Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol163.Issue2.Feb2000:

A mathematical model of cancer treatment by

immunotherapy

q

Frank Nani

a

, H.I. Freedman

a,b,*,1

a

Department of Mathematical Sciences, Applied Mathematics Institute, 632 Central Academic Building, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

b

School of Mathematical Sciences, Swinburne University of Technology, Hawthorn, Victoria, Australia

Received 12 September 1998; received in revised form 19 October 1999; accepted 20 October 1999

Abstract

In this paper, a detailed mathematical study of cancer immunotherapy will be presented. General principles of cancer immunotherapy and the model equations and hypotheses will be discussed. Mathe-matical analyses of the model equations with regard to dissipativity, boundedness of solutions, invariance of non-negativity, nature of equilibria, persistence, extinction and global stability will be analyzed. It will also be shown that bifurcations can occur, and criteria for total cure will also be derived. Ó _{2000 Elsevier} Science Inc. All rights reserved.

Keywords: Cancer treatment; Competition; Dynamic modelling; Hopf bifurcation; Immunotherapy; Periodicity; Persistence and extinction; Stability

1. Introduction: The immune system and cancer

When cancer cells proliferate to a detectable threshold number in a given physiological space of the human anatomy, the body's own natural immune system is triggered into a search-and-de-stroy mode. The spontaneous immune response is possible if the cancer cells possess distinctive surface markers called tumor-speci®c antigens. Tumor cells which possess such antigens are called

www.elsevier.com/locate/mbs

q

This paper is derived from a thesis submitted to the Faculty of Graduate Studies and Research of the University of Alberta in partial ful®llment of the PhD requirements.

*

Corresponding author. Tel.: +1-403 492 3396; fax: +1-403 492 6826.

E-mail address:hfreedma@math.ualberta.ca, mathsci@math.ualberta.ca (H.I. Freedman).

1

Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant NSERC OGP4823.

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

immunogenic cancers [1±5]. The immune response against cancer cells can be categorized into two types: the cellularand thehumoral immune response.

The cellular natural immune response is provided by (i) lymphocytes (ii) lymphokines/cytokines and (iii) antigen-presenting cells.

The e€ector lymphocytes which are involved in anti-cancer mechanisms are T-cells, natural killer (NK) cells, lymphokine activated killer (LAK) cells and K cells. The lymphokines or cy-tokines are biological response modi®ers or growth-stimulating substances biosynthesized by certain immune cells. These include the interleukins and the interferons. In particular, interleukin-2 (IL-interleukin-2) is biosynthesized by an antigen-sensitized subset of the T-cells called Helper T-cells. IL-interleukin-2 is responsible for stimulating antigen-sensitized NK cells, cytotoxic T-cells and LAK cells to develop into mature anti-cancer e€ector lymphocytes and also provides the growth stimulus for these lymphocytes to proliferate into a high enough cell number capable of mounting an e€ective attack against the cancer cells. The antigen-presenting cells include macrophages and dendritic cells. These cells are responsible for presenting cancer antigens to the T-cells such as to trigger the immune response. The detailed description of the morphology and roles of the lymphocytes, lymphokines and antigen-presenting cells can be obtained from the following Refs. [3±5].

The humoral immune response to cancer is provided by (i) B-lymphocytes and (ii) Immuno-globulins/Antibodies. There exists a mechanism in which both cellular and humoral responses cooperate in providing an anti-cancer activity. This mechanism is called antibody-dependent cancer cell destruction: cf. [1,3,5], and it involves K and NK cells as well as IL-2 and immuno-globulin-G.

The cellular response is most proli®c against most cancers and usually the ®rst line of action. The basic steps involved in a cellular immune response are listed as follows:

s1: Cancer cells develop in a given physiological space of the human anatomy. The cancer cells could be immunogenic or non-immunogenic.

s2: The cancer cells subvent the immuno-surveillance activity provided by NK cells (which can kill cancer cells whether immunogenic or not).

s3: The cancer cells proliferate above the subclinical threshold of 103 cells and reach 109 cells which is the X-ray detectable threshold. Some cancer cells might have metastasized to other physiological regions of the human anatomy.

s4: The antigen-presenting cells, particularly macrophages, encounter the cancer cells. They in-ternalize the cancer cells, dissolve them into fragments called epitopes. These epitopes bear the cancer-associated antigens. The macrophages then exhibit the cancer antigens on their surfaces and circulate into the vicinity of T-cells (particularly helper T-cells) and mechanistically present these cancer antigens to them (cf. [4,5]).

s5: The antigen-sensitized helper T-cells then release the immuno-stimulatory growth substance called IL-2. This lymphokine, IL-2, then stimulates the cancer killing subset of the T-cells called the cytotoxic T-cells, to mature and proliferate. In particular, the IL-2 also enhances the pro-liferation of NK and LAK cells.

s6: The activated lymphocytes (LAK, T, NK cells) then engage in a search-and-destroy anti-cancer activity.

r1: The initial numbers of the cancer-killing lymphocytes at the time of tumor diagnosis or ini-tiation of therapy are insigni®cant and easily overwhelmed by the rapidly proliferating tumor cells.

r2: The cancer cells eventually evade the immune recognition mechanism by shedding, altering or re-distributing its surface tumor-associated antigens, cf. [4,5]. The `stealth' tumor then be-comes non-immunogenic.

r3: Some of the shedded tumor antigens and receptors bind to form circulating immune com-plexes which interact destructively with the surface receptors on the cancer cells and thereby e€ectively block the cancer-killing lymphocytes from getting access to the cancer cells [7].

r4: Cancer cells release inhibitory substances which e€ectively reduce the therapeutic ecacy of the cancer-killing lymphocytes (cf. [4]).

In view of the processes r1±r4 it is observed by clinical investigators and medical oncologists that the natural immune system cannot provide a sustained and therapeutically successful anti-cancer attack (see [1,4,5]).

Further research by clinical oncologists including those at the National Cancer Institute led to the development of several techniques and methodologies to enhance the natural immune re-sponse against cancer. Some of these novel approaches include (see [1,2,8±11]):

(i) non-speci®c cancer immunotherapy,

(ii) speci®c passive cancer immunotherapy (adoptive cancer immunotherapy), (iii) speci®c active cancer immunotherapy,

(iv) gene therapy of cancer,

(v) monoclonal antibody mediated anti-tumor immunization of host via induction of idiotype-anti-idiotypic immune network.

In this paper, we present a model of cancer treatment by immunotherapy, treating normal cells and cancer cells as competitors for common resources. The anti-cancer cells are thought of as predators on the cancer cells.

This paper is organized as follows. In Section 2 we state the principles upon which our model is based. In Section 3 our model will be derived, and some elementary properties discussed in Section 4. The equilibria and their stabilities are given in Section 5 followed by a global stability analysis of subsystems in Section 6. Section 7 deals with persistence and interior equilibria while in Section 8 Hopf bifurcation is discussed. The question of cancer extinction is addressed in Section 9. A short discussion is given Section 10.

2. Clinical principles of adoptive cellular immunotherapy

Adoptive cellular immunotherapy is a relatively new immunotherapeutic modality for treating advanced and metastatically disseminated human solid tumors (see [11±17]). It involves the use of tumor-killing lymphocytes and lymphokines such as natural or cloned IL-2 (n-IL-2, or r-IL-2), NK cells, tumor in®ltrating lymphocytes (TIL), interferon-c activated killer monocytes (AKM) and LAK.

lympthocytes are clinically extracted from the cancer patients' body by a process called cytapheresis. The LAK cells are then incubated (outside the patient's body for at least 48 h) using high dose IL-2. Two phenotypes of LAK cells are called NK-LAK or A-LAK and T-LAK depending on their precursors are produced. The NK-LAK cells have been used for adoptive immunotherapy of metastatic cancers (cf. [12±14]). T-LAK cells have also been used in adoptive immunotherapy of ovarian cancer and malignant brain tumors [14,18]. In LAK ACI, the LAK cells are incubated with high dose IL-2 until the number of LAK cells is of the order of 107_±108 _{cells. They are then re-transfused by intravenous injection infusion into the} patient in addition to continuous infusion of IL-2 in the order of 105 _units=m2

=day or 106 units/kg/day of r-IL-2 (cf. [11,14]).

The choice of ACI is based on its current status as the most clinically successful and promising during clinical trials and applications to advanced cancers [12,13,16,18±20].

3. The mathematical model of ACI for solid tumors

In this section, the mathematical model for ACI will be presented.

Notation

x1: The concentration of normal/non-cancer cells in the physiologic space or organ of the hu-man anatomy where cancer cells are localized.

x2: The concentration of cancer cells in a given physiologic space or organ of the human anatomy.

w: The concentration of cancer-killing lymphocyte binding sites such as LAK cells in the neigh-borhood of the cancer cells and normal cells.

z: The concentration of lymphokine (e.g., IL-2) in the neighborhood of the cancer cells and nor-mal cells.

Q1: The rate of external (adoptive) intravenous re-infusion of lymphocyte (LAK cells) into the cancer patient.

Q2: The rate of external (adoptive) intravenous re-infusion of lymphokines (IL-2) into the can-cer patient.

S1: The rate of internal production of lymphocytes (LAK cells).

S2: The rate of internal production of lymphokines (IL-2).

The model equations are as follows:

_

x1 ˆB1…x1† ÿD1…x1† ÿx1x2q1…x1;x2†;

_

x2 ˆB2…x2† ÿD2…x2† ÿx1x2q2…x1;x2† ÿh…x2;w†;

_

wˆS1‡Q1ÿa1e1…w† ‡f…x;z† ÿbh…x2;w†;

_

zˆS2‡Q2ÿa2e2…z† ÿgf…w;z†;

xi…t0† ˆxi0P0; iˆ1;2;

w…t0† ˆw0P0;

where Bi…xi† and Di…xi†, iˆ1;2; are, respectively, the birth and death rates of xi;, qi…x1;x2† the speci®c natural competition functions between cancer and normal cells, i.e., the functions rep-resenting suppression of growth, f…w;z† the rate of lymphocyte (LAK) proliferation due to in-duction by lymphokine (IL-2), h…x2;w† the rate of cancer cell destruction by (cancer killing) lymphocytes,e1…w†;e2…z† the rates of degradation or elimination of lymphocytes (LAK) or lym-phokine (IL-2), respectively, g;b are constants depicting binding stoichiometry, andai are

elim-ination coecients.

We assume that all functions are suciently smooth so that solutions to initial value problems exist uniquely for all positive time.

We shall assume that QiSi are such that the process relies solely on the rate of adoptive

transfer of LAK cells and IL-2. Then Si is negligible and will subsequently be omitted.

Fur-thermore, the toxicity to normal cells is assumed to be minimal and hence not represented in the models. This can be achieved in practice by use of low dose IL-2 (see [14,19]).

Thus the ®nal form of the model equations are

_

x1ˆB1…x1† ÿD1…x1† ÿx1x2q1…x1;x2†;

_

x2ˆB2…x2† ÿD2…x2† ÿx1x2q2…x1;x2† ÿh…x2;w†;

_

wˆQ1ÿa1e1…w† ‡f…w;z† ÿbh…x2;w†;

_

zˆQ2ÿa2e2…z† ÿgf…w;z†;

xi…t0† ˆxi0P0 foriˆ1;2;

w…t0† ˆw0P0; z…t0† ˆz0P0: …3:2†

The following additional hypotheses are assumed to hold: P₁: The initial conditions are such that…x10;x20;w0;z0† 2R4_‡: P_{2: (a)f…w;z† 2C}1_R

‡R‡;R‡†,

(b)fw…w;z†>0; w>0; z>0,

(c) fz…w;z†>0; w>0; z>0,

(d)f…0;z† ˆ0; f…w;0† ˆ0:

P₃: (a)h…x2;w† 2C1…R‡R‡;R‡†,

(b)hx2…x2;w†>0; x2 >0; w>0, (c) hw…x2;w†>0; x2 >0; w>0, (d)hw…0;w† ˆ0; w>0,

(e) hx2…0;w† 6ˆ0; w>0, (f) h…0;w† ˆ0; h…x2;0† ˆ0:

(Some plausible expressions for h…x2;w† include: (i)_aa1x2 2‡x2

b1w

b2‡wand (ii)

c1x2w

c2‡c3x2‡c4w.) P₄: (a)ei2C1…R‡;R†,

(b)ei…0† ˆ0,

(c) e0…w†>0; w>0, (d)e0…z†>0; z>0:

P₅: Bi…0† ˆDi…0† ˆ0;B0i…xi†>0; D0i…xi†>0; B0i…0†>D0i…0†; there exists Ki>0 such that

Bi…Ki† ˆDi…Ki† and B0_i…Ki†<D0_i…Ki†; iˆ1;2:

P_{6: qi…0;0†>0,}oqi

4. Boundedness, invariance of non-negativity, and dissipativity

In this section, we shall show that the model equations are bounded, positively (non-negatively) invariant with respect to a region in R4

‡; and dissipative.

Theorem 4.1. LetB_{be the region defined by}

B_{ˆ …x1;x2;w;z† 2}R4

‡

06x16K1; 06x26K2;

06w6 ÿQ1

d1; where d1 <0; 06z6Q2

d2; where d2 >0

8

> < > :

9 > = >

;: …4:1†

Then

(i) Bis positively invariant.

(ii)All solutions of system(3.1)with initial values inR4

‡are eventually uniformly bounded and are

attracted into the region B_: (iii) System(3.1)is dissipative.

Proof.Letx10>0: Consider

_

x1 ˆB1…x1† ÿD1…x1† ÿx1x2q1…x1;x2†

)x_1 <B1…x1† ÿD1…x1†:

But there existsK1 such that B1…K1† ˆD1…K1† by hypothesis. Thus

x1…t†6max…K1;x10†:

Note that x_1 <0 forx1 >K1 and hence lim sup

t!1

x1…t†6K1:

For

_

x2 ˆB2…x2† ÿD2…x2† ÿx1x2q2…x1;x2† ÿh…x2;w†;

a similar analysis gives

x2…t†6max…K2;x20†;

and

lim sup

t!1

x2…t†6K2:

Now consider

_

wˆQ1ÿa1e1…w† ‡f…w;z† ÿbh…x2;w†;

)w_ <Q1‡f…w;z† ÿa1e1…w†;

)w_ <Q1‡w max

w;z2B

e

f…w;z† ÿwa1 min

where

f…w;z† ˆwfe…w;z†; e1…w† ˆwee1…w†:

Now

_

w<Q1‡w max

w;z2B maxw;z2B

e

f…w;z†

ÿa1min

w;z2B ee1…w†

:

Let

d1ˆmax

w;z2B maxw;z2B

e

f…w;z†

ÿa1min

w;z2B ee1…w†

: …4:2†

We shall henceforth assume that

max

w;z2B ef…w;z†<a1minw;z2Bee1…w†;

and consequentlyd1<0. Thenw6 ÿQ1=d1‡w0ed1t. Thus

w6 max

ÿQ1

d1 ;w0

;

)lim sup

t!1

w6 ÿQ1

d1

; d1 <0; w0P0:

…4:3†

Similarly we consider thez equation

_

zˆQ2ÿa2e2…z† ÿgf…w;z†

)z_ <Q2ÿa2e2…z†

<Q2ÿa2zmin

w;z2Bee2…z†;

wheree2…z† ˆzee2…z†.

We shall henceforth assume that

d2ˆa2 min

w;z2Bee2…z†>0: …4:4†

Then z6Q2=d2‡z0eÿd2t. Thus

z6max Q2

d2

;z0

and lim

t!1 sup z6 Q2

d2

: 4:5†

5. The equilibria: existence and local stability

B1…x1† ÿD1…x1† ÿx1x2q1…x1;x2† ˆ0;

B2…x2† ÿD2…x2† ÿx1x2q2…x1;x2† ˆ0;

Q1ÿa1e1…w† ‡f…w;z† ÿbh…x2;w† ˆ0;

Q2ÿa2e2…z† ÿgf…w;z† ˆ0;

…5:1†

subject to the hypotheses P₁±P₆, (4.2), (4.4) and (4.5). The possible equilibria are of the form

…i† E0…0;0;w

;z†; …ii† E1…x1;0;w;z†;

…iii† E2…0;bx2;wb;bz†;

…iv† E3…x1;x

2;w

_;z†:

…5:2†

The existence and local stability of the prospective equilibria are analysed as follows. Existence and local stability ofE0…0;0;w

;z†: The system of equation (3.1) is restricted to R‡

wz:

This leads to the system

_

wˆQ1‡f…w;z† ÿa1e1…w†;

_

zˆQ2ÿa2e2…z† ÿgf…w;z†; …5:3†

w…0† ˆw0P0; z…0† ˆz0P0:

Theorem 5.1. Let

L1 ˆmax

w;z2B f…w;z†>0;

L2 ˆmin

w;z2B a1minw2Bee1…w†;a2minz2B ee2…z†

>0: …5:4†

Then

…w‡z†6 Q1‡Q2ÿ …gÿ1†L1

L2

‡ …w0‡z0†eÿL2t;

and

lim

t!1 sup …w‡z†6

Q1‡Q2ÿ …gÿ1†L1

L2

: …5:5†

Proof.Using the system of equations (5.3) we obtain the di€erential equation

…w‡z†0 ˆQ1‡Q2‡f…w;z† ÿgf…w;z† ÿa1e1…w† ÿa2e2…z†

6Q1‡Q2‡ …1ÿg†max

w;z2B f…w;z† ÿa1wminw2B ee1…w† ÿa2zminz2B ee2…z†

6Q1‡Q2ÿ …gÿ1†L1ÿ …w‡z†min

w;z2B a1minw2B ee1…w†;a2minz2B ee2…z†

and

…w‡z†6Q1‡Q2ÿ …gÿ1†L1

L2

‡ …w0‡z0†eÿL2t;

and hence

lim

t!1 sup…w‡z†6

Q1‡Q2ÿ …gÿ1†L1

L2

:

Lemma 5.2. Suppose there exists …w;z† 2R‡

wz such that

Q1ÿa1e1…w

† ‡1

g Q2

ÿa2e2…z

†ˆ0

ast! 1. ThenE0…0;0;w

;z† exists.

Proof.By equating the right-hand side of system (5.3) to zero, we have the two surfaces

C1 : Q1ÿa1e1…w† ‡f…w;z† ˆ0;

C2 : Q2ÿa2e2…z† ÿgf…w;z† ˆ0:

We have shown by Theorem 5.1 that system (5.3) is dissipative under the stated conditions of the theorem. Now

a1e1…w† ÿQ1 ˆf…w;z†;

1

g…Q2ÿa2e2…z†† ˆf…w;z†:

Then

a1e1…w† ÿQ1 ˆ 1

g…Q2ÿa2e2…z††;

()a1e1…w† ÿQ1ÿ

1

g…Q2ÿa2e2…z†† ˆ0;

()Q1ÿa1e1…w† ‡

1

g…Q2ÿa2e2…z†† ˆ0:

The lemma now follows immediately.

We now discuss the (local) linearized stability of system (3.1) restricted to a neighborhood of the equilibriumE0…0;0;w

;z†:

The Jacobian matrix due to the linearization of (3.1) about an arbitrary equilibrium

JE…x1;x2;w;t† ˆ

Using hypotheses P₁±P₆, the Jacobian matrix due to linearization of (3.1) about the rest point

E0…0;0;w

Henceforth we letM22 de®ne the matrix

M22ˆ fw…w

The eigenvalues ofJ_E

0…0;0;w

and the eigenvalues ofM22 which are given by

r…M22† ˆ fkijdet…M22ÿkI† ˆ0; iˆ3;4g

ˆ fkijk

2

ÿ …trace M22†k‡detM22ˆ0; iˆ3;4g: …5:10†

By the Routh±Hurwitz criteria, the eigenvalues of M22 have negative real parts, i.e., Rer…M22†<0, i.e., if ÿTraceM22>0, and detM22>0.

Proof.The proof is by inspection of the eigenvalues of the Jacobian matrix forE0…0;0;w

;z† and the qualitative theory of di€erential equations (cf. [21±24]).

Theorem 5.4. Suppose (i) B0₁…0† ÿD0₁…0†>0, (ii)B0₂…0† ÿD0₂…0† ÿhx2…0;w

†>0; and

(iii) TraceM22<0 with detM22>0. Then the rest point E0…0;0;w

;z† is a hyperbolic saddle and is repelling in both x1 andx2 directions locally.In particular,the dimensions of the stable manifoldW‡ _{and unstable manifoldW}ÿ_{are given,}

respectively, by

Dim W‡ E0…0;0;w

;z†

ˆ2; Dim Wÿ E0…0;0;w

;z†

ˆ2:

Proof.This result follows directly from inspection of the eigenvalues of the Jacobian matrix for

E0…0;0;w

;z†and examples from Freedman and Mathsen [22].

Remark. Clinically the rest point E0…0;0;w

;z† is not therapeutically feasible since it has neither normal nor cancer cells. It is also highly unstable.

5.1. Existence and local stability analysis ofE1[x1, 0,w,z]

Consider system (3.1) restricted toR‡

x1wz as represented by

_

x1ˆB1…x1† ÿD1…x1†;

_

wˆQ1ÿa1e1…w† ‡f…w;z†;

_

zˆQ2ÿa2e2…z† ÿgf…w;z†; …5:11†

x1…0† ˆx10P0; z…0† ˆz0P0; w…0† ˆw0P0:

The possible equilibria in R‡

x1wz are eE1‰0;w;zŠ and Ee1‰x1;w;zŠ: In particular, the existence of

~

E1‰x1;w;zŠ(which will be shown by persistence analysis) will imply the existence of E1‰x1;0;w;zŠ. Using methods similar to the previous section, we can conclude thatE~1‰0;w;zŠexists inR‡x1wz if there existsw;z such that

Q1ÿa1e1…w† ‡ 1 g Q2

ÿa2e2…z†

ˆ0:

We now linearize system (5.11) in the neighborhood of Ee1‰0;w;zŠ. This procedure leads to the result

_

nˆDF Ee1…0;w;z†

whereDF…Ee1…0;w;z†† is the Jacobian matrix of the linearization and is given by

DF Ee1…0;w;z†

ˆ B0

1…0† ÿD01…0† 0 0 0 fw…w;z† ÿa1e10…w† fz…w;z†

0 ÿgfw…w;z† ÿa2e02…z† ÿgfz…w;z†

2 4

3

5: …5:13†

The eigenvalues ofDF…Ee1…0;w;z†† are given by k1 ˆB0₁…0† ÿD0₁…0†;

and

r…M22† ˆ fkijdet…M22ÿkI† ˆ0; iˆ2;3g;

whereM22is de®ned as in (5.8), with…w

;z†replaced by…w;z†. Note that Rek2 <0 and Rek3<0 if TraceM22<0 and detM22>0.

Theorem 5.5. The rest point eE1‰0;w;zŠ 2R‡_x1wz is (i) a hyperbolic saddle if

k1 ˆB01…0† ÿD

0

1…0†>0;

andTrace M22<0with detM22>0.In particular Ee1‰0;w;zŠ is repelling in the x1-direction, (ii)a hyperbolic source if

k1 ˆB0₁…0† ÿD0₁…0†>0 and Re ki>0 for iˆ2;3;

(iii) asymptotically stable(sink) if

k1 ˆB0₁…0† ÿD0₁…0†<0 and Trace M22<0 with detM22>0:

Proof.Similar to those of the previous subsection.

De®nition 5.6. A set A_{S is a strong attractor with respect toS if}

lim

t!1 supq u…t†;

A

… † ˆ0; …5:14†

whereu…t†is an orbit such that u…t0† S and qis the Euclidean distance function.

Lemma 5.7. The invariance box

A_{1 ˆ …x1;w;z† 2}R‡

x1wz 0

6x16K1; 06w6 ÿ

Q1 d1

; 06z6Q2

d2

; …5:15†

where

d1ˆmax

w;z maxw;z

~

f…w;z†

ÿa1 min

w ee1…w†

<0;

d2ˆa2 min

z e~2…z†>0;

is a strong attractor set with respect toR‡

w1wz.

Remark.SinceA_{1is a strong attractor, it implies that all solutions of (5.11) with initial conditions} inR‡

x1wz are dissipative, uniformly bounded, and eventually enter the region A_1.

Theorem 5.8. Existence of E1‰x1;0;w;zŠ. Suppose (i) Lemma5.2holds.

(ii)E~1…0;w;z†is a unique hyperbolic rest point inR‡_x1wzand repelling locally in thex1-direction(see Theorem5.5 (i)).

(iii)No periodic nor homoclinic orbits exist in the planes of R‡

x1wz;…

In particular, the subsystem in R‡

x1wz exhibits uniform persistence and consequently, the rest point

E1‰x1;0;w;zŠ exists.

Theorem 5.9. Let (i) B0

2…0† ÿD02…0† ÿxq2…x;0† ÿhx2…0;w†>0 (ii)B0₁…x† ÿD0₁…x†<0

(iii) TraceM22<0and detM22>0.

Then the equilibrium E1‰x1;0;w;zŠ is a hyperbolic saddle point which is repelling in thex2-direction locally.In particular,the stable manifold,W‡…E1…x1;0;w;z††is thex1ÿwÿzspace and the unstable manifoldWÿ…E1…x1;0;w;z††is the x2-direction, withDim Wÿ…E1† ˆ1:

Theorem 5.10. The rest point E1‰x1;0;w;zŠ is locally asymptotically stable (hyperbolic sink) if B0

2…0† ÿD02…0† ÿx1q2…x1;0† ÿhx2…0;w†<0; B

0

1…x1† ÿD0…x1†<0 and Trace M22<0 with detM22>0.

The proofs of Theorems 5.8 and 5.10 follow from an inspection of the Jacobian matrix of linearization in the neighborhood of E1‰x1;0;w;zŠ and using the Routh±Hurwitz criteria.

5.2. The existence and stability ofE2[0;bx2;wb;bz]

We now establish criteria for the existence and stability of the rest pointE2‰0;bx2;wb;bzŠ: When system (3.1) is restricted toR‡

x2wz;we obtain the following subsystem:

_

x2 ˆB2…x2† ÿD2…x2† ÿh…x2;w†;

_

wˆQ1ÿa1e1…w† ‡f…w;z† ÿbh…x2;w†;

_

zˆQ2ÿa2e2…z† ÿgf…w;z†; …5:17†

x2…0† ˆx20P0; w…0† ˆw0P0; z…0† ˆz0P0:

The possible equilibria corresponding to system (5.7) are inR‡

x2wz (i) eE2‰0;wb;bzŠ and

(ii) Ee2‰bx2;wb;bzŠ:

Using the arguments from Lemma 5.2 we can conclude that the rest pointEe2‰0;wb;bzŠexists if there exist…wb;bz† such that

Q1ÿa1e1…bw† ‡ 1

g…Q2ÿa2e2…bz†† ˆ0

ast! 1.

The existence of eE2‰bx2;wb;bzŠ and hence E2‰0;bx2;wb;bzŠ will be established similar to the previous section using persistence analyses.

The Jacobian matrix DF…eE2‰0;wb;bzŠ† due to linearization of (5.17) in the neighborhood of e

E2‰0;wb;bzŠinR‡x2wz satis®es the ordinary di€erential equation

_

gˆDF Ee2‰0;wb;bzŠ

where

DF…eE2‰0;wb;bzŠ† ˆ

B0₂…0† ÿD0₂…0† ÿhx2…0;wb†

0 0

ÿbhx2…0;wb†

ÿa1e0₁…wb† ‡fw…wb;bz†

fz…wb;bz†

0 ÿgfw…wb;bz† ÿ

a2e02…bz†

ÿgfz…wb;bz†

2 6 6 6 6 6 6 4

3 7 7 7 7 7 7 5

: …5:19†

The hypotheses H1±H4 andP1±P6are again used in the computation of the entries of the Jacobian matrix together with expression (5.6).

The eigenvalues of DF…Ee2‰0;wb;bzŠ†are given by

k1ˆB02…0† ÿD

0

2…0† ÿhx2…0;w† and k2;k32r…M22†:

Theorem 5.11. The rest point Ee2‰0;wb;bzŠ of (5.17)is such that

(i) Ee2‰0;wb;bzŠ is a hyperbolic saddle point (repelling, in the x2-direction) if

B0₂…0† ÿD0₂…0† ÿhx2…0;bw†>0 andTrace M22<0with detM22>0: (ii) eE2‰0;bw;bzŠis a hyperbolic source if B02…0† ÿD

0

2…0† ÿhx2…0;w†>0and Reki >0; iˆ2;3. (iii) _Ee2‰0;wb;bzŠ is a hyperbolic sink and hence locally asymptotically stable if

B0₂…0† ÿD₂0…0† ÿhx2…0;bw†<0 andTrace M22<0with detM22>0.

Proof. These results follow immediately from inspection of the Jacobian matrix due to lineari-zation of (5.17) around eE2‰0;wb;bzŠ and applying the qualitative theory of ordinary di€erential equations.

Lemma 5.12. The(non-negatively) invariant set

A_{2 ˆ …x2;w;z† 2}R‡

x2wz j0

6x26K2; 06w6 ÿ

Q1

d1; 06z6

Q2 d2

; …5:20†

whered1 andd2 are defined as in (4.2)and(4.4), respectively, is a strong attractor with respect to solutions initiating fromint R‡

x2wz with non-negative initial conditions.

Proof.Similar to the previous section's proof for the invariant set A_1:

Remark.Since the compact setA_{2 is a strong attractor, it therefore means that, all solutions of} (5.17) with initial conditions in intR‡

x2wz are dissipative, uniformly bounded, and eventually enter the regionA_2.

Theorem 5.13. Existence ofE2‰0;bx2;wb;bzŠ. Suppose (i) Lemma5.11holds.

(ii)Ee2‰0;wb;bzŠis a unique hyperbolic saddle repelling in thex2direction ofR‡x2wz(see Theorem3.7 (i)). (iii)There are no periodic nor homo/hetero-clinic trajectories in the planes of R‡

x2wz Z T

0

B0₂…0†

ÿD0₂…0† ÿhx2…0;w†

dt>0

Then the subsystem(5.17)exhibits uniform persistence and the interior equilibriumEe‰bx2;wb;bzŠexists

The eigenvalues of J_E

2‰0;bx2;bw;bzŠ are given by

k1 ˆB0₁…0† ÿD0₁…0† ÿbx2q1…0;bx2† and k2;k3;k42r…M33†:

In particularM33 is the matrix de®ned by

where

b

a1 ˆ ÿ …Trace M33† ˆ ÿ…mb11‡mb22‡mb33†;

b

a2 ˆ det b

m22 mb23 b

m32 mb33

‡det m_mb_b11₃₁ m_mb_b13₃₃

‡det m_mb_b11₂₁ m_mb_b12₂₂

; b

a3 ˆ ÿdetM33:

…5:24†

Lemma 5.14. The eigenvalues ofM33have negative real parts if

b

a1 >0; ba3 >0 and ba1ba2 >ba3:

Proof.The proof uses the Routh±Hurwitz criterion.

Theorem 5.15. Let

(i) B0₁…0† ÿD0₁…0† ÿbx2q1…0;bx2†>0, (ii)_ba1 >0; ba3>0 and ba1ba2>ba3.

ThenE2‰0;bx2;wb;bzŠ is a hyperbolic saddle point and repelling in the x1-direction. In particular, the stable manifoldW‡…E2†is thex2ÿwÿzspace and the unstable manifoldWÿ…E2†is thex1-direction, such thatDim W‡_E

2† ˆ3 andDim Wÿ…E2† ˆ1.

Theorem 5.16. E2‰0;bx2;wb;bzŠ is locally asymptotically stable(hyperbolic sink) if (i) B0₁…0† ÿD0₁…0† ÿbx2q1…0;bx2†<0,and

(ii)_ba1 >0; ba3>0; ba1ba2>ba3 hold concurrently.

The proofs of Theorems 5.15 and 5.16 follow directly from linearized stability analysis and application of the Routh±Hurwitz criteria.

Remark.The equilibriumE2‰0;bx2;wb;bzŠcorresponds to the scenario in which thenormalcells in the cancer-a€ected tissue or organ are all destroyed. This will eventually lead to the demise of the cancer patient unless a transplant of a new organ is implemented. Thus E2‰0;bx2;wb;bzŠ is highly clinically unstable.

5.3. Existence ofE3[x1;x2;w;z]

In this section, we shall establish sucient conditions for the existence of a positive interior equilibriumE3‰x1;x2;w;zŠ. This will be done by showing that system (3.1) is uniformly persistent (see [21,24,30]).

To show uniform persistence inR‡

x1x2wz we must assume or verify the following hypotheses for system (3.1).

H0: All dynamics are trivial onoR‡x1x2wz.

H1: All invariant sets (equilibria/rest points) are hyperbolic and isolated.

H2: No invariant sets onoR‡x1x2wz are asymptotically stable.

H4: IfM is an invariant set onoR‡x1x2wz, and W

‡_{M† is its strong stable manifold, then}

W…M† \ intR_x

1x2wz ˆ ;: …5:25†

H5: The given system of di€erential equations is dissipative and eventually uniformly bounded fort2R_‡ with respect to a strong (compact) attractor set.

H6: All invariant sets are acyclic.

Remark.H1±H5 gives persistence and H6 is required for uniform persistence.

6. Global stability of subspace equilibria

6.1. Global asymptotic stability of E1,[x1, w,x]

In this section, we derive criteria for the global stability hypothesisH3 to be valid. First criteria for the global asymptotic stability ofE1‰x1;0;w;zŠ with respect to solutions initiating in int R‡_x1wz will be established.

InR‡

x1wz we choose the Liapunov function,

V…x1;w;z† ˆx1ÿx1ÿx1ln

x1

x1

‡1

2k1…wÿw† 2

‡1

2k2…zÿz† 2

; …6:1†

whereki2R‡ for iˆ1;2.

The derivative of (6.1) along the solution curves of (5.11) inR‡

x1wz is given by the expression

_

V ˆ …x1ÿx1†g1…x1† ‡k1…wÿw†‰Q1ÿa1e1…w† ‡f…w;z†Š ‡k2…zÿz†‰Q2ÿa2e2…z† ÿgf…w;z†Š;

…6:2†

where we set

Bi…xi† ÿDi…xi† xigi…xi†; iˆ1;2: …6:3†

Thus

_

V ˆ …x1ÿx1†g1…x1†

‡k1…wÿw†a1‰e1…w† ÿe1…w†Š ‡k1…wÿw†‰f…w;z† ÿf…w;z†Š

‡k2…zÿz†a2‰e2…z† ÿe2…z†Š ‡k2…zÿz†g‰f…w;z† ÿf…w;z†Š: …6:4†

Let

X ˆ

v1

v2

v3 0 @

1

A such that

v1 ˆx1ÿx1

v2 ˆwÿw

and set

In particular,A is symmetric and real such thatAˆ1 2…A‡A

t_{† where t denotes transpose.}

Lemma 6.1. Negative Definiteness of V_. (i) V_ is negative ifXTAX is negative definite. (ii)XT_{AX is negative if A is negative definite.}

(iii)A is negative definite if the (eigenvalues) zeros of the polynomial

p…k;A† ˆdet…AÿkIn† ˆ0

have negative real parts.

A complete discussion and proofs of the lemma can be found in Refs. [26,27].

Lemma 6.2 (Frobenius 1876). Let

X ˆ

matrix A to be negative definite is that the principal minors ofA,starting with that of the first-order, be alternatively negative and positive.

The discussion of Lemma 6.2 is found in Howard Eves book (see [26]). We now state additional hypotheses.

Q: Let V_ ˆXTAX; Aˆ faijgnn, where Ais a real symmetric nn matrix. Then the aij's are

such that

(i) aij2C0…R‡R‡R‡;R†,

(ii) limx!x aij exist as a ®nite number, wherex is rest point,

(iii) the aij are bounded.

Let the matrix Abe given as in (6.8). Then

p…k;A† ˆ det…AÿkI†

Hence by the Routh±Hurwitz criterion and Lemma 6.1 (iii), the matrix A is negative de®-nite if

m1>0; m3 >0 and m1m2 >m3: …6:11†

A re®nement of the criteria (6.11) leads to the following theorem.

Theorem 6.3. The rest point Ee1‰x1;w;zŠ 2R‡_x1wz is globally asymptotically stable with respect to solution trajectories initiating from intR‡

x1wz if (i) a11<0; a22<0; a33<0, and

(ii)a22a33ÿ1₄a223>0.

Thus Ais negative de®nite if

a11<0; det

a11 1₂a12 1

2a12 a22

>0; …6:12†

and detA<0, by Lemma 6.2.

Since a12ˆa13ˆ0, we arrive at re®ned criteria for the negative de®niteness of Aas

…i† a11<0; a22<0; a33<0 and

…ii† a22a33ÿ1₄a223>0:

…6:13†

This agrees with Theorem 6.3.

6.2. Global asymptotic stability of Ee2[bx2;wb;bz]

In this section, criteria for global asymptotic stability of the 3-dimensional equilibrium e

E2‰bx2;wb;bzŠor equivalently E2‰0;bx2;wb;bzŠwith respect to solutions initiating from intR‡_x2wz will be established.

We consider the subsystem (5.17) and choose the Liapunov function

V ˆx2ÿbx2ÿbx2lnx2 bx2

‡1

2 k2…wÿwb† 2

‡1

2 k3…zÿbz† 2

: …6:14†

Let

h…x2;w† ˆx2h1…x2;w† and h1…x2;w† ˆwh2…x2;w†: …6:15†

Then using (3.33) and (3.43) we have

_

V ˆ …x2ÿbx2†‰g2…x2† ÿh1…x2;w†Š

‡k2…wÿwb†‰Q1ÿa1e1…w† ‡f…w;z† ÿbh…x2;w†Š

‡k3…zÿbz†‰Q2ÿa2e2…z† ÿgf…w;z†Š: …6:16†

Simplifying (3.44) leads to

_

V ˆ …x2ÿbx2†g2…x2† ÿ …x2ÿbx2†‰wh2…x2;w† ÿwhb 2…x2;wb†Š ÿ …x2ÿwb2†whb 2…x2;wb†

‡k2…wÿwb†‰a1…e1…wb† ÿe1…w††Š ‡k2…wÿwb†‰f…w;z† ÿf…wb;bz†Š

‡bk2…wÿwb†‰h…bx2;wb† ÿh…x2;w†Š

‡k3…zÿbz†‰a2…e2…bz† ÿe2…z††Š ‡k3g…zÿbz†‰f…bw;bz† ÿf…w;bz†Š: …6:17†

We now set V_ ˆXTBX with

X ˆ

v1

v2

v3 0 @

1 Aˆ

x2ÿbx2

wÿwb

zÿbz

0 @

1 A;

where

Bˆ

b11 1₂b12 1₂b13 1

2b12 b22 1 2b23 1

2b13 1

2b23 b33 0

@

1

Note that bijˆbji. Thus B is a real and symmetric 33 matrix, such that

Bˆ1

2…B‡B t_†:

In particular thebij's are de®ned as

b11ˆg2…x2† ÿwhb 2…x2;wb†;

b12ˆb21ˆ

wh2…x2;w† ÿwhb 2…x2;bw†

…wÿwb† ‡bk2

‰h…bx2;wb† ÿh…x2;w†Š

…x2ÿbx2†

;

b13ˆb31ˆ0;

b22ˆk2a1

‰e1…wb† ÿe1…w†Š

…wÿwb† ;

b33ˆk3a2

‰e2…bz† ÿe2…z†Š

…zÿbz† :

…6:19†

The leading principal minors of Bare

b11; det

b11 1₂b12 1

2b12 b22

and det B:

By Frobenius' theorem, Bwill be negative de®nite if

b11<0; det

b11 1₂b12 1

2b12 b22

>0 and det B<0: …6:20†

But b13ˆb31ˆ0 and hence (3.50) simpli®es the criteria

…i† b11<0; b22<0; b33<0; b12<0;

…ii† b11b22ÿ1₄b212>0;

…iii† b22b33ÿ1₄b223>0:

…6:21†

This leads to the following theorem.

Theorem 6.4. The rest point Ee2‰bx2;wb;bzŠ 2R‡x2wz is globally asymptotically stable with respect to solution trajectories initiating from intR‡

x2wz if (i) b11<0; b22<0; b33<0; b12<0,and (ii)b11b22ÿ1₄b212>0,

(iii) b22b33ÿ1₄b223>0.

6.3. Global asymptotic stability of E0[0;0,w,z] in R‡wz

Consider system (3.1) restricted toR‡

wz as depicted by (5.11).

We have shown that the 2-dimensional equilibriumeE0‰w

;zŠand consequentlyE0‰0;0;w

;zŠexists if Lemma 5.2 holds. In this section, we shall establish criteria for the global asymptotic stability of

E0‰0;0;w

;zŠ with respect to solutions emanating from the interior of R‡

Let G be a neighborhood of any point in R‡

wz. We choose the Liapunov function V such

that

(iii)V is a Liapunov function for (5.11) inG. (iv)V 2C0_R2

along the solution trajectories of (3.2). From (6.23) we obtain the expression

De®ne

A_{0 ˆ …w;z† 2}R‡

wz 0

6w6 ÿQ1

d1 ; 06z6

Q2

d2 ;d1 <0;d2 >0

; …6:28†

whered1 and d2 are as de®ned by expressions (4.2) and (4.4). We now de®ne the sets

S1 ˆ …w;z† 2A0\int R‡_wz V…w;z†

ˆ0 ; …6:29†

S2 ˆ …w;z† 2intR‡wz

w

n

ˆw;zˆz

o

: …6:30†

By inspection, we see immediately that

S1 S2:

Now de®ne the set E_{as follows:}

E_{ˆ …w;z† 2}R‡

wz

_V_

n

ˆ0o\G: …6:31†

Then the largest invariant setinE isE0…0;0;w

;z† restricted toR‡

wz.

Hence by LaSalle's Invariant Principle, cf. [27±29], we conclude that Ee0‰w

;zŠ or consequently

E0‰0;0;w

;zŠis globally asymptotically stable with respect to solutions initiating from int R‡

wz if the

matrix Cis negative de®nite.

Theorem 6.5. The equilibrium E0…0;0;w

;z† 2R‡

wz is globally asymptotically stable with respect to

solution trajectories emanating from int R‡

wz if

(i) c11<0; c22<0,and (ii)c11c22ÿ1₄c212>0.

Proof. The proof follows from computing the leading principal minors of (6.26) and using the Frobenius theorem, see Lemma 6.2 or alternatively by means of Lemma 6.1.

7. Persistence, uniform persistence and existence ofE₃[x 1;x

2;w

_;z]

In this section, we shall present results on persistence, uniform persistence and ®nally give sucient criteria for the existence of a positive interior equilibrium E3‰x1;x2;w;zŠ.

Theorem 7.1. Assume system (3.1)is such that (i)E0…0;0;w

;z†is a hyperbolic saddle point and is repelling in thex1 andx2-directions locally(see Theorem5.4)

(ii) E1…x1;0;w;z† is a hyperbolic saddle point and is repelling in the x2-direction locally (see Theorem5.8)

(iii) E2…0;bx2;bw;bz† is a hyperbolic saddle point and is repelling in the x1-direction locally (see Theorem5.10)

(iv) system (3.1) is dissipative and solutions initiating in int R‡

(v)the equilibriaE0…0;0;w

;z†, E1…x1;0;w;z†andE2‰0;bx2;wb;bzŠare globally asymptotically stable with respect toR‡

wz,R

‡

x1wz and

R‡

x2wz, respectively, (see Theorems6.3±6.5). Then system(3.1) exhibits (robust) persistence.

Proof.The proof will be done using the Butler±McGehee Lemma (cf. [30]). Let

B_{ˆ …x1;x2;w;z† 2}R‡

x1x2wz 0

6x16K1; 06x26K2; 06w6 ÿ

Q1 d1

; 06z6Q2

d2

R4

‡;

where d1, and d2 are as de®ned by (4.2) and (4.4).We have shown in Theorem 5.1 that B _is positively invariant and any solution of system (3.1) initiating at a point inBR4

‡ is eventually

bounded. However E0ˆE0…0;0;w

;z†, E1ˆE1…x1;0;w;z† and E2ˆE2‰0;bx2;bw;bzŠ are the only compact invariant sets onoR4

‡. LetM ˆE3‰x₁;x₂;w;zŠ be such thatM 2int R4_‡.

The proof is completed by showing that no point Qi2oR4‡ belongs to X…M†. The proof is

divided into ®ve steps. Step1. We show that

E062X…M†:

Suppose E0 2X…M†. Since E0 is hyperbolic, E0 6ˆX…M†. By the Butler±McGehee lemma, there exists a point Q‡₀ 2W‡…E0†nfE0g† such that Q‡0 2X…M†. But W‡…E0† \ …R4_‡nfE0g† ˆ ;. This contradicts the positive invariance property ofBR4

‡. Thus E0 62X…M†. Step2. We show that

E162X…M†:

If E12X…M†; then there exists a point Q‡1 2W‡…E1†nfE1g such that Q‡1 2X…M† by the Butler± McGehee lemma. But W‡…E1† \ …int R4_‡† ˆ ; and E1‰x1;0;w;zŠ is globally asymptotically stable with respect toR‡

x1wz. This implies that the closure of the orbitO…Q

‡

1†throughQ‡1 either contains

E0 or is unbounded. This is a contradiction. Hence E1 62X…M†: Step3. We show that

E262X…M†:

The proof is similar to Step 2. Step4. We show that

oR4_‡\X…M† ˆ ;:

SupposeoR4

‡\X…M† 6ˆ ;. LetQ2oR

4

‡andQ2X…M†. Then, the closure of the orbit throughQ,

i.e.,O…Q† must either contain E0;E1;E2 or is unbounded. This gives a contradiction.

Step5. Thus we see that if E0 is unstable then

W‡…E0† \ …R4_‡ j fE0g† ˆ ;:

Also, we deduce that ifE1 is unstable, then

W‡…E1† \ …int R4‡† ˆ ;;

Similarly if E2 is unstable, then

W‡…E2† \ …int R4‡† ˆ ;;

Wÿ…E1† \ …R4nR4‡† 6ˆ ;;

and the persistence result follows sinceX…M† must be in intR4

‡.

Remark.The global asymptotic stability of the equilibriaE0;E1;andE2 with respect toR‡wz,R

‡

x1wz and R‡

x2wz, respectively, implies that the boundary ¯ow isisolatedand acyclic with respect to C_.

Theorem 7.2. Let the conditions of Theorem 7.1 hold. Then system(3.1) exhibits uniform persis-tence. In particular, a positive interior equilibrium of the form E3‰x₁;x₂;w;zŠ exists.

Proof.The results follow directly from the results obtained in [25].

8. Hopf±Andronov±Poincar bifurcation

In this section, the Hopf±Andronov±Poincare bifurcation (cf. [31]) will be performed on the system of equations (3.1) with bifurcation parameter l. In particular the parameter l is chosen such that the activation±proliferation function fis a function ofl. The system of equations now takes the form

_

x1 ˆB1…x1† ÿD1…x1† ÿx1x2q1…x1;x2†;

_

x2 ˆB2…x2† ÿD2…x2† ÿx1x2q2…x1;x2† ÿh…x2;w†;

_

wˆQ1ÿa1e1…w† ‡f…w;z;l† ÿbh…x2;w†; …8:1†

_

zˆQ2ÿa2e2…z† ÿgf…w;z;l†;

x1…0† ˆx10P0; x2…0† ˆx20P0;

w…0† ˆw0P0; z…0† ˆz0P0:

System (3.62) can be recast into the form

_

xˆF…x;l†; x…0† ˆx0;

…8:2†

where

x2R4 _ˆ

x1

x2

w z

0 B B @

1 C C A:

l2R1is the bifurcation parameter.F…x;l†is aCr_{rP5†function on an open set inR}4_R1_{. Let}

P_{lˆ ‰E0…0;0;w;z;l†;E1…x1;0;w;z;l†;E2…0;bx2;wb;bz;l†;E3…x}

be the set of rest (®xed) points of (8.1) such that

F…P_{l† ˆ}0 for some l2R1

on a suciently large open set Gcontaining each member ofP_l:

The linear vector ®eld obtained by linearizing (3.6 2) about any lis given by

_

nˆDF…P_l†n; n2R4_:

We are interested in studying how the orbit structure nearP_l changes as lis varied.

S: [The technicality criterion] Let

d

dl‰fw…w;z;l† ÿgfz…w;z;l†Š>0 …8:4†

forlˆl₀ at…w;z† ˆ …w;z†:

Theorem 8.1. Suppose Theorems5.4and5.15hold.Then Hopf bifurcation cannot occur atE0andE2:

Proof.This follows from the fact that when Theorems 5.4 and 5.15 hold, respectively, thenE0and

E2 are hyperbolic saddle points and their stable manifolds lie along an axis.

Next we have to compute stability criteria forE1‰x1;0;w;z;lŠandE3‰x1;x2;w;z;lŠand then vary l in order to obtain the desired Hopf bifurcation for lˆm1 and lˆm2 around E1 and E2, respectively.

Hopf bifurcation analysis forE1‰x1;0;w;z;lŠ Let

Jl…P1† ˆDF E1…x1;0;w;z;l†

ˆ A11…l† 0

A21…l† A22…l† !

; A21…l† ˆ

0 ÿbhx2…0;w†

0 0

;

where

A11ˆ

B0₁…x1† ÿD01…x1† ÿx1q1…x1;0†

0 B0

2…0† ÿD02…0† ÿx1q2…x1;0† ÿhx2…0;w†

" #

;

A22ˆ

fw…w;z;l† ÿa1e01…w† f2…w;z;l†

ÿgfw…w;z;l† ÿa2e02…z† ÿgfz…w;z;l†

" #

:

The eigenvalues ofJl…P1† are

k1 ˆB0₁…x1† ÿD01…x1†;

k2 ˆB0₂…0† ÿD0₂…0† ÿx1q2…x1;0† ÿhx2…0;w†;

…8:5†

and the solution of

p…k;A22† ˆ detjA22ÿkI2 jˆ0

Letk3;4ˆ fkj p…k;A22† ˆ0g. Then

k3;4 ˆ 1

2 TraceA22



…Trace A22† 2

ÿ4 det A22

q

: …8:7†

Theorem 8.2. The Jacobian matrix Jl…P_{1† has two negative real roots and two purely imaginary} roots if the following criteria hold concurrently:

(i) k1 ˆB0₁…x1† ÿD01…x1†<0,

(ii)k2ˆB0₂…0† ÿD0₂…0† ÿx1q2…x1;0† ÿhx2…0;w†<0, (iii) there exists a lˆm1 with TraceA22ˆ0, (iv) det A22>0with k3;k42 im C.

Thusd=dl‰Re ki…m1†Š>0.

Proof. The proof follows directly from inspecting Eqs. (8.5) and (8.7). Re ki ˆ1₂ Trace A22 for

iˆ3;4. But

TraceA22ˆfw…w;z;l† ÿa1e01…w† ÿa2e

0

2…z† ÿgfz…w;z;l†:

Thus

d

dl‰Re kiŠ ˆ Re d dlki

lˆm1

ˆ d

dl fw…w;z;l† h

ÿa1e01…w† ÿa2e02…z† ÿgfz…w;z;l†

i

lˆm1

ˆ d

dl fw…w;z;l† h

ÿa1e01…w† ÿa2e02…z† ÿgfz…w;z;l†

i

lˆm1

ˆfwl…w;z;m1† ÿgfzl…w;z;m1†>0 …8:8†

by S.

Theorem 8.3. Suppose

(i) Tr A22ˆ0 for some m1 2R‡.

(ii)Technicality criterion S holds, i.e.,

fwl…w;z;m1† ÿgfzl…w;z;m1†>0 (iii) k1 ˆB01…x1† ÿD01…x1†<0

k2 ˆB0₂…0† ÿD0₂…0† ÿx1q2…x1;0†<0:

Then Hopf bifurcation occurs at lˆm1 for the equilibriumE_{1…x1;0;w;z†for system (8.1).}

This leads to the following theorem.

Now, consider the steady state E3‰x1;x2;w;zŠ. The Jacobian matrix corresponding to

The characteristic equation corresponding toJl…Sl† is given by

By the Routh±Hurwitz criteria, necessary and sucient conditions for all the roots of

p…k;Jl…Sl†† ˆ0 to have negative real parts are

R1 : a₁ >0; a₃ >0; a₄ >0 and R2 : a₁a₂a₃ >…a₃†2‡ …a₁†2a₄:

Now in order to have Hopf bifurcation, we must violate eitherR1 andR2.P11. Suppose eachai>0

foriˆ1;2;3;4 such that (i) a

2a3ÿa4>0, and (ii)R2 is violated such that

a₁a₂a₃ ˆ …a₃†2‡ …a₁†2a₄;

()a₂a₃ ˆ …a

3† 2

a

1

‡a₁a₄ >a₄: …8:11†

Lemma 8.5. Let

(i) Eachai >0 for iˆ1;2;3;4,

(ii)a₂a₃ÿa₄ >0,

(iii) conditionR2 be violated.

Thenq…k;Jl…Sl†† ˆ0can be factored into the form

…k2‡k1†…k‡k2†…k‡k3†; ki>0; iˆ1;2;3;4;

where

a₁ˆk2k3;

a₂ˆk2k3‡k1;

a₃ˆk1…k2‡k3†;

()k1ˆ

a₄ a

1

; k2 ˆ

a₁…a₂a₃ÿa₄† a

2a4

;

k3ˆ

a₁a₄ a

2a3ÿa4

:

…8:12†

In particular, the spectrum of Jl…Sl† is given by

r…Jl…Sl†† ˆ i k1 p

;

n

ÿi k1 p

; ÿk2;ÿk3 o

: …8:13†

But for l2 …m2ÿe;m2‡e†; the roots are in general of the form

k1…l† ˆa…l† ‡ib…l†;

k2…l† ˆa…l† ÿib…l†;

k3…l† ˆ ÿk26ˆ0; …8:14†

k4…l† ˆ ÿk36ˆ0:

We now apply Hopf's transversality criterion to q…k;Jl…Sl†† in order to obtain the required conditions for Hopf bifurcation to occur for this system. Hopf's transversality criterion is given by

Re dkj dl

lˆm2

6ˆ0 for jˆ1;2: …8:15†

Now substituting kj…l† ˆa…l† ‡ib…l† into

q k;Jl…P2†

ÿ

ˆk4‡a₁k3‡a₂k2‡a₃k‡a₄ˆ0; …8:16†

and computing the derivatives with respect tol, we obtain

a0‰a…a2ÿb2† ÿ2ab2Š ‡i4a0‰b…a2ÿb2† ‡2a2bŠ

ÿ4b0‰b…a2ÿb2† ‡2a2bŠ ‡i4b0‰a…a2ÿb2† ÿ2ab2Š

‡a0₁‰a…a2_ÿb2_{† ÿ2ab}2_{Š ‡i}

a0₁‰b…a2_ÿb2_{† ‡2a}2_bŠ

‡3a0a1…a2ÿb2† ‡ia06a1ab

ÿ6b0aba1‡i3a1b0…a2ÿb2† …8:17†

‡a0₂…a2ÿb2† ‡i2aba0₂‡2a2aa0‡i2a2a0b

ÿb0a22b‡i2a2ab0‡aa03‡iba03

‡a0a3‡ia3b0‡a04ˆ0:

Comparing the real and imaginary parts we have

A…l†a0…l† ÿB…l†b0…l† ‡C…l† ˆ0;

B…l†a0…l† ‡A…l†b0…l† ‡D…l† ˆ0; …

8:18†

where

A…l† ˆ4a…a2ÿb2† ÿ8ab2‡3a1…a2ÿb2† ‡2a2a‡a2;

B…l† ˆ4b…a2ÿb2† ‡8a2b‡6aba1‡2a2b;

C…l† ˆa0₁‰a…a2ÿb2† ÿ2ab2Š ‡a0₂…a2ÿb2† ‡aa0₃‡a0₄;

D…l† ˆa0₁‰b…a2_ÿb2

† ‡2a2_{bŠ ‡2aba}0

2‡ba

0

Thus

Re dkj dl

lˆm2

ˆa0…m2† ˆ

det ÿC…l† ÿB…l†

ÿD…l† A…l†

det A…l† ÿB…l†

B…l† A…l†

lˆm2

ˆ ÿ…AC‡BD†

A2_‡B2 lˆm2 6ˆ0

; …8:19†

since

A…m2†C…m2† ‡B…m2†D…m2† 6ˆ0: …8:20†

Theorem 8.6. Suppose

(i) System(5.17) is uniformly persistent. (ii)E_{ˆ ‰x}

1;x2;w;zŠ exists. (iii) Lemma5.2holds.

Then system (5.17) exhibits a (small amplitude) Hopf±Andronov±Poincare bifurcation in the first orthant,leading to a family of periodic solutions that bifurcate fromEfor suitable values oflin the neighborhood of lˆm2:

It has been established that system (3.1) has solutions which are eventually bounded in the future. Also the equilibrium E2‰0;x2;w;zŠ is constrained as a hyperbolic saddle point. Further-more, Theorems 8.4 and 8.6 establishlˆm1andlˆm2as two bifurcation values. In particular, a periodic solution bifurcates from E1‰x1;0;w;zŠ when l passes through m1. We denote

S1 ˆ ‰x1…t;m1†;x2…t;m1†;w…t;m1†;z…t;m1†Š. Another periodic solution S2ˆ ‰x2…t;m2†;x2…t;m2†; w…t;m2†;

z…t;m2†Šbifurcates from E when l passes throughm2.

Theorems 8.1±8.4 and 8.6 imply that during adoptive cancer immunotherapy as described in this chapter, it is possible under the speci®ed criteria, that Hopf bifurcations occur from the clinically preferred steady state E1‰x1;0;w;zŠ leading to periodic orbits. This phenomenon com-plicates the nature and outcome of therapy as a second subsequent bifurcation may lead to pe-riodic orbits near the less preferred rest point E‰x₁;x₂;w;zŠ.

Theorems 7.1 and 7.2 imply that the normal cells and cancer cells persist under continuous infusion of LAK cells and IL-2. Thus the therapy is unable to annihilate the cancer cells but is able to control the lethal proliferation of the cancer cells.

If the cancer cell number during therapy is either below the detection threshold of 109 _{cells or} the subclinical threshold of 103 _{cells, then the outcome of the therapy may be declared a `partial'} therapeutic success. The only problem associated with this result is that therapy must be con-tinued inde®nitely and the cancer cell number must be preferably subclinical.

However the much preferred objective of cancer immunotherapy is the total annihilation of the cancer cells. This will require that the rest point E1‰x1;0;w;zŠ be globally asymptotically stable with respect to solutions initiating from int R_x

9. Criteria for a `total cure'

In this section, necessary and sucient criteria for total or absolute elimination of all cancer cells will be derived. Usually during cancer therapy the time domain of therapeutic ecacy of the anti-cancer drug is very restricted and the cancer cells eventually repopulate leading to the death of the patient. In such scenarios, the rest point E1‰x1;0;w;zŠ is locally asymptotically stable during therapy and eventually becomes unstable during the repopulation of cancer cells.

In order to obtain the conditions for `total cure', we must, by means of a Liapunov function, establish criteria forglobalasymptotic stability of the rest point E1‰x1;0;w;zŠ:

9.1. Global asymptotic stability ofE1 [x 1, 0,w,z] with respect to R‡x1x2wz

We choose the Liapunov function

V ˆx1ÿx1ÿx1ln

x1

x1

‡k1x2‡1₂k2…wÿw†2‡1₂k3…zÿz†2: …9:1†

The derivative of (9.1) along the solution trajectories of system (3.1) and using (6.15) leads to the following algebraic equations:

_

V ˆ …x1ÿx1†‰g1…x1† ÿx2q1…x1;x2†Š

‡k1x2‰g2…x2† ÿx1q2…x1;x2† ÿh1…x2;w†Š

‡k2…wÿw†‰Q1ÿa1e1…w† ‡f…w;z† ÿbh…x2;w†Š

‡k3…zÿz†‰Q2ÿa2e2…z† ÿgf…w;z†Š: …9:2†

_

V ˆ …x1ÿx1†g1…x1† ÿ …x1ÿx1†x2q1…x1;x2†

‡k1x2‰g2…x2† ÿbx1q2…bx1;x2†Š ÿk1x2‰wh2…x2;w†

ÿwh 2…x2;w†Š ÿk1x2wh 2…x2;w†

‡a1k2…wÿs†‰e1…w† ÿe1…w†Š ‡k2…wÿw†‰f…w;z† ÿf…w;z†Š

ÿbk2…wÿw†x2‰wh2…x2;w† ÿwh 2…x2;w†Š ÿbk2…wÿw†x2h2…x2;w†

‡a2k3…zÿz†‰e2…z† ÿe2…z†Š ‡gk3…zÿz†‰f…w;z† ÿf…w;z†Š: …9:3†

Now

_

V ˆXTDX; …9:4†

where

X ˆ

v1

v2

v3

v4 0 B B @

1 C C Aˆ

x1ÿx1

x2

wÿw

zÿz

0 B B @

1 C C A;

and Dis a real symmetric matrix such that

and

The explicit form of matrixDis as given by (9.6).

Dˆ

Theorem 9.1. The matrix D and consequently the quadratic form (9.4) is negative definite if the following criteria hold:

Proof.The proof follows directly from Frobenius' theorem, and the Hermiticity of D.

Theorem 9.2.LetE1‰x1;0;w;zŠdenote the cancer-free equilibrium for the cancer immunotherapy in system (3.1). Then E1‰x1;0;w;zŠ is globally asymptotically stable or equivalently, there is total annihilation of the cancer cells, if conditions (9.7)of Theorem9.1are satisfied.

Remark.If the persistence criteria speci®ed in Theorems 7.1 and 7.2 are violated, then either the cancer cells or normal cells will be outcompeted locally.

It is possible also, that the normal cell density in the organ may ¯uctuate periodically or chaotically before eventual extinction, according to the principles of Dynamical Disease as mentioned in the introduction.

On the other hand if the adoptively transferred lymphocytes (LAK cells) and lymphokines (IL-2) are very potent and therapeutically ecacious, then the cancer cells may be driven to ex-tinction if persistence fails.

9.2. Criteria for the extinction of normal cells

In this section, we shall derive sucient conditions for the extinction of the normal cells in the a‚icted anatomical organ and in the case of extinction, the probable consequent death of the cancer patient. This will be done by using a Liapunov function to provide criteria for the rest point E2‰0;bx2;wb;bzŠ to be globally asymptotically stable and hence unique. We choose the Liapunov function Vas given by (9.8)

V ˆc1x1‡x2ÿbx2ÿx2ln

x2 bx2

‡1

2c2…wÿwb† 2

‡1

2c3…zÿbz† 2

: …9:8†

The derivative of (9.8) along the solution curves of (3.1) is given by the equation

_

V ˆc1z_1‡

…x1ÿbx2†

x2

_

x2‡c2…wÿwb†w_ ‡c3…zÿbz†z_

ˆc1x1‰g1…x1† ÿx2q1…x1;x2†Š

‡ …x2ÿbx2†‰g2…x2† ÿx1q1…x1;x2† ÿh1…x2;w†Š

‡c2…wÿwb†‰Q1ÿa1e1…w† ‡f…w;z† ÿbh…x2;w†Š

‡c3…zÿbz†‰Q2ÿa2e2…z† ÿgf…w;z†Š: …9:9†

We then use (6.15) and the equilibrium algebraic expressions and tacit rearrangement to obtain a modi®ed form ofV_ as follows:

_

V ˆc1x1g1…x1† ÿc1x1‰x2q1…x1;x2† ÿbx2q1…x1;bx2†Š ÿc1x1bx2q1…x1;bx2†

‡ …x1ÿbx2†g2…x2† ÿ …x2ÿbx2†x1q1…x1;x2†

ÿ …x2ÿbx2†‰wh2…x2;w† ÿwhb 2…x2;wb†Š ÿ …x2ÿbx2†whb 2…x2;wb†

‡c2…wÿwb†a1‰e1…wb† ÿe1…w†Š ‡c2…wÿwb†b‰h…bx1;wb† ÿh…x2;w†Š

‡c2…wÿwb†‰f…w;z† ÿf…wb;bz†Š

‡c3…zÿbz†a2‰e2…bz† ÿe2…z†Š ‡c3…zÿbz†g‰f…wb;bz† ÿf…w;z†Š: …9:10†

Now

_

where

Theorem 9.3. The real symmetric matrix E and consequently the quadratic form(9.11)is negative definite if the following criteria hold for the leading principal minor matrices Di; iˆ f1;2;3;4g:

D2 ˆ det e11

Proof. The proof follows directly from the Frobenius theorem and the fact that E is a real symmetric matrix and consequently Hermitian‰Eˆ …E†TŠ.

The results of the preceding theorem leads to the following theorem which is the manifestation of therapeutic failure of the anti-cancer immunotherapy drugs (LAK and IL-2).

Theorem 9.4. Suppose the criteria (9.14) of Theorem 9.3 hold. Then clinically the cancer immu-notherapy drug protocol will be strongly non-efficacious and eventually the normal cells in the af-flicted organ will be annihilated, leading to probable death of the cancer patient.

Remark.During the clinical administration of LAK and IL-2, it is therapeutically prudent to use continuous infusionsinterspacedwith days of no infusions (cf. [11]). This will enable the patient's physiological system to recover from drug toxicities and associated ``therapeutic stress''. The advantages of such procedure may be circumvented by the phenomenon of repopulations of the tumor by drug-resistant cancer cells. In Section 9.3, we shall analyse the local stability of periodic infusion of LAK and IL-2. The phenomenom of drug resistance and optimal therapy policies will not be addressed in this section but deferred to future research.

9.3. Periodic treatment case

We now consider the treatment scenario in which the LAK and IL-2 are given by periodic adoptive transfusions. The input functionsQ1…t† andQ2…t†are in the form of Heaviside-type step functions with equal or unequal dose rates of periodw:

LetB_{wbe the Banach space of real valued, continuousx-periodic functions of a real variablet.} Consider the periodic system

Q1…t† ˆQ1…t‡x†;

We make the following assumptions: a1.Q1;Q2 2B1w.

a2. The system (9.15) in R‡

x1wz, has a positive x-periodic solutionUsuch that 0<UˆU0 ˆ …~x1;w~;~z† 2B3_w:

a3. LetX4 denote an open set inR4_‡. Then …0;U0† 2X4 for all t2 ‰0;1†:

The assumptions a1±a3 imply the existence of a non-trivialx-periodic solution…x2;U† ˆ …0;U0† on the boundary of the positive cone inB_wB3

w:

a4. The solutionUˆU0 is such thatU0 is non-critical. This implies that all the Floquet expo-nents of the linearization of (9.15) inR‡

x1wzin the neighborhood ofU0 have non-zero real parts. a5. System (9.15) satis®es the invariance of non-negativity criteria.

Let us de®neX4_w…U0† as follows:

X4_w…U0† ˆ f…x2;U† 2BwB3w j …x2…t†; U…t† ‡U0…t†† 2X4 for alltg: …9:16†

The assumption a5 implies thatX4_w…U0† is an open set inB_wB3

w which contains…0;U0…t††:

and q₃;q₄ are Floquet multipliers of the matrix

M…t† ˆ fw…we;ez† ÿa1e 0

1…we† fz…we;ez†

ÿgfw…we;ez† ÿa2e02…we† ÿgz…we;ez†

ˆ fmbij…t†g16i;j62: …9:19†

In particular, the matrix (9.18) corresponds to the Jacobian matrix of linearization of (9.15) in

R‡

wz in the neighborhood of…ew…t†;ez…t††.

We de®ne the following with respect to the matrixM…t†:

kxk ˆsup

k

jxkj;

kM…t†k ˆsup

k

X

jmikj;

l…M…t†† ˆsup

k

Remkk

(

‡X

i;k6ˆi

jmikj

)

:

…9:20†

The Lozinski matrix measure ofM…t† takes the simpli®ed form

l…M…t†† ˆsupfm11‡ jm21j;m22‡ jm12jg:

Theorem 9.5.Let

b

l…t†,max…g‡1†fw…we…t†;ez…t†† ÿa1e01…we…t††;…1ÿg†fz…we…t†;ez…t†† ÿa2e02…ez…t†† ; …9:21†

where…we…t†;ez…t††is the periodic orbit specified by assumptiona2.

If_bl…M…t††6 ÿa;a>0;then…we…t†;ez…t††is uniformly asymptotically stable and consequently the Floquet multipliers q₃;q₄ ofM…t†; are such that jq_ij<1; iˆ3;4:

Proof. The function _bl…M…t†† de®ned by (9.21) is the simpli®ed form of the Lozinskii matrix measure l…M…t††. The theorem therefore follows from Lozinskii matrix stability criteria and Floquet theory.

Remark.Ifg>1 and a1e10…ew…t††>…g‡1†fw…we…t†;ez…t††; then thea in Theorem 9.5 exists.

Theorem 9.6. Suppose

(i) R₀x‰B0₁…ex1…s†† ÿD01…ex1…s††Šds<0. (ii)R₀x‰B0

2…0† ÿD02…0† ÿex1…s†q2…ex1…s†;0† ÿhx2…0;we…s††Šds<0: (iii)The conditions of Theorem 9.5hold.

Then the periodic solution…0;U0…t††is locally uniformly asymptotically stable.

Proof.The hypotheses (i), (ii) and (iii) imply thatjq_i j<1; iˆ1;2;3;4: Hence the result follows from Floquet theory.

(®nite) time domain of therapeutic ecacy of drug. But on the global time scale, the prospect of a `total cure' cannot be guaranteed because bifurcations of the periodic orbit…0;U0…t††may occur, leading to the re-emergence of the cancer as the (interior) periodic branch …x2;U…t†† manifests. This type of bifurcations from periodic orbits have been studied by several authors.

10. Discussion and conclusions

In the preceeding subsections we used mathematical models to discuss the cancer immuno-therapy regimen called Adoptive Cancer (cellular) Immunoimmuno-therapy (ACI). We employed the mathematical tools of di€erential analysis, persistence theory, Hopf±Andronov±Poincare bifur-cation, and linear systems theory to give generalized criteria for the therapeutic ecacy of ACI. When the input functionsQ1 and Q2 are constant and continuous, we derived criteria for the existence, local stability and global asymptotic stability of the clinically preferred rest point

E1‰x1;0;w;zŠ: WhenE1‰x1;0;w;zŠ does not exist, then the patient has persistent cancer.

We established the circumstances under which E1‰x1;0;w;zŠ exists but is unstable, and also criteria for the existence of a positive interior rest pointE‰x₁;x₂;w;zŠ:If the value ofx₂ is below the subclinical threshold of 103 _{cells or for the worst case scenario between 10}3 _{and 10}9 _cells,

E‰x₁;x₂;w;zŠmay persist. But clinically these criteria for the existence ofE‰x₁;x₂;w;zŠare too stringent and can hardly be attained in a human cancer patient.

When periodic infusions are used, Theorem 9.5 and later Theorem 10.5 give criteria for local uniform asymptotic stability of the clinically preferred periodic orbit …0;U0† ˆ …0;ex1;we…t†;ez…t††:

Complications in therapy always occur. Some of these could be explained by the phenomena of chaos and dynamical diseases. In Theorems 8.3, 8.4, 8.6, we presented generalized criteria for Hopf bifurcation to occur at E1‰x1;0;w;zŠ leading to periodic orbits. These rami®cations pose complications for the cancer patient and the clinical oncologists. We presented theoretical criteria under which a second bifurcation occurs leading to a family of periodic solutions from the interior equilibrium E‰x₁;x₂;w;zŠ:

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