (z)| < |'(z)| is true for any z in the right half plane, and Rouché’s theorem implies that the

characteristic equation (9) does not have any root in the right half-plane (or on the imaginary axis). Therefore, all the roots

of (9) are in the open left half plane, and it follows that the equilibrium E is asymptotically stable.

Remark 3. Assume that (I0) holds and that the delay kernels ℎi(t), i ∈ {1, 2, 31, 32, 34} are chosen. If the equilibrium pointE of system (1) is unstable, Theorem 1 implies that inequality (I3) holds.

5 BIFURCATION ANALYSIS

In this section, we explore the possibility of the occurrence of limit cycles in a neighborhood of E, due to Hopf bifurcations,that reflect the ultradian rhythm of the HPA axis.

For simplicity, we further assume thatH32(z) = H34(z) = H1(z)H31(z),

and we denoteH(z) = H2(z)H32(z) = H2(z)H34(z) = H1(z)H2(z)H31(z).

We emphasize that H(z) is the Laplace transform of the convolution of ℎ2 and ℎ32:

ℎ(t) =

t

∫0

ℎ2(s)ℎ32(t − s)ds,

with the average time-delay

� =

∞

∫0

sℎ(s)ds = �2 + �32 = �2 + �34 = �1 + �2 + �31, (14)


where �i represent the average delays of the kernels ℎi, for any i ∈ {1, 2, 31, 32, 34}.The characteristic equation (9) is

(z +w1)(z +w2)(z +w3)(z + w4) + [a(z +w1)(z +w4) + b(z + w4)]H(z) = 0,

which can be rewritten as:

H(z)−1 = Q(z), (15)

where

Q(z) = −a(z +w1)(z +w4) + b(z + w4)

(z +w1)(z +w2)(z +w3)(z + w4).

The properties of the function Q(z) are given in the following Lemma.

Lemma 2. Assume that (I0) holds.

a. The function

! → |Q(i!)| =√

(bw4 + aw1w4 − a!2)2 + !2(a(w1 +w4) + b)

2

(!2 +w21)(!2 +w2

2)(!2 +w2

3)(!2 + w4

2)

defined on [0,∞) is strictly decreasing.

b. A unique positive real root !0 exists for the equation |Q(i!)| = 1 if and only if inequality (I3) holds.

c. The function Q satisfies the following inequality:

ℑ

(Q′(i!)

Q(i!)

)> 0 ∀! > 0.

Proof. To prove, a. it is easy to see that

|Q(i!)|2 = 1

(!2 +w22)(!2 +w2

3)

[a2 +

d1

(!2 +w21)+

d2

(!2 + w42)

]

where

d1 =b2 + 2ab

w1(w1 +w4)

w1 + w4

> 0

d2 =2abw4(w4 − w4)

w1 + w4

> 0

Therefore, ! → |Q(i!)| is strictly decreasing on [0,∞), and tends to 0 as ! → ∞. Therefore, the equation |Q(i!)| = 1

admits a unique positive solution if and only if |Q(0)| > 1. This implies w1w2w3w4 < aw1w4 + bw4, which in turn, is

equivalent to (I3), and b. is proved.

Point c. follows from (12).

For the bifurcation analysis, due to the complexity of the problem, we restrict our attention to Dirac kernels and Gammakernels.

5.1 Dirac kernels

If all delay kernels are of Dirac type: ℎ1(t) = �(t−�1), ℎ2(t) = �(t−�2), ℎ31(t) = �(t−�31), ℎ32(t) = �(t−�32), ℎ34(t) = �(t−�34)

where �1, �2, �31, �32, �34 ≥ 0 satisfy the property

�2 + �32 = �2 + �34 = �1 + �2 + �31 = � > 0, (16)

then, the characteristic equation (15) becomes:

e�z = Q(z). (17)

Choosing � as bifurcation parameter and following the same proof as in (12), we have:


Theorem 2 (Hopf bifurcations; Dirac kernels). If inequalities (I0), (I1), (I2) and (I3) hold, considering !0 > 0 given by

Lemma 2 and

�p =arccos

[ℜ(Q(i!0))

]+ 2p�

!0

, p ∈ ℤ+, (18)

the equilibrium point E is asymptotically stable if any only if � ∈ [0, �0). For any p ∈ ℤ+, at � = �p, a Hopf bifurcation

takes place in a neighborhood of the equilibrium point E of system (1).

5.2 Gamma kernels

If all delay kernels are of Gamma type: ℎ1(t) =tp1−1e−t∕�

�p1(p1 − 1)!, ℎ2(t) =

tp2−1e−t∕�

�p2(p2 − 1)!, ℎ31(t) =

tp31−1e−t∕�

�p31(p31 − 1)!, ℎ32(t) =

tp32−1e−t∕�

�p32(p32 − 1)!, ℎ34(t) =

tp34−1e−t∕�

�p34(p34 − 1)!, where � > 0 and p1, p2, p31, p32, p34 ∈ ℤ

+ ⧵ {0} satisfy:

p2 + p32 = p2 + p34 = p1 + p2 + p31 = p ≥ 2,

the characteristic equation (9) is:(�z + 1)p = Q(z). (19)

Choosing � as bifurcation parameter, as in (12), the following result holds:

Theorem 3 (Hopf bifurcations; Gamma kernels). If inequalities (I0), (I1), (I2) and (I3) hold and !p is the largest real root

from the interval (0, !0) of the equation

Tp

(1

|Q(i!)|1∕p)

=ℜ(Q(i!))

|Q(i!)| (20)

where Tp denotes the Chebyshev polynomial of the first kind of order p, considering

�p =1

!p

√|Q(i!p)|2∕p − 1. (21)

the equilibrium point E is asymptotically stable if � ∈ (0, �p). At � = �p, system (1) undergoes a Hopf bifurcation at the

equilibrium point E.

6 NUMERICAL SIMULATIONS

The literature values of the elimination constants wi, i ∈ {1, 2, 3} are given by wi =ln(2)

Ti, where Ti is the plasma half-life of

hormones: T1 ≈ 4 min, T2 ≈ 19.9 min, T3 ≈ 76.4 min (9, 11). We choose w4 = 0.001 min−1 as in (22).For simplicity, let � = � = 1 and hence, the considered feedback functions are:

f1(x) =c�1

c�1+ x�

, f2(x) =c�2

c�2+ x�

, f3(x) =x�

c�

3+ x�

with � = 4 and � = 5 as in (22), c1 = 2 ng/ml as in (12) and c2 = c3 = 0.8 ng/ml.The normal equilibrium state E should reflect the normal mean values of the hormones: xn

1= 7.659 pg/ml (24-h mean value

of CRH), xn2= 21 pg/ml (24-h mean value of ACTH) and xn

3= 3.055 ng/ml (24-h mean value of free CORT) (11). In accordance

with (20), we assume xn4= 0.1. Choosing � = 0.1, from system (7) we deduce:

k1 = w1

xn1

f1(xn3)= 8.55261

pgml ⋅ min

;

k2 = w2

xn2

xn1f2(x

n3xn4)= 0.09753 min−1;

k3 = w3

xn3

xn2

= 1.31985 min−1;

k4 = w4

xn4

� + f3(xn3x4)

= 0.00092545 min−1.


For these values of the system parameters, the following equilibrium states exist:

En = (7.659 pg/ml, 21 pg/ml, 3.055 ng/ml, 0.1) normal state

Ed = (38.425 pg/ml, 10.04 pg/ml, 1.4606 ng/ml, 0.967) diseased state

Eu = (8.3097 pg/ml, 20.495 pg/ml, 2.981 ng/ml, 0.16) unstable state

The low level of cortisol in the case of the equilibrium state Ed can be associated with hypocortisolism, and hence, Ed isregarded as the "diseased" state. In the non-delayed case, the normal equilibrium state En and the diseased equilibrium stateEd are both asymptotically stable, as inequalities (I0), (I1) and (I2) are satisfied (see Theorem 1). On the hand, the equilibriumstate Eu is unstable, therefore, it is not significant from the biological point of view.

It is important to emphasize that for both equilibria En and Ed , inequality (I3) is satisfied, which implies that when delaysare introduced in the mathematical model, for sufficiently high average time delays bifurcations will occur, causing the loss ofstability the En and Ed .

As for the choice of mean time delays, firstly, as CRH travels from the hypothalamus to the pituitary through the hypophysealportal blood vessels in an extremely short time (6), we assume �1 = 0. Moreover, the human inhibitory time course for thenegative feedback of cortisol on the secretion of ACTH has been described as anything between 15 and 60 min (28, 29), thereforewe consider a mean delay �32 ∈ (0, 60]. In our numerical simulations, we additionally assume that �31 = �32 = �34. In (30),a 30-min delay has been given for the positive-feedforward effect of ACTH on plasma cortisol levels, therefore, we assume�2 ∈ (0, 30].

6.1 Dirac kernels

In the case of discrete time delays, choosing the bifurcation parameter � = �2 + �32, we find the following critical valuescorresponding to Hopf bifurcations, based on Theorem 2 and equation (18): �n

0= 49.8505 (min) for En and �d

0= 37.8362 (min)

forEd , respectively. For � < �d0

, both equilibriaEn andEd are asymptotically stable. When � crosses the critical value �d0

, a Hopfbifurcation occurs in a neighborhood of the equilibriumEd , which causes this equilibrium to become unstable and generates anasymptotically stable limit cycle in its neighborhood. The equilibrium state En remains asymptotically stable whenever � < �n

0.

However, when the bifurcation parameter � passes through the critical value �n0, a supercritical Hopf bifurcation takes place at

En. Numerical simulations show that for � > �n0

two asymptotically stable limit cycles coexist, one corresponding to the normalultradian rythm of the HPA axis and the other one reflecting a diseased hypocortisolic ultradian rythm. Considering � = 50

(min), the coexisting limit cycles are presented in Figures 1 , 2 and 3 .

FIGURE 1 Two asymptotically stable limit cycles coexist in (1) with discrete delays: �1 = 0, �2 = 30 (min), �31 = �32 = �34 =

20 (min).


FIGURE 2 Evolution of the state variables of (1) with discrete delays: �1 = 0, �2 = 30 (min), �31 = �32 = �34 = 20 (min) andan initial condition in a neighborhood of En.

FIGURE 3 Evolution of the state variables of (1) with discrete delays: �1 = 0, �2 = 30 (min), �31 = �32 = �34 = 20 (min) andan initial condition in a neighborhood of Ed .

6.2 Strong Gamma kernels

We now consider system (1) with strong Gamma kernels with the same parameter � and p2 = p31 = p32 = p34 = 2 andp1 = 0. Choosing the bifurcation parameter �, we find the following critical values corresponding to Hopf bifurcations, based onTheorem 3 and equation (21): �d

4= 12.625 (min) for Ed and �n

4= 18.9 (min) forEn, respectively. As in the previous case, when

� passes one of the critical values �d4

or �n4, a supercritical Hopf bifurcation takes place in a neighborhood of the corresponding

equilibrium Ed or En. For � > �n4, numerical simulations show the coexistence of two asymptotically stable limit cycles, one

corresponding to the normal ultradian rythm of the HPA axis and the other one reflecting a diseased hypocortisolic ultradianrythm. Considering � = 19 (min) (i.e. a total average time delay � = 76 (min)), the coexisting limit cycles are presented inFigures 4 , 5 and 6 .

7 CONCLUSIONS

This paper presents an analysis of a four-dimensional mathematical model describing the hypothalamus-pituitary-adrenal axiswith the influence of the GR concentration, considering general feedback functions (which include as a special case Hill-typefunctions frequently used in the literature) to account for the interactions within the HPA axis. Due to the fact that the involvedprocesses are not instantaneous, general distributed delays have been included. This is a more realistic approach to the mod-eling of the biological processes, as it takes into account the whole past history of the variables, efficiently capturing the vitalmechanisms of the HPA system.

The positivity of the solutions and the existence of a positively invariant bounded region are proved. It is shown that theconsidered four-dimensional system has at least one equilibrium state and a detailed local stability and Hopf bifurcation analysis


FIGURE 4 Two asymptotically stable limit cycles coexist in (1) with strong gamma kernels (p2 = p31 = p32 = p34 = 2, p1 = 0)with mean time delay � = 19 (min).

FIGURE 5 Evolution of the state variables of (1) with strong gamma kernels (p2 = p31 = p32 = p34 = 2, p1 = 0) with meantime delay � = 19 (min) and an initial condition in a neighborhood of En.

is given. Sufficient conditions expressed in terms of inequalities involving the system’s parameters are found which guarantee thelocal asymptotic stability of an equilibrium. On the other hand, a necessary condition has also been obtained for the occurrence ofbifurcations in a neighborhood of an equilibrium, when time delays are present. For the Hopf bifurcation analysis, two particulartypes of delays have been considered, given by Dirac and Gamma kernels, respectively.

Numerical simulations reflect the importance of the theoretical results. They exemplify the fact that an appropriate choice ofthe system’s parameters leads to the coexistence of two asymptotically stable equilibria in the non-delayed case. When the totalaverage time delay of the system passes through critical values which are computed according to the theoretical findings, theasymptotically stable equilibria loose their stability due to Hopf bifurcations and stable limit cycles are born in their neighbor-hoods. The coexistence of two stable limit cycles is revealed for a sufficiently large average time delay, which successfully modelthe ultradian rhythm of the HPA axis both in a normal disease-free situation and in a diseased hypocortisolim state, respectively.

As a direction for future research, a fractional-order formulation of the mathematical model will be analyzed.

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| Mihaela Neamţu* arXiv:1709.08936v1 [math.DS] 26 Sep 2017 · Eva Kaslik1,2 | Mihaela Neamţu*1 1West University of Timişoara, Romania 2Institute e-Austria Timişoara, Romania Correspondence

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