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.
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ABSTRACT: Human T-cell Lymphotropic Virus Type I (HTLV-I) primarily infects CD4+ helper T cells. HTLV-I infection is clinically linked to the development of Adult T-cell Leukemia/Lymphoma and of HTLV-I Associated Myelopathy/Tropical Spastic Paraparesis, among other illnesses. HTLV-I transmission can be either horizontal through cell-to-cell contact, or vertical through mitotic division of infected CD4+ T cells. It has been observed that HTLV-I infection has a high proviral load but a low rate of proviral genetic variation. This suggests that vertical transmission through mitotic division of infected cells may play an important role. We consider and analyze a mathematical model for HTLV-I infection of CD4+ T cells that incorporates both horizontal and vertical transmission. Among interesting dynamical behaviors of the model is a backward bifurcation which raises many new challenges to effective infection control.Bulletin of Mathematical Biology 02/2005; 67(1):101-14. · 2.02 Impact Factor
Article: The immune response to HTLV-I.[show abstract] [hide abstract]
ABSTRACT: A strong cytotoxic T lymphocyte response to HTLV-I protects against the associated inflammatory disease of the central nervous system, HAM/TSP (HTLV-I-associated myelopathy/tropical spastic paraparesis), by reducing the proviral load of HTLV-I; however, when the proviral load exceeds a threshold level, HTLV-I-specific cytotoxic T lymphocytes could contribute to inflammation.Current Opinion in Immunology 09/2000; 12(4):397-402. · 8.77 Impact Factor
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ABSTRACT: The treatment of HAM/TSP is a challenge. No agent has shown to significantly modify the long-term disability associated with HAM/TSP. Advances in our understanding of the pathogenesis of HAM/TSP have led to the identification of several biomarkers and therapeutic targets. Clinical trials in HAM/TSP continue to be opportunities for further qualification and refinement of biomarkers and therapeutic targets. The validation of HAM/TSP relevant biomarkers and the identification of new targets remain key challenges in the development of effective targeted therapy in HAM/TSP.Neurologic Clinics 08/2008; 26(3):781-97, ix-x. · 1.34 Impact Factor
J. Math. Biol.
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
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
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
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
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
The cytotoxic T lymphocyte (CTL) response to the infection of CD4+
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
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
J. Lang, M. Y. Li
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
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).
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).
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
2000; Jacobson 2002).
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;
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
Stable and transient periodic oscillations in a mathematical model
Fig. 1 Transfer diagram for the
HTLV-I infection in vivo and the
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+awith a > 0 (Wodarz et al.
˙ x = λ − βxy − μ1x
˙ y = σβxy − γyz − μ2y
˙ z = νyf (z) − μ3z.
System (1.1) has three types of equilibria:
P1= (¯ x, ¯ y,0),
P∗= (x∗, y∗,z∗),
carrier equilibrium, and
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+a, the final outcomes of the system are determined by values of
μ2(μ1ν + βμ3),
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.
J. Lang, M. Y. Li
Table 1 Global dynamics of model (1.1) for f (z) =
z+a(Gomez-Acevedo et al. 2010)
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
in both healthy and proviral CD4+T cells:
˙ x = (λ + rx)
1 −x + y
− βxy − μ1x
˙ y = βxy + sy
1 −z + y
− γyz − μ2y
˙ z = ν
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) =
a > 0, n ≥ 2.
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
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:
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
(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
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.