Skip to main content

Advertisement

SpringerLink
Log in

An HIV stochastic model with cell-to-cell infection, B-cell immune response and distributed delay

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

In this study, a delayed HIV stochastic model with virus-to-cell infection, cell-to-cell transmission and B-cell immune response is proposed. We first transform the stochastic differential equation with distributed delay into a high-dimensional degenerate stochastic differential equation, and then theoretically analyze the dynamic behaviour of the degenerate model. The unique global solution of the model is given by rigorous analysis. By formulating suitable Lyapunov functions, the existence of the stationary Markov process is obtained if the stochastic B-cell-activated reproduction number is greater than one. We also use the law of large numbers theorem and the spectral radius analysis method to deduce that the virus can be cleared if the stochastic B-cell-inactivated reproduction number is less than one. Through uncertainty and sensitivity analysis, we obtain key parameters that determine the value of the stochastic B-cell-activated reproduction number. Numerically, we examine that low level noise can maintain the number of the virus and B-cell populations at a certain range, while high level noise is helpful for the elimination of the virus. Furthermore, the effect of the cell-to-cell infection on model behaviour, and the influence of the key parameters on the size of the stochastic B-cell-activated reproduction number are also investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Adler PB, Drake JM (2008) Environmental variation, stochastic extinction, and competitive coexistence. Am Nat 172:186–195

    Google Scholar 

  • Agosto LM, Zhong P, Munro J, Mothes W (2014) Highly active antiretroviral therapies are effective against HIV-1 cell-to-cell transmission. PLoS Pathog 10:e1003982

    Google Scholar 

  • Alvarez LHR (2001) Does increased stochasticity speed up extinction? J Math Biol 43:534–544

    MathSciNet  MATH  Google Scholar 

  • Baum LL (2010) Role of humoral immunity in host defense against HIV. Curr HIV-AIDS Rep 7:11–18

    Google Scholar 

  • Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int Stat Rev 62:229–243

    MATH  Google Scholar 

  • Cai Y, Kang Y, Wang W (2017) A stochastic SIRS epidemic model with nonlinear incidence rate. Appl Math Comput 305:221–240

    MathSciNet  MATH  Google Scholar 

  • Culshaw RV, Ruan S, Webb G (2003) A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay. J Math Biol 46:425–444

    MathSciNet  MATH  Google Scholar 

  • Cagigi A, Mowafi F, Phuong Dang LV, Tenner-Racz K, Atlas A, Grutzmeier S, Racz P, Chiodi F, Nilsson A (2008) Altered expression of the receptor–ligand pair CXCR5/CXCL13 in B cells during chronic HIV-1 infection. Blood 112(12):4401–4410

    Google Scholar 

  • Dalal N, Greenhalgh D, Mao X (2008) A stochastic model for internal HIV dynamics. J Math Anal Appl 341:1084–1101

    MathSciNet  MATH  Google Scholar 

  • De Milito A, Mörch C, Sönnerborg A, Chiodi F (2001) Loss of memory (CD27) B lymphocytes in HIV-1 infection. AIDS 15(8):957–964

    Google Scholar 

  • Deans JA, Cohen S (1983) Immunology of malaria. Microbiology 37:25–49

    Google Scholar 

  • Duncan CJ, RuRussell RA, Sattentau QJ (2013) High multiplicity HIV-1 cell-to-cell transmission from macrophages to CD4\(^+\) T cells limits antiretroviral efficacy. AIDS 27:2201–2206

    Google Scholar 

  • Elaiw AM, Raezah AA, Alofi AS (2016) Effect of humoral immunity on HIV-1 dynamics with virus-to-target and infected-to-target infections. AIP Adv 6:085204

    Google Scholar 

  • Feng T, Qiu Z, Meng X, Rong L (2019) Analysis of a stochastic HIV-1 infection model with degenerate diffusion. Appl Math Comput 348:437–455

    MathSciNet  MATH  Google Scholar 

  • Gupta P, Balachandran R, Ho M, Enrico A, Rinaldo C (1989) Cell-to-cell transmission of human immunodeficiency virus type 1 in the presence of azidothymidine and neutralizing antibody. J Virol 63:2361–2365

    Google Scholar 

  • Hill AL, Rosenbloom DI, Fu F, Nowak MA, Siliciano RF (2014) Predicting the outcomes of treatment to eradicate the latent reservoir for HIV-1. Proc Natl Acad Sci 111(37):13475–13480

    Google Scholar 

  • Heffernan JM, Wahl LM (2005) Monte Carlo estimates of natural variation in HIV infection. J Theor Biol 236:137–153

    MathSciNet  MATH  Google Scholar 

  • Heffernan JM, Wahl LM (2006) Natural variation in HIV infection: monte carlo estimates that include CD8 effector cells. J Theor Biol 243:191–204

    MathSciNet  MATH  Google Scholar 

  • Herz AVM, Bonhoeffer S, Anderson RM, May RM, Nowak MA (1996) Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc Natl Acad Sci USA 93:7247–7251

    Google Scholar 

  • Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 43:525–546

    MathSciNet  MATH  Google Scholar 

  • Ikeda N, Watanabe S (1977) A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J Math 14:619–633

    MathSciNet  MATH  Google Scholar 

  • Iwami S, Takeuchi JS, Nakaoka S, Mammano F, Clavel F, Inaba H, Kobayashi T, Misawa N, Aihara K, Koyanagi Y, Sato K (2015) Cell-to-cell infection by HIV contributes over half of virus infection. Elife 4:e08150

    Google Scholar 

  • Khasminskii R (1980) Stochastic stability of differential equations. Sijthoff & Noordhoff, Alphen aan den Rijn

    MATH  Google Scholar 

  • Kutoyants AY (2003) Statistical inference for ergodic diffusion processes. Springer, London

    MATH  Google Scholar 

  • Lahrouz A, Settati A, Akharif A (2017) Effects of stochastic perturbation on the SIS epidemic system. J Math Biol 74:469–498

    MathSciNet  MATH  Google Scholar 

  • Lai X, Zou X (2014) Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission. SIAM J Appl Math 74:898–917

    MathSciNet  MATH  Google Scholar 

  • Lai X, Zou X (2015) Modeling cell-to-cell spread of HIV-1 with logistic target cell growth. J Math Anal Appl 426:563–584

    MathSciNet  MATH  Google Scholar 

  • Lan G, Yuan S, Song B (2021) The impact of hospital resources and environmental perturbations to the dynamics of SIRS model. J Frankl Inst 358:2405–2433

    MathSciNet  MATH  Google Scholar 

  • Li F, Wang J (2015) Analysis of an HIV infection model with logistic target-cell growth and cell-to-cell transmission. Chaos Soliton Fract 81:136–145

    MathSciNet  MATH  Google Scholar 

  • Li D, Cui J, Liu M, Liu S (2015) The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate. Bull Math Biol 77:1705–1743

    MathSciNet  MATH  Google Scholar 

  • Lin J, Xu R, Tian X (2017) Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity. Appl Math Comput 315:516–530

    MathSciNet  MATH  Google Scholar 

  • Macdonald N (1978) Time lags in biological models. Springer, Heidelberg

    MATH  Google Scholar 

  • Mao X (2008) Stochastic differential equations and applications. Horwood, Chichester

    Google Scholar 

  • Mathez D, Schinazi RF, Liotta DC, Leibowitch J (1993) Infectious amplification of wild-type human immunodeficiency virus from patients’ lymphocytes and modulation by reverse transcriptase inhibitors in vitro. Antimicrob Agents Chemother 37:2206–2211

    Google Scholar 

  • Marino S, Hogue IB, Ray CJ, Kirschner DE (2009) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 254(1):178–196

    MathSciNet  MATH  Google Scholar 

  • MATLAB (2017) Version 9.3.0.713579 (R2017b), The MathWorks Inc, Natick

  • McCune JM (2001) The dynamics of CD4\(^{+}\) T-cell depletion in HIV disease. Nature 410(6831):974–979

    Google Scholar 

  • Meng X, Zhao S, Feng T, Zhang T (2016) Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. J Math Anal Appl 433:227–242

    MathSciNet  MATH  Google Scholar 

  • Mittler JE, Sulzer B, Neumann AU, Perelson AS (1998) Influence of delayed virus production on viral dynamics in HIV-1 infected patients. Math Biosci 152:143–163

    MATH  Google Scholar 

  • Murase A, Sasaki T, Kajiwara T (2005) Stability analysis of pathogen–immune interaction dynamics. J Math Biol 51:247–267

    MathSciNet  MATH  Google Scholar 

  • Nijhuis M, Boucher CAB, Schipper P, Leitner T, Schuurman R, Albert J (1998) Stochastic processes strongly influence HIV-1 evolution during suboptimal protease-inhibitor therapy. Proc Natl Acad Sci USA 95:14441–14446

    Google Scholar 

  • Nowak MA, Bangham C (1996) Population dynamics of immune response to persistent viruses. Science 272:74–79

    Google Scholar 

  • Nowak MA, May RM (2000) Virus dynamics: mathematical principles of immunology and virology. Oxford University, Oxford

    MATH  Google Scholar 

  • Okoye AA, Picker LJ (2013) CD4\(^{+}\) T-cell depletion in HIV infection: mechanisms of immunological failure. Immunol Rev 254(1):54–64

    Google Scholar 

  • Pedro KD, Henderson AJ, Agosto LM (2019) Mechanisms of HIV-1 cell-to-cell transmission and the establishment of the latent reservoir. Virus Res 265:115–121

    Google Scholar 

  • Perelson AS, Nelson PW (1999) Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev 41:3–44

    MathSciNet  MATH  Google Scholar 

  • Perelson A, Neumann A (1996) HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271:1582–1586

    Google Scholar 

  • Permanyer M, Ballana E, Ruiz A, Badia R, Riveira-Munoz E, Gonzalo E, Clotet B, Esté JA (2012) Antiretroviral agents effectively block HIV replication after cell-to-cell transfer. J Virol 86:8773–8780

    Google Scholar 

  • Reyes-Silveyra J, Mikler AR (2016) Modeling immune response and its effect on infectious disease outbreak dynamics. Theor Biol Med Model 13(1):1–21

    Google Scholar 

  • Qesmi R, Wu J, Wu J, Heffernan JM (2010) Influence of backward bifurcation in a model of hepatitis B and C viruses. Math Biosci 224:118–125

    MathSciNet  MATH  Google Scholar 

  • Qesmi R, Elsaadany S, Heffernan JM, Wu J (2011) A hepatitis B and C virus model with age since infection that exhibits backward bifurcation. SIAM J Appl Math 71:1509–1530

    MathSciNet  MATH  Google Scholar 

  • Qi H, Meng X (2021) Mathematical modeling, analysis and numerical simulation of HIV: the influence of stochastic environmental fluctuations on dynamics. Math Comput Simul 187:700–719

    MathSciNet  MATH  Google Scholar 

  • Rouzine IM, Razooky BS, Weinberger LS (2014) Stochastic variability in HIV affects viral eradication. Proc Natl Acad Sci USA 111:13251–13252

    Google Scholar 

  • Sanchez MA, Blower SM (1997) Uncertainty and sensitivity analysis of the basic reproductive rate: tuberculosis as an example. Am J Epidemiol 145(12):1127–1137

    Google Scholar 

  • Shi P, Dong L (2019) Dynamical behaviors of stochastic virus dynamic models with saturation responses. Math Biosci 308:20–26

    MathSciNet  MATH  Google Scholar 

  • Shu H, Chen Y, Wang L (2018) Impacts of the cell-free and cell-to-cell infection modes on viral dynamics. J Dyn Differ Equ 30:1817–1836

    MathSciNet  MATH  Google Scholar 

  • Sigal A, Kim JT, Balazs AB, Dekel E, Mayo A, Milo R, Baltimore D (2011) Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy. Nature 477:95–98

    Google Scholar 

  • Singh A, Weinberger LS (2009) Stochastic gene expression as a molecular switch for viral latency. Curr Opin Microbiol 12(4):460–466

    Google Scholar 

  • Singh A, Razooky B, Cox CD, Simpson ML, Weinberger LS (2010) Transcriptional bursting from the HIV-1 promoter is a significant source of stochastic noise in HIV-1 gene expression. Biophys J 98:L32–L34

    Google Scholar 

  • Smith HL, De Leenheer P (2003) Virus dynamics: a global analysis. SIAM J Appl Math 63:1313–1327

    MathSciNet  MATH  Google Scholar 

  • Tan Y, Ning L, Tang S, Cheke RA (2019) Optimal threshold density in a stochastic resource management model with pulse intervention. Nat Resour Model 32:e12220

    MathSciNet  Google Scholar 

  • Tan Y, Cai Y, Sun X, Wang K, Yao R, Wang W, Peng Z (2022) A stochastic SICA model for HIV/AIDS transmission. Chaos SolitonS Fractals 165:112768

    MathSciNet  MATH  Google Scholar 

  • Teng Z, Wang L (2016) Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate. Physica A 451:507–518

    MathSciNet  MATH  Google Scholar 

  • Tuckwell HC, Lecorfec E (1998) A stochastic model for early HIV-1 population dynamics. J Theor Biol 195:451–463

    Google Scholar 

  • Wang Y, Zhou Y, Wu J, Heffernan J (2009) Oscillatory viral dynamics in a delayed HIV pathogenesis model. Math Biosci 219:104–112

    MathSciNet  MATH  Google Scholar 

  • Wang Y, Zhou Y, Brauer F, Heffernan JM (2013) Viral dynamics model with CTL immune response incorporating antiretroviral therapy. J Math Biol 67:901–934

    MathSciNet  MATH  Google Scholar 

  • Wang J, Guo M, Liu X, Zhao Z (2016) Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay. Appl Math Comput 291:149–161

    MathSciNet  MATH  Google Scholar 

  • Wang X, Tang S, Song X, Rong L (2017a) Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission. J Biol Dyn 11:455–483

  • Wang Y, Jiang D, Hayat T, Ahmad B (2017b) A stochastic HIV infection model with T-cell proliferation and CTL immune response. Appl Math Comput 315:477–493

  • Wang Y, Jiang D, Alsaedi A, Hayat T (2018) Modelling a stochastic HIV model with logistic target cell growth and nonlinear immune response function. Physica A 501:276–292

    MathSciNet  MATH  Google Scholar 

  • Wang Y, Jiang D, Hayat T, Alsaedi A (2019a) Stationary distribution of an HIV model with general nonlinear incidence rate and stochastic perturbations. J Frankl Inst 356:6610–6637

  • Wang Y, Zhao T, Liu J (2019b) Viral dynamics of an HIV stochastic model with cell-to-cell infection, CTL immune response and distributed delays. Math Biosci Eng 16:7126–7154

  • Wei F, Wang K (2007) The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J Math Anal Appl 331:516–531

    MathSciNet  MATH  Google Scholar 

  • Willey S, Marlén MIA (2008) Humoral immunity to HIV-1: neutralisation and antibody effector functions. Trends Microbiol 16:596–604

    Google Scholar 

  • Xu J, Zhou Y (2017) Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Math Biosci Eng 13:343–367

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees and editor for their constructive criticisms and valuable comments, which helped us to improve our study. This work is supported by Shandong Provincial Natural Science Foundation (Nos. ZR2020MA039, ZR2021MA020), the Fundamental Research Funds for the Central Universities (No. 22CX03016A), National Natural Science Foundation of China (Nos. 11871473, 11801566), and the Natural Science and Engineering Research Council of Canada (NSERC).

Author information

Authors and Affiliations

  1. College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China

    Yan Wang, Jun Liu & Xinhong Zhang

  2. Modelling Infection and Immunity Lab, Centre for Disease Modelling, Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada

    Jane M. Heffernan

Authors
  1. Yan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinhong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jane M. Heffernan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jane M. Heffernan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Liu, J., Zhang, X. et al. An HIV stochastic model with cell-to-cell infection, B-cell immune response and distributed delay. J. Math. Biol. 86, 35 (2023). https://doi.org/10.1007/s00285-022-01863-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00285-022-01863-8

Keywords

Mathematics Subject Classification

Search

Navigation