A mathematical model of HTLVI infection with two time delays

1.
Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 YiKuang street, Harbin, 150080

2.
Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 YiKuang Street, Harbin, 150080

3.
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899

Received:
01 December 2013
Accepted:
29 June 2018
Published:
01 January 2015


MSC :
Primary: 34K20, 34K60; Secondary: 34K18.


In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxicT lymphocytes (CTLs) response to the Human Tcell leukaemia virus type I (HTLVI) infection,where one is the intracellular infection delay and the other is the immune delay to account for aseries of immunological events leading to the CTL response. We show that the global dynamicsof the model system are determined by two threshold values $R_0$, the correspondingreproductive number of a viral infection, and $R_1$, the corresponding reproductive numberof a CTL response, respectively. If $R_0<1$, the infectionfree equilibrium is globallyasymptotically stable, and the HTLVI viruses are cleared. If $R_1 < 1 < R_0$, the immunefreeequilibrium is globally asymptotically stable, and the HTLVI infection is chronic but with nopersistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLVIinfection becomes chronic with a persistent CTL response. Moreover, we show that the immunedelay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numericalsimulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize theHAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, thestability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing"effects from the intracellular delay with those ``destabilizing" influences from immune delay.
Citation: Xuejuan Lu, Lulu Hui, Shengqiang Liu, Jia Li. A mathematical model of HTLVI infection with two time delays[J]. Mathematical Biosciences and Engineering, 2015, 12(3): 431449. doi: 10.3934/mbe.2015.12.431

Abstract
In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxicT lymphocytes (CTLs) response to the Human Tcell leukaemia virus type I (HTLVI) infection,where one is the intracellular infection delay and the other is the immune delay to account for aseries of immunological events leading to the CTL response. We show that the global dynamicsof the model system are determined by two threshold values $R_0$, the correspondingreproductive number of a viral infection, and $R_1$, the corresponding reproductive numberof a CTL response, respectively. If $R_0<1$, the infectionfree equilibrium is globallyasymptotically stable, and the HTLVI viruses are cleared. If $R_1 < 1 < R_0$, the immunefreeequilibrium is globally asymptotically stable, and the HTLVI infection is chronic but with nopersistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLVIinfection becomes chronic with a persistent CTL response. Moreover, we show that the immunedelay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numericalsimulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize theHAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, thestability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing"effects from the intracellular delay with those ``destabilizing" influences from immune delay.
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