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2016, 13(1): 135-157. doi: 10.3934/mbe.2016.13.135

## A delayed HIV-1 model with virus waning term

 1 Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China 2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada 3 Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080

Received  April 2015 Revised  July 2015 Published  October 2015

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.
Citation: Bing Li, Yuming Chen, Xuejuan Lu, Shengqiang Liu. A delayed HIV-1 model with virus waning term. Mathematical Biosciences & Engineering, 2016, 13 (1) : 135-157. doi: 10.3934/mbe.2016.13.135
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