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Mathematical Biosciences and Engineering (MBE)
 

A delayed HIV-1 model with virus waning term

Pages: 135 - 157, Volume 13, Issue 1, February 2016      doi:10.3934/mbe.2016.13.135

 
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Bing Li - Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China (email)
Yuming Chen - Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada (email)
Xuejuan Lu - Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China (email)
Shengqiang Liu - Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080, China (email)

Abstract: In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD$8^+$ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number $R_0$. If $R_0<1$, then the infection-free steady state is globally asymptotically stable and the infection is cleared from the T-cell population; whereas if $R_0>1$, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when $R_0>1$ is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage, the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.

Keywords:  HIV-1 infection, immune response, delay, virus waning, CTLs, stability, permanence.
Mathematics Subject Classification:  Primary: 34K20, 92D30; Secondary: 34K18.

Received: April 2015;      Accepted: July 2015;      Available Online: October 2015.

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