Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

Special Issues: Non-smooth biological dynamical systems and applications

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.
  Article Metrics

Keywords 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. Mathematical Biosciences and Engineering, 2019, 16(6): 7126-7154. doi: 10.3934/mbe.2019358


  • 1. M. A. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74–79.
  • 2. H. L. Smith and P. D. Leenheer, Virus dynamics: a global analysis, SIAM J. Appl. Math., 63 (2003), 1313–1327.
  • 3. G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693–2708.
  • 4. M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434–2448.
  • 5. M. A. Nowak and R. M. May, Virus dynamics: mathematical principles of immunology and virology, Oxford University, Oxford, 2000.
  • 6. A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3–44.
  • 7. Y. Wang, Y. Zhou, F. Brauer, et al., Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901–934.
  • 8. Y. Wang, Y. Zhou, J. Wu, et al., Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104–112.
  • 9. P. Zhong, L. M. Agosto, J. B. Munro, et al., Cell-to-cell transmission of viruses, Curr. Opin. Virol., 3 (2013), 44–50.
  • 10. S. Gummuluru, C. M. Kinsey and M. Emerman, An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virus, J. Virol., 74 (2000), 10882–10891.
  • 11. H. Sato, J. Orenstein, D. Dimitrov, et al., Cell-to-cell spread of HIV-1 occurs within minutes and may not involve the participation of virus particles, Virology, 186 (1992), 712–724.
  • 12. A. Sigal, J. T. Kim, A. B. Balazs, et al., Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95–98.
  • 13. R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425–444.
  • 14. X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898–917.
  • 15. X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563–584.
  • 16. F. Li and J. Wang, Analysis of an HIV infection model with logistic target-cell growth and cell-to-cell transmission, Chaos. Soliton. Fract., 81 (2015), 136–145.
  • 17. X. Wang, S. Tang, X. Song, et al., Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission, J. Biol. Dynam., 11 (2017), 455–483.
  • 18. S. S. Chen, C. Y. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642–672.
  • 19. Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays. J. Math. Anal. Appl., 375 (2011), 14–27.
  • 20. T. Nicoleta, Drug therapy model with time delays for HIV infection with virus-to-cell and cell-to- cell transmissions, J. Appl. Math. Comput., 59 (2019), 677–691.
  • 21. J. Xu and Y. Zhou, Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay, Math. Biosci. Eng., 13 (2017), 343–367.
  • 22. H. Shu, Y. Chen and L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, J. Dyn. Differ. Equ., 30 (2018), 1817–1836.
  • 23. Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183–191.
  • 24. J. Wang, M. Guo, X. Liu, et al., Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149–161.
  • 25. Y. Cai, Y. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221–240.
  • 26. D. Li, J. Cui, M. Liu, et al., The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate, Bull. Math. Biol., 77 (2015), 1705–1743.
  • 27. X. Meng, S. Zhao, T. Feng, et al., Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227–242.
  • 28. Z. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physica A., 451 (2016), 507–518.
  • 29. Y. Tan, L. Ning, S. Tang, et al., Optimal threshold density in a stochastic resource management model with pulse intervention, Nat. Resour. Model., (2019), e12220.
  • 30. H. H. Mcadams and A. Arkin, Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci. USA, 94 (1997), 814–819.
  • 31. K. Millerjensen, R. Skupsky, P. S. Shah, et al., Genetic selection for context-dependent stochastic phenotypes: Sp1 and TATA mutations increase phenotypic noise in HIV-1 gene expression, PLos. Comp. Biol., 9 (2013), e1003135.
  • 32. X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95–110.
  • 33. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084–1101.
  • 34. Z. Huang, Q. Yang and J. Cao, Complex dynamics in a stochastic internal HIV model, Chaos. Soliton. Fract., 44 (2011), 954–963.
  • 35. H. C. Tuckwell and E. Lecorfec, A stochastic model for early HIV-1 population dynamics, J. Theor. Biol., 195 (1998), 451–463.
  • 36. Y. Wang, D. Jiang, T. Hayat, et al., A stochastic HIV infection model with T-cell proliferation and CTL immune response, Appl. Math. Comput., 315 (2017), 477–493.
  • 37. C. Ji, Q. Liu and D. Jiang, Dynamics of a stochastic cell-to-cell HIV-1 model with distributed delay, Physica A., 492 (2018), 1053–1065.
  • 38. T. Feng, Z. Qiu, X. Meng, et al., Analysis of a stochastic HIV-1 infection model with degenerate diffusion, Appl. Math. Comput., 348 (2019), 437–455.
  • 39. Q. Liu, D. Jiang, N. Shi, et al., Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay, Discrete Cont. Dyn-B, 22 (2017), 2479–2500.
  • 40. W. Zuo, D. Jiang, X. Sun, et al., Long-time behaviors of a stochastic cooperative Lotka CVolterra system with distributed delay, Physica A. 506 (2018), 542–559.
  • 41. X. Ji, S. Yuan, T. Zhang, et al., Stochastic modeling of algal bloom dynamics with delayed nutrient recycling, Math. Biosci. Eng., 16 (2018), 1–24.
  • 42. N. Macdonald, Time lags in biological models, Lecture Notes in Biomathematics, Springer-Verlag, Heidelberg, 1978.
  • 43. J. Mittler, B. Sulzer, A. Neumann, et al., Influence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143–163.
  • 44. X. Mao, Stochastic differential equations and applications, 2nd edition, Horwood, Chichester, UK, 2008.
  • 45. R. Khasminskii, Stochastic stability of differential equations, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.
  • 46. N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka. J. Math., 14 (1977), 619–633.
  • 47. D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM. Rev., 43 (2001), 525–546.


This article has been cited by

  • 1. A. M. Elaiw, S. F. Alshehaiween, A. D. Hobiny, I. A. Abbas, Global properties of latent virus dynamics with B-cell impairment, AIP Advances, 2019, 9, 9, 095035, 10.1063/1.5108890
  • 2. Hernán Darío Toro-Zapata, Carlos Andrés Trujillo-Salazar, Edwin Mauricio Carranza-Mayorga, Mathematical Model Describing HIV Infection with Time-Delayed CD4 T-Cell Activation, Processes, 2020, 8, 7, 782, 10.3390/pr8070782
  • 3. Igor Sazonov, Dmitry Grebennikov, Mark Kelbert, Andreas Meyerhans, Gennady Bocharov, Viral Infection Dynamics Model Based on a Markov Process with Time Delay between Cell Infection and Progeny Production, Mathematics, 2020, 8, 8, 1207, 10.3390/math8081207
  • 4. Tongqian Zhang, Junling Wang, Yi Song, Zhichao Jiang, Dynamical Analysis of a Delayed HIV Virus Dynamic Model with Cell-to-Cell Transmission and Apoptosis of Bystander Cells, Complexity, 2020, 2020, 1, 10.1155/2020/2313102

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved