In this article, we investigate the dynamics of a stochastic HIV model with a Hill-type infection rate and distributed delay, which are better choices for mass action laws. First, we transform a stochastic system with weak kernels into a degenerate high-dimensional system. Then the existence of a stationary distribution is obtained by constructing a suitable Lyapunov function, which determines a sharp critical value $ R_0^s $ corresponding to the basic reproduction number for the determined system. Moreover, the sufficient condition for the extinction of diseases is derived. More importantly, the exact expression of the probability density function near the quasi-equilibrium is obtained by solving the Fokker-Planck equation. Finally, numerical simulations are illustrated to verify the theoretical results.
Citation: Wenjie Zuo, Mingguang Shao. Stationary distribution, extinction and density function for a stochastic HIV model with a Hill-type infection rate and distributed delay[J]. Electronic Research Archive, 2022, 30(11): 4066-4085. doi: 10.3934/era.2022206
In this article, we investigate the dynamics of a stochastic HIV model with a Hill-type infection rate and distributed delay, which are better choices for mass action laws. First, we transform a stochastic system with weak kernels into a degenerate high-dimensional system. Then the existence of a stationary distribution is obtained by constructing a suitable Lyapunov function, which determines a sharp critical value $ R_0^s $ corresponding to the basic reproduction number for the determined system. Moreover, the sufficient condition for the extinction of diseases is derived. More importantly, the exact expression of the probability density function near the quasi-equilibrium is obtained by solving the Fokker-Planck equation. Finally, numerical simulations are illustrated to verify the theoretical results.
| [1] |
N. Bairagi, D. Adak, Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay, Appl. Math. Model., 38 (2014), 5047–5066. https://doi.org/10.1016/j.apm.2014.03.010 doi: 10.1016/j.apm.2014.03.010
|
| [2] | N. Macdonald, Time Lags in Biological Models, in: Lecture Notes in Biomathematics, Springer-Verlag, Heidelberg, 27 (1978). |
| [3] |
R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 2799–2805. https://doi.org/10.1016/j.camwa.2011.03.050 doi: 10.1016/j.camwa.2011.03.050
|
| [4] |
X. Zhang, Q. Yang, Threshold behavior in a stochastic SVIR model with general incidence rates, Appl. Math. Lett., 121 (2021), 107403. https://doi.org/10.1016/j.aml.2021.107403 doi: 10.1016/j.aml.2021.107403
|
| [5] |
W. Zuo, Y. Zhou, Density function and stationary distribution of a stochastic SIR model with distributed delay, Appl. Math. Lett., 129 (2022), 107931. https://doi.org/10.1016/j.aml.2022.107931 doi: 10.1016/j.aml.2022.107931
|
| [6] | X. Mu, D. Jiang, A. Alsaedi, Analysis of a Stochastic Phytoplankton-Zooplankton Model under Non-degenerate and Degenerate Diffusions, J. Nonlinear Sci., 32 (2022). https://doi.org/10.1007/s00332-022-09787-9 |
| [7] |
Y. Wang, D. Jiang, H. Tasawar, A. Alsaedi, Stationary distribution of an HIV model with general nonlinear incidence rate and stochastic perturbations, J. Franklin Inst., 356 (2019), 6610–6637. https://doi.org/10.1016/j.jfranklin.2019.06.035 doi: 10.1016/j.jfranklin.2019.06.035
|
| [8] |
Q. Liu, D. Jiang, Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation, Appl. Math. Lett., 73 (2017), 8–15. https://doi.org/10.1016/j.aml.2017.04.021 doi: 10.1016/j.aml.2017.04.021
|
| [9] |
M. Song, W. Zuo, D. Jiang, H. Tasawar, Stationary distribution and ergodicity of a stochastic cholera model with multiple pathways of transmission, J. Franklin Inst., 357 (2020), 10773–10798. https://doi.org/10.1016/j.jfranklin.2020.04.061 doi: 10.1016/j.jfranklin.2020.04.061
|
| [10] | R. Khasminskii, Stochastic Stability of Differential Equations, Springer, Heidelberg, 1980. https://doi.org/10.1007/978-3-642-23280-0 |
| [11] |
W. Guo, Q. Zhang, Explicit numerical approximation for an impulsive stochastic age-structured HIV infection model with Markovian switching, Math. Comput. Simulat., 182 (2021), 86–115. https://doi.org/10.1016/j.matcom.2020.10.015 doi: 10.1016/j.matcom.2020.10.015
|
| [12] |
T. Feng, Z. Qiu, X. Meng, L. Rong, Analysis of a stochastic HIV-1 infection model with degenerate diffusion, Appl. Math. Comput., 348 (2019), 437–455. https://doi.org/10.1016/j.amc.2018.12.007 doi: 10.1016/j.amc.2018.12.007
|
| [13] |
Y. Emvudu, D. Bongor, R. Koïna, Mathematical analysis of HIV/AIDS stochastic dynamic models, Appl. Math. Model., 40 (2016), 9131–9151. https://doi.org/10.1016/j.apm.2016.05.007 doi: 10.1016/j.apm.2016.05.007
|
| [14] |
D. Nguyen, G. Yin, C. Zhu, Long-term analysis of a stochastic SIRS model with general incidence rates, SIAM J. Appl. Math., 80 (2020), 814–838. https://doi.org/10.1137/19M1246973 doi: 10.1137/19M1246973
|
| [15] |
Y. Lin, D. Jiang, P. Xia, Long-time behavior of a stochastic SIR model, Appl. Math. Comput., 236 (2014), 1–9. https://doi.org/10.1016/j.amc.2014.03.035 doi: 10.1016/j.amc.2014.03.035
|
| [16] |
D. Xu, Y. Huang, Z. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005–1023. https://doi.org/10.3934/dcds.2009.24.1005 doi: 10.3934/dcds.2009.24.1005
|
| [17] |
N. Du, N. Dang, G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187–202. https://doi.org/10.1017/jpr.2015.18 doi: 10.1017/jpr.2015.18
|
| [18] |
S. P. Meyn, R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Probab., 25 (1993), 518–548. https://doi.org/10.2307/1427522 doi: 10.2307/1427522
|
| [19] | H. Roozen, An asymptotic solution to a two-dimensional exit problem arising in population dynamics, SIAM J. Appl. Math., 49 (1989). https://doi.org/10.1137/0149110 |
| [20] |
J. Ge, W. Zuo, D. Jiang, Stationary distribution and density function analysis of a stochastic epidemic HBV model, Math. Comput. Simulat., 191 (2022), 232–255. https://doi.org/10.1016/j.matcom.2021.08.003 doi: 10.1016/j.matcom.2021.08.003
|
| [21] |
D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302
|
| [22] |
F. Evirgen, S. Ucar, N. $\ddot{O}$zdemir, System analysis of HIV infection model with CD4+ T under Non-Singular kernel derivative, Appl. Math. Nonlinear Sci., 5 (2020), 139–146. https://doi.org/10.2478/amns.2020.1.00013 doi: 10.2478/amns.2020.1.00013
|
| [23] |
M. Umar, Z. Sabir, M. A. Z. Raja, H. M. Baskonus, S. W. Yao, E. Ilhan, A novel study of Morlet neural networks to solve the nonlinear HIV infection system of latently infected cells, Results Phys., 25 (2021), 104235. https://doi.org/10.1016/j.rinp.2021.104235 doi: 10.1016/j.rinp.2021.104235
|
| [24] |
M. Dewasurendra, Y. Zhang, N. Boyette, I. Islam, K. Vajravelu, A method of directly defining the inverse mapping for a HIV infection of CD4+ T-cells model, Appl. Math. Nonlinear Sci., 6 (2021), 469–482. https://doi.org/10.2478/amns.2020.2.00035 doi: 10.2478/amns.2020.2.00035
|
| [25] |
D. Ho, A. Neumann, A. Perelson, W. Chen, J. Leonard, M. Markowitz, Rapid turnover of plasma virons and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123–126. https://doi.org/10.1038/373123a0 doi: 10.1038/373123a0
|
Figures(3)
Wenjie Zuo, Mingguang Shao. Stationary distribution, extinction and density function for a stochastic HIV model with a Hill-type infection rate and distributed delay[J]. Electronic Research Archive, 2022, 30(11): 4066-4085. doi: 10.3934/era.2022206
DownLoad: