Abstract
In this paper, we propose a stochastic HIV-1 infection model with distributed delay and logistic growth. Firstly, we transfer the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Then, we establish sufficient conditions for the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Moreover, we obtain sufficient criteria for extinction of the infected cells; that is, the uninfected cells are survival and the infected cells are extinct. Our results show that the smaller white noise can ensure the existence of a stationary distribution when the basic reproduction number \(R_{0}^{S}\) of the stochastic system is bigger than one, while the larger white noise can lead to the extinction of the infected cells when the basic reproduction number \(R_{0}\) of the deterministic system is smaller than one.
Similar content being viewed by others
References
Anderson, R.M.: Mathematical and statistical studies of the epidemiology of HIV. AIDS 3, 333–346 (1989)
Anderson, R.M., May, R.M., Goldbeter, A.: Complex dynamical behavior in the interaction between HIV and the immune system. In: Goldbeter, A. (ed.) Cell to Cell Signalling: From Experiments to Theoretical Models, p. 335. Academic Press, New York (1989)
Bailey, J.J., Fletcher, J.E., Chuck, E.T., Shrager, R.I.: A kinetic model of CD4+ lymphocytes with the human immunodeficiency virus (HIV). Biosystems 26, 177–183 (1992)
Bonhoeer, S., May, R.M., Shaw, G.M., Nowak, M.A.: Virus dynamics and drug therapy. Proc. Natl. Acad. Sci. USA 94, 69–71 (1997)
Culshaw, R.V., Ruan, S.: A delay-differential equation model of HIV infection of \(CD4^{+}\) T-cells. Math. Biosci. 165, 27–39 (2000)
Culshaw, R.V., Ruan, S., Webb, G.: A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay. J. Math. Biol. 46, 425–444 (2003)
Dalal, N., Greenhalgh, D., Mao, X.: A stochastic model for internal HIV dynamics. J. Math. Anal. Appl. 341, 1084–1101 (2008)
De Boer, R.J., Perelson, A.S.: Target cell limited and immune control models of HIV infection: a comparison. J. Theor. Biol. 190, 201–214 (1998)
Emvudu, Y., Bongor, D., Koïna, R.: Mathematical analysis of HIV/AIDS stochastic dynamic models. Appl. Math. Model. 40, 9131–9151 (2016)
Grossman, Z., Feinberg, M.B., Kuznetsov, V., Dimitrov, D., Paul, W.: HIV infection: how effective is drug combination treatment? Immunol. Today 19, 528–532 (1998)
Grossman, Z., Polis, M., Feinberg, M.B., et al.: Ongoing HIV dissemination during HAART. Nat. Med. 5, 1099–1104 (1999)
Hasminskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen aan den Rijn (1980)
Herz, A.V.M., Bonhoeffer, S., Anderson, R.M., May, R.M., Nowak, M.A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc. Natl. Acad. Sci. USA 93, 7247–7251 (1996)
Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)
Hraba, T., Dolezal, J., Celikovsky, S.: Model-based analysis of CD4+ lymphocyte dynamics in HIV infected individuals. Immunobiology 181, 108–118 (1990)
Huang, Z., Yang, Q., Cao, J.: Complex dynamics in a stochastic internal HIV model. Chaos Solitons Fractals 44, 954–963 (2011)
Intrator, N., Deocampo, G.P., Cooper, L.N.: Analysis of immune system retrovirus equations. In: Perelson, A.S. (ed.) Theoretical Immunology II, p. 85. Addison-Wesley, Redwood City (1988)
Ji, C., Liu, Q., Jiang, D.: Dynamics of a stochastic cell-to-cell HIV-1 model with distributed delay. Physica A 492, 1053–1065 (2018)
Jiang, D., Liu, Q., Shi, N., Hayat, T., Alsaedi, A., Xia, P.: Dynamics of a stochastic HIV-1 infection model with logistic growth. Physica A 469, 706–717 (2017)
Kirschner, D.E.: Using mathematics to understand HIV immune dynamics. Not. Am. Math. Soc. 43, 191–202 (1996)
Kirschner, D.E., Perelson, A.S., Arino, O., Axelrod, D., Kimmel, M., Langlais, M.: A model for the immune system response to HIV: AZT treatment studies. In: Mathematical Population Dynamics: Analysis of Heterogeneity. Theory of Epidemics Series, vol. 1, p. 295. Wuerz, Winnipeg, Canada (1995)
Kirschner, D.E., Lenhart, S., Serbin, S.: Optimal control of the chemotherapy of HIV. J. Math. Biol. 35, 775–792 (1997)
Kutoyants, A.Y.: Statistical Inference for Ergodic Diffusion Processes. Springer, London (2003)
Liu, Q.: Asymptotic behaviors of a cell-to-cell HIV-1 infection model perturbed by white noise. Physica A 467, 407–418 (2017)
Liu, H., Yang, Q., Jiang, D.: The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences. Automatica 48, 820–825 (2012)
Macdonald, N.: Time Lags in Biological Models, Lecture Notes in Biomathematics. Springer, Heidelberg (1978)
Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)
Peng, S., Zhu, X.: Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations. Stoch. Process. Appl. 116, 370–380 (2006)
Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41, 3–44 (1999)
Perelson, A.S., Kirschner, D.E., Deboer, R.: Dynamics of HIV infection of \(CD4^{+}\) T cells. Math. Biosci. 114, 81–125 (1993)
Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586 (1996)
Pinto, C.M.A., Carvalho, A.R.M.: Stochastic model for HIV dynamics in HIV specific helper cells. IFAC-Papers On Line 48–1, 184–185 (2015)
Sánchez-Taltavull, D., Alarcón, T.: Stochastic modelling of viral blips in HIV-1-infected patients: effects of inhomogeneous density fluctuations. J. Theor. Biol. 371, 79–89 (2015)
Sánchez-Taltavull, D., Vieiro, A., Alarcón, T.: Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy. J. Math. Biol. 73, 919–946 (2016)
Tam, J.: Delay effect in a model for virus replication. IMA J. Math. Appl. Med. Biol. 16, 29–37 (1999)
Tuckwell, H.C., Le Corfec, E.: A stochastic model for early HIV-1 population dynamics. J. Theor. Biol. 195, 451–463 (1998)
Wang, X., Liu, X., Xu, W., Zhang, K.: Stochastic dynamics of HIV models with switching parameters and pulse control. J. Frankl. Inst. 352, 2765–2782 (2015)
Xu, D., Huang, Y., Yang, Z.: Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete Contin. Dyn. Syst. 24, 1005–1023 (2009)
Acknowledgements
We are very grateful to the editor and the anonymous referees for their careful reading and valuable comments, which greatly improved the presentation of the paper. We also thank the National Natural Science Foundation of P.R. China (No. 11871473), Natural Science Foundation of Guangxi Province (No. 2016GXNSFBA380006) and the Fundamental Research Funds for the Central Universities (No. 15CX08011A), KY2016YB370 and 2016CSOBDP0001.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dr. Alain Goriely.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, Q., Jiang, D., Hayat, T. et al. Stationary Distribution and Extinction of a Stochastic HIV-1 Infection Model with Distributed Delay and Logistic Growth. J Nonlinear Sci 30, 369–395 (2020). https://doi.org/10.1007/s00332-019-09576-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-019-09576-x