Article

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: 2.39). 07/2011; 65(1):181-99. DOI: 10.1007/s00285-011-0455-z
Source: PubMed

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.

Full-text

Available from: John Lang, Apr 23, 2014
0 Followers
 · 
105 Views
  • [Show abstract] [Hide abstract]
    ABSTRACT: With the consideration of mechanism of prevention and control for the spread of viral diseases, in this paper, we propose two novel virus dynamics models where state feedback control strategies are introduced. The first model incorporates the density of infected cells (or free virus) as control threshold value; we analytically show the existence and orbit stability of positive periodic solution. Theoretical results imply that the density of infected cells (or free virus) can be controlled within an adequate level. The other model determines the control strategies by monitoring the density of uninfected cells when it reaches a risk threshold value. We analytically prove the existence and orbit stability of semi-trivial periodic solution, which show that the viral disease dies out. Numerical simulations are carried out to illustrate the main results.
    Nonlinear Dynamics 09/2014; 77(4):1223-1236. DOI:10.1007/s11071-014-1372-7 · 2.42 Impact Factor
  • [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; 13(3). DOI:10.1016/j.nonrwa.2011.02.026 · 2.34 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    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.
    Mathematical biosciences and engineering: MBE 06/2015; 12(3):431–449. DOI:10.3934/mbe.2015.12.431 · 0.87 Impact Factor