Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection

London School of Economics-LSE, London, UK.
Journal of Mathematical Biology (Impact Factor: 1.85). 07/2011; 65(1):181-99. DOI: 10.1007/s00285-011-0455-z
Source: PubMed


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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Available from: Michael Li, Jan 16, 2016
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    • "At any given proviral load, the Tax expression is significantly higher in the HAM/TSP individuals than that in the ACs[19], thus a high rate of Tax expression is linked to a large increase in the prevalence and the risk of HAM/TSP[20]. Exploring the dynamics of HTLV-I infection has attracted much attention, with many mathematical models being proposed[7,13,19,20,21,22,23,24,25,26,27,28]. Wodarz et al.[21]described a basic three-dimensional mathematical model, took CTL response, infectious and mitotic transmission into account. "

    Preview · Article · Jan 2016 · Discrete and Continuous Dynamical Systems - Series B
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    • "The evidence implies that cytotoxicity of the CTL is ultimately responsible for the demyelination of the central nervous system resulting in HAM/TSP [5]. Mathematical models have been formulated to describe the in-vivo infection process with the humoral immune response to HTLV-I infections [2] [6] [9] [11] [12] [16] [26] [25] [14] as well as to the human immunodeficiency virus (HIV) [13] [15] [19] [21] [24] [27]. In [6], Gómez-Acevedo et al consider the following mathematical model for the HTLV-I infection of CD4 + T cells that incorporates the CD8 + cytotoxic T-cell response: "
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    ABSTRACT: In this paper, we include two time delays in a mathematical model for the CD8+ cytotoxic T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection, where one is the intracellular infection delay and the other is the immune delay to account for a series of immunological events leading to the CTL response. We show that the global dynamics of the model system are determined by two threshold values R0, the corresponding reproductive number of a viral infection, and R1, the corresponding reproductive number of a CTL response, respectively. If R_0 < 1, the infectionfree equilibrium is globally asymptotically stable, and the HTLV-I viruses are cleared. If R_1 < 1 < R_0, the immune-free equilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with no persistent CTL response. If 1 < R1, a unique HAM/TSP equilibrium exists, and the HTLV-I infection becomes chronic with a persistent CTL response. Moreover, we show that the immune delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numerical simulations suggest that if 1 < R_1, an increase of the intracellular delay may stabilize the HAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, the stability of the HAM/TSP equilibrium may generate rich dynamics combining the “stabilizing” effects from the intracellular delay with those “destabilizing” influences from immune delay.
    Full-text · Article · Jun 2015 · Mathematical biosciences and engineering: MBE
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    • "Chaotic like behavior is observed in Wang et al. (2007) for a three-dimensional delayed model, but with no bistability. We point out that sustained periodic oscillations and transient oscillations induced by nonlinearity have also been reported in ordinary differential equation models (Lang and Li 2012). The rest of this paper is organized as follows. "
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    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.
    Full-text · Article · Jan 2013 · Journal of Mathematical Biology
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