Mathematical Biosciences and Engineering, 2019, 16(1): 292-319. doi: 10.3934/mbe.2019015

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Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses

Jiazhe Lin, Rui Xu, Xiaohong Tian

1 Institute of Applied Mathematics, Army Engineering University, Shijiazhuang 050003, Hebei, P.R. China
2 Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, P.R. China
3 Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan 030006, Shanxi, P.R. China

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Human specific immunity consists of two branches: humoral immunity and cellular immunity. To protect us from pathogens, cell-mediated and humoral immune responses work together to provide the strongest degree of e cacy. In this paper, we propose an HIV-1 model with cell-mediated and humoral immune responses, in which both virus-to-cell infection and cell-to-cell transmission are considered. Five reproduction ratios, namely, immunity-inactivated reproduction ratio, cellmediated immunity-activated reproduction ratio, humoral immunity-activated reproduction ratio, cellmediated immunity-competed reproduction ratio and humoral immunity-competed reproduction ratio, are calculated and verified to be sharp thresholds determining the local and global properties of the virus model. Numerical simulations are carried out to illustrate the corresponding theoretical results and reveal the e ects of some key parameters on viral dynamics.
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