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J. Math. Biol.

DOI 10.1007/s00285-011-0455-z

MathematicalBiology

Stable and transient periodic oscillations

in a mathematical model for CTL response

to HTLV-I infection

John Lang · Michael Y. Li

Received: 11 March 2011

© Springer-Verlag 2011

Abstract

T cells by human T cell leukemia virus type I (HTLV-I) has previously been modelled

using standard response functions, with relatively simple dynamical outcomes. In this

paper,weinvestigatetheconsequencesofamoregeneralCTLresponseandshowthata

sigmoidal response function gives rise to complex behaviours previously unobserved.

Multiple equilibria are shown to exist and none of the equilibria is a global attractor

duringthechronicinfectionphase.Coexistenceoflocalattractorswiththeirownbasin

of attractions is the norm. In addition, both stable and unstable periodic oscillations

can be created through Hopf bifurcations. We show that transient periodic oscillations

occurwhenasaddle-typeperiodicsolutionexists.Asaconsequence,transientperiodic

oscillations can be robust and observable. Implications of our findings to the dynam-

ics of CTL response to HTLV-I infections in vivo and pathogenesis of HAM/TSP are

discussed.

The cytotoxic T lymphocyte (CTL) response to the infection of CD4+

Keywords

Stable periodic oscillations · Transient periodic oscillations · Hopf bifurcation

HTLV-I infection · HAM/TSP · CTL response · Bi-stability ·

Mathematics Subject Classification (2000) 92D30 · 92D25

J. Lang (B )

London School of Economics (LSE), London, UK

e-mail: j8lang@uwaterloo.ca

Present Address:

J. Lang

Department of Applied Mathematics, University of Waterloo,

Waterloo, Ontario N2L 3G1, Canada

M. Y. Li

Department of Mathematical and Statistical Sciences,

University of Alberta, Edmonton, AB T6G 2G1, Canada

e-mail: mli@math.ualberta.ca

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J. Lang, M. Y. Li

1 Introduction

Human T cell leukemia virus type I (HTLV-I) is the etiologic agent for the HTLV-I

associated myelopathy (HAM), a chronic inflammatory disease of the central nervous

system, also called tropical spastic paraparesis (TSP) (Gout et al. 1990; Osame et al.

1990). HTLV-I infection can also lead to adult T cell leukemia (ATL) (Kubota et al.

2000; Gallo 2005). Approximately 20–40 million people are infected by HTLV-I

worldwide. Endemic areas include the Caribbean, southern Japan, Central and South

America, the Middle East, Melanesia, and equatorial Africa (Coffin et al. 1997). Most

infectedindividualsremainlifelongasymptomaticcarriers.Approximately0.25–3.8%

develop HAM/TSP and 2–3% of infected individuals develop ATL (Hollsberg and

Hafler 1993; Bangham 2000).

In the peripheral blood, HTLV-I preferentially infects CD4+helper T cells

(Richardson et al. 1990; Bangham 2000; Jacobson 2002). HTLV-I does not exist

as free virions in vivo and infection of healthy CD4+T cells is achieved through

cell-to-cell contact with infected CD4+T cells (Okochi et al. 1984; Bangham 2003).

InfectionalsospreadsverticallythroughmitosisofCD4+TcellsthatharbourHTLV-I

provirus (Wattel et al. 1996). The vertical transmission allows for viral propagation

without expression of the HTLV-I genome and explains the low rate of mutation in

the HTLV-I genome (Wattel et al. 1996). It is known that the horizontal transmission

of HTLV-I replication is active. The immune system reacts to HTLV-I infection with

a strong cytotoxic T lymphocyte (CTL) response (Bangham 2000; Jacobson 2002).

WhileCTLhasaprotectiverolebyregulatingtheproviralload,evidencesuggeststhat

cytotoxicity of the CTL is ultimately responsible for the demyelination of the central

nervous system resulting in HAM/TSP (Greten et al. 1998). The precise reason for the

autoimmune response is unknown. One leading hypothesis is that the CTL response is

reactingtocellsintheCNSmyelinwhichhavebecomeinfectedbyHTLV-I(Bangham

2000; Jacobson 2002).

UnderstandingthepathogenesisoftheHTLV-Iwithinthehosthasimportantimpli-

cations for the development of therapeutic measures and for the identification of risk

factors for HAM/TSP. Mathematical models have been developed to capture the inter-

action in vivo among HTLV-I, its target cells, and the CTL immune response in or-

der to explain the pathogenesis of HTLV-I-associated diseases (Wodarz et al. 1999;

Nowak and May 2000; Wodarz and Bangham 2000; Gomez-Acevedo and Li 2002;

AsquithandBangham2007;Gomez-Acevedoetal.2010).Thesimplestmathematical

modelonecandevelopforthispurposeconsistsofthreecompartments:healthyCD4+

Tcellsx,proviralCD4+Tcells y,andCTLsz.Themodelcanbeschematicallyshown

using a transfer diagram as in Fig. 1.

As shown in Fig. 1, it is assumed in the model that healthy CD4+T cells are pro-

duced at a constant rate λ > 0. Compartments x, y, and z have turn-over rates μ1,μ2,

and μ3, respectively. The infection of healthy CD4+T cells is through direct cell-

to-cell contact with a proviral CD4+T cell. This interaction is modeled by the mass

action term βxy, where β > 0 is the transmission coefficient. The proviral CD4+

T cells are constantly expressing the HTLV-I genome (Bangham 2000), and they are

constantly subject to both antibody and CTL responses. We assume that a fraction

σ of the newly infected CD4+T cells survive the antibody response. The loss of

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Stable and transient periodic oscillations in a mathematical model

Fig. 1 Transfer diagram for the

HTLV-I infection in vivo and the

CTL response

proviral CD4+T cells due to CTL lysis is given by γyz. The term νyf (z) represents

the production of CTLs in response to HTLV-I, where f (z) is the CTL response func-

tion.Intheliterature,CTLresponsefunctionhastakenalinearform f (z) = z (Nowak

and May 2000) or a density dependent form f (z) =

1999; Gomez-Acevedo et al. 2010). These assumptions and the schematic diagram in

Fig. 1 lead to the following system of differential equations for the model

z

z+awith a > 0 (Wodarz et al.

˙ x = λ − βxy − μ1x

˙ y = σβxy − γyz − μ2y

˙ z = νyf (z) − μ3z.

(1.1)

System (1.1) has three types of equilibria:

P0=

P1= (¯ x, ¯ y,0),

P∗= (x∗, y∗,z∗),

?

λ

μ1,0,0

?

,

infection-free equilibrium,

carrier equilibrium, and

HAM/TSP equilibrium.

Here ¯ x, ¯ y,x∗, y∗,z∗are all positive. At equilibrium P0the virus is cleared and all tar-

get CD4+T cells are healthy. At equilibrium P1the HTLV-I infection is chronic but

the CTL response is absent, so are its cytotoxic effects on the central nervous system

and the risk for developing HAM/TSP; this corresponds to the asymptomatic carrier

state. At the positive equilibrium P∗the HTLV-I infection is chronic and there is a

persistent CTL response and constant cytotoxic effect on the central nervous system;

this corresponds to the HAM/TSP state. It is shown in Gomez-Acevedo et al. (2010)

that, for f (z) =

two threshold parameters,

z

z+a, the final outcomes of the system are determined by values of

R0=

λσβ

μ1μ2,

and R1=

λσβν

μ2(μ1ν + βμ3),

(1.2)

and they are called the basic reproduction numbers for HTLV-I infection and the CTL

response, respectively. The global dynamics of system (1.1) is completely determined

in Gomez-Acevedo et al. (2010). Their results can be summarized in Table 1.

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J. Lang, M. Y. Li

Table 1 Global dynamics of model (1.1) for f (z) =

z

z+a(Gomez-Acevedo et al. 2010)

Threshold value Equilibria

P0

P1

P∗

R1< R0< 1

R1< 1 < R0

1 < R1< R0

aGlobally Asymptotically Stable. By this we mean asymptotically stable within the interior of ?, see (2.1)

bDoes not exist

GASa

Unstable

Unstable

DNEb

GAS

Unstable

DNE

DNE

GAS

Wodarzetal.(1999)consideredthefollowingmodelthatincludesmitoticdivisions

in both healthy and proviral CD4+T cells:

˙ x = (λ + rx)

?

1 −x + y

?

k

?

?

− βxy − μ1x

˙ y = βxy + sy

1 −z + y

k

− γyz − μ2y

(1.3)

˙ z = ν

yz

z + 1− μ3z.

Model (1.3) not only has equilibria as outcomes; it can have stable periodic oscil-

lations in certain parameter regions, which are not present in the models of Gomez-

Acevedo et al. (2010). The existence of stable periodic oscillations is used in Wodarz

et al. (1999) to explain patient data that shows treatment-induced transient oscilla-

tions. In a simpler version of model (1.3), Wodarz and Bangham (2000) showed that

model (1.3) can possess a bistability phenomenon: both P0and P∗exist and are both

stable when R0is below threshold 1. In this case, the outcome of system is critically

dependent on the initial conditions. This is also related to the backward bifurcation

observed in a simple HTLV-I model in Gomez-Acevedo and Li (2005).

HTLV-I specific CTL response typically occurs after a time lag from weeks to

months after seroconversion. To better describe this lagged response during the early

stage of the infection when z is small, as well as the saturation effect when z is large,

we propose to approximate the response function using a sigmoidal function of form

f (z) =

zn

zn+ a,

a > 0, n ≥ 2.

(1.4)

Functions of this form are commonly used for modeling of enzyme kinetics as well as

in the ecological modeling literature. Such a response function when n = 2 was used

in a model for general autoimmune disorders in Iwami et al. (2007). The objective of

our study is to investigate whether sustained oscillations and bistability in model (1.1)

can be the result of immune response alone, without mitosis. Our analysis of model

(1.1)withgeneralresponsefunctionsin(1.4)revealsawidearrayofpossibleoutcomes

of the dynamics, many of which have not been observed before or are distinct from

those observed in earlier models. Of particular interest are the following new results:

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Stable and transient periodic oscillations in a mathematical model

(1) The carrier equilibrium P1is always asymptotically stable when R0> 1. This

finding is different from those in Wodarz and Bangham (2000) and Gomez-

Acevedo et al. (2010), where P1can lose stability when the value of R0is suffi-

ciently large. Our result can better explain the fact that most of HTLV-I infected

people remain as life-long asymptomatic carriers.

(2) Bistability. We show that, for a large region of parameter values, it is possible for

the carrier equilibrium P1and a HAM/TSP equilibrium P∗to coexist and both

be stable, when R0is above threshold 1. In this case, a solution remains close to

the stable carrier equilibrium until perturbations force it to cross into the basin

of attraction of the HAM/TSP equilibrium. This provides an explanation why

an infected person can remain asymptomatic for a long time before developing

HAM/TSP.

(3) Existence of stable periodic oscillations. We show that stable periodic solutions

exist through supercritical Hopf bifurcations near the HAM/TSP equilibrium.

Stable periodic solutions have also been shown to exist in Wodarz and Bangham

(2000) in a model with both CTL response and mitosis of proviral target cells.

Our result demonstrates that stable periodic solutions can be the result of CTL

response alone without mitosis. This is significant for dynamics of immune

responses to infection of viruses such as HIV-I or HBV, for which mitosis may

not be as important for viral replication. Because the carrier equilibrium P1is

always stable when HTLV-I infection is chronic, it is possible for the coexistence

of a stable equilibrium P1and a stable periodic solution. This suggests that when

HAM/TSP develops, the proviral load can either approach an equilibrium level

or appear as sustained oscillations.

(4) Existence of transient periodic oscillations. We show that subcritical Hopf bifur-

cations can also occur in model (1.1) for a large range of parameter values. The

resulting periodic solutions are of saddle type. They are unstable and thus tran-

sient. Nonetheless, they are robust with respect to small perturbations because

of the saddle property and thus observable. These robust but transient periodic

solutions are not commonly observed in mathematical models in biological con-

text and have not received much research attention. Biological implications of

their existence to HTLV-I infection and development of HAM/TSP need to be

further investigated. HAM/TSP patients undergoing treatments are known to

exhibit episodes of transient oscillations in their proviral loads and CTL fre-

quency (Wodarz et al. 1999). Large-amplitude transient oscillations have con-

sistently been observed in equine infectious anemia virus (EIAV) infection data

(Leroux et al. 2004). Our results suggest that the lagged CTL response may be

responsible for the occurrence of transient oscillations.

(5) Hysteresis. Bistability in cases (2) and (3) leads to hysteresis behaviours; as a

parameterisincreasedordecreased,thesystemmayfollowtwodifferentbranches

of stable equilibria. In our case, it is possible that the system follows a branch of

stable equilibria as a parameter decreases and follows a branch of stable periodic

solutions as the parameter increases, resulting in completely different dynamical

behaviours (see Fig. 3b). Hysteresis behaviours have significant implications for

potential treatment and intervention measures.

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In Sects. 2–4, we present mathematical and numerical evidence for our findings.

Further discussions are given in Sect. 5. Mathematical proofs are provided in the

Appendix.

2 Equilibria and bistability

Model (1.1) will be investigated in the following bounded feasible region

? =

?

(x, y,z) ∈ R3

+: x ≤

λ

μ1,x + y ≤λ

¯ m,z ≤

λν

¯ mμ3

?

,

(2.1)

where ¯ m = min{μ1,μ2}.Itcanbeshownthatallsolutionsof(1.1)eventuallyenter?,

and that ? is positively invariant with respect to system (1.1).

From equilibrium equations

λ − βxy − μ1x = 0,

σβxy − γyz − μ2y = 0,

νyf (z) − μ3z = 0,

(2.2)

we know that the infection-free equilibrium P0 = (λ/μ1,0,0) always exists. The

carrier equilibrium P1= (¯ x, ¯ y,0) satisfies equations

λ − β ¯ x ¯ y − μ1¯ x = 0, σβ ¯ x = μ2,

and it exists if and only if R0=

P∗= (x∗, y∗,z∗),x∗, y∗,z∗> 0, we can show that z∗is a positive solution of the

equation

λσβ

μ1μ2> 1. For a positive (HAM/TSP) equilibrium

f (z) = h(z),

(2.3)

where f (z) is the response function and

h(z) =βμ3

ν

μ2z + γz2

λσβ − μ1μ2− μ1γz.

Graphically, a solution z∗of Eq. (2.3) corresponds to an intersection in the first quad-

rant of graphs of f (z) and h(z). For 0 ≤ z ≤μ2

while f (z)canchangeitsconcavitywhenn ≥ 2,theirgraphscanhaveno,exactlyone,

or two intersections, as we demonstrate in Fig. 2. As a consequence, when R0> 1,

there can be no, exactly one, or two HAM/TSP equilibria. We summarize these results

in the next theorem. Detailed proof will be given in the Appendix.

Let

γ(R0−1), h(z) ≥ 0 and is concave up,

g(z) = βμ3γzn+1+ (νμ1γ + βμ2μ3)zn− ν(λσβ − μ1μ2)zn−1

+aβμ3γz + aβμ2μ3,

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Fig. 2 Intersections of graphs of f (z) and h(z)

and

m = min

?

g(z) : 0 ≤ z ≤μ2

γ(R0− 1)

?

.

Theorem 2.1 (a) Theinfection-freeequilibrium P0= (˜ x,0,0) =

exists.

(b) The carrier equilibrium

?

λ

μ1,0,0

?

always

P1= (¯ x, ¯ y,0) =

?μ2

σβ,λσβ − μ1μ2

βμ2

,0

?

exists if and only if R0> 1.

(c) If R0> 1 and if m > 0, then there is no HAM/TSP equilibrium.

(d) If R0> 1 and if m = 0, then there is exactly one HAM/TSP equilibrium

P∗= (x∗, y∗,z∗),

x∗, y∗,z∗> 0.

(e) If R0> 1 and if m < 0, then there are two HAM/TSP equilibria,

P∗= (x∗, y∗,z∗)

and P∗∗= (x∗∗, y∗∗,z∗∗),

where z∗∗> z∗> 0.

Stabilityresultsofequilibriaaresummarizedinthefollowingtheorem,whoseproof

is given in the Appendix.

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J. Lang, M. Y. Li

Theorem 2.2 (a) If R0≤ 1,thentheinfection-freeequilibrium P0isgloballyasymp-

totically stable in ?, and the virus is cleared. If R0> 1, P0is unstable and the

HTLV-I infection becomes chronic.

(b) If R0> 1, then the carrier equilibrium P1comes to exist and is always locally

asymptotically stable.

From Theorem 2.2, we know that the basic reproduction number R0for asymp-

tomatic carriers and HAM/TSP patients should be above the threshold 1. The fact that

carrier equilibrium P1remains asymptotically stable implies that an infected person

tends to remain as a carrier for a long time. This offers an explanation why HTLV-I

infection has a long latent period and why most of HTLV-I infected people remain as

life-long asymptomatic carriers.

The phenomenon that carrier equilibrium P1remains asymptotically stable for

R0> 1 opens up opportunities for bistability in the system when HTLV-I infection

is chronic. We show through numerical simulations that, for large range of parameter

values, a stable HAM/TSP equilibrium P∗or a stable periodic solution can coexist

with P1(see Figs. 3, 6). An important implication of bistability is that outcomes of the

systemcriticallydependsontheinitialconditions.Itishighlylikelyinsuchasituation

that a solution remains in the basin of attraction of the carrier equilibrium and then

is forced by system perturbations into the basin of attraction of the stable HAM/TSP

equilibrium. This provides a plausible mechanism for the development of HAM/TSP

after a long incubation period.

3 Stable oscillations

AsprovedinTheorem2.1andshowninFig.2,whenparametersvary,system(1.1)may

change from having no HAM/TSP equilibrium to have two branches of HAM/TSP

equilibria P∗and P∗∗. We show through numerical simulations that stability change

may occur to the branch of P∗∗, leading to a Hopf bifurcation. We also note that, as

parametersincrease,thebranchofHAM/TSPequilibrium P∗∗canchangefromstable

to unstable or vice versa, and the Hopf bifurcation can be either supercritical or sub-

critical. Correspondingly, the resulting periodic solutions from the Hopf bifurcation

can be either stable or unstable. In this section, we show numerical evidence for stable

periodic solutions. In the next section, we discuss unstable periodic solutions.

Bifurcation diagrams and numerical solutions to (1.1) are generated in MATLAB.

Stable and unstable periodic solutions are detected using XPPAUTO. Scales used in

figures are arbitrary for ease of display. We note that although the simulations shown

below are for the case n = 2, similar results can be observed for integer values of

n ≥ 2. We use a vector p to include all the parameters in (1.1).

p = (λ,β,σ,γ,ν,a,μ1,μ2,μ3).

Consider parameter values given by

p = (2,1,1,1.99,3.85,0.45,1,1,0.5).

(3.1)

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Stable and transient periodic oscillations in a mathematical model

Fig. 3 Supercritical Hopf bifurcation with p as in (3.1). Solid lines indicate stable equilibria, dash lines

unstable equilibria, and dotted lines stable periodic solutions

We have observed a super-critical Hopf bifurcation when we change each of parame-

ters.Weincluderepresentativebifurcationdiagramswithrespecttoparametersλandν

in Fig. 3. Solid lines indicate stable equilibria, dash lines indicate unstable equilibria,

and dotted lines are for stable periodic solutions.

From bifurcation diagrams in Fig. 3, we see that supercritical Hopf bifurcations

occur when we vary parameter λ or ν. In both cases, a stable periodic solution persists

for a large range of the bifurcation parameter. However, there is a distinct difference

in the nature of the λ and ν Hopf bifurcations: the Hopf branch with respect to λ is

bounded and eventually vanishes as λ increases; in contrast, the Hopf branch with

respect to ν persists as ν increases. This difference can be further demonstrated in

two-parameter bifurcation analysis. In Fig. 4, stability region of the HAM/TSP equi-

librium P∗∗is shown in the λμ-parameter plane. The dark region contains values of

(λ,ν) for which P∗∗is stable, while the light region corresponds to parameter values

for which P∗∗is unstable. If we let λ vary along a horizontal line across the light

region, P∗∗loses its stability when the line first crosses from the dark region into the

light region, creating the Hopf bifurcation. When the line exists the light region into

the dark region on the right, P∗∗regains its stability, a backward Hopf bifurcation

occurs, and the Hopf branch terminates. In a similar fashion, if we let ν vary along a

vertical line through the light region, Hopf bifurcation only occurs once and the Hopf

branch persists.

From the bifurcation diagrams in Fig. 3, we also see that bistability occurs in two

different forms: a stable carrier equilibrium P1and a stable HAM/TSP equilibrium

P∗∗coexist, or a stable P1and a stable periodic solution near P∗∗coexist. In these

situations, different initial conditions may lead to different outcomes, for the same

set of parameters. We demonstrate these phenomena in Fig. 5. We fix the parameter

vector p as in (3.1). We select two set of initial conditions which are very close to each

other:

(x(0), y(0),z(0)) = (2,0.5,0.5) and

(x(0), y(0),z(0)) = (2,0.25,0.5) ≈ P∗∗

(3.2a)

(3.2b)

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J. Lang, M. Y. Li

Fig. 4 Region of stability of P∗∗in the λν-plane

Fig. 5 Two orbits showing bistability with p as in (3.1)

As shown in Fig. 5, one solution converges to the carrier equilibrium P1while the

other converges to the stable periodic orbit.

4 Transient oscillations that are robust

Consider the parameter vector

p = (0.25,1,0.5,4,1.25,0.01,0.01,0.1,1).

(4.1)

A different kind of Hopf bifurcation occurs when the parameters are varied from this

value of p. We show the bifurcation diagrams for parameters λ and ν in Fig. 6, and

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Stable and transient periodic oscillations in a mathematical model

Fig. 6 Subcritical Hopf bifurcation with p as in (4.1). Solid lines indicate stable equilibria, dashed lines

unstable equilibria, and circles indicate unstable periodic solutions

Fig. 7 Stability region of P∗∗in the λν plane, with p as in (4.1)

the corresponding two-parameter bifurcation analysis in Fig. 7. Similar bifurcation

diagrams are observed when other parameters are varied.

In Fig. 6, as the parameter λ or ν is increased, stability of the lower branch of the

HAM/TSP equilibrium P∗∗changes from unstable (dashed line) to stable (solid line),

andabranchofunstableperiodicsolutions(circles)arecreated.Thisiscalledasubcrit-

ical Hopf bifurcation, a phenomenon often ignored in biological models, largely due

to a misconception that an unstable periodic solution is not detectable or observable.

We show in Figs. 8 and 9 that these unstable periodic solutions are robust and

observable, and can be easily detected in numerical simulations. In Fig. 8, solutions

track a periodic orbit for sometime before converging to one of the stable equilibria.

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J. Lang, M. Y. Li

Fig. 8 Two transient periodic oscillations converging to different equilibria with p as in (4.1)

Fig. 9 Long-lasting transient periodic solutions with p as in (4.1)

These oscillations are different from damped oscillations, since, for the same set of

parameter values, by choosing different initial conditions, we can have solutions that

track the unstable periodic solution for as long as we wish (see Fig. 9). These oscil-

lations are also transient since they are all tracking an unstable periodic solution.

The mathematical explanation for these robust transient oscillations is the following:

the unstable periodic solution from the Hopf bifurcation is of a saddle type; it has a

Floquet multiplier greater than 1 and another less than 1. As a result, the periodic

orbit has a two-dimensional center-unstable manifold and a two-dimensional center-

stable manifold. Solutions starting near the center-stable manifold will be quickly

attracted to a neighborhood of the periodic solution, stay a long time in the neighbor-

hood while behaving like a periodic solution, then leave the neighborhood along the

center-unstable manifold. To observe such a saddle-type periodic orbit, one need only

choose initial conditions close to the center-stable manifold which may contain points

that are far from the periodic orbit itself.

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Stable and transient periodic oscillations in a mathematical model

Given the existence of bistable equilibria P1and P∗∗, it is not hard to see that the

unstable periodic solution is located on the boundary of basins of attractions of P1

and P∗∗. Naturally, one expects that the transient oscillations could converge either to

P1or to P∗∗. Indeed, in Fig. 8, we use the parameter vector in (4.1) and choose two

initial conditions that are close to each other:

(x(0), y(0),z(0)) = (0.83040,0.51130,0.14677),

(x(0), y(0),z(0)) = (0.83042,0.51124,0.14675).

and(4.2a)

(4.2b)

The two resulting transient periodic oscillations converge to two different equilibria.

In Fig. 9 we use the parameter vector in (4.1) and choose a different set of initial

conditions:

(x(0), y(0),z(0)) = (0.8240345,0.2502873,0.1107864),

(x(0), y(0),z(0)) = (0.8240676,0.2420150,0.1108380).

and(4.3a)

(4.3b)

We observe that the two transient periodic oscillations converge to different equilibria

as in Fig. 8. Furthermore, both oscillations in Fig. 9 appear periodic for much longer

a time than those in Fig. 8. In fact, by selecting initial conditions, we can produce

transient periodic oscillations that last for any desired length of time.

5 Discussions

In this paper, using a sigmoidal response function to model the CTL response to the

human T cell leukaemia/lymphoma virus type I (HTLV-I) infection in vivo, we have

shownthatthatthedynamicinteractionsbetweentheimmuneresponseandviralinfec-

tions can be very complex and multi-faceted. The outcomes of our model (1.1) can

be summarized into four main types: bistability, stable periodic solutions, unstable

periodic solutions, and hysteresis. Our model analysis have revealed new dynamics

that have not been previously observed for this type of models. In comparison to the

study in Wodarz et al. (1999), we have intentionally neglected the effect of mitosis of

the CD4+target cells, and we are able to show that Hopf bifurcations and complex

behaviourscanbetheresultofinterplaybetweenCTLresponseandtheviralinfection,

whether or not mitosis is playing a role in the process. We comment that our findings

are based on the assumption that viral transmission is through cell-to-cell contact and

virological synapse. Recent research has shown that this route of transmission also

occurs in the infection of other retroviruses including HIV-I (Feldmann and Schwartz

2010; Hübner et al. 2009). In this light, our findings may have important implications

for modeling immune response to infections from viruses such as HIV-I, HCV and

HBV,wheremitosismaynotplayasanimportantroleasbelievedinHTLV-Iinfection.

5.1 Bistability and HAM/TSP development

Previously documented bistability in these type of models (Wodarz and Bangham

2000) results from the coexistence of a stable infection-free equilibrium and a stable

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J. Lang, M. Y. Li

HAM/TSPequilibriumwhenthebasicreproductionnumber R0isbelowthethreshold

of 1. We have shown in our model that, when HTLV-I infection is chronic (R0> 1),

the carrier equilibrium remains stable and can coexist with a stable HAM/TSP steady-

state either in the form of equilibrium or in the form of periodic oscillations. This is a

different and more general phenomenon than the bistability in Wodarz et al. (1999).

Our results provide new insights for the long incubation period and development of

HAM/TSP. Since the HTLV-I infection is life-long, we may assume that infected indi-

viduals have R0> 1. In such a case, if the initial infection is in the basin of attraction

of the stable carrier equilibrium P1, then the individual will remain a carrier for a

long period of time. Because of the coexistence of a stable HAM/TSP steady state,

perturbations to the system that might result from subsequent exposure to HTLV-I or

physiological changes can cause the trajectory of the system to move into the basin of

attraction of the HAM/TSP steady state, resulting in the development of HAM/TSP.

How close the trajectory is to the boundary of the two basins of attraction determines

how easy it is for the HAM/TSP to develop. Compared to previous modelling studies

of HTLV-I and HAM/TSP, our result reveals that it is possible to change the course

of HAM/TSP development through perturbations of a trajectory rather than to the

physiological or immunological parameters. Our result implies that the initial dos-

age of HTLV-I, subsequent exposure to HTLV-I, and routes of HTLV-I infection can

all be important risk factors of HAM/TSP development. This is supported by clini-

cal HAM/TSP studies using animal models. Kato et al. (1998) has demonstrated that

adultratsorallyinoculatedwithHTLV-1-producingcellsdevelopapersistentHTLV-1

infection with no humoral and cellular immune responses, while adult rats intrave-

nously or intraperitoneally inoculated with the same cell line developed significant

CTLresponsespecifictoHTLV-1.Osameetal.(1990)hasshownthatHTLV-Iinfection

through blood transfusion is positively correlated with higher numbers of HAM/TSP

patients. Seto et al. (1995) has shown that chronic progressive myeloneuropathy can

be developed in rats intraperitoneally inoculated with HTLV-I producing cells, after a

longincubationperiod,andthatahighdoseofinoculationcansignificantlyaccelerate

the disease onset.

5.2 Transient and robust periodic oscillations

Another significant new phenomenon our study has revealed is the existence of tran-

sient periodic oscillations that are robust and observable. These unstable periodic

solutions are created through a subcritical Hopf bifurcation when parameter values

are changed (Figs. 6, 7). A surprising property of these periodic solutions is that

they are of saddle type, namely, they have both stable and unstable manifolds. In the

terminology of nonlinear dynamics, the corresponding fixed point of the associated

Poincaré map is a saddle with one-dimensional stable manifold and one-dimensional

unstable manifold. This property has created many interesting features for this type of

oscillations: (1) they are transient since the periodic solution is unstable, and they will

convergetooneofthestableequilibriaastimegoeson(Fig.8);(2)theyarerobustwith

respect to small perturbations of parameters and initial conditions due to the normal

hyperbolicity of the periodic solutions, for the same reason that a saddle-type fixed

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Stable and transient periodic oscillations in a mathematical model

point is robust with respect to small perturbations; (3) they are easily observable and

detectable in model simulations. This is because nearby solutions tend to stay close to

andtracktheperiodicsolutionforsometime,andhenceappearasperiodicoscillations

(Figs. 8, 9). We also note that these “nearby” solutions can be produced by choos-

ing initial conditions close to the center-stable manifold of the periodic solution, not

necessarily close to the periodic solution itself; and (4) for the same set of parameter

values, these transient oscillations may last for a variable length of time depending

on how close the initial condition is from the center-stable manifold (Figs. 8, 9). We

comment that subcritical Hopf bifurcations have received little attention in biological

modelling literature, likely due to a misconception that unstable periodic solutions

are not detectable nor observable. Another reason for the neglect of subcritical Hopf

bifurcations is an often misguided belief that they are supercritical Hopf bifurcations

with a reverse change of parameters. Our discovery that saddle-type periodic solu-

tions can be created through a Hopf bifurcation in a biological model demonstrates

that these common beliefs are faulty, and biological significance of the transient and

robust periodic solutions ought to be further explored.

SomeHTLV-Idatahasshowntransientperiodicoscillationsaftertreatment(Wodarz

et al. 1999). Clinical data from other viral infections such as EIAV, a retrovirus related

to HIV, also show episodes of transient and sustained oscillations in both viral load

and CTL abundance (Leroux et al. 2004). In the study of Wodarz et al. (1999), the

route to periodic oscillations is described as a switching of drug therapies which per-

turbs model parameters back and forth, creating and then destroying a stable periodic

solution within a finite time. Our result offers a different explanation: therapies can

produce small perturbations to the orbit, and switch it to the neighborhood of an

unstable periodic solution and produce transient periodic oscillations.

5.3 Treatment implications of hysteresis behaviours

Currently, there is no established treatment program for HAM/TSP. Several agents

includingcorticosteroids,plasmapheresis,danazol,pentoxifylline,andinterferonhave

been reported to show short-term clinical improvements, but none have been con-

clusively shown to alter the long-term disability of HAM/TSP (Oh and Jacobson

2008). Clinical studies suggest that interferon-α provides benefits over short periods.

The long-term benefits of interferon, however, has not been established. Complicated

dynamics in our model, especially when hysteric behaviours are present, may provide

an explanation for why long-term benefits of interferon treatment are hard to achieve.

HAM/TSP in our model is represented by solutions lying in the basin of attraction of

thestableHAM/TSPequilibrium P∗∗,whileACsarerepresentedbysolutionsstaying

in the basin of attraction of the stable AC equilibrium P1. Interferon-α can interfere

with viral activities and viral entry of cells and hence reduce the transmission coeffi-

cient β. For interferon-α treatments to be effective, it has to shift a solution from the

basin of attraction of the HAP/TSP equilibrium P∗∗into that of the AC equilibrium

P1. Once this is achieved, the treatment must then be terminated in a continuous fash-

ion to ensure that the solution will remain in the basin of attraction of P1, as shown in

Fig.10a.Discontinuingthetreatmentinagradualfashioniscriticallyimportant,since

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J. Lang, M. Y. Li

Fig. 10 Hysteresis behaviours that may impact treatments of HAM/TSP

a rapid change in parameters may result in the solution passing back into the basin

of attraction of the HAM/TSP equilibrium, with short-term gains of the treatment

completely lost (see Fig. 10b).

Acknowledgments

Research Council (NSERC) of Canada and Canada Foundation of Innovation (CFI). MYL acknowledges

the support of NCE-MITACS project “Transmission Dynamics and Spatial Spread of Infectious Diseases:

Modelling, Prediction and Control.” JL acknowledges the support of an NSERC Postgraduate Scholarship.

TheresearchissupportedinpartbygrantsfromtheNaturalScienceandEngineering

Appendix: Proofs

Feasible region ? and its positive invariance

It can be verified from (1.1) that R3

are non-negative quantities, we restrict any further analysis of this model to R3

the first equation of (1.1) we get ˙ x ≤ λ−μ1x and this implies x(t) ≤

if x(0) ≤

+is positively invariant. Since populations of cells

+. From

λ

μ1for all t ≥ 0

λ

μ1and

limsup

t→∞

x(t) ≤

λ

μ1.

A similar treatment of x(t) + y(t) allows us to show

x(t) + y(t) ≤λ

¯ m

for t ≥ 0 if x(0) + y(0) ≤λ

¯ m,

and

limsup

t→∞

x(t) + y(t) ≤λ

¯ m,

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Stable and transient periodic oscillations in a mathematical model

where ¯ m = min{μ1,μ2}. The bounds on x and x + y and a similar argument for the

z equation allow us to show z(t) ≤ λν/ ¯ mμ3for t ≥ 0 if z(0) ≤ λν/ ¯ mμ3, and

limsup

t→∞

z(t) ≤

λν

¯ mμ3.

These relations establish the boundedness and positive invariance of the feasible re-

gion ?.

Theorem 2.1: Number of HAM/TSP equilibria

From equilibrium equation (2.2) we know that a HAM/TSP equilibrium with non-

zero z must satisfy

x =μ2+ γz

y =λσβ − μ1μ2− μ1γz

βμ2+ βγz

σβ

and

.

Substituting these equations into the last equation of (2.2) we arrive at

f (z) =μ3

ν

z

y=βμ3

ν

μ2z + γz2

λσβ − μ1μ2− μ1γz:= h(z).

Thisallowsustodemonstratethenumberofsolutions z∗asintersectionsofthegraphs

of f (z)andh(z)inFig.2.Algebraically,thisequationisequivalenttog(z) = 0,where

g(z)=βμ3γzn+1+(νμ1γ +βμ2μ3)zn− ν(λσβ − μ1μ2)zn−1+aβμ3γz+aβμ2μ3.

and the methods of single variable calculus can now be applied to complete the proof.

Theorem 2.2: Stability of P0and P1

Part (1). Let R0≤ 1. Consider the Lyapunov function V0(x, y,z) = y for the equi-

librium P0and differentiate along the differential equations (1.1). This yields

˙ V0(x, y,z) = σβxy − μ2y − γyz ≤ (σβx − μ2)y

≤

μ1

?λσβ

− μ2

?

y = (R0− 1)μ2y ≤ 0 if R0≤ 1.

Here we have used the fact that (x, y,z) ∈ ? ?⇒ x ≤

pact invariant set in {(x, y,z) ∈ ? : ˙ V0(x, y,z) = 0} is the singleton {P0}, LaSalle’s

Invariance Principle gives that P0is globally asymptotically stable.

λ

μ1. Since the largest com-

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J. Lang, M. Y. Li

Part (2). Let R0> 1. Consider the Jacobian matrix of (1.1), J(x, y,z).

⎡

⎢

0

J(x, y,z) =

⎢

⎢

⎣

−μ1− βy

σβy

−βx

0

σβx − μ2− γz

νzn

zn+a

−γy

anνyzn−1

(zn+a)2− μ3

⎤

⎥

⎥

⎥

⎦.

Therefore, at P0,

J(P0) =

⎡

⎢

⎢

⎣

−μ1

0

−β ˜ x

0

σβ ˜ x − μ2

0

0

0

−μ3

⎤

⎥

⎥

⎦.

The eigenvalues of J(P0) are −μ1,σβ ˜ x −μ2, and −μ3. Thus, P0is unstable because

σβ ˜ x − μ2= μ2(R0− 1) > 0.

At P1,

J(P1) =

⎡

⎢

⎢

⎣

−μ1− β ¯ y −β ¯ x

σβ ¯ y

0

0

0

−γ ¯ y

−μ3

0

⎤

⎥

⎥

⎦.

The trace, determinant, and sum of 2×2 principal minors of J(P1) can be calculated

as, respectively,

tr(J(P1)) = −(μ1+ μ3+ β ¯ y) < 0,

det(J(P1)) = −σβ2μ3¯ x ¯ y < 0,

M(J(P1)) = σβ2¯ x ¯ y + μ1μ3+ βμ3¯ y.

and

Observe that

[M · tr − det](J(P1)) = −(μ1+ β ¯ y)(σβ2¯ x ¯ y + μ1μ3+ βμ3¯ y) < 0.

Routh–Hurwitz conditions imply that P1is locally asymptotically stable.

? ?

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