Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection.
ABSTRACT The cytotoxic T lymphocyte (CTL) response to the infection of CD4+ 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, we investigate the consequences of a more general CTL response and show that a 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 during the chronic infection phase. Coexistence of local attractors with their own basin 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 occur when a saddle-type periodic solution exists. As a consequence, transient periodic oscillations can be robust and observable. Implications of our findings to the dynamics of CTL response to HTLV-I infections in vivo and pathogenesis of HAM/TSP are discussed.
- [Show abstract] [Hide abstract]
ABSTRACT: Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4+ T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8+ cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R0 and R1, basic reproduction numbers for viral infection and for CTL response, respectively. If R0≤1, the infection-free equilibrium P0 is globally asymptotically stable, and the HTLV-I viruses are cleared. If R1≤1R0, the asymptomatic-carrier equilibrium P1 is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R1>1, a unique HAM/TSP equilibrium P2 exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.Nonlinear Analysis Real World Applications 06/2012; · 2.20 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.Journal of Mathematical Biology 01/2013; · 2.37 Impact Factor -
Article: Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection.
[Show abstract] [Hide abstract]
ABSTRACT: Stable periodic oscillations have been shown to exist in mathematical models for the CTL response to HTLV-I infection. These periodic oscillations can be the result of mitosis of infected target CD4(+) cells, of a general form of response function, or of time delays in the CTL response. In this study, we show through a simple mathematical model that time delays in the CTL response process to HTLV-I infection can lead to the coexistence of multiple stable periodic solutions, which differ in amplitude and period, with their own basins of attraction. Our results imply that the dynamic interactions between the CTL immune response and HTLV-I infection are very complex, and that multi-stability in CTL response dynamics can exist in the form of coexisting stable oscillations instead of stable equilibria. Biologically, our findings imply that different routes or initial dosages of the viral infection may lead to quantitatively and qualitatively different outcomes.Bulletin of Mathematical Biology 10/2010; 73(8):1774-93. · 2.02 Impact Factor
Page 1
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
123
Page 2
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
123
Page 3
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.
123
Page 4
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 valueEquilibria
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:
123
Page 5
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.
123
Page 6
J. Lang, M. Y. Li
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,
123
Page 7
Stable and transient periodic oscillations in a mathematical model
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.
123
Page 8
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)
123
Page 9
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)
123
Page 10
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
123
Page 11
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.
123
Page 12
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.
123
Page 13
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
123
Page 14
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
123
Page 15
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
123