Viral dynamics of an HIV stochastic model with cell-to-cell infection, CTL immune response and distributed delays

Yan Wang; Tingting Zhao; Jun Liu; Yan Wang; Tingting Zhao; Jun Liu

doi:10.3934/mbe.2019358

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7126-7154. doi: 10.3934/mbe.2019358

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Viral dynamics of an HIV stochastic model with cell-to-cell infection, CTL immune response and distributed delays

1.
College of Science, China University of Petroleum (East China), Qingdao, Shandong 266580, P.R. China
2.
School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, P.R. China

Received: 31 May 2019 Accepted: 31 July 2019 Published: 06 August 2019

Recent studies have demonstrated that both virus-to-cell infection and cell-to-cell transmission play an important role in the process of HIV infection. In this paper, stochastic perturbation is introduced into HIV model with virus-to-cell infection, cell-to-cell transmission, CTL immune response and three distributed delays. The stochastic integro-delay differential equations is transformed into a degenerate stochastic differential equations. Through rigorous analysis of the model, we obtain the solution is unique, positive and global. By constructing appropriate Lyapunov functions, the existence of the stationary Markov process is derived when the critical condition is bigger than one. Furthermore, the extinction of the virus for sufficiently big noise intensity is established. Numerically, we investigate that the small noise intensity of fluctuations could help to sustain the number of virions and CTL immune response within a certain range, while the big noise intensity may be beneficial to the extinction of the virus. We also examine that the influence of random fluctuations on model dynamics may be more significant than that of the delay.
- HIV infection model,
- cell-to-cell infection,
- delay,
- stochastic differential equation,
- stationary Markov process,
- extinction
Citation: Yan Wang, Tingting Zhao, Jun Liu. Viral dynamics of an HIV stochastic model with cell-to-cell infection, CTL immune response and distributed delays[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7126-7154. doi: 10.3934/mbe.2019358

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Abstract

Recent studies have demonstrated that both virus-to-cell infection and cell-to-cell transmission play an important role in the process of HIV infection. In this paper, stochastic perturbation is introduced into HIV model with virus-to-cell infection, cell-to-cell transmission, CTL immune response and three distributed delays. The stochastic integro-delay differential equations is transformed into a degenerate stochastic differential equations. Through rigorous analysis of the model, we obtain the solution is unique, positive and global. By constructing appropriate Lyapunov functions, the existence of the stationary Markov process is derived when the critical condition is bigger than one. Furthermore, the extinction of the virus for sufficiently big noise intensity is established. Numerically, we investigate that the small noise intensity of fluctuations could help to sustain the number of virions and CTL immune response within a certain range, while the big noise intensity may be beneficial to the extinction of the virus. We also examine that the influence of random fluctuations on model dynamics may be more significant than that of the delay.

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