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

Xuejuan Lu; Lulu Hui; Shengqiang Liu; Jia Li

doi:10.3934/mbe.2015.12.431

Article Contents

2015, Volume 12, Issue 3: 431-449. Doi: 10.3934/mbe.2015.12.431

This issue Previous Article An SIRS model with differential susceptibility and infectivity on uncorrelated networks Next Article A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain

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

1.
Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080
2.
Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080
3.
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899

Received: November 30, 2013

Revised: November 30, 2014

Published: December 2014

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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 $R_0$, the corresponding reproductive number of a viral infection, and $R_1$, the corresponding reproductive number of a CTL response, respectively. If $R_0<1$, the infection-free 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 $1 < R_1$, 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.

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Mathematics Subject Classification: Primary: 34K20, 34K60; Secondary: 34K18.

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