### Mathematical Biosciences and Engineering

2015, Issue 3: 431-449. doi: 10.3934/mbe.2015.12.431

# A mathematical model of HTLV-I infection with two time delays

• 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 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 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 infection-free equilibrium is globallyasymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-freeequilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with nopersistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-Iinfection 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 HTLV-I infection with two time delays[J]. Mathematical Biosciences and Engineering, 2015, 12(3): 431-449. doi: 10.3934/mbe.2015.12.431

### Related Papers:

• In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxicT 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 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 infection-free equilibrium is globallyasymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-freeequilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with nopersistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-Iinfection 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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