Skip to main content
SpringerLink
Log in

Qualitative Analysis for an HIV Infection Model with Caspase-1-Mediated Pyroptosis of the Predominance: Threshold Dynamics and Traveling Waves

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we investigate the global threshold-type result and traveling waves for a general HIV infection model involving CD\(4^+\) T cell death caused by caspase-1-mediated pyroptosis of the predominance. We first give the well-posedness of the model and establish the existence of a global attractor. In a bounded domain, the basic reproduction number, denoted by \(\Re _0\), is identified as a threshold parameter for indicating whether the infection occurs or not. Specifically, if \(\Re _0<1\), then the system admits a globally asymptotically stable infection-free steady state; if \(\Re _0>1\), the system is uniformly persistent. In an unbounded domain and homogeneous environment, we find that if \(\Re _0>1\) and wave speed is large enough, the system admits traveling wave solutions.

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

Similar content being viewed by others

References

  1. Bai, N., Xu, R.: Modelling of HIV viral load and 2-LTR dynamics during high active antiretroviral therapy in a heterogeneous environment. Commun. Nonlinear Sci. Numer. Simul. 116, 106874 (2023)

    MathSciNet  MATH  Google Scholar 

  2. Beltman, J.B., Maree, A.F.M., Lynch, J.N., Miller, M.J., de Boer, R.J.: Lymph node topology dictates T cell migration behavior. J. Exp. Med. 204, 771–780 (2007)

    Google Scholar 

  3. Chen, W., Teng, Z., Zhang, L.: Global dynamics for a drug-sensitive and drug-resistant mixed strains of HIV infection model with saturated incidence and distributed delays. Appl. Math. Comput. 406, 126284 (2021)

    MathSciNet  MATH  Google Scholar 

  4. Cox, A.L., Siliciano, R.F.: HIV: not-so-innocent bystanders. Nature 505, 492–493 (2014)

    Google Scholar 

  5. Culshaw, R.V., Ruan, S.: A delay-differential equation model of HIV infection of CD\(4^+\) T-cells. Math. Biosci. 165, 27–39 (2000)

    MATH  Google Scholar 

  6. Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio \(r_0\) in the models for infectious disease in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Doitsh, G., Galloway, N.L.K., Geng, X., Yang, Z., Monroe, K.M., Zepeda, O., Hunt, P.W., Hatano, H., Sowinski, S., Muñoz-Arias, I., Greene, W.C.: Pyroptosis drives CD4 T-cell depletion in HIV-1 infection. Nature 505, 509–514 (2014)

    Google Scholar 

  8. Gao, M., Jiang, D., Hayat, T.: Qualitative analysis of an HIV/AIDS model with treatment and nonlinear perturbation. Qual. Theor. Dyn. Syst. 21, 85 (2022)

    MathSciNet  MATH  Google Scholar 

  9. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, vol. 224. Springer, Berlin (2001)

    MATH  Google Scholar 

  10. Gourley, S.A., Britton, N.F.: A predator–prey reaction–diffusion system with nonlocal effects. J. Math. Biol. 34, 297–333 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Graw, F., Perelson, A.S.: Spatial aspects of HIV infection. In: Ledzewicz, U., Schättler, H., Friedman, A., Kashdan, E. (eds.) Mathematical Methods and Models in Biomedicine. Lecture Notes on Mathematical Modelling in the Life Sciences, pp. 3–31. Springer, New York (2013)

  12. Graziano, F.M., Kettoola, S.Y., Munshower, J.M., Stapleton, J.T., Towfic, G.J.: Effect of spatial distribution of T-Cells and HIV load on HIV progression. Bioinformatics 24, 855–860 (2008)

    Google Scholar 

  13. Guo, Z., Wang, F.B., Zou, X.: Threshold dynamics of an infective disease model with a fixed latent period and non-local infections. J. Math. Biol. 65, 1387–1410 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence (1988)

    Google Scholar 

  15. Huang, G., Liu, X., Takeuchi, Y.: Lyapunov functions and global stability for age-structured HIV infection model. SIAM J. Appl. Math. 72, 25–38 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Iwami, S., Miura, T., Nakaoka, S., Takeuchi, Y.: Immune impairment in HIV infection: existence of risky and immunodeficiency thresholds. J. Theor. Biol. 260, 490–501 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Korobeinikov, A.: Global properties of basic virus dynamics models. Bull. Math. Biol. 66, 879–883 (2004)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Liu, Q.: Analysis of a stochastic HIV model with cell-to-cell transmission and Ornstein–Uhlenbeck process. J. Math. Phys. 64, 012702 (2023)

    MathSciNet  MATH  Google Scholar 

  20. Lou, Y., Zhao, X.Q.: A reaction–diffusion malaria model with incubation period in the vector population. J. Math. Biol. 62, 543–568 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Magal, P., Zhao, X.Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Martin, R.J., Smith, H.L.: Abstract functional differential equations and reaction–diffusion systems. T. Am. Math. Soc. 321, 1–44 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Murphy, K.M., Weaver, C., Berg, L.J.: Janeway’s Immunobiology, 10th edn. W. W. Norton & Company, New York (2022)

    Google Scholar 

  24. Nowak, M.A., Bonhoeffer, S., Hill, A.M., Boehme, R., Thomas, H.C., McDade, H.: Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. USA 93, 4398–4402 (1996)

    Google Scholar 

  25. Pazy, A.: Semigroups of Linear Operators and Application to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    MATH  Google Scholar 

  26. Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41, 3–44 (1999)

    MathSciNet  MATH  Google Scholar 

  27. Ren, X., Tian, Y., Liu, L., Liu, X.: A reaction–diffusion within-host HIV model with cell-to-cell transmission. J. Math. Biol. 76, 1831–1872 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Schroder, K., Tschopp, J.: The inflammasomes. Cell 140, 821–832 (2010)

    Google Scholar 

  29. Shu, H., Ma, Z., Wang, X.S.: Threshold dynamics of a nonlocal and delayed cholera model in a spatially heterogeneous environment. J. Math. Biol. 83, 41 (2021)

    MathSciNet  MATH  Google Scholar 

  30. Smith, H.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. American Mathematical Society, Providence (1995)

    Google Scholar 

  31. Smith, H.L., Zhao, X.Q.: Robust persistence for semidynamical systems. Nonlinear Anal. 47, 6169–6179 (2001)

    MathSciNet  MATH  Google Scholar 

  32. Thieme, H.R.: Convergence results and a Poincar\(\acute{\rm e }\)–Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30, 755–763 (1992)

    MathSciNet  MATH  Google Scholar 

  33. Thieme, H.R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211 (2009)

    MathSciNet  MATH  Google Scholar 

  34. Thieme, H.R., Zhao, X.Q.: A non-local delayed and diffusive predator–prey model. Nonlinear Anal. Real World Appl. 2, 145–160 (2001)

    MathSciNet  MATH  Google Scholar 

  35. UNAIDS: In Danger: UNAIDS Global AIDS Update 2022. Joint United Nations Programme on HIV/AIDS, Geneva (2022)

  36. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    MathSciNet  MATH  Google Scholar 

  37. Wang, J., Guo, M., Liu, X., Zhao, Z.: 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 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Wang, J., Zhang, R., Gao, Y.: Global threshold dynamics of an infection age-space structured HIV infection model with Neumann boundary condition. J. Dyn. Diff. Equ. (in press). https://doi.org/10.1007/s10884-021-10086-2

  39. Wang, J., Zhang, R., Kuniya, T.: Global dynamics for a class of age-infection HIV models with nonlinear infection rate. J. Math. Anal. Appl. 432, 289–313 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Wang, W., Feng, Z.: Global dynamics of a diffusive viral infection model with spatial heterogeneity. Nonlinear Anal. RWA 72, 103763 (2023)

    MathSciNet  MATH  Google Scholar 

  41. Wang, W., Zhang, T.: Caspase-1-mediated pyroptosis of the predominance for driving CD4\(^{+}\) T cells death: a nonlocal spatial mathematical model. Bull. Math. Biol. 80, 540–582 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Wang, W., Zhao, X.Q.: A nonlocal and time-delayed reaction-diffusion model of dengue transmission. SIAM J. Appl. Math. 71, 147–168 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Wang, W., Zhao, X.Q.: Basic reproduction numbers for reaction–diffusion epidemic model. SIAM J. Appl. Dyn. Syst. 11, 1652–1673 (2012)

    MathSciNet  MATH  Google Scholar 

  44. Wang, Z.C., Wu, J.: Travelling waves of a diffiusive Kermack–McKendrick epidemic model with non-local delayed transmission. Proc. Roy. Soc. A Math. 466, 237–261 (2010)

    MATH  Google Scholar 

  45. Wang, Z.C., Wu, J., Liu, R.: Traveling waves of Avian influenza spread. Proc. Am. Math. Soc. 140, 3931–3946 (2012)

    MATH  Google Scholar 

  46. Wu, P., Zhao, H.: Dynamical analysis of a nonlocal delayed and diffusive HIV latent infection model with spatial heterogeneity. J. Franklin I(358), 5552–5587 (2021)

    MathSciNet  MATH  Google Scholar 

  47. Wu, P., Zhao, H.: Mathematical analysis of an age-structured HIV/AIDS epidemic model with HAART and spatial diffusion. Nonlinear Anal. RWA 60, 103289 (2021)

    MathSciNet  MATH  Google Scholar 

  48. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences, vol. 119. Springer, New York (1996)

    Google Scholar 

  49. Yang, Y., Zou, L., Ruan, S.: Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions. Math. Biosci. 270, 183–191 (2015)

    MathSciNet  MATH  Google Scholar 

  50. Zhao, L., Wang, Z.C., Ruan, S.: Traveling wave solutions in a two-group epidemic model with latent period. Nonlinearity 30, 1287–1325 (2017)

    MathSciNet  MATH  Google Scholar 

  51. Zhao, X.-Q.: Dynamical Systems in Population Biology. CMS Books in Mathematics, Springer, Cham (2017)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their valuable comments which led to a significant improvement of this study. The research was supported by the National Natural Science Foundation of China (Nos. 12071115, 12101309), the Heilongjiang Natural Science Funds for Distinguished Young Scholar (No. JQ2023A005), the Fundamental Research Funds for the Colleges and Universities in Heilongjiang Province (No. 2022-KYYWF-1113), and the Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems.

Author information

Authors and Affiliations

  1. Engineering Research Center of Agricultural Microbiology Technology, Ministry of Education and Heilongjiang Provincial Key Laboratory of Ecological Restoration and Resource Utilization for Cold Region and School of Mathematical Science, Heilongjiang University, Harbin, 150080, China

    Ran Zhang, Jiangxue Xu & Jinliang Wang

Authors
  1. Ran Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiangxue Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinliang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinliang Wang.

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

Zhang, R., Xu, J. & Wang, J. Qualitative Analysis for an HIV Infection Model with Caspase-1-Mediated Pyroptosis of the Predominance: Threshold Dynamics and Traveling Waves. Qual. Theory Dyn. Syst. 22, 131 (2023). https://doi.org/10.1007/s12346-023-00828-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00828-6

Keywords

Mathematics Subject Classification

Search

Navigation