Dynamical behaviors of a stochastic HIV epidemic model with different infection stages driven by Ornstein-Uhlenbeck process

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Abstract

This paper presents a stochastic human immunodeficiency vrius transmission model driven by a log-normal Ornstein–Uhlenbeck process, designed to capture the stochastic fluctuations in the transmission rates across multiple stages of infection while incorporating co-infection characteristics. First, by constructing suitable Lyapunov functions, the existence and uniqueness of the global posi- tive solution to the model are proved. Subsequently, it is shown that the system admits a stationary distribution when R^S 0 > 1; under this condition, an analytical expression for the probability density function around the quasi-equilibrium state is derived. Furthermore, a sufficient condition for disease extinction is established, namely R^E 0 < 1. Finally, the theoretical results are validated through numer- ical simulations, and the impacts of the stochastic process’s parameters on disease transmission are investigated. The results indicate that a smaller reversion speed and a larger fluctuation intensity of the Ornstein–Uhlenbeck process tend to destabilize the epidemic system.

Keywords

stochastic HIV epidemic model / log-normal Ornstein–Uhlenbeck process / stationary distribution / probability density function / extinction

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References

[1] E. H. Moylett, W. T. Shearer, HIV: Clinical manifestations, J. Allergy Clin. Immunol., 110 (2002), 3-16. https://doi.org/10.1067/mai.2002.125978

[2] H. Nakamura, K. Yamamoto, Mpox in people with HIV: A narrative review, HIV Med., 25 (2024), 910-918. https://doi.org/10.1111/hiv.13661

[3] N. M. Zetola, C. D. Pilcher, Diagnosis and management of acute HIV infection, Infect. Dis. Clin. N. Am., 21 (2007), 19-48. https://doi.org/10.1016/j.idc.2007.01.008

[4] J. L. Weinberg, C. L. Kovarik, The WHO clinical staging system for HIV/AIDS, AMA J. Ethics, 12 (2010), 202-206. https://doi.org/10.1001/virtualmentor.2010.12.3.cprl1-1003

[5] V. Simon, D. D. Ho, Q. Abdool Karim, HIV/AIDS epidemiology, pathogenesis, prevention, and treatment, Lancet, 368 (2006), 489-504. https://doi.org/10.1016/s0140-6736(06)69157-5

[6] F. Inayaturohmat, N. Anggriani, A. Supriatna, M. H. A. Biswas, A systematic literature review of mathematical models for coinfections: Tuberculosis, malaria, and HIV/AIDS, J. Multidiscip. Healthc., 17 (2024), 1091-1109. https://doi.org/10.2147/JMDH.S446508

[7] C. C. Chang, M. Crane, J. Zhou, M. Mina, J. J. Post, B. A. Cameron, et al., HIV and co-infections, Immunol. Rev., 254 (2013), 114-142. https://doi.org/10.1111/imr.12063

[8] J. Bruchfeld, M. Correia-Neves, G. Källenius, Tuberculosis and HIV coinfection, Cold Spring Harb. Perspect. Med., 5 (2015), a017871. https://doi.org/10.1101/cshperspect.a017871

[9] Q. Deng, T. Guo, Z. Qiu, Y. Chen, A mathematical model for HIV dynamics with multiple infec- tions: implications for immune escape, J. Math. Biol., 89 (2024), 6. https://doi.org/10.1007/s00285-024-02104-w

[10] UNAIDS, Global HIV & AIDS statistics - Fact sheet, 2025. Available from: https://www.unaids.org/en/resources/fact-sheet.

[11] H. F. Huo, L. X. Feng, Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Appl. Math. Modell., 37 (2013), 1480-1489. https://doi.org/10.1016/j.apm.2012.04.013

[12] S. Sharma, G. P. Samanta, Dynamical behaviour of an HIV/AIDS epidemic model, Differ. Equations Dyn. Syst., 22 (2014), 369-395. https://doi.org/10.1007/s12591-013-0173-7

[13] E. A. Hernandez-Vargas, R. H. Middleton, Modeling the three stages in HIV infection, J. Theor. Biol., 320 (2013), 33-40. https://doi.org/10.1016/j.jtbi.2012.11.028

[14] K. R. Cheneke, K. P. Rao, G. Kenassa Edessa, Bifurcation and stability analysis of HIV transmission model with optimal control, J. Math., 2021 (2021), 7471290. https://doi.org/10.1155/2021/7471290

[15] N. Chen, S. Chen, X. Li, Z. Li, Modelling and analysis of the HIV/AIDS epidemic with fast and slow asymptomatic infections in China from 2008 to 2021, Math. Biosci. Eng., 20 (2023), 20770-20794. https://doi.org/10.3934/mbe.2023919

[16] X. Wang, C. Wang, K. Wang, Extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission, Adv. Differ. Equations, 2021 (2021), 260. https://doi.org/10.1186/s13662-021-03392-y

[17] F. Rao, Y. Tan, X. Lian, Stochastic analysis of an HIV model with various infection stages, Adv. Contin. Discrete Models, 2025 (2025), 38. https://doi.org/10.1186/s13662-025-03908-w

[18] Y. Wang, D. Jiang, A. Alsaedi, T. Hayat, Modelling a stochastic HIV model with logistic target cell growth and nonlinear immune response function, Physica A, 501 (2018), 276-292. https://doi.org/10.1016/j.physa.2018.02.040

[19] Y. Luo, J. Huang, Z. Teng, Q. Liu, Role of ART and PrEP treatments in a stochastic HIV/AIDS epidemic model, Math. Comput. Simul., 221 (2024), 337-357. https://doi.org/10.1016/j.matcom.2024.03.010

[20] N. Dalal, D. Greenhalgh, X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084-1101. https://doi.org/10.1016/j.jmaa.2007.11.005

[21] Q. Liu, Analysis of a stochastic HIV model with cell-to-cell transmission and Ornstein-Uhlenbeck process, J. Math. Phys., 64 (2023), 012702. https://doi.org/10.1063/5.0127775

[22] Y. Hao, Y. Luo, J. Huang, L. Zhang, Z. Teng, Analysis of a stochastic HIV/AIDS model with commercial heterosexual activity and Ornstein-Uhlenbeck Process, Math. Comput. Simul., 234 (2025), 50-72. https://doi.org/10.1016/j.matcom.2025.02.020

[23] Q. Dong, Y. Wang, D. Jiang, Dynamic analysis of an HIV model with CTL immune response and logarithmic Ornstein-Uhlenbeck process, Chaos Solitons Fractals, 191 (2025), 115789. https://doi.org/10.1016/j.chaos.2024.115789

[24] X. Zhang, X. Jiao, Analyzing HIV transmission through a stochastic system with the log-normal Ornstein-Uhlenbeck process, Appl. Math. Lett., 170 (2025), 109616. https://doi.org/10.1016/j.aml.2025.109616

[25] J. X. Shang, W. H. Li, Dynamical behaviors of a stochastic HIV/AIDS epi- demic model with Ornstein-Uhlenbeck process, Int. J. Biomath., 19 (2024), 2450055. https://doi.org/10.1142/S1793524524500554

[26] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. https://doi.org/10.1533/9780857099402

[27] H. Roozen, An asymptotic solution to a two-dimensional exit problem arising in population dynamics, SIAM J. Appl. Math., 49 (1989), 1793-1810. https://doi.org/10.1137/0149110

[28] Q. Liu, D. Jiang, Analysis of a stochastic within-host model of dengue infection with immune response and Ornstein-Uhlenbeck process, J. Nonlinear Sci., 34 (2024), 28. https://doi.org/10.1007/s00332-023-10004-4

[29] S. Wei, D. Jiang, Y. Zhou, Dynamic behavior of a stochastic HIV model with latent infection and Ornstein-Uhlenbeck process, Math. Comput. Simul., 225 (2024), 731-759. https://doi.org/10.1016/j.matcom.2024.06.011

[30] V. Henner, A. Nepomnyashchy, T. Belozerova, M. Khenner, Qualitative Methods and Stability of ODE Solutions, in Ordinary Differential Equations: Analytical Methods and Applications (eds. V. Henner et al. ), Springer International Publishing, Cham, (2023), 173-217.

[31] B. Zhou, D. Jiang, Y. Dai, T. Hayat, Threshold dynamics and probability density function of a stochastic avian influenza epidemic model with nonlinear incidence rate and psychological effect, J. Nonlinear Sci., 33 (2023), 29. https://doi.org/10.1007/s00332-022-09885-8

[32] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. https://doi.org/10.1137/S0036144500378302